Controllability of Singularly Perturbed Linear Time Delay Systems (Systems & Control: Foundations & Applications) 303065950X, 9783030659509

Table of contents :
Contents
1 Introduction
1.1 Real-Life Models
1.1.1 Neurosystem Model
1.1.2 Sunflower Equation
1.1.3 Model of Nuclear Reactor Dynamics
1.1.4 Model of Controlled Coupled-Core Nuclear Reactor
1.1.5 Car-Following Model: Lane as a Simple Open Curve
1.1.6 Car-Following Model: Lane as a Simple Closed Curve
References
2 Singularly Perturbed Linear Time Delay Systems
2.1 Introduction
2.2 Singularly Perturbed Systems with Small Delays
2.2.1 Original System
2.2.2 Slow–Fast Decomposition of the Original System
2.2.3 Fundamental Matrix Solution
2.2.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Small Delays
2.2.5 Example 1
2.2.6 Example 2: Tracking Model with Delay
2.2.7 Example 3: Analysis of Neurosystem Model
2.2.8 Example 4: Analysis of Sunflower Equation
2.2.9 Proof of Lemma 2.2
2.2.10 Proof of Theorem 2.1
2.2.10.1 Technical Proposition
2.2.10.2 Main Part of the Proof
2.3 Singularly Perturbed Systems with Delays of Two Scales
2.3.1 Original System
2.3.2 Slow–Fast Decomposition of the Original System
2.3.3 Fundamental Matrix Solution
2.3.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Delays of Two Scales
2.3.5 Example 5
2.3.6 Example 6: Dynamics of Nuclear Reactor
2.3.7 Example 7: Analysis of Car-Following Model in a Simple Closed Lane
2.3.8 Proof of Theorem 2.2
2.4 One Class of Singularly Perturbed Systems with NonsmallDelays
2.4.1 Original System
2.4.2 Slow–Fast Decomposition of the Original System
2.4.3 Fundamental Matrix Solution
2.4.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Nonsmall Delays
2.4.5 Example 8
2.4.6 Proof of Lemma 2.4
2.4.7 Proof of Theorem 2.4
2.5 Concluding Remarks and Literature Review
References
3 Euclidean Space Output Controllability of Linear Systems with State Delays
3.1 Introduction
3.2 Systems with Small Delays: Main Notions and Definitions
3.2.1 Original System
3.2.2 Asymptotic Decomposition of the Original System
3.3 Auxiliary Results
3.3.1 Output Controllability of a System with State Delays: Necessary and Sufficient Conditions
3.3.2 Linear Control Transformation in Systems with Small Delays
3.3.2.1 Control Transformation in the Original System
3.3.2.2 Asymptotic Decomposition of the Transformed System (3.30)–(3.31), (3.3)
3.3.3 Hybrid Set of Riccati-Type Matrix Equations
3.3.4 Proof of Lemma 3.1
3.3.4.1 Sufficiency
3.3.4.2 Necessity
3.3.5 Proof of Lemma 3.5
3.3.6 Proof of Lemma 3.7
3.3.7 Proof of Lemma 3.8
3.3.8 Proof of Lemma 3.9
3.4 Parameter-Free Controllability Conditions for Systems with Small Delays
3.4.1 Case of the Standard System (3.1)–(3.2)
3.4.2 Case of the Nonstandard System (3.1)–(3.2)
3.4.3 Proofs of Theorems 3.1, 3.2, and 3.3
3.4.3.1 Proof of Theorem 3.1
3.4.3.2 Proof of Theorem 3.2
3.4.3.3 Proof of Theorem 3.3
3.5 Special Cases of Controllability for Systems with Small Delays
3.5.1 Complete Euclidean Space Controllability
3.5.2 Controllability with Respect to x(t)
3.5.3 Controllability with Respect to y(t)
3.6 Examples: Systems with Small Delays
3.6.1 Example 1
3.6.2 Example 2
3.6.3 Example 3
3.6.4 Example 4
3.6.5 Example 5
3.6.6 Example 6: Pursuit-Evasion Engagement with Constant Speeds of Participants
3.6.7 Example 7: Pursuit-Evasion Engagement with Variable Speeds of Participants
3.6.8 Example 8: Analysis of Controlled Coupled-Core Nuclear Reactor Model
3.7 Systems with Delays of Two Scales: Main Notionsand Definitions
3.7.1 Original System
3.7.2 Asymptotic Decomposition of the Original System
3.8 Linear Control Transformation in Systems with Delays of Two Scales
3.8.1 Control Transformation in the Original System
3.8.2 Asymptotic Decomposition of the Transformed System (3.196)–(3.197), (3.187)
3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two Scales
3.9.1 Case of the Validity of the Assumption (AIII)
3.9.2 Case of the Validity of the Assumption (AIV)
3.9.3 Special Cases of Controllability
3.9.3.1 Complete Euclidean Space Controllability
3.9.3.2 Controllability with Respect to x(t)
3.9.3.3 Controllability with Respect to y(t)
3.9.4 Example 9
3.9.5 Example 10
3.9.6 Example 11: Controlled Car-Following Model in a Simple Open Lane
3.10 Concluding Remarks and Literature Review
References
4 Complete Euclidean Space Controllability of Linear Systems with State and Control Delays
4.1 Introduction
4.2 System with Small State Delays: Main Notions and Definitions
4.2.1 Original System
4.2.2 Asymptotic Decomposition of the Original System
4.3 Preliminary Results
4.3.1 Auxiliary System with Small State Delays and Delay-Free Control
4.3.2 Output Controllability of the Auxiliary System and Its Slow and Fast Subsystems: Necessary and Sufficient Conditions
4.3.2.1 Equivalent Forms of the Auxiliary System
4.3.2.2 Output Controllability of the Auxiliary System
4.3.2.3 Output Controllability of the Slow and Fast Subsystems Associated with the Auxiliary System
4.3.3 Linear Control Transformation in the Original System with Small State Delays
4.3.4 Stabilizability of a Parameter-Dependent System with State and Control Delays by a Memory-Less Feedback Control
4.3.5 Proof of Lemma 4.8
4.4 Parameter-Free Controllability Conditions for Systems with Small State Delays
4.4.1 Case of the Standard System (4.1)–(4.2)
4.4.2 Case of the Nonstandard System (4.1)–(4.2)
4.4.3 Proof of Main Lemma (Lemma 4.9)
4.4.3.1 Auxiliary Propositions
4.4.3.2 Main Part of the Proof
4.4.4 Alternative Approach to Controllability Analysis of the Nonstandard System (4.1)–(4.2)
4.4.4.1 Linear Control Transformation in the Auxiliary System (4.40)–(4.42)
4.4.4.2 Proof of Lemma 4.10
4.4.4.3 Hybrid Set of Riccati-Type Matrix Equations
4.4.4.4 Parameter-Free Controllability Conditions of the Nonstandard System (4.1)–(4.2)
4.5 Examples: Systems with Small State and Control Delays
4.5.1 Example 1
4.5.2 Example 2
4.5.3 Example 3
4.6 Systems with State Delays of Two Scales: Main Notions and Definitions
4.6.1 Original System
4.6.2 Asymptotic Decomposition of the Original System
4.7 Auxiliary System with State Delays of Two Scales and Delay-Free Control
4.7.1 Description of the Auxiliary System and Some of Its Properties
4.7.2 Asymptotic Decomposition of the Auxiliary System (4.180)–(4.181)
4.7.3 Linear Control Transformation in the Auxiliary System (4.180)–(4.181)
4.8 Parameter-Free Controllability Conditions for Systems with Delays of Two Scales
4.8.1 Case of the Validity of the Assumption (AV)
4.8.2 Case of the Validity of the Assumption (AVI)
4.8.3 Example 4
4.8.4 Example 5
4.8.5 Example 6: Analysis of Car-Following Model with State and Control Delays
4.9 Concluding Remarks and Literature Review
References
5 First-Order Euclidean Space Controllability Conditions for Linear Systems with Small State Delays
5.1 Introduction
5.2 Singularly Perturbed System: Main Notions and Definitions
5.2.1 Original System
5.2.2 Asymptotic Decomposition of the Original System
5.3 Auxiliary Results
5.3.1 Estimates of Solutions to Some Singularly Perturbed Linear Time Delay Matrix Differential Equations
5.3.2 Proof of Lemma 5.1
5.3.2.1 Technical Proposition
5.3.2.2 Main Part of the Proof
5.3.3 Complete Controllability of the Original System and Its Slow Subsystem: Necessary and SufficientConditions
5.4 Parameter-Free Controllability Conditions
5.4.1 Formulation of Main Assertions
5.4.2 Proof of Theorem 5.1
5.4.3 Proof of Lemma 5.2
5.4.4 Proof of Theorem 5.2
5.4.4.1 Euclidean Space Controllability of a Pure Fast System
5.4.4.2 Main Part of the Proof
5.5 Examples
5.5.1 Example 1
5.5.2 Example 2
5.5.3 Example 3
5.5.4 Example 4
5.5.5 Example 5
5.5.6 Example 6
5.5.7 Example 7: Analysis of Controlled Car-Following Model in a Simple Open Lane
5.6 Concluding Remarks and Literature Review
References
6 Miscellanies
6.1 Introduction
6.2 Euclidean Space Controllability of Linear Time Delay Systems with High Gain Control
6.2.1 High Gain Control System: Main Notionsand Definitions
6.2.1.1 Initial System
6.2.1.2 Transformation of the System (6.1)
6.2.2 High Dimension Controllability Condition for the System (6.5)
6.2.3 Asymptotic Decomposition of the System (6.5)
6.2.4 Auxiliary Results
6.2.4.1 Linear Control Transformation in the System (6.13)–(6.14) and Some of its Properties
6.2.4.2 Asymptotic Decomposition of the Transformed System (6.13), (6.21)
6.2.4.3 Block-Wise Estimate of the Solution to the Terminal-Value Problem (6.23)
6.2.5 Lower Dimension Parameter-Free Controllability Condition for the System (6.5)
6.2.6 Example
6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input Delay
6.3.1 Original System
6.3.2 Discussion on the Slow–Fast Decomposition of the Original System
6.3.3 Auxiliary Results
6.3.3.1 Necessary and Sufficient Controllability Conditions of the Original System
6.3.3.2 Block-Wise Estimate of the Solution to the Terminal-Value Problem (6.58)
6.3.3.3 Asymptotic Analysis of the Controllability Matrix W(tc,)
6.3.4 Parameter-Free Controllability Conditions
6.3.5 Example 1
6.3.6 Example 2
6.4 Functional Null Controllability of Some Nonlinear Systems with Small State Delays
6.4.1 Problem Formulation
6.4.2 Preliminary Results
6.4.2.1 Euclidean Space Controllability of Singularly Perturbed Linear Time Delay System
6.4.2.2 Euclidean Space Controllability of the Original System (6.113)–(6.114)
6.4.3 System of the First Type: Formulation and Some Auxiliary Results
6.4.3.1 Asymptotic Decomposition of the System (6.124)–(6.125)
6.4.3.2 Controllability Conditions for the Slow and Fast Subsystems
6.4.3.3 Parameter-Free Conditions for the Complete Euclidean Space Controllability of the Linear System (6.124)–(6.125)
6.4.3.4 Euclidean Space Null Controllability of the Nonlinear System (6.122)–(6.123): Parameter-Free Conditions
6.4.4 System of the First Type: Parameter-Free Conditions for the Functional Null Controllability
6.4.5 System of the Second Type: Formulation and Some Auxiliary Results
6.4.5.1 Asymptotic Decomposition of the System (6.155)–(6.156)
6.4.5.2 Parameter-Free Conditions for the Complete Euclidean Space Controllability of the Linear System (6.155)–(6.156)
6.4.5.3 Euclidean Space Null Controllability of the Nonlinear System (6.153)–(6.154): Parameter-Free Conditions
6.4.6 System of the Second Type: Parameter-Free Conditions for the Functional Null Controllability
6.4.7 Example 1
6.4.8 Example 2: Analysis of Controlled Sunflower Equation
6.4.9 Example 3
6.5 Some Open Problems
6.5.1 Complete Euclidean Space Controllability of Linear Systems with State Delays and Nonsmall ControlDelays
6.5.2 Euclidean Space Controllability of Linear Systems with Nonsmall Delays
6.5.3 Complete Euclidean Space Controllability of One Class of Nonlinear Systems with Small State Delays
6.6 Concluding Remarks and Literature Review
References
Index

Citation preview

Systems & Control: Foundations & Applications

Valery Y. Glizer

Controllability of Singularly Perturbed Linear Time Delay Systems

Systems & Control: Foundations & Applications Series Editor Tamer Ba¸sar, University of Illinois at Urbana-Champaign, Urbana, IL, USA

Editorial Board Karl Johan Åström, Lund Institute of Technology, Lund, Sweden Han-Fu Chen, Academia Sinica, Beijing, China Bill Helton, University of California, San Diego, CA, USA Alberto Isidori, Sapienza University of Rome, Rome, Italy Miroslav Krstic, University of California, San Diego, La Jolla, CA, USA H. Vincent Poor, Princeton University, Princeton, NJ, USA Mete Soner, ETH Zürich, Zürich, Switzerland; Swiss Finance Institute, Zürich, Switzerland Former Editorial Board Member Roberto Tempo, (1956–2017), CNR-IEIIT, Politecnico di Torino, Italy

More information about this series at http://www.springer.com/series/4895

Valery Y. Glizer

Controllability of Singularly Perturbed Linear Time Delay Systems

Valery Y. Glizer Department of Applied Mathematics ORT Braude College of Engineering Karmiel, Israel

ISSN 2324-9749 ISSN 2324-9757 (electronic) Systems & Control: Foundations & Applications ISBN 978-3-030-65950-9 ISBN 978-3-030-65951-6 (eBook) https://doi.org/10.1007/978-3-030-65951-6 Mathematics Subject Classification: 34K26, 34K35, 93B05, 93C23 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of my parents, Anna & Yakov To the memory of my grandparents, Liza & Semyon and Roza & Moisei

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Real-Life Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Neurosystem Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Sunflower Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Model of Nuclear Reactor Dynamics . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Model of Controlled Coupled-Core Nuclear Reactor . . . . . . 1.1.5 Car-Following Model: Lane as a Simple Open Curve . . . . . 1.1.6 Car-Following Model: Lane as a Simple Closed Curve . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5 5 6 7 9 11 14 17

2

21 21 22 22 23 25

Singularly Perturbed Linear Time Delay Systems . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Singularly Perturbed Systems with Small Delays . . . . . . . . . . . . . . . . . . . . 2.2.1 Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Slow–Fast Decomposition of the Original System . . . . . . . . . 2.2.3 Fundamental Matrix Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Small Delays . . . . . . . . . . . . 2.2.5 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Example 2: Tracking Model with Delay . . . . . . . . . . . . . . . . . . . . 2.2.7 Example 3: Analysis of Neurosystem Model . . . . . . . . . . . . . . . 2.2.8 Example 4: Analysis of Sunflower Equation . . . . . . . . . . . . . . . 2.2.9 Proof of Lemma 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.10 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Singularly Perturbed Systems with Delays of Two Scales. . . . . . . . . . . 2.3.1 Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Slow–Fast Decomposition of the Original System . . . . . . . . . 2.3.3 Fundamental Matrix Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Delays of Two Scales . . .

27 32 36 41 46 49 51 57 57 58 60 66

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2.3.5 2.3.6 2.3.7

Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Example 6: Dynamics of Nuclear Reactor . . . . . . . . . . . . . . . . . . 73 Example 7: Analysis of Car-Following Model in a Simple Closed Lane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.3.8 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.4.1 Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.4.2 Slow–Fast Decomposition of the Original System . . . . . . . . . 87 2.4.3 Fundamental Matrix Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.4.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Nonsmall Delays. . . . . . . . 91 2.4.5 Example 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.4.6 Proof of Lemma 2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.4.7 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.5 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3

Euclidean Space Output Controllability of Linear Systems with State Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Systems with Small Delays: Main Notions and Definitions . . . . . . . . . 3.2.1 Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Asymptotic Decomposition of the Original System . . . . . . . . 3.3 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Output Controllability of a System with State Delays: Necessary and Sufficient Conditions. . . . . . . . . . . . . . . 3.3.2 Linear Control Transformation in Systems with Small Delays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Hybrid Set of Riccati-Type Matrix Equations . . . . . . . . . . . . . . 3.3.4 Proof of Lemma 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Proof of Lemma 3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Proof of Lemma 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.7 Proof of Lemma 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.8 Proof of Lemma 3.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Parameter-Free Controllability Conditions for Systems with Small Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Case of the Standard System (3.1)–(3.2). . . . . . . . . . . . . . . . . . . . 3.4.2 Case of the Nonstandard System (3.1)–(3.2) . . . . . . . . . . . . . . . 3.4.3 Proofs of Theorems 3.1, 3.2, and 3.3. . . . . . . . . . . . . . . . . . . . . . . . 3.5 Special Cases of Controllability for Systems with Small Delays . . . 3.5.1 Complete Euclidean Space Controllability . . . . . . . . . . . . . . . . . 3.5.2 Controllability with Respect to x(t) . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Controllability with Respect to y(t) . . . . . . . . . . . . . . . . . . . . . . . .

111 111 112 112 113 116 116 119 125 127 130 133 141 148 149 149 152 154 164 164 165 166

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3.6

Examples: Systems with Small Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.6 Example 6: Pursuit-Evasion Engagement with Constant Speeds of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.7 Example 7: Pursuit-Evasion Engagement with Variable Speeds of Participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.8 Example 8: Analysis of Controlled Coupled-Core Nuclear Reactor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Systems with Delays of Two Scales: Main Notions and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Asymptotic Decomposition of the Original System . . . . . . . . 3.8 Linear Control Transformation in Systems with Delays of Two Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Control Transformation in the Original System . . . . . . . . . . . . 3.8.2 Asymptotic Decomposition of the Transformed System (3.196)–(3.197), (3.187) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Case of the Validity of the Assumption (AIII) . . . . . . . . . . . . . . 3.9.2 Case of the Validity of the Assumption (AIV). . . . . . . . . . . . . . 3.9.3 Special Cases of Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.4 Example 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.5 Example 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.6 Example 11: Controlled Car-Following Model in a Simple Open Lane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Complete Euclidean Space Controllability of Linear Systems with State and Control Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 System with Small State Delays: Main Notions and Definitions . . . . 4.2.1 Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Asymptotic Decomposition of the Original System . . . . . . . . 4.3 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Auxiliary System with Small State Delays and Delay-Free Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Output Controllability of the Auxiliary System and Its Slow and Fast Subsystems: Necessary and Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3.3

Linear Control Transformation in the Original System with Small State Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Stabilizability of a Parameter-Dependent System with State and Control Delays by a Memory-Less Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Proof of Lemma 4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Parameter-Free Controllability Conditions for Systems with Small State Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Case of the Standard System (4.1)–(4.2). . . . . . . . . . . . . . . . . . . . 4.4.2 Case of the Nonstandard System (4.1)–(4.2) . . . . . . . . . . . . . . . 4.4.3 Proof of Main Lemma (Lemma 4.9) . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Alternative Approach to Controllability Analysis of the Nonstandard System (4.1)–(4.2) . . . . . . . . . . . . . . . . . . . . . 4.5 Examples: Systems with Small State and Control Delays . . . . . . . . . . . 4.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Systems with State Delays of Two Scales: Main Notions and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Asymptotic Decomposition of the Original System . . . . . . . . 4.7 Auxiliary System with State Delays of Two Scales and Delay-Free Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Description of the Auxiliary System and Some of Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Asymptotic Decomposition of the Auxiliary System (4.180)–(4.181). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Linear Control Transformation in the Auxiliary System (4.180)–(4.181) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Parameter-Free Controllability Conditions for Systems with Delays of Two Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Case of the Validity of the Assumption (AV) . . . . . . . . . . . . . . . 4.8.2 Case of the Validity of the Assumption (AVI) . . . . . . . . . . . . . . 4.8.3 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.5 Example 6: Analysis of Car-Following Model with State and Control Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

First-Order Euclidean Space Controllability Conditions for Linear Systems with Small State Delays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Singularly Perturbed System: Main Notions and Definitions . . . . . . . 5.2.1 Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Asymptotic Decomposition of the Original System . . . . . . . .

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5.3

Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Estimates of Solutions to Some Singularly Perturbed Linear Time Delay Matrix Differential Equations. . . . . . . . . . 5.3.2 Proof of Lemma 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Complete Controllability of the Original System and Its Slow Subsystem: Necessary and Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Parameter-Free Controllability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Formulation of Main Assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Proof of Lemma 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 Example 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.7 Example 7: Analysis of Controlled Car-Following Model in a Simple Open Lane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Miscellanies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Euclidean Space Controllability of Linear Time Delay Systems with High Gain Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 High Gain Control System: Main Notions and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 High Dimension Controllability Condition for the System (6.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Asymptotic Decomposition of the System (6.5). . . . . . . . . . . . 6.2.4 Auxiliary Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Lower Dimension Parameter-Free Controllability Condition for the System (6.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input Delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Discussion on the Slow–Fast Decomposition of the Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Auxiliary Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Parameter-Free Controllability Conditions . . . . . . . . . . . . . . . . . 6.3.5 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.4

Functional Null Controllability of Some Nonlinear Systems with Small State Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 System of the First Type: Formulation and Some Auxiliary Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 System of the First Type: Parameter-Free Conditions for the Functional Null Controllability. . . . . . . . . . . . . . . . . . . . . . 6.4.5 System of the Second Type: Formulation and Some Auxiliary Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.6 System of the Second Type: Parameter-Free Conditions for the Functional Null Controllability . . . . . . . . . 6.4.7 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.8 Example 2: Analysis of Controlled Sunflower Equation . . . 6.4.9 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Some Open Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Complete Euclidean Space Controllability of Linear Systems with State Delays and Nonsmall Control Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Euclidean Space Controllability of Linear Systems with Nonsmall Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Complete Euclidean Space Controllability of One Class of Nonlinear Systems with Small State Delays . . . . . . 6.6 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

382 382 384 386 390 395 398 399 401 402 404

404 409 412 413 415

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Chapter 1

Introduction

Singularly perturbed differential systems, i.e., the systems with a small multiplier ε > 0 for a part of the highest order derivatives, are adequate mathematical models for various real-life processes with two-time-scale dynamics. In real-life problems, the small multiplier, called the parameter of singular perturbation, can be a time constant, a mass, a capacitance, a geotropic reaction, and another parameters in physics, chemistry, engineering, biology, medicine, etc (see e.g. [34, 40, 41] and references therein). The parameter of singular perturbation also can appear in a mathematical model as a ratio between two values of the same physical (or another) dimension, one of which is considerably smaller than the other (see e.g. [12, 13, 48, 50] and references therein). An important class of singularly perturbed differential systems represents the systems with time delays. Among such systems there are the systems with all the delays of order of ε (small delays), the systems with the delays of two scales (of order of ε and of order of 1), the systems with all delays of order of 1 (nonsmall delays). In real-life problems, such delays can be the following: (1) a delay, generated by a sampled-data control; (2) a communication delay in optical, electronic, or another kind of a network; (3) a propagation delay of action potentials between neurons in a neurosystem; (4) a feedback power reactivity delay in a nuclear reactor dynamics; (5) a neutrons transport delay in a coupledcore nuclear reactor; (6) a driver reaction delay in a car-following problem; (7) delays in a feedback control, caused by filtering and estimation of noise corrupted measurements of state variables; and some others. Singularly perturbed differential systems with time delays, modeling various real-life problems, can be found in the works [7, 21, 38, 42, 43, 45, 48, 49] and in references therein. Different aspects in theory and applications of singularly perturbed controlled systems without/with delays in the state and control variables, including various properties of such systems, are extensively investigated in the literature (see e.g. [5, 11, 18, 27, 34, 37, 39, 52] and references therein).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Y. Glizer, Controllability of Singularly Perturbed Linear Time Delay Systems, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-65951-6_1

1

2

1 Introduction

Controllability of a controlled system is one of its basic properties. This property means the ability to transfer the system from any position of a given set of initial positions to any position of a given set of terminal positions in a finite time by a proper choice of the control function. The controllability is used in many issues of control theory and applications, for instance, in solution of optimal control problems with given sets of initial and terminal positions [44] (in particular, in solution of a minimum energy control problem), in solution of pursuit problems [26], in control problems of various industrial processes and in many other issues [32]. Conditions of different types of controllability for various controlled systems without and with delays were extensively studied in the literature (see e.g. [1, 4, 9, 28–32] and references therein). To check whether a singularly perturbed system is controllable in a proper sense, the corresponding controllability conditions can be directly applied for any specified value of the small parameter ε of singular perturbation. However, the stiffness, as well as a possible high dimension of the singularly perturbed system, can considerably complicate this application. Moreover, such an application depends on the value of ε, i.e., it is not robust with respect to this parameter, while in most of real-life problems this value is unknown. Controllability of singularly perturbed systems was studied in the literature in a number of works. Thus, in [33, 34, 46, 47], the complete controllability of some linear and nonlinear systems without delays was analyzed using the separation of time-scales (slow–fast decomposition) concept, [34]. In [35], a class of linear standard singularly perturbed time-invariant systems with a single point-wise delay in the state variables was considered. For these systems, based on the separation of time-scales concept, the robust complete Euclidean space controllability was studied in the case of nonsmall delay. In [14, 15], based on this concept, parameter-free conditions of the complete Euclidean space controllability, robust with respect to ε, were obtained for linear standard singularly perturbed time-dependent systems with point-wise and distributed small delays in the state variables. In [16], such a result was obtained for nonstandard singularly perturbed systems with multiple point-wise and distributed small delays in the state variables. In [21], a singularly perturbed linear time-dependent system with small state delays (multiple point-wise and distributed) was studied. Along with the set of time delay differential equations describing the dynamics of this system, a set of delay-free algebraic equations, describing the system’s output, also was considered. Based on the separation of time-scales concept, different parameter-free sufficient conditions for the Euclidean space output controllability of this system were established. In [17], another type of parameter-free conditions of the complete Euclidean space controllability, which are not based on the separation of time-scales concept, were derived for a class of linear singularly perturbed systems with small delays (multiple point-wise and distributed) in the state variables. In these systems, the slow mode is controlled directly, while the fast mode is controlled through the slow one. In [19, 22], singularly perturbed linear time-dependent controlled systems with small pointwise and distributed delays in the state and control variables were considered. Two types of these systems, standard and nonstandard, were analyzed. For each type, parameter-free conditions of the complete Euclidean space controllability

1 Introduction

3

were established. These results were obtained by a transformation of the complete Euclidean space controllability of the original system to an equivalent output controllability of a larger dimension singularly perturbed system with only state delays. In [23], like in [19, 22], a singularly perturbed linear time-dependent system with point-wise and distributed delays in the state and control was studied. However, in contrast with the latter, in [23] the case of the state delays of two scales (nonsmall of order of 1 and small of order of ε) was analyzed. In [20], the complete Euclidean space controllability for one class of singularly perturbed linear systems with nonsmall delays (point-wise and distributed) in the state variables was studied. In [25], the complete Euclidean space controllability of a singularly perturbed linear system with a nonsmall point-wise delay only in the control was analyzed. This analysis, not being based on the slow–fast decomposition of the original system, yields ε-free conditions providing its controllability robust with respect to ε for all sufficiently small values of this parameter. In [36], the defining equations approach (see e.g. [10]) was applied for analysis of the complete Euclidean space controllability of a linear singularly perturbed neutral type system with a single nonsmall point-wise delay. In [51], the functional null controllability was investigated for a singularly perturbed linear constant coefficients system with nonsmall point-wise commensurate delays only in the slow state variable. Parameter-independent sufficient conditions for the functional null controllability of this system were derived. In [24], two types of singularly perturbed nonlinear timedependent systems with small state delays (multiple non-commensurate point-wise and distributed) were considered. For each type of these systems, parameter-free functional null controllability conditions were derived. The objective of this book is to summarize and generalize the author’s results on different kinds of controllability for singularly perturbed systems with delays in the state/control variables, which are spread in a number of the journal and conference proceedings papers [14–17, 19–25]. Of course, the book does not cover all problems, which arise in the topic of controllability of singularly perturbed time delay systems. However, it presents in a systematic and detailed form the study of main issues of this topic. The book consists of the introduction chapter (the present one) followed by five body chapters (Chaps. 2–6). In Chap. 2, various classes of singularly perturbed linear nonautonomous time delay systems are considered. These systems do not contain control variables. Main notions, connected with these systems, are introduced. Main properties of the considered systems are studied. In Chap. 3, singularly perturbed linear time-dependent controlled systems with multiple pointwise and distributed state delays are considered. The cases of the small delays and the delays of two scales are treated. For these systems, the Euclidean space output controllability with respect to linear algebraic output equations is studied. Various parameter-free conditions of this controllability are established. More general singularly perturbed systems, namely the systems with state and control delays, are considered in Chap. 4. The complete Euclidean space controllability for these systems, robust with respect to the parameter of singular perturbations, is analyzed. In Chap. 5, a singularly perturbed linear time-dependent controlled

4

1 Introduction

system with small state delays, in which the fast mode does not contain a control variable, is considered. First-order Euclidean space controllability conditions are derived for this system. Chapter 6 is a miscellaneous one. It consists of several sections. One of these sections deals with a class of singularly perturbed systems with nonsmall delays in both, slow and fast, state variables. The systems of this class are obtained from high gain control systems. Conditions of the complete Euclidean space controllability for these systems, robust with respect to the parameter of singular perturbation, are derived. In the third section of Chap. 6, a singularly perturbed linear time-dependent system with a point-wise delay only in the control is considered. Parameter-free conditions of its complete Euclidean space controllability are derived in the case where the delay is nonsmall. Some singularly perturbed nonlinear controlled systems with small state delays are considered in the fourth section of Chap. 6. Euclidean space and functional space types of controllability for these systems, robust with respect to the parameter of singular perturbation, are studied. In the fifth section of Chap. 6, some open problems in the topic of controllability of singularly perturbed time delay systems are discussed. Theoretical results, obtained in the book, are illustrated by numerous examples, including the examples based on real-life models. Technically complicated proofs are placed into separate sections/subsections. For the sake of a better readability, each chapter is organized to be self-contained as much as possible. In particular, for this purpose, some assertions are placed in each chapter where they are considerably used. In such a case, the assertion is proven in the chapter where it appears for the first time. In the other chapters the assertion appears in a brief form and without its proof but with a proper reference to the corresponding chapter/section. Moreover, each chapter contains an introduction, a list of notations used in the chapter, concluding remarks with a brief literature review on the chapter’s topic, and a corresponding bibliography. Due to such an organization of the book’s chapters, each chapter can be studied independently of the others. The book can be helpful for researchers and engineers, working in such areas as Applied Mathematics, Systems Science, Control Engineering, Electrical Engineering, Mechanical and Aerospace Engineering, and even in Biology and Medicine. Also, the book can be useful for graduate students in these areas. Moreover, due to the detailed and systematic presentation of the theoretical material and the numerous illustrative examples, the book can be used as a textbook in various courses on Systems Science and Control Theory. In completion of this chapter, we present singularly perturbed time delay models of several real-life problems. In this chapter, we present these models in a brief form. Much more detailed analysis of these and some other real-life models is presented in subsequent chapters based on theoretical results obtained in these chapters.

1.1 Real-Life Models

5

1.1 Real-Life Models 1.1.1 Neurosystem Model The noise-free system of two mutually coupled neurons can be modeled by the following nonlinear singularly perturbed time delay differential equations (see [48] and references therein): dX1 (t) = Y1 (t) + b, dt dX2 (t) = Y2 (t) + b, dt ε1

  Y 3 (t) dY1 (t) = −X1 (t) + Y1 (t) − 1 + C Y2 (t − tdel ) − Y1 (t) , dt 3

ε2

  Y 3 (t) dY2 (t) = −X2 (t) + Y2 (t) − 2 + C Y1 (t − tdel ) − Y2 (t) , dt 3

(1.1)

where the scalar variables X1 (t) and X2 (t) correspond to quantities connected with the electrical conductance of the relevant ion currents; the scalar variables Y1 (t) and Y2 (t) correspond to the transmembrane voltage; the values ε1 > 0 and ε2 > 0 are small time-scale parameters showing that the actuator variables Y1 (t) and Y2 (t) are much faster than the inhibitor variables X1 (t) and X2 (t); the positive constant b is an excitability parameter; the positive constant C is the coupling strength connected with the distribution of the information between neurons; the constant time delay tdel ≥ 0 in the coupling terms means the propagation delay of action potentials between the two neurons. As it was  the equilibrium state of the system (1.1) has the form  shown in [48], Z ∗ = col X1∗ , X2∗ , Y1∗ , Y2∗ , where X1∗ = b3 /3 − b, X2∗ = b3 /3 − b, Y1∗ = −b, Y2∗ = −b. Similarly to the results of [48], the linearization of the nonlinear system (1.1) around this equilibrium state, along with the assumption on the symmetry of the time scales (ε1 = ε2 = ε), yields the following singularly perturbed linear time delay system: dx1 (t) = y1 (t), dt dx2 (t) = y2 (t), dt dy1 (t) = −x1 (t) + ωy1 (t) + Cy2 (t − tdel ), dt dy2 (t) = −x2 (t) + ωy2 (t) + Cy1 (t − tdel ), ε dt

ε

(1.2)

6

1 Introduction

where xj (t) = Xj (t) − Xj∗ ,

yj (t) = Yj (t) − Yj∗ ,

j = 1, 2,

ω = 1 − b2 − C.

(1.3) (1.4)

If tdel = εh, where h > 0 is a constant independent of ε, the system (1.2) becomes a singularly perturbed system with a small delay of order of a small positive multiplier for a part of the derivatives. Such a case is an intermediate one between the following two cases considered in [48]: (1) tdel = 0; (2) the delay tdel is nonsmall (of order of 1).

1.1.2 Sunflower Equation The nonlinear sunflower equation has the form (see [43] and references therein) ε

  d 2 x(t) dx(t) + c2 sin x(t − ε) = 0, + c1 2 dt dt

(1.5)

where x(t) is the angle between the plant and the vertical; the parameter ε > 0 is the geotropic reaction; c1 > 0 and c2 > 0 are constants, which are obtained experimentally for a specific kind of the plant. Like in [43], we assume the parameter ε to be small.   If the absolute value of the angle x(t) is small such that sin x(t − ε) ≈ x(t − ε), the nonlinear equation (1.5) can be replaced (approximately) with the following simpler linear equation: ε

d 2 x(t) dx(t) + c2 x(t − ε) = 0. + c1 dt dt 2

(1.6)

This second order differential equation can be rewritten equivalently in the form of the following system of two first-order differential equations: dx(t) = y(t), dt ε

dy(t) = −c1 y(t) − c2 x(t − ε), dt

(1.7)

which is a singularly perturbed linear system with a small delay of order of the small multiplier ε for the derivative dy(t)/dt. If the sunflower equation (1.5) is subject to an external force, a forced sunflower equation is obtained (see e.g., [3]). Moreover, if the external force is not a known function of time but it is a control u(t), then we obtain the controlled sunflower equation

1.1 Real-Life Models

ε

7

  dx(t) d 2 x(t) + c2 sin x(t − ε) = u(t). + c1 2 dt dt

(1.8)

This second order controlled differential equation can be rewritten equivalently as follows: dx(t) = y(t), dt   dy(t) ε = −c1 y(t) − c2 sin x(t − ε) + u(t). dt

(1.9)

The system (1.9) is a singularly perturbed nonlinear controlled system with a small delay of order of ε.

1.1.3 Model of Nuclear Reactor Dynamics The normalized and linearized model of a nuclear reactor dynamics proposed in [2] is   dZN (t) = −γN δZC (t) + ZN (t − hN ) , dt   dZC (t) = γC ZN (t) − ZC (t − hC ) . dt

(1.10)

This model corresponds to the case of a single group of delayed neutrons, where ZN (t) =

N(t) − 1, Nst

ZC (t) =

C(t) − 1; Cst

N (t) is the current density of neutrons, while Nst is its steady state; C(t) is the current density of delayed neutrons, while Cst is its steady state; γN > 0 is the coefficient of the linear growth of the neutrons’ density; γC > 0 is the coefficient of the linear growth of the delayed neutrons’ density; hN > 0 is the delay in the feedback power reactivity; hC > 0 is the generation time of the delayed neutrons; δ ∈ (−1, 0] is a nondimensional parameter, which regulates the power of the reactor. Here, we consider the model (1.10) in the time interval [0, tc ], and we consider the following two cases of its data: (I) γC  γN , γC tc ∼ O(1); (II) γN  γC , γN tc ∼ O(1). Also, we assume that in both cases γN hN ∼ O(1), where O(1) is a value of order of 1.

γC hC ∼ O(1),

8

1 Introduction

Case I (γC  γN ) Let us make the following transformations of the independent variable and the unknown functions in the system (1.10):     θ θ θ = y(θ ), ZC = x(θ ), , ZN t= γC γC γC where θ is a new independent variable (the nondimensional time); y(θ ) and x(θ ) are new unknown functions. Due to these transformations, the system (1.10) becomes as dx(θ ) = y(θ ) − x(θ − g), dθ dy(θ ) = −δx(θ ) − y(θ − εh), ε dθ

θ ∈ [0, θc ], θ ∈ [0, θc ],

(1.11)

where ε=

γC , γN

g = γC h C ,

h = γN h N ,

θc = γC tc ,

and ε > 0 is a small parameter. Case II (γN  γC ) Let us make the following transformations of the independent variable and the unknown functions in the system (1.10):     θ θ θ = x(θ ), ZC = y(θ ). , ZN t= γN γN γN Like in Case I, θ is a new independent variable (the nondimensional time); x(θ ) and y(θ ) are new unknown functions. Due to these transformations, the system (1.10) becomes as dx(θ ) = −δy(θ ) − x(θ − g), dθ dy(θ ) = x(θ ) − y(θ − εh), ε dθ

θ ∈ [0, θc ], θ ∈ [0, θc ],

(1.12)

where ε=

γN , γC

g = γN h N ,

h = γC h C ,

θc = γN tc ,

and ε > 0 is a small parameter. Both systems, (1.11) and (1.12), are singularly perturbed systems with a nonsmall delay (of order of 1) in the slow state variable x(·) and with a small delay (of order of ε) in the fast state variable y(·).

1.1 Real-Life Models

9

1.1.4 Model of Controlled Coupled-Core Nuclear Reactor In [45] the following model of coupled-core nuclear reactor was proposed: dX1 (t) dt dX2 (t) dt dX3 (t) dt dX4 (t) dt dX5 (t) dt dX6 (t) dt dY1 (t) ε dt dY2 (t) ε dt

= a11 (ε)X1 (t) + X2 (t − ε)+a13 εX3 (t) + c15 εX1 (t)X5 (t) + X1 (t)Y1 (t), = a22 (ε)X2 (t) + X1 (t − ε)+a24 εX4 (t) + c26 εX2 (t)X6 (t) + X2 (t)Y2 (t),   = ε a31 X1 (t) + a33 X3 (t) ,   = ε a42 X2 (t) + a44 X4 (t) ,   = ε a51 X1 (t) + a55 X5 (t) ,   = ε a62 X2 (t) + a66 X6 (t) , = −Y1 (t) + U1 (t), = −Y2 (t) + U2 (t),

(1.13)

where X1 (t) and X2 (t) are the current levels of the power in the first and second core, respectively; X3 (t) and X4 (t) are the current levels of the power of the delayed neutrons in the first and second core, respectively; X5 (t) and X6 (t) are the current values of the temperature in the first and second core, respectively; Y1 (t) and Y2 (t) are the current values of the normalized external control reactivity in the first and second core, respectively; U1 (t) and U2 (t) are the current values of the control inputs to the servomotors associated with the first and second core, respectively; the value ε > 0 is a small time-scale parameter indicating that the values Y1 (t) and Y2 (t) are much faster than the values Xi (t). (i = 1, . . . , 6); the small delay ε appears in X1 (·) and X2 (·) due to the neutrons transport; the coefficients a11 (ε) and a22 (ε) are linear functions of ε; a13 , a24 , a31 , a33 , a42 , a44 , a51 , a55 , a62 , a66 , c15 , and c26 are constants independent of ε; a11 (0) = 0,

a22 (0) = 0.

(1.14)

As it is mentioned in [45], there are desirable (nominal) values of the power levels for the first and second cores, X1d > 0 and X2d > 0. Substituting X1d and X2d instead of X1 (·) and X2 (·) into the right-hand sides of the first six equations of the system (1.13), as well as replacing the derivatives in these equations with zero, we obtain the following algebraic system for the equilibrium point

10

1 Introduction

  col X3∗ , X4∗ , X5∗ , X6∗ , Y1∗ , Y2∗ , corresponding to these desirable values of the power levels: a13 εX3∗ + c15 εX1d X5∗ + X1d Y1∗ = −a11 (ε)X1d − X2d , a24 εX4∗ + c26 εX2d X6∗ + X2d Y2∗ = −a22 (ε)X2d − X1d , a33 X3∗ = −a31 X1d , a44 X4∗ = −a42 X2d , a55 X5∗ = −a51 X1d , a66 X6∗ = −a62 X2d . This system has the unique solution X3∗ = −

a31 a42 a51 a62 X1d , X4∗ = − X2d , X5∗ = − X1d , X6∗ = − X2d , a33 a44 a55 a66

Y1∗ = Y1∗ (ε) = −

a11 (ε)X1d + X2d + a13 εX3∗ + c15 εX1d X5∗ , X1d

Y2∗ = Y2∗ (ε) = −

a22 (ε)X2d + X1d + a24 εX4∗ + c26 εX2d X6∗ . X2d

(1.15)

The (1.13) around the Euclidean state vector  linearization of the nonlinear system  col X1d , X2d , X3∗ , X4∗ , X5∗ , X6∗ , Y1∗ , Y2∗ yields the following singularly perturbed linear controlled system with the small (of order of ε) state delay: dx1 (t) ∗ = a11 (ε)x1 (t) + x2 (t − ε) + a13 εx3 (t) + c15 εX1d x5 (t) + X1d y1 (t), dt dx2 (t) ∗ (ε)x2 (t) + x1 (t − ε) + a24 εx4 (t) + c26 εX2d x6 (t) + X2d y2 (t), = a22 dt   dx3 (t) = ε a31 x1 (t) + a33 x3 (t) , dt   dx4 (t) = ε a42 x2 (t) + a44 x4 (t) , dt   dx5 (t) = ε a51 x1 (t) + a55 x5 (t) , dt   dx6 (t) = ε a62 x2 (t) + a66 x6 (t) , dt dy1 (t) = −y1 (t) + u1 (t), ε dt dy2 (t) ε = −y2 (t) + u2 (t), dt (1.16)

1.1 Real-Life Models

11

where x1 (t)=X1 (t)−X1d , x2 (t)=X2 (t)−X2d , xj (t)=Xj (t)−Xj∗ , j = 3, . . . , 6; yi (t)=Yi (t)−Yi∗ ,

i = 1, 2; (1.17)

∗ ∗ a11 (ε)=a11 (ε)+c15 εX5∗ +Y1∗ (ε), a22 (ε)=a22 (ε) + c26 εX6∗ + Y2∗ (ε);

(1.18)

ui (t), (i = 1, 2) are the shifted controls having the form ui (t) = Ui (t) − Yi∗ ,

i = 1, 2.

(1.19)

1.1.5 Car-Following Model: Lane as a Simple Open Curve Consider the model of vehicular traffic flow or the car-following model (see e.g. [6, 8] and references therein). Here, we treat the case of three vehicles, which follow each other in one lane having the geometric shape of a simple open curve (for instance, a straight line). For this shape of the lane, the vehicular traffic flow’s model is the following system of time delay differential equations:  dZL (t − ηL ) dZF 1 (t − η1 )  − τ1 , dt dt   d 2 ZF 2 (t) dZF 1 (t − η1 ) dZF 2 (t − η2 )  − τ2 , = dt dt dt 2 d 2 ZF 1 (t) = dt 2



(1.20)

where dZL (t)/dt is the current speed of the leading vehicle; d 2 ZF 1 (t)/dt 2 and dZF 1 (t)/dt are the current acceleration value and speed, respectively, of the first following vehicle; d 2 ZF 2 (t)/dt 2 and dZF 2 (t)/dt are the current acceleration value and speed, respectively, of the second following vehicle; τ1 > 0 and τ2 > 0 are the time constants of the first and second following vehicles; ηL > 0 is the delay in the reaction of the driver of the leading vehicle; η1 > 0 and η2 > 0 are the delays in the reaction of the drivers of the first and second following vehicles. Now, we transform the variables in the system (1.20) as dZL (t) = XL (t), dt

dZF 1 (t) = XF 1 (t), dt

dZF 2 (t) = XF 2 (t), dt

which yields  dXF 1 (t)  = XL (t − ηL ) − XF 1 (t − η1 ) /τ1 , dt  dXF 2 (t)  = XF 1 (t − η1 ) − XF 2 (t − η2 ) /τ2 . dt

(1.21)

12

1 Introduction

Remark 1.1 We consider the system (1.20) in the time interval [0, tc ], and we assume that the speed XL (t − ηL ) of the leading vehicle is a known continuous function in this interval: XL (t − ηL )/dt = f (t),

t ∈ [0, tc ].

(1.22)

It should be noted the following. If t varies in the interval [0, tc ], the delayed argument in the speed of the leading vehicle varies in the interval [−ηL , tc − ηL ]. Thus, the above made assumption means that we know the actual speed of the leading vehicle in the interval [−ηL , tc − ηL ]. Here, we consider the following two cases: (I) the first following vehicle is much more agile than the second following vehicle (τ1 /τ2  1), and η1 /τ1 ∼ O(1), η2 /τ2 ∼ O(1), tc /τ2 ∼ O(1); (II) the second following vehicle is much more agile than the first following vehicle (τ2 /τ1  1), and η1 /τ2 ∼ O(1), η2 /τ2 ∼ O(1), tc /τ1 ∼ O(1). Case I (τ1 /τ2  1, η1 /τ1 ∼ O(1), η2 /τ2 ∼ O(1), tc /τ2 ∼ O(1)) Let us make the following transformations of the independent variable and the unknown functions in the system (1.21): t = τ2 θ,

XF 1 (τ2 θ ) = y(θ ),

XF 2 (τ2 θ ) = x(θ ),

where θ is a new independent variable (the nondimensional time); y(θ ) and x(θ ) are new unknown functions. Due to these transformations and the Eq. (1.22), the system (1.21) becomes as dx(θ ) = −x(θ − g) + y(θ − εh), dθ dy(θ ) ε = −y(θ − εh) + f (τ2 θ ), dθ

θ ∈ [0, θc ], θ ∈ [0, θc ],

(1.23)

where ε=

τ1 , τ2

g=

η2 , τ2

h=

η1 , τ1

θc =

tc , τ2

and ε > 0 is a small parameter. The system (1.23) is a singularly perturbed system with a nonsmall delay (of order of 1) in the slow state variable x(·) and with a small delay (of order of ε) in the fast state variable y(·). Case II (τ2 /τ1  1, η1 /τ2 ∼ O(1), η2 /τ2 ∼ O(1), tc /τ1 ∼ O(1)) In this case, we transform the independent variable and the unknown functions in the system (1.21) as t = τ1 θ,

XF 1 (τ1 θ ) = x(θ ),

XF 2 (τ1 θ ) = y(θ ),

1.1 Real-Life Models

13

where, like in Case I, θ is a new independent variable (the nondimensional time); x(θ ) and y(θ ) are new unknown functions. These transformations, along with the Eq. (1.22), convert the system (1.21) to the form dx(θ ) = −x(θ − εh1 ) + f (τ1 θ ), dθ dy(θ ) = x(θ − εh1 ) − y(θ − εh2 ), ε dθ

θ ∈ [0, θc ], θ ∈ [0, θc ],

(1.24)

where ε=

τ2 , τ1

h1 =

η1 , τ2

h2 =

η2 , τ2

θc =

tc , τ1

and ε > 0 is a small parameter. The system (1.24) is a singularly perturbed system with small delays (of order of ε) in both state variables x(·) and y(·). Remark 1.2 Let us assume that the speed of the leading vehicle is not a known function f (t) but it is a control v(t) at the disposal of its driver in the entire interval [0, tc ]. Due to Remark 1.1, this means that the control v(t) is the actual speed of the leading vehicle in the entire interval [−ηL , tc − ηL ]. Thus, the systems (1.23) and (1.24) become the following singularly perturbed controlled systems with state delays: dx(θ ) = −x(θ − g) + y(θ − εh), dθ dy(θ ) = −y(θ − εh) + u(θ ), ε dθ

θ ∈ [0, θc ], θ ∈ [0, θc ],

(1.25)

and dx(θ ) = −x(θ − εh1 ) + u(θ ), dθ dy(θ ) = x(θ − εh1 ) − y(θ − εh2 ), ε dθ

θ ∈ [0, θc ], θ ∈ [0, θc ].

(1.26)

In (1.25) the control is u(θ ) = v(τ2 θ ), while in (1.26) the control is u(θ ) = v(τ1 θ ). Now, let us consider a situation, which differs from that described in Remark 1.2. Namely, we assume that tc > ηL . Also, we assume the following. In order to provide a desirable behavior of the system (1.21), the driver of the leading vehicle can properly choose its current speed only in the time interval (ηL , tc ], and therefore, its actual speed only in the time interval (0, tc − ηL ]. However, the current speed in the interval [0, ηL ] (the actual speed in the interval [−ηL , 0]) can

14

1 Introduction

be any known continuous function. We treat this situation in the following case of the system’s (1.21) data: τ1 /τ2  1,

η1 /τ1 ∼ O(1),

η2 /τ2 ∼ O(1),

ηL /τ1 ∼ O(1),

tc /τ2 ∼ O(1).

We make the following transformations of the independent variable and the unknown functions in the system (1.21): XF 1 (τ2 θ ) = y(θ ),

t = τ2 θ,

XF 2 (τ2 θ ) = x(θ ),

XL (τ2 θ ) = u(θ ),

where θ is a new independent variable (the nondimensional time); y(θ ) and x(θ ) are new unknown functions (Euclidean state variables); u(θ ) is a control. Due to these transformations, the system (1.21) becomes as dx(θ ) = −x(θ − g) + y(θ − εh1 ), dθ dy(θ ) = −y(θ − εh1 ) + u(θ − εh2 ), ε dθ

θ ∈ [0, θc ], θ ∈ [0, θc ],

(1.27)

where ε=

τ1 , τ2

g=

η2 , τ2

h1 =

η1 , τ1

h2 =

ηL , τ1

θc =

tc , τ2

and ε > 0 is a small parameter. The system (1.27) is a singularly perturbed controlled system with state and control delays. The state delays are of two scales: a nonsmall delay (of order of 1) in the slow state variable x(·) and a small delay (of order of ε) in the fast state variable y(·). The delay in the control is small (of order of ε).

1.1.6 Car-Following Model: Lane as a Simple Closed Curve Here, we consider another than in Sect. 1.1.5 car-following model of three vehicles. Namely, we consider the case where the vehicles follow each other in one lane having the geometric shape of a simple closed curve (for instance, a circle). For this shape of the lane, there are not a leading vehicle and following vehicles, and the vehicular traffic flow is modeled by the following system of time delay differential equations:  dZ3 (t − η3 ) dZ1 (t − η1 )  − τ1 , dt dt   d 2 Z2 (t) dZ1 (t − η1 ) dZ2 (t − η2 )  τ2 , = − dt dt dt 2

d 2 Z1 (t) = dt 2



1.1 Real-Life Models

15

d 2 Z3 (t) = dt 2



 dZ2 (t − η2 ) dZ3 (t − η3 )  − τ3 , dt dt

(1.28)

where d 2 Zi (t)/dt 2 and dZi (t)/dt, (i = 1, 2, 3) is the current acceleration value and speed, respectively, of the corresponding vehicle; τi > 0, (i = 1, 2, 3) is the time constant of the corresponding vehicle; ηi > 0, (i = 1, 2, 3) is the delay in the reaction of the driver of the corresponding vehicle. Note that in [8], the stability of a car-following model in a ring was studied. Here and in the next chapter, we study another property of the system (1.28). We consider the system (1.28) in the time interval [0, tc ], and we transform the unknown functions of this system as dZi (t) = Xi (t), dt

i = 1, 2, 3,

yielding  dX1 (t)  = X3 (t − η3 ) − X1 (t − η1 ) τ1 , dt  dX2 (t)  = X1 (t − η1 ) − X2 (t − η2 ) τ2 , dt  dX3 (t)  = X2 (t − η2 ) − X3 (t − η3 ) τ3 . dt

(1.29)

Now, we consider the following two cases: (A) the first vehicle is much more agile than the second and third vehicles (τ1 /τ2  1, τ1 /τ3  1), and tc /τ2 ∼ O(1); (B) the second and third vehicles are much more agile than the first vehicle (τ2 /τ1  1, τ3 /τ1  1), and tc /τ1 ∼ O(1). We also assume that in both cases the agilities of the second and third vehicles are comparable with each other: τ2 ∼ O(1), τ3 and each delay ηi and the corresponding time constant τi are comparable with each other: η1 ∼ O(1), τ1

η2 ∼ O(1), τ2

η3 ∼ O(1). τ3

Case A (τ1 /τ2  1, τ1 /τ3  1, tc /τ2 ∼ O(1)) In this case we make the following transformations of the independent variable and the unknown functions in the system (1.29): t = τ2 θ,

X1 (τ2 θ ) = y(θ ),

X2 (τ2 θ ) = x1 (θ ),

X3 (τ2 θ ) = x2 (θ ),

16

1 Introduction

where θ is a new independent variable (the nondimensional time); y(θ ), x1 (θ ), and x2 (θ ) are new unknown functions. Due to these transformations, the system (1.29) becomes as dx1 (θ ) = −x1 (θ − g1 ) + y(θ − εh), dθ   dx2 (θ ) = τ x1 (θ − g1 ) − x2 (θ − g2 ) , dθ dy(θ ) = x2 (θ − g2 ) − y(θ − εh), ε dθ

θ ∈ [0, θc ], θ ∈ [0, θc ], θ ∈ [0, θc ],

(1.30)

where ε=

τ1 , τ2

g1 =

η2 , τ2

g2 =

η3 , τ2

h=

η1 , τ1

τ=

τ2 , τ3

θc =

tc , τ2

and ε > 0 is a small parameter. The system (1.30) is a singularly perturbed system consisting of two slow modes (the first two equations) and one fast mode (the third equation). This system has two nonsmall delays (of the order of 1) in the slow state variables x1 (·) and x2 (·), and it has the small delay (of the order of ε) in the fast state variable y(·). Case B (τ2 /τ1  1, τ3 /τ1  1, tc /τ1 ∼ O(1)) In this case we make the following transformations of the independent variable and the unknown functions in the system (1.29): t = τ1 θ,

X1 (τ1 θ ) = x(θ ),

X2 (τ1 θ ) = y1 (θ ),

X3 (τ1 θ ) = y2 (θ ),

where θ is a new independent variable (the nondimensional time); x(θ ), y1 (θ ), and y2 (θ ) are new unknown functions. These transformations convert the system (1.29) to the system dx(θ ) = −x(θ − g) + y2 (θ − εh2 ), dθ dy1 (θ ) = x(θ − g) − y1 (θ − εh1 ), ε dθ   dy2 (θ ) = τ y1 (θ − εh1 ) − y2 (θ − εh2 ) , ε dθ

θ ∈ [0, θc ], θ ∈ [0, θc ], θ ∈ [0, θc ],

(1.31)

where ε=

τ2 , τ1

g=

η1 , τ1

h1 =

and ε > 0 is a small parameter.

η2 , τ2

h2 =

η3 , τ2

τ=

τ2 , τ3

θc =

tc , τ1

References

17

The system (1.31) is a singularly perturbed system consisting of one slow mode (the first equation) and two fast modes (the second and third equations). This system has one nonsmall delay (of the order of 1) in the slow state variable x(·), and it has two small delays (of the order of ε) in the fast state variables y1 (·) and y2 (·).

References 1. Bensoussan, A., Da Prato, G., Delfour, M.C., Mitter, S.K.: Representation and Control of Infinite Dimensional Systems. Birkhuser, Boston (2007) 2. Buˇcys, K., Švitra, D.: Modelling of nuclear reactors dynamics. Math. Model Anal. 4, 26–32 (1999) 3. Casal, A., Somolinos, A.: Forced oscillations for the sunflower equation, entrainment. Nonlinear Anal. 6, 397–414 (1982) 4. Curtain, R.F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York (1995) 5. Dmitriev, M.G., Kurina, G.A.: Singular perturbations in control problems. Autom. Remote Control 67, 1–43 (2006) 6. Erneux, T.: Applied Delay Differential Equations. Springer, New York (2009) 7. Fridman E.: Robust sampled-data H∞ control of linear singularly perturbed systems. IEEE Trans. Autom. Control 51, 470–475 (2006) 8. Fridman, E.: Introduction to Time-Delay Systems. Birkhauser, New York (2014) 9. Gabasov, R., Kirillova, F.M.: The Qualitative Theory of Optimal Processes. Marcel Dekker, New York (1976) 10. Gabasov, R., Kirillova, F.M., Krakhotko, V.V.: Controllability of multiloop systems with lumped parameters. Autom. Remote Control 32, 1710–1717 (1971) 11. Gajic, Z., Lim, M.T.: Optimal Control of Singularly Perturbed Linear Systems and Applications. High Accuracy Techniques. Marsel Dekker, New York (2001) 12. Glizer, V.Y.: Optimal planar interception with fixed end conditions: a closed form solution. J. Optim. Theory Appl. 88, 503–539 (1996) 13. Glizer, V.Y.: Optimal planar interception with fixed end conditions: approximate solutions. J. Optim. Theory Appl. 93, 1–25 (1997) 14. Glizer, V.Y.: Euclidean space controllability of singularly perturbed linear systems with state delay. Syst. Control Lett. 43, 181–191 (2001) 15. Glizer, V.Y.: Controllability of singularly perturbed linear time-dependent systems with small state delay. Dynam. Control 11, 261–281 (2001) 16. Glizer, V.Y.: Controllability of nonstandard singularly perturbed systems with small state delay. IEEE Trans. Automat. Control 48, 1280–1285 (2003) 17. Glizer, V.Y.: Novel controllability conditions for a class of singularly perturbed systems with small state delays. J. Optim. Theory Appl. 137, 135–156 (2008) 18. Glizer, V.Y.: L2 -stabilizability conditions for a class of nonstandard singularly perturbed functional-differential systems. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms. 16, 181–213 (2009) 19. Glizer, V.Y.: Controllability conditions of linear singularly perturbed systems with small state and input delays. Math. Control Signals Syst. 28(1), 1–29 (2016) 20. Glizer, V.Y.: Euclidean space controllability conditions and minimum energy problem for time delay system with a high gain control. J. Nonlinear Var. Anal. 2, 63–90 (2018) 21. Glizer, V.Y.: Euclidean space output controllability of singularly perturbed systems with small state delays. J. Appl. Math. Comput. 57, 1–38 (2018) 22. Glizer, V.Y.: Euclidean space controllability conditions for singularly perturbed linear systems with multiple state and control delays. Axioms. 8, Paper No. 36 (2019)

18

1 Introduction

23. Glizer, V.Y.: Euclidean space controllability conditions of singularly perturbed systems with multiple state and control delays. In: 2019 IEEE 15th International Conference on Control and Automation (ICCA), pp. 1144–1149, Edinburgh, Scotland (2019) 24. Glizer, V.Y.: Conditions of functional null controllability for some types of singularly perturbed nonlinear systems with delays. Axioms. 8, Paper No. 80 (2019) 25. Glizer, V.Y.: Novel conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numer. Algebra Control Optim. (2020). https://doi.org/10.3934/naco. 2020027 26. Glizer, V.Y., Turetsky, V.: Robust Controllability of Linear Systems. Nova Science, New York (2012) 27. Glizer, V.Y., Fridman, E., Feigin, Y.: A novel approach to exact slow-fast decomposition of linear singularly perturbed systems with small delays. SIAM J. Control Optim. 55, 236–274 (2017) 28. Halanay, A.: On the controllability of linear difference-differential systems. In: Kuhn, H.W., Szegö, G.P. (eds.) Mathematical Systems Theory and Economics. Lecture Notes in Operations Research and Mathematical Economics Book Series, vol. 12, pp. 329–336, 2nd edn. Springer, Berlin (1969) 29. Kaczorek T.: Linear Control Systems. Research Studies Press and Wiley, New York (1993) 30. Kalman, R.E.: Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana 5, 102–119 (1960) 31. Klamka, J.: Controllability of Dynamical Systems. Kluwer Academic Publishers, Dordrecht (1991) 32. Klamka, J.: Controllability of dynamical systems. A survey. Bull. Pol. Acad. Sci.: Tech. 61, 335–342 (2013) 33. Kokotovic, P.V., Haddad, A.H.: Controllability and time-optimal control of systems with slow and fast modes. IEEE Trans. Automat. Control 20, 111–113 (1975) 34. Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic, London (1986) 35. Kopeikina, T.B.: Controllability of singularly perturbed linear systems with time-lag. Differ. Equ. 25, 1055–1064 (1989) 36. Kopeikina, T.B.: Unified method of investigating controllability and observability problems of time-variable differential systems. Funct. Differ. Equ. 13, 463–481 (2006) 37. Kuehn, C.: Multiple Time Scale Dynamics. Springer, New York (2015) 38. Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary-value problems for differential-difference equations. Part V: small shifts with layer behavior. SIAM J. Appl. Math. 54, 249–272 (1994) 39. Naidu, D.S.: Singular perturbations and time scales in control theory and applications: an overview. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 9, 233–278 (2002) 40. Naidu, D.S., Calise, A.J.: Singular perturbations and time scales in guidance and control of aerospace systems: a survey. J. Guid. Control Dyn. 24, 1057–1078 (2001) 41. O’Malley, R.E., Jr.: Historical Developments in Singular Perturbations. Springer, New York (2014) 42. Pavel, L.: Game Theory for Control of Optical Networks. Birkhauser, Basel (2012) 43. Pena, M.L.: Asymptotic expansion for the initial value problem of the sunflower equation. J. Math. Anal. Appl. 143, 471–479 (1989) 44. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience, New York (1962) 45. Reddy, P.B., Sannuti, P.: Optimal control of a coupled-core nuclear reactor by singular perturbation method. IEEE Trans. Automat. Control 20, 766–769 (1975) 46. Sannuti, P.: On the controllability of singularly perturbed systems. IEEE Trans. Automat. Control 22, 622–624 (1977) 47. Sannuti, P.: On the controllability of some singularly perturbed nonlinear systems. J. Math. Anal. Appl. 64, 579–591 (1978)

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48. Schöll, E., Hiller, G., Hövel, P., Dahlem, M.A.: Time-delayed feedback in neurosystems. Phil. Trans. R. Soc. A 367, 1079–1096 (2009) 49. Stefanovic, N., Pavel, L.: A Lyapunov-Krasovskii stability analysis for game-theoretic based power control in optical links. Telecommun. Syst. 47, 19–33 (2011) 50. Stefanovic, N., Pavel, L.: Robust power control of multi-link single-sink optical networks with time-delays. Automatica J. IFAC 49, 2261–2266 (2013) 51. Tsekhan, O.: Complete controllability conditions for linear singularly-perturbed time-invariant systems with multiple delays via Chang-type transformation. Axioms 8, Paper No. 71 (2019) 52. Zhang, Y., Naidu, D.S., Cai, C., Zou, Y.: Singular perturbations and time scales in control theories and applications: an overview 2002–2012. Int. J. Inf. Syst. Sci. 9, 1–36 (2014)

Chapter 2

Singularly Perturbed Linear Time Delay Systems

2.1 Introduction In this chapter, some basic notions and results in the topic of singularly perturbed linear time delay systems are presented. Namely, we consider the singularly perturbed systems of three kinds: (1) with small delays; (2) with delays of two scales (small and nonsmall); and (3) with nonsmall delays. In the first and second kinds of the systems, the small delays are proportional to a small positive parameter multiplying a part of the derivatives. The nonsmall delays in the second and third kinds of the systems are independent of this parameter. We consider the asymptotic slow–fast decomposition of each kind of the systems. Also, we study some properties and estimates of the fundamental matrix solutions to these systems. The following main notations are applied in this chapter: 1. E n is the n-dimensional real Euclidean space. 2. The Euclidean norm of either a matrix or a vector is denoted by · . 3. The upper index T denotes the transposition either of a vector x (x T ) or of a matrix A (AT ). 4. In denotes the identity matrix of dimension n. 5. L2 [t1 , t2 ; E n ] denotes the linear space of all functions x(·) : [t1 , t2 ] → E n square integrable in the interval [t1 , t2 ]. 6. col(x, y), where x ∈ E n , y ∈ E m , denotes the column block vector of the dimension n + m with the upper block x and the lower block y, i.e., col(x, y) = (x T , y T )T . 7. Reλ denotes the real part of a complex number λ. 8. Imλ denotes the imaginary part of a complex number λ.   9. M [θ1 , θ2 ; n] denotes the linear space of the pairs f (θ ) = fE , fL (θ ) , where fE ∈ E n , fL (θ ) ∈ L2 [θ1 , θ2 ; E n ].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Y. Glizer, Controllability of Singularly Perturbed Linear Time Delay Systems, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-65951-6_2

21

22

2 Singularly Perturbed Linear Time Delay Systems

2.2 Singularly Perturbed Systems with Small Delays 2.2.1 Original System Consider the system

dx(t)  = A1j (t, ε)x(t − εhj ) + A2j (t, ε)y(t − εhj ) dt N

 +

j =0

0 −h



G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)y(t + εη) dη + b1 (t, ε), t ≥ 0, (2.1)

dy(t)  = A3j (t, ε)x(t − εhj ) + A4j (t, ε)y(t − εhj ) dt N

ε  +

j =0

0 −h



G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη + b2 (t, ε), t ≥ 0, (2.2)

where x(t) ∈ E n , y(t) ∈ E m ; ε > 0 is a small parameter; N ≥ 1 is an integer; 0 = h0 < h1 < h2 < . . . < hN = h are some given constants independent of ε; Aij (t, ε), Gi (t, η, ε) (i = 1, . . . , 4; j = 0, . . . , N) are matrix-valued functions of corresponding dimensions, given for t ≥ 0, η ∈ [−h, 0] and ε ∈ [0, ε0 ] (ε0 > 0); bk (t, ε), (k = 1, 2) are vector-valued functions of corresponding dimensions, given for t ≥ 0, ε ∈ [0, ε0 ]. The system (2.1)–(2.2) is a differential system with the point-wise and distributed delays in the state variables x(·) and y(·). Such a type of differential systems is called a functional-differential system. These The  systems are infinite-dimensional.  state variable of (2.1)–(2.2) has the form z(t), z(t + εη) , η ∈ [−h, 0), where     z(t) = col x(t), y(t) and z(t + εη) = col x(t + εη), y(t + εη) . The component z(t) of the state variable is called its Euclidean part, while the component z(t + εη) is called the functional part of the state variable. For any given t ≥ 0 and ε ∈ (0, ε0 ], we consider the space M [−εh, 0; n + m] as a state space for the system (2.1)–(2.2). More details on a functional-differential system and its state space can be found for instance in [5] and references therein. Since a part of the derivatives in the system (2.1)–(2.2), namely dy(t)/dt, is multiplied by the small positive parameter ε, this system is singularly perturbed (see, e.g., [6, 11, 17, 22, 24, 26, 34, 37] and references therein). The parameter ε is called a parameter of singular perturbation. The important feature of (2.1)–(2.2) is that all the delays are proportional to the small parameter ε. This system is singularly perturbed not only because of the presence of the small multiplier for dy(t)/dt, but also because of the small delays (see, e.g., [26] (Section 18.4) and references therein). Equation (2.1) is called a slow mode, and the Euclidean part x(t) of the   state variable x(t), x(t +εη) is called a slow Euclidean part of the state variable (or

2.2 Singularly Perturbed Systems with Small Delays

23

state variable). Equation (2.2) and the entire state variable simply, a slow Euclidean  y(t), y(t + εη) are called a fast mode and a fast state variable, respectively, while y(t) is called a fast Euclidean state variable. Remark 2.1 Note that,  although (2.1)  is a slow mode, the functional part x(t + εη) of the state variable x(t), x(t +εη) is fast. Indeed, for any ε ∈ (0, ε0 ], this function satisfies the partial first-order differential equation ε

∂x(t + εη) ∂x(t + εη) − = 0, ∂t ∂η

η ∈ [−h, 0],

t + εη > 0.

The partial derivative with respect to time t in this equation is multiplied by the small parameter ε, meaning that the functional part x(t + εη) is fast.

2.2.2 Slow–Fast Decomposition of the Original System One of the main approaches to the study of the system (2.1)–(2.2) is based on its asymptotic decomposition into two much simpler ε-free subsystems. The first one, called a slow subsystem, is obtained from (2.1)–(2.2) by setting there formally ε = 0. Thus, the slow subsystem, associated with (2.1)–(2.2), has the form dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) + b1 (t, 0), t ≥ 0, dt 0 = A3s (t)xs (t) + A4s (t)ys (t) + b2 (t, 0), t ≥ 0,

(2.3)

where xs (t) ∈ E n and ys (t) ∈ E m are the state variables; Ais (t) =

N  j =0

 Aij (t, 0) +

0 −h

Gi (t, η, 0)dη,

i = 1, . . . , 4.

(2.4)

The slow subsystem (2.3) is an ε-free descriptor (differential-algebraic) system without delays. If det A4s (t) = 0,

t ≥ 0,

(2.5)

then the original singularly perturbed system (2.1)–(2.2) is called standard. Otherwise, this system is called nonstandard. In the case of the standard original system, its slow subsystem (2.3), being a differential-algebraic system, can be converted into an equivalent system. This new system consists of the explicit expression for ys (t) −1 ys (t) = −A−1 4s (t)A3s (t)xs (t) − A4s (t)b2 (t, 0), t ≥ 0,

24

2 Singularly Perturbed Linear Time Delay Systems

and the differential equation with respect to xs (t) dxs (t) = A¯ s (t)xs (t) + b¯s (t), t ≥ 0, dt

(2.6)

where −1 ¯ A¯ s (t) = A1s (t) − A2s (t)A−1 4s (t)A3s (t), bs (t) = b1 (t, 0) − A2s (t)A4s (t)b2 (t, 0). (2.7) Note that Eq. (2.6) is also called the slow subsystem, associated with the system (2.1)–(2.2). The second subsystem, obtained by the asymptotic decomposition of (2.1)–(2.2) and called a fast subsystem, isderived from (2.2)  in the following way: (a) the terms containing the state variable x(t), x(t + εη) are removed from (2.2) and (b) the

transformations of the variables t = t1 + εξ , y(t1 + εξ ) = yf (ξ ) are made in the resulting system, where t1 ≥ 0 is any fixed time instant. Thus, we obtain the system dyf (ξ )  = A4j (t1 + εξ, ε)yf (ξ − hj ) dξ N

j =0

 +

0 −h

G4 (t1 + εξ, η, ε)yf (ξ + η)dη + b2 (t1 + εξ, ε).

Finally, setting formally ε = 0 in this system and replacing t1 with t yield the fast subsystem dyf (ξ )  = A4j (t, 0)yf (ξ − hj ) dξ N

 +

j =0

0 −h

G4 (t, η, 0)yf (ξ + η)dη + b2 (t, 0), ξ ≥ 0,

(2.8)

  where t ≥ 0 is a parameter; yf (ξ ) ∈ E m ; yf (ξ ),  yf (ξ + η) , η ∈ [−h, 0), is a state variable; for any given ξ ≥ 0, we consider yf (ξ ), yf (ξ + η) in the space M [−h, 0; m]. The new independent variable ξ , called the stretched time, is expressed by the original time t in the form ξ = (t − t1 )/ε. For any t > t1 , ξ → +∞ as ε → +0. The fast subsystem is a differential equation with state delays. This equation is of a lower Euclidean dimension than the original system (2.1)–(2.2), and it is ε-free.

2.2 Singularly Perturbed Systems with Small Delays

25

2.2.3 Fundamental Matrix Solution Consider the block vectors:   z(t) = col x(t), y(t) ,

  1 b(t, ε) = col b1 (t, ε), b2 (t, ε) , ε

and the block matrices:   A (t, ε) A2j (t, ε) Aj (t, ε) = 1 1j , j = 0, 1, . . . , N, 1 ε A3j (t, ε) ε A4j (t, ε)   G1 (t, η, ε) G2 (t, η, ε) . G(t, η, ε) = 1 1 ε G3 (t, η, ε) ε G4 (t, η, ε)

(2.9)

Using the above introduced vectors and matrices, we can rewrite the system (2.1)–(2.2) in the form dz(t)  = Aj (t, ε)z(t − εhj ) + dt N

j =0



0

−h

G(t, η, ε)z(t + εη)dη + b(t, ε), t ≥ 0. (2.10)

Let us give the following initial conditions for the system (2.10): z(τ ) = ϕz (τ ), τ ∈ [−ε0 h, 0);

z(0) = z0 .

(2.11)

Along with Eq. (2.10), for a given σ ≥ 0, we consider the initial-value problem for the (n + m) × (n + m)-matrix-valued function Φ(t) dΦ(t)  = Aj (t, ε)Φ(t − εhj ) + dt N

j =0



0 −h

G(t, η, ε)Φ(t + εη)dη, t > σ,

Φ(τσ ) = 0, σ − εh ≤ τσ < σ ;

Φ(σ ) = In+m . (2.12)

Let tc > 0 be a given time instant independent of ε. Let ε ∈ (0, ε0 ] be given. Based on the results of [4, 22], we have the following assertions. Proposition 2.1 Let the matrix-valued functions Aj (t, ε) (j = 0, 1, . . . , N) be continuous in t ∈ [0, tc ]. Let the matrix-valued function G(t, η, ε) be piecewise continuous with respect to η ∈ [−h, 0] for each t ∈ [0, tc ] and be continuous with respect to t ∈ [0, tc ] uniformly in η ∈ [−h, 0]. Then, for any given σ ∈ [0, tc ], the initial-value problem (2.12) has the unique solution Φ(t) = Φ(t, σ, ε), 0 ≤ σ ≤ t ≤ tc . Proposition 2.2 Let the conditions of Proposition 2.1 be valid. Let b(·, ε) ∈ L2 [0, tc ; E n+m ] and ϕz (·) ∈ L2 [−ε0 h, 0; E n+m ]. Then, the initial-value prob-

26

2 Singularly Perturbed Linear Time Delay Systems

lem (2.10)–(2.11) has the unique absolutely continuous solution z(t) = z(t, ε), t ∈ [0, tc ], and this solution has the form  z(t, ε) = Φ(t, 0, ε)z0 + 

0 −εh

0, τ, ε)ϕz (τ )dτ Φ(t,

t

+

Φ(t, σ, ε)b(σ, ε)dσ, 0

σ, τ, ε) is where the (n + m) × (n + m)-matrix-valued function Φ(t, σ, τ, ε) = Φ(t,

 N  Φ(t, σ + τ + εhj , ε)Aj (σ + τ + εhj , ε), σ + τ − t ≤ −εhj ≤ τ 0, otherwise j =1

 +

τ/ε −h

Φ(t, σ + τ − εχ , ε)G(σ + τ − εχ , χ , ε)dχ .

The matrix-valued function Φ(t, σ, ε) is called the fundamental matrix solution of the homogeneous system, corresponding to the system (2.10) (and to the system (2.1)–(2.2)). This matrix-valued function plays an important role in analysis of (2.1)–(2.2). In the sequel, for the sake of the brevity, we call Φ(t, σ, ε) the fundamental matrix of the system (2.10) (and of the system (2.1)–(2.2)). Similarly to the fundamental matrix Φ(t, σ, ε) of the system (2.1)–(2.2), we introduce the fundamental matrices Φs (t, σ ) and Φf (ξ, t) of the slow (2.6) and fast (2.8) subsystems, respectively. For a given σ ≥ 0, the n × n-matrix-valued function Φs (t, σ ) is the unique solution of the initial-value problem dΦs (t, σ ) = A¯ s (t)Φs (t, σ ), t > σ, dt

Φs (σ, σ ) = In ,

(2.13)

while, for any t ≥ 0, the m × m-matrix-valued function Φf (ξ, t) is the unique solution of the initial-value problem dΦf (ξ, t)  = A4j (t, 0)Φf (ξ − hj , t) dξ N

 +

j =0

0 −h

G4 (t, η, 0)Φf (ξ + η, t)dη,

Φf (ξ, t) = 0,

ξ < 0,

ξ > 0,

Φf (0, t) = Im .

Similarly to Proposition 2.2, we have the following two assertions.

(2.14)

2.2 Singularly Perturbed Systems with Small Delays

27

Proposition 2.3 Let the inequality in (2.5) be valid for t ∈ [0, tc ]. Let the matrix-valued function A¯ s (t) be continuous in the interval [0, tc ], while the vectorvalued function b¯s (t) belongs to L2 [0, tc ; E n ]. Then, for any given xs0 ∈ E n , the system (2.6) subject to the initial condition xs (0) = x0 has the unique absolutely continuous solution  t Φs (t, σ )b¯s (σ )dσ, t ∈ [0, tc ]. xs (t) = Φs (t, 0)xs0 + 0

Proposition 2.4 Let t ∈ [0, tc ] be any given value. Let, for this t, the matrixvalued function G4 (t, η, 0), η ∈ [−h, 0], be piecewise continuous. Then, for any given vector-valued function ϕf 0 (·) ∈ L2 [−h, 0; E m ] and vector yf 0 ∈ E m , the system (2.8) subject to the initial conditions yf (κ) = ϕf 0 (κ), κ ∈ [−h, 0), yf (0) = yf 0 has the unique absolutely continuous solution  yf (ξ, t) = Φf (ξ, t)yf 0 + 

ξ

+

0

−h

f (ξ, κ, t)ϕf 0 (κ)dκ Φ

Φf (ξ − ω, t)b2 (t, 0)dω,

ξ ≥ 0,

0

where f (ξ, κ, t) = Φ

 N  Φf (ξ − κ − hj , t)A4j (κ + hj , 0), κ − ξ ≤ −hj ≤ κ 0, otherwise j =1

 +

κ −h

Φf (ξ − κ + η, t)G4 (t, η, 0)dη.

2.2.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Small Delays In this subsection, we obtain estimates of the solutions to initial/terminal-value problems for some singularly perturbed linear time delay matrix differential systems in a given time interval [0, tc ], where tc > 0 is independent of the small parameter of singular perturbations ε. We consider the system in (2.12) and some other systems where all the delays are proportional to this parameter. In what follows of this subsection, we assume that:

28

2 Singularly Perturbed Linear Time Delay Systems

The matrix-valued functions Aij (t, ε) (i = 1, . . . , 4; j = 0, 1, . . . , N) are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ]. The matrix-valued functions Gi (t, η, ε) (i = 1, . . . , 4) are piecewise continuous with respect to η ∈ [−h, 0] for each (t, ε) ∈ [0, tc ] × [0, ε0 ], and they are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in η ∈ [−h, 0]. All roots λ(t) of the equation

(AI-1) (AII-1)

(AIII-1)

⎡ det ⎣λIm −

N 

 A4j (t, 0) exp(−λhj ) −

j =0

⎤ 0

−h

G4 (t, η, 0) exp(λη)dη⎦ = 0

(2.15) satisfy the inequality Reλ(t) < −2β for all t ∈ [0, tc ], where β > 0 is some constant. Remark 2.2 For any given t ∈ [0, tc ], Eq. (2.15) is a quasi-polynomial equation with respect to λ. This equation is called the characteristic equation of the homogeneous equation corresponding to the fast subsystem (2.8). In the sequel, for the sake of brevity, we call (2.15) the characteristic equation of the fast subsystem (2.8). Let for any given ε ∈ (0, ε0 ] and σ ∈ [0, tc ], Y (t, σ, ε) be the m × m-matrixvalued function satisfying the following differential equation and initial conditions: dY (t, σ, ε)  = A4j (t, ε)Y (t − εhj , σ, ε) dt N

ε  +

j =0

0 −h

G4 (t, η, ε)Y (t + εη, σ, ε)dη, σ < t ≤ tc ,

Y (t, σ, ε) = 0, σ − εh ≤ t < σ,

Y (σ, σ, ε) = Im .

(2.16)

Along with the problem (2.16), we consider the following quasi-polynomial equation with respect to μ: ⎡ det ⎣μIm −

N  j =0

 A4j (t, ε) exp(−μhj ) −

⎤ 0

−h

G4 (t, η, ε) exp(μη)dη⎦ = 0. (2.17)

Lemma 2.1 Let the assumptions (AI-1)–(AIII-1) be valid. Then, there exists a positive number ε1 (ε1 ≤ ε0 ) such that, for all (t, ε) ∈ [0, tc ] × [0, ε1 ], all roots μ(t, ε) of Eq. (2.17) satisfy the inequality Reμ(t, ε) < −2β. Proof We prove the lemma by contradiction, i.e., we assume that the statement of the lemma is wrong. This means the existence of three sequences {tl }, {εl }, and {μl } with the following properties: (a) tl ∈ [0, tc ] (l = 1, 2, . . .); (b) εl > 0 (l =

2.2 Singularly Perturbed Systems with Small Delays

29

1, 2, . . .), and liml→+∞ εl = 0;  (c) Reμl ≥ −2β (l = 1, 2, . . .); (d) Eq. (2.17) is satisfied for any triplet t, ε, μ = (tl , εl , μl ) (l = 1, 2, . . .). Due to the property (a), there exists a convergent subsequence of {tl }. For the sake of simplicity (but without loss of generality), we assume that the sequence {tl } itself is convergent, and t¯ = liml→+∞ tl . It is clear that t¯ ∈ [0, tc ]. The following two cases can be distinguished with respect to the sequence {μl }: (i) {μl } is bounded and (ii) {μl } is unbounded. We start with the first case. In this case, there exists a convergent subsequence of {μl }. For the sake of simplicity (but without loss of generality), we assume that the sequence {μl } itself is such a subsequence. Let μ¯ = liml→+∞  μl .Due to the abovementioned property (c), Reμ¯ ≥ −2β. Substitution of t, ε, μ = (tl| , εl , μl ) into (2.17), followed by calculation of the limit of the resulting equality for l → +∞, yields ⎡ det ⎣μI ¯ m−

N  j =0

A4j (t¯, 0) exp(−μh ¯ j) −



⎤ 0 −h

⎦ = 0. G4 (t¯, η, 0) exp(μη)dη ¯

The latter means that μ¯ is a root of Eq. (2.15) for t = t¯. Thus, due to the assumption (AIII-1), Reμ¯ < −2β, which contradicts the above obtained inequality Reμ¯ ≥ −2β. Proceed to the case (ii) where the sequence {μl } is unbounded. In this case, there exists a subsequence of {μl }, modules of elements of which tend to infinity. Similarly to the case (i), we assume  that {μl } itself is such a subsequence, i.e., liml→+∞ |μl | = +∞. Substituting t, ε, μ = (tl , εl , μl ) into (2.17), dividing the resulting equality by (μl )m , and then calculating the limit of the last equality for l → +∞, one obtains the contradiction 1 = 0. The contradictions, obtained in the cases (i) and (ii), prove the lemma.   The following proposition is a direct consequence of Lemma 2.1 and the results of [3, 15] on the estimate of the fundamental matrix solution of a purely fast singularly perturbed differential system with small delays. Proposition 2.5 Let the assumptions (AI-1)–(AIII-1) be valid. Then, there exists a positive number ε2 (ε2 ≤ ε1 ) such that, for all ε ∈ (0, ε2 ], the solution of the initialvalue problem (2.16) satisfies the inequality  

Y (t, σ, ε) ≤ a exp − β(t − σ )/ε ,

0 ≤ σ ≤ t ≤ tc ,

(2.18)

where a > 0 is some constant independent of ε. Let us partition the solution Φ(t, σ, ε) of the initial-value problem (2.12) into blocks as   Φ1 (t, σ, ε) Φ2 (t, σ, ε) Φ(t, σ, ε) = , (2.19) Φ3 (t, σ, ε) Φ4 (t, σ, ε)

30

2 Singularly Perturbed Linear Time Delay Systems

where the matrices Φ1 (t, σ, ε), Φ2 (t, σ, ε), Φ3 (t, σ, ε), and Φ4 (t, σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Lemma 2.2 Let the assumptions (AI-1)–(AIII-1) be valid. Then, there exists a positive number ε3 (ε3 ≤ ε2 ) such that, for all ε ∈ (0, ε3 ], the following inequalities are satisfied:

Φp (t, σ, ε) ≤ a, p = 1, 3,

Φ2 (t, σ, ε) ≤ aε,

 

Φ4 (t, σ, ε) ≤ a ε + exp − β(t − σ )/ε , 0 ≤ σ ≤ t ≤ tc , where a > 0 is some constant independent of ε. Proof of the lemma is presented in Sect. 2.2.9. Let, for any given ε ∈ (0, ε3 ], the (n + m) × (n + m)-matrix-valued function Ψ (σ, ε), σ ∈ [0, tc ], be a solution of the terminal-value problem  T dΨ (σ, ε) Aj (σ + εhj , ε) Ψ (σ + εhj , ε) =− dσ N

 −

j =0

0 −h

 T G(σ − εη, η, ε) Ψ (σ − εη, ε)dη, σ ∈ [0, tc ), Ψ (tc , ε) = In+m ,

Ψ (σ, ε) = 0, σ > tc .

(2.20)

In this problem, it is assumed that the blocks of the matrices Aj (t, ε) (j = 0, 1, . . . , N) and G(t, η, ε) satisfy the following equalities for all ε ∈ [0, ε3 ]: Aij (t, ε)=Aij (tc , ε), Gi (t, η, ε)=Gi (tc , η, ε), t>tc , η∈[−h, 0], i=1, . . . , 4. (2.21) Remark 2.3 By virtue of the results of [22] (Section 4.3), Ψ (σ, ε) exists and is unique for σ ∈ [0, tc ], ε ∈ (0, ε3 ]. Moreover, the following equality is valid: Φ(tc , σ, ε) = Ψ T (σ, ε), σ ∈ [0, tc ], ε ∈ (0, ε3 ]. In the sequel, we call the matrix Ψ (σ, ε) the adjoint matrix of the fundamental matrix Φ(t, σ, ε) at the time tc , or briefly the adjoint matrix. Let us partition the matrix Ψ (σ, ε) into blocks as  Ψ (σ, ε) =

Ψ1 (σ, ε) Ψ3 (σ, ε)

 Ψ2 (σ, ε) , Ψ4 (σ, ε)

where the matrices Ψ1 (σ, ε), Ψ2 (σ, ε), Ψ3 (σ, ε), and Ψ4 (σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively.

2.2 Singularly Perturbed Systems with Small Delays

31

Let the matrix-valued functions Ψ1s (σ ) and Ψ4f (ξ ) be the solutions of the following problems:  T dΨ1s (σ ) = − A¯ s (σ ) Ψ1s (σ ), σ ∈ [0, tc ), Ψ1s (tc ) = In , dσ

(2.22)

T dΨ4f (ξ )   = A4j (tc , 0) Ψ4f (ξ − hi ) dξ N

 +

j =0

0

−h

 T G4 (tc , η, 0) Ψ4f (ξ + η)dη, ξ > 0, Ψ4f (ξ ) = 0, ξ < 0,

Ψ4f (0) = Im .

(2.23)

Remember that the matrix A¯ s (t) is defined in (2.7). From the assumption (AIII-1) and the results of [23], we directly obtain

Ψ4f (ξ ) ≤ a exp(−2βξ ), ξ ≥ 0,

(2.24)

where a > 0 is some constant. Remark 2.4 By virtue of the results of [22] (Section 4.3), we have Ψ1s (σ ) = ΦsT (tc , σ ),

σ ∈ [0, tc ];

Ψ4f (ξ ) = ΦfT (ξ, tc ),

ξ ∈ (−∞, +∞),

where Φs (t, σ ) is defined by the initial-value problem (2.13); Φf (ξ, t) is defined by the initial-value problem (2.14). Theorem 2.1 Let the assumptions (AI-1)–(AIII-1) be valid. Then, there exists a positive number ε4 (ε4 ≤ ε3 ) such that, for all ε ∈ (0, ε4 ], the following inequalities are satisfied:

Ψ1 (σ, ε) − Ψ1s (σ ) ≤ aε,

Ψ2 (σ, ε) ≤ a,

 

Ψ3 (σ, ε) − εΨ3s (σ ) ≤ aε ε + exp − β(tc − σ )/ε ,  

Ψ4 (σ, ε) − Ψ4f (tc − σ )/ε ≤ aε, where σ ∈ [0, tc ],  T T Ψ3s (σ ) = − A−1 4s (σ ) A2s (σ )Ψ1s (σ ), a > 0 is some constant independent of ε. Proof of the theorem is presented in Sect. 2.2.10.

(2.25)

32

2 Singularly Perturbed Linear Time Delay Systems

2.2.5 Example 1 Consider the following system, a particular case of (2.1)–(2.2), dx(t) = x(t) − 4y(t) − 2x(t − ε) + 5y(t − ε) + t − 5, dt dy(t) ε = 3x(t)+(t−4)y(t)−2x(t − ε)+y(t − ε)+t − 3, dt

t ≥ 0, t ≥ 0,

(2.26)

where x(t) and y(t) are scalars, i.e., n = m = 1. Setting formally ε = 0 in this system, we obtain its slow subsystem in the differential-algebraic form dxs (t) = −xs (t) + ys (t) + t − 5, t ≥ 0, dt 0 = xs (t) + (t − 3)ys (t) + t − 3, t ≥ 0.

(2.27)

Consider the original system (2.26) in the interval [0, tc ] = [0, 2]. For all t ∈ [0, 2], the second equation of (2.27) can be resolved with respect to ys (t), yielding ys (t) =

1 xs (t) − 1, 3−t

t ∈ [0, 2].

Substitution of this expression for ys (t) into the first equation of (2.27) yields the slow subsystem of (2.26) in the pure differential form dxs (t) t −2 = xs (t) + t − 6, dt 3−t

t ∈ [0, 2].

The fast subsystem, associated with the original system (2.26), is dyf (ξ ) = (t − 4)yf (ξ ) + yf (ξ − 1) + t − 3, dξ

ξ ≥ 0,

(2.28)

where t ∈ [0, 2] is a parameter. The fundamental matrix Φ(t, σ, ε) of the system (2.26) is the unique solution of the initial-value problem dΦ(t) = A0 (t, ε)Φ(t) + A1 (t, ε)Φ(t − ε), t ∈ (σ, 2], dt Φ(τσ ) = 0, σ − ε ≤ τσ < σ ; Φ(σ ) = I2 , where σ ∈ [0, 2],

(2.29)

2.2 Singularly Perturbed Systems with Small Delays

 A0 (t, ε) =

1 −4 3 ε

t−4 ε

33



 ,

A1 (t, ε) =

−2 5 − 2ε 1ε

 ,

and  Φ(t, σ, ε) =

 Φ1 (t, σ, ε) Φ2 (t, σ, ε) . Φ3 (t, σ, ε) Φ4 (t, σ, ε)

The assumptions (AI-1) and (AII-1) are fulfilled for the coefficients of the differential equation in (2.29). Let us check up the fulfillment of the assumption (AIII-1). For any t ∈ [0, 2], the characteristic equation of the fast subsystem (2.28) has the form λ − t + 4 − exp(−λ) = 0.

(2.30)

Let us estimate the real part of the left-hand side of this equation. We have     Re λ − t + 4 − exp(−λ) = Re(λ) − t + 4 − Re exp(−λ) , t ∈ [0, 2].       Taking that = exp − Reλ cos Imλ , we obtain that   into account   Re exp(−λ)  Re exp(−λ)  ≤ exp − Reλ . Therefore,     Re λ − t + 4 − exp(−λ) ≥ Re(λ) + 2 − exp − Reλ , t ∈ [0, 2]. By a direct calculation, we obtain that, for all complex numbers λ with Reλ ≥ −0.4,  the following inequality is satisfied: Re(λ) + 2 − exp − Reλ > 0.1. The latter, along with the above obtained inequality for the real part of the expression in the left-hand side of (2.30), means that any complex number λ with Reλ ≥ −0.4 does not satisfy this equation for any t ∈ [0, 2]. Thus, all roots λ(t) of the characteristic equation (2.30) satisfy the inequality Reλ(t) < −2β, t ∈ [0, 2], β = 0.2, i.e., the assumption (AIII-1) is satisfied for the problem (2.29). Applying Lemma 2.2 to the initial-value problem (2.29), we directly have that, for all sufficiently small ε > 0, the blocks of the fundamental matrix Φ(t, σ, ε) satisfy the inequalities   Φp (t, σ, ε) ≤ a, p = 1, 3,

  Φ2 (t, σ, ε) ≤ aε,

   

Φ4 (t, σ, ε) ≤ a ε + exp − 0.2(t − σ )/ε , where 0 ≤ σ ≤ t ≤ 2; a > 0 is some constant independent of ε. Now, let us proceed with the problems (2.20), (2.22), and (2.23). In the present example, the first problem becomes as  T  T dΨ (σ, ε) = − A0 (σ, ε) Ψ (σ, ε) − A1 (σ + ε, ε) Ψ (σ + ε, ε), dσ

σ ∈ [0, 2),

34

2 Singularly Perturbed Linear Time Delay Systems

Ψ (2, ε) = I2 ;

Ψ (σ, ε) = 0, σ > 2, (2.31)

where  Ψ (σ, ε) =

 Ψ1 (σ, ε) Ψ2 (σ, ε) . Ψ3 (σ, ε) Ψ4 (σ, ε)

The second and third problems become as: dΨ1s (σ ) σ −2 = Ψ1s (σ ), σ ∈ [0, 2), dσ σ −3

Ψ1s (2) = 1,

(2.32)

and dΨ4f (ξ ) = −2Ψ4f (ξ ) + Ψ4f (ξ − 1), ξ > 0, dξ Ψ4f (ξ ) = 0,

ξ < 0;

Ψ4f (0) = 1.

(2.33)

Solving the problem (2.32), we obtain immediately Ψ1s (σ ) = (3 − σ ) exp(σ − 2),

σ ∈ [0, 2].

(2.34)

Proceed to the problem (2.33). Let us show that the following function is the solution of this problem: Ψ4f (ξ ) =

k  i=0

  (ξ − i)i , exp − 2(ξ − i) i!

ξ ∈ [k, k + 1), k = 0, 1, 2, . . . .

(2.35) For k = 0, we obtain Ψ4f (ξ ) = exp(−2ξ ), ξ ∈ [0, 1), which is the solution of the problem (2.33) in the interval [0, 1). Let us assume that the function (2.35) is the solution of (2.33) in the interval [q, q + 1) (q > 0 is some integer). Based on this assumption, let us show that the function (2.35) is the solution of (2.33) in the interval [q + 1, q + 2). Indeed, in this interval, the problem (2.33) becomes    (ξ − 1 − i)i dΨ4f (ξ ) = −2Ψ4f (ξ ) + exp(2) , exp − 2(ξ − i) dξ i! q

i=0

Ψ4f (q + 1) = exp(−2)

q  i=0

  (q + 1 − i)i . exp − 2(q − i) i!

This is an initial-value problem for a linear nonhomogeneous differential equation. Solving this problem, we have the following expression for Ψ4f (ξ ) in the interval [q + 1, q + 2]:

2.2 Singularly Perturbed Systems with Small Delays

 Ψ4f (ξ ) = exp(−2)

q 





ξ

+ exp(2)

 (q + 1 − i)i

exp − 2(q − i)

i=0



exp − 2(ξ − ρ)

q+1

35



i!

 q  



  exp − 2(ξ − q − 1)

exp − 2(ρ − i)

i=0

 (ρ − 1 − i)i



i!

dρ.

Using the equalities       exp(−2) exp − 2(q − i) exp − 2(ξ − q − 1) = exp − 2(ξ − i) ,       exp(2) exp − 2(ξ − ρ) exp − 2(ρ − i) = exp − 2(ξ − i − 1) , and calculating the integral in the right-hand side of the expression for Ψ4f (ξ ), we can rewrite this expression as Ψ4f (ξ ) ==

q  i=0

+

q  i=0

  (q + 1 − i)i exp − 2(ξ − i) i!

   (ξ − i − 1)i+1 (q − i)i+1 − , exp − 2(ξ − i − 1) (i + 1)! (i + 1)! 

ξ ∈ [q + 1, q + 2].

(2.36)

Let us treat separately the addends in the right-hand side of this equation. The first addend can be rewritten as q  i=0

   (q + 1 − i)i   (q + 1 − i)i = exp(−2ξ ) + . exp − 2(ξ − i) exp − 2(ξ − i) i! i! q

i=1

The second addend can be represented as q  i=0

=

q  i=0

   (ξ − i − 1)i+1 (q − i)i+1 − exp − 2(ξ − i − 1) (i + 1)! (i + 1)! 

  (ξ − i − 1)i+1    (q − i)i+1 exp − 2(ξ − i − 1) exp − 2(ξ − i − 1) − (i + 1)! (i + 1)!

=

q−1 i=0

q+1  j =1

 (ξ − j )j    (q − j + 1)j − . exp − 2(ξ − j ) exp − 2(ξ − j ) (j )! j! 

q

j =1

Substitution of these representations into (2.36) yields

36

2 Singularly Perturbed Linear Time Delay Systems

Ψ4f (ξ ) = exp(−2ξ ) +

q+1  j =1

=

q+1  j =0

  (ξ − j )j exp − 2(ξ − j ) (j )!

  (ξ − j )j , exp − 2(ξ − j ) (j )!

which coincides with the expression in (2.35) for k = q + 1. Thus, due to the principle of the mathematical induction, the function Ψ4f (ξ ) given by Eq. (2.35) is the solution of the problem (2.33). The characteristic equation of the differential equation in (2.33) is obtained from (2.30) by setting there t = 2. Therefore, using the above obtained estimate for the roots λ(t) of this equation and the inequality (2.24), we have the inequality for the solution of the problem (2.33)   Ψ4f (ξ ) ≤ a exp(−0.4ξ ),

ξ ≥ 0,

where a > 0 is some constant. Moreover, by virtue of Theorem 2.1, we have the following inequalities for all sufficiently small ε > 0 and all σ ∈ [0, 2]:   Ψ1 (σ, ε) − Ψ1s (σ ) ≤ aε,

  Ψ2 (σ, ε) ≤ a,

   

Ψ3 (σ, ε) − εΨ3s (σ ) ≤ aε ε + exp − 0.2(2 − σ )/ε ,    Ψ4 (σ, ε) − Ψ4f (2 − σ )/ε  ≤ aε, where Ψ1 (σ, ε), Ψ2 (σ, ε), Ψ3 (σ, ε), and Ψ4 (σ, ε) are the corresponding blocks of the solution Ψ (σ, ε) to the problem (2.31); Ψ1s (σ ) and Ψ4f (ξ ) are given by Eqs. (2.34) and (2.35), respectively; Ψ3s (σ ) = exp(σ − 2); a > 0 is some constant independent of ε.

2.2.6 Example 2: Tracking Model with Delay In this example, we consider a simple tracking model with a delay. Such a model appears in analysis of various real-life problems (see, e.g., [7, 10] and references therein). Namely, let the continuous function ynom (t) be a nominal (desirable) function given in the interval [0, tc ]. Let y(t), t ∈ [0, tc ] be a function tracking the nominal

2.2 Singularly Perturbed Systems with Small Delays

37

one. Let hd > 0 be a delay in the current value of y(·). This delay can be, for instance, a delay in information on the current value of y(·) (see, e.g., [10]). Then, the dynamics of the tracking can be described by the following time delay differential equation:   dy(t) = −γ (t) y(t − hd ) − ynom (t) , dt

t ∈ [0, tc ],

(2.37)

where γ (t) > 0 is a variable gain of the tracking. In the present example, we assume that the gain γ (t) is high, while the delay hd is small. We also assume that the small values 1/γ (t) and hd are comparable with each other, i.e., they are of the same order of smallness. More precisely, we assume that γ (t) =

γ0 (t) , ε

hd = εh,

where ε > 0 is a small parameter; γ0 (t) > 0, t ∈ [0, tc ] is a known continuously differentiable function, and h > 0 is a known number, both independent of ε. Thus, Eq. (2.37) can be rewritten in the form ε

  dy(t) = −γ0 (t) y(t − εh) − ynom (t) , dt

t ∈ [0, tc ].

(2.38)

This equation is a singularly perturbed differential equation with a small delay, and it is a fast mode equation. In what follows of this example, we assume that hγ0 (t)
0 is a given number), there exists no more than a finite number of roots λ(t) of (2.41). This observation, along with the continuity with respect to t ∈ [0, tc ] of all roots λ(t) of this equation, yields the existence of

2.2 Singularly Perturbed Systems with Small Delays

39

a positive constant β such that these roots satisfy the inequality Reλ(t) < −2β, t ∈ [0, tc ]. Proceed with the estimate of the fundamental matrix Φ(t, σ, ε), which in this example becomes the scalar function satisfying the initial-value problem dΦ(t) γ0 (t) =− Φ(t − εh), dt ε Φ(τσ ) = 0, σ − εh ≤ τσ < σ ;

t ∈ [σ, tc ], Φ(σ ) = 1,

where σ ∈ [0, tc ]. Since all roots λ(t) of Eq. (2.41) satisfy the inequality Reλ(t) < −2β, t ∈ [0, tc ], then the solution Φ(t,    σ, ε) of this  initial-value problem satisfies the inequality Φ(t, σ, ε) ≤ a exp − β(t − σ ) , where 0 ≤ σ ≤ t ≤ tc ; a > 0 is some constant independent of ε. Now, let us proceed with the problems (2.20) and (2.23). In the present example, the first problem becomes as dΨ (σ, ε) γ0 (σ + εh) =− Ψ (σ + εh), σ ∈ [0, tc ), dσ ε Ψ( tc , ε) = 1, Ψ( σ, ε) = 0, σ > tc ,

(2.45)

while the second problem has the form dΨ4f (ξ ) = −γ0 (tc )Ψ4f (ξ − h), dξ Ψ4f (ξ ) = 0,

ξ < 0;

ξ > 0,

Ψ4f (0) = 1.

(2.46)

It is important to note that the problem (2.22) does not exists in this example because the differential equation in (2.45) is purely fast. Let us show that the solution of the problem (2.46) has the form Ψ4f (ξ ) =

k   j (ξ − j h)j , (−1)j γ0 (tc ) j!

 ξ ∈ kh, (k + 1)h , k = 0, 1, 2, . . . .

j =0

(2.47) Indeed, for k = 0, we have from (2.47) that Ψ4f (ξ ) = 1, ξ ∈ [0, 1), which is the solution of the problem (2.46) in the interval [0, 1). Let us assume  that the function (2.47) is the solution of (2.46) in the interval qh, (q + 1)h (q > 0 is some integer). Using this assumption, we are going to show  that the function (2.47) is the solution of (2.46) in the interval (q + 1)h, (q + 2)h . Namely, in this interval, the problem (2.46) becomes

40

2 Singularly Perturbed Linear Time Delay Systems

  j (ξ − h − j h)j dΨ4f (ξ ) (−1)j γ0 (tc ) = −γ0 (tc ) , dξ j! q

j =0

  Ψ4f (q + 1)h0 =

 j j (q + 1)h − j h . (−1) γ0 (tc ) j!

q  j =0

j



Solving this problem, we obtain  j q  j (q + 1)h − j h  j Ψ4f (ξ ) = (−1) γ0 (tc ) j! j =0

+

q  j =0

−

q 

(−1)

 j +1 (ξ − h − j h)j +1 (−1)j +1 γ0 (tc ) (j + 1)!

j +1

j =0

  j +1 (q + 1)h − h − j h)j +1 . γ0 (tc ) (j + 1)!

(2.48)

Comparison of the first and third addends in the right-hand side of this equation directly yields  j q   j (q + 1)h − j h j =1 (−1) γ0 (tc ) j! j =0

 q   j +1 (q + 1)h − h − j h)j +1 j +1 . + γ0 (tc ) (−1) (j + 1)! j =0

Due to this equality, Eq. (2.48) becomes as Ψ4f (ξ ) = 1 +

q   j +1 (ξ − h − j h)j +1 (−1)j +1 γ0 (tc ) (j + 1)! j =0

=

q+1  j =0

 j (ξ − j h)j , (−1)j γ0 (tc ) j!

 ξ ∈ (q + 1)h, (k + 2)h .

The latter coincides with the expression in (2.47) for k = q + 1. Therefore, due to the principle of the mathematical induction, the function Ψ4f (ξ ) given by Eq. (2.47) is the solution of the problem (2.46). The characteristic equation of the differential equation in (2.46) is obtained from (2.41) by setting there t = tc . Hence, using the above obtained estimate for the roots λ(t) of this equation and the inequality (2.24), we have the inequality for the

2.2 Singularly Perturbed Systems with Small Delays

41

solution of the problem (2.46)   Ψ4f (ξ ) ≤ a exp(−2βξ ),

β > 0,

ξ ≥ 0,

where a > 0 is some constant. Furthermore, using Theorem 2.1, we obtain the following inequality for all sufficiently small ε > 0 and all σ ∈ [0, tc ]:    Ψ4 (σ, ε) − Ψ4f (tc − σ )/ε  ≤ aε, where a > 0 is some constant independent of ε.

2.2.7 Example 3: Analysis of Neurosystem Model In this example, we analyze a linear model of a neurosystem consisting of two neurons. This model is presented in Sect. 1.1.1 (see Eq. (1.2)). In this example, we deal with a generalized version of this model. Namely, we assume that the coupling strength C connected with the distribution of the information between neurons is not a positive constant, but it is a positive function of time, i.e., C = C(t). Thus, here we treat the system dx1 (t) = y1 (t), dt dx2 (t) = y2 (t), dt dy1 (t) = −x1 (t) + ω(t)y1 (t) + C(t)y2 (t − tdel ), dt dy2 (t) = −x2 (t) + ω(t)y2 (t) + C(t)y1 (t − tdel ), ε dt

ε

(2.49)

where ω(t) = 1 − b2 − C(t).

(2.50)

We consider the system (2.49) in the time interval [0, tc ], and we assume that the function C(t) is continuously differentiable in this interval. Remark 2.5 In the work [29], two cases with respect to values of the delay tdel were considered: (i) tdel = 0 and (ii) the delay tdel is nonsmall (of order of 1). For such cases and for the constant C, the stability analysis of the system (2.49) was carried out in [29]. In the present example, we consider an intermediate case. Namely, we consider the case where tdel is nonzero but small of order of ε, i.e.,

42

2 Singularly Perturbed Linear Time Delay Systems

tdel = εh,

(2.51)

where h > 0 is some constant independent of ε. For this case, we are going to derive the slow and fast subsystems associated with the system (2.49), as well as the estimates for the fundamental matrix of (2.49) and its adjoint matrix. In what follows of this example, we assume (similarly to [29]) b > 1, which, due to (2.50), yields ω(t) < −C(t) < 0,

t ∈ [0, tc ].

(2.52)

Setting ε = 0 in (2.49) and taking into account (2.51), we obtain the slow subsystem of this system in the differential-algebraic form dxs1 (t) = ys1 (t), dt dxs2 (t) = ys2 (t), dt 0 = −xs1 (t) + ω(t)ys1 (t) + C(t)ys2 (t), 0 = −xs2 (t) + ω(t)ys2 (t) + C(t)ys1 (t).

(2.53)

Solving the system of the third and fourth equations in (2.53) with respect to   ys1 (t), ys2 (t) and taking into account the inequality (2.52), we obtain ys1 (t)=

ω(t)xs1 (t)−C(t)xs2 (t) , ω2 (t)−C 2 (t)

ys2 (t)=

−C(t)xs1 (t)+ω(t)xs2 (t) , t ∈ [0, tc ]. ω2 (t)−C 2 (t)

Substitution of these expressions into the first and second equations of (2.53) yields the slow subsystem of the system (2.49) in the pure differential form dxs1 (t) ω(t)xs1 (t) − C(t)xs2 (t) = , dt ω2 (t) − C 2 (t)

t ∈ [0, tc ],

−C(t)xs1 (t) + ω(t)xs2 (t) dxs2 (t) = , dt ω2 (t) − C 2 (t)

t ∈ [0, tc ].

The fast subsystem, associated with the system (2.49), is dyf 1 (ξ ) = ω(t)yf 1 (ξ ) + C(t)yf 2 (ξ − h), dξ

ξ ≥ 0,

2.2 Singularly Perturbed Systems with Small Delays

43

dyf 2 (ξ ) = ω(t)yf 2 (ξ ) + C(t)yf 1 (ξ − h), dξ

ξ ≥ 0,

where t ∈ [0, tc ] is a parameter. For any t ∈ [0, tc ], the characteristic equation of the fast subsystem has the form  2 λ − ω(t) − C 2 (t) exp(−2λh) = 0.

(2.54)

If for any given t ∈ [0, tc ], λ(t) is a root of this equation, then it is a root of one of the following equations: λ − ω(t) = C(t) exp(−λh),

λ − ω(t) = −C(t) exp(−λh).

(2.55)

Vice versa, if for any given t ∈ [0, tc ], λ(t) is a root of one of the equations in (2.55), then it is a root of (2.54). Let us show that, for any t ∈ [0, tc ], these equations do not have roots λ(t) with nonnegative real parts. Let us assume the opposite, i.e., for some t ∈ [0, tc ], there exists of one of the equations in (2.55) such  a root λ(t)  that Reλ(t) ≥ 0. Hence, Re exp(−λ(t)h)  ≤ 1. Using this inequality and the equations in (2.55), we obtain      Reλ(t) − ω(t) = C(t)Re exp(−λ(t)h)  ≤ C(t).

(2.56)

Taking into account that Reλ(t) ≥ 0 and ω(t) < 0, we have from (2.56) Reλ(t) − ω(t) ≤ C(t), which, along with the inequality (2.52), yields Reλ(t) < 0. This inequality contradicts to our assumption on the nonnegativeness of Reλ(t). Therefore, the equations in (2.55) (and, therefore, Eq. (2.54)) do not have roots λ(t) with nonnegative real parts for all t ∈ [0, tc ], i.e., real parts of all roots λ(t) of (2.54) are negative for all t ∈ [0, tc ]. Now, let us show the existence of a number β > 0 such that all roots λ(t) of the equations in (2.55) (and, therefore, of Eq. (2.54)) satisfy the inequality Reλ(t) < −2β,

t ∈ [0, tc ].

(2.57)

For this purpose, we assume that Cmax < ωmin ,





Cmax = max C(t), ωmin = min |ω(t)|. t∈[0,tc ]

t∈[0,tc ]

(2.58)

Consider the following inequality with respect to β: − 2β + ωmin − Cmax exp(−2βh) > 0.

(2.59)

44

2 Singularly Perturbed Linear Time Delay Systems

Subject to the assumption (2.58), the inequality (2.59) has a positive solution. Indeed, it is verified directly that β = (ωmin − Cmax )/2 > 0 satisfies this inequality. Let β > 0 be any given solution of the inequality (2.59). We are going to show that the inequality (2.57) is valid with this β. Assume the opposite, i.e., there exist a value t ∈ [0, tc ] such that some root λ(t) of one of the equations in (2.55)   satisfies the inequality Reλ(t) ≥ −2β. In such a case ±Re exp(−λ(t)h) ≤      Re exp(−λ(t)h)  ≤ exp − Reλ(t)h ≤ exp(−2βh). Using this inequality and the equations in (2.55), we obtain Reλ(t) − ω(t) ≤ C(t) exp(−2βh), which, along with the notations in (2.58) and the inequalities ω(t) < 0, Reλ(t) ≥ −2β, yields −2β + ωmin ≤ Cmax exp(−2βh). This inequality contradicts the inequality (2.59). Hence, the inequality (2.57) is fulfilled for all roots λ(t) of the equations in (2.55) (and, therefore, of Eq. (2.54)). Now, consider the fundamental matrix Φ(t, σ, ε) of the system (2.49). This fundamental matrix is the unique solution of the initial-value problem dΦ(t) = A0 (t, ε)Φ(t) + A1 (t, ε)Φ(t − εh), t ∈ (σ, tc ], dt Φ(τσ ) = 0, σ − εh ≤ τσ < σ ; Φ(σ ) = I4 , where σ ∈ [0, tc ], ⎛

0 0

0 ⎜ 0 A0 (t, ε) = ⎜ ⎝−1 0 ε 0 − 1ε

1 0 ω(t) ε

0

⎞ 0 1 ⎟ ⎟, 0 ⎠ ω(t) ε

⎛

0 ⎜0 A1 (t, ε) = ⎜ ⎝0 0

0 0 0 0

0 0 0 C(t) ε

⎞ 0 0 ⎟ ⎟ C(t) ⎠ , ε

0

and  Φ(t, σ, ε) =

 Φ1 (t, σ, ε) Φ2 (t, σ, ε) . Φ3 (t, σ, ε) Φ4 (t, σ, ε)

Note that all the blocks in this representation of the matrix Φ(t, σ, ε) are of the dimension 2 × 2. The assumptions (AI-1), (AII-1), and (AIII-1) are fulfilled for the system (2.49). Therefore, by virtue of Lemma 2.2, the blocks of its fundamental matrix Φ(t, σ, ε) satisfy the following inequalities for all sufficiently small ε > 0: Φp (t, σ, ε) ≤ a, p = 1, 3,

Φ2 (t, σ, ε) ≤ aε,

2.2 Singularly Perturbed Systems with Small Delays

45

 

Φ4 (t, σ, ε) ≤ a ε + exp − 0.2(t − σ )/ε , where 0 ≤ σ ≤ t ≤ tc ; a > 0 is some constant independent of ε. Further, let us consider the problems (2.20), (2.22), and (2.23) associated with the system (2.49). The first problem has the form  T dΨ (σ, ε) = − A0 (σ, ε) Ψ (σ, ε) dσ  T − A1 (σ + εh, ε) Ψ (σ + εh, ε), σ ∈ [0, tc ), Ψ (tc , ε) = I4 ;

Ψ (σ, ε) = 0, σ > tc ,

(2.60)

where  Ψ (σ, ε) =

 Ψ1 (σ, ε) Ψ2 (σ, ε) , Ψ3 (σ, ε) Ψ4 (σ, ε)

and all the blocks in this representation of the matrix Ψ (σ, ε) are of the dimension 2 × 2. The second problem has the form dΨ1s (σ ) = −A¯ s (σ )Ψ1s (σ ), σ ∈ [0, tc ), Ψ1s (tc ) = I2 , dσ   1 ω(σ ) − C(σ ) A¯ s (σ ) = 2 , σ ∈ [0, tc ]. ω(σ ) ω (σ ) − C 2 (σ ) −C(σ )

(2.61)

The third problem has the form dΨ4f (ξ ) = A40 (tc )Ψ4f (ξ ) + A41 (tc )Ψ4f (ξ − h), dξ  A40 (tc , 0) =

ξ > 0,

Ψ4f (ξ ) = 0, ξ < 0, Ψ4f (0) = I2 ,    ω(tc ) 0 0 C(tc ) , A41 (tc ) = . 0 ω(tc ) C(tc ) 0

(2.62)

Note that in contrast with (2.22) and (2.23), the matrices A¯ s (σ ) and A40 (tc ), A41 (tc ) appear in (2.61) and (2.62) without the sign of the transposition because here these matrices are symmetric. Since  σ   σ A¯ s (σ ) A¯ s (ρ)dρ = A¯ s (ρ)dρ A¯ s (σ ), σ ∈ [0, tc ], tc

tc

then the unique solution of the terminal-value problem (2.61) is

46

2 Singularly Perturbed Linear Time Delay Systems

  Ψ1s (σ ) = exp −

σ

 ¯ As (ρ)dρ ,

σ ∈ [0, tc ].

tc

Now, let us solve the initial-value problem (2.62). Similarly to Examples 1 and 2, by application of the principle of the mathematical induction, we obtain the solution of this problem in the form k     j (ξ − j h)j exp − ω(tc )j h A41 (tc ) , Ψ4f (ξ ) = exp ω(tc )ξ j! j =0

 ξ ∈ kh, (k + 1)h , k = 0, 1, 2, . . .

This solution satisfies the inequality Ψ4f (ξ ) ≤ a exp(−2βξ ),

ξ ≥ 0,

where a > 0 is some constant, while the constant β > 0 is defined in the inequality (2.57). Moreover, due to Theorem 2.1, we have the following inequalities for all sufficiently small ε > 0 and all σ ∈ [0, tc ]: Ψ1 (σ, ε) − Ψ1s (σ ) ≤ aε,

Ψ2 (σ, ε) ≤ a,

 

Ψ3 (σ, ε) − εΨ3s (σ ) ≤ aε ε + exp − β(tc − σ )/ε ,   Ψ4 (σ, ε) − Ψ4f (tc − σ )/ε ≤ aε, where Ψ3s (σ ) = −A¯ s (σ )Ψ1s (σ ); a > 0 is some constant independent of ε.

2.2.8 Example 4: Analysis of Sunflower Equation In this example, we analyze the system (1.7) (see Sect. 1.1.2). This system is equivalent to the linearized sunflower equation (1.6). For the sake of the book’s reading convenience, we write this system here once again dx(t) = y(t), dt

2.2 Singularly Perturbed Systems with Small Delays

ε

47

dy(t) = −c1 y(t) − c2 x(t − ε), dt

(2.63)

where ε > 0 is a small parameter; c1 and c2 are the positive constants. We consider this system in the time interval [0, tc ]. Setting formally ε = 0 in the system (2.63), we obtain its slow subsystem in the differential-algebraic form dxs (t) = ys (t), t ∈ [0, tc ], dt 0 = −c1 ys (t) − c2 xs (t), t ∈ [0, tc ].

(2.64)

Elimination of ys (t) from this system yields the slow equation of (2.63) in the pure differential form c2 dxs (t) = − xs (t), dt c1

t ∈ [0, tc ].

The fast subsystem, associated with the system (2.63), is dyf (ξ ) = −c1 yf (ξ ), dξ

ξ ≥ 0.

The corresponding characteristic equation is λ + c1 = 0, having the single root λ = −c1 < 0. The fundamental matrix Φ(t, σ, ε) of the system (2.63) is the unique solution of the initial-value problem dΦ(t) = A0 (ε)Φ(t) + A1 (ε)Φ(t − ε), t ∈ (σ, tc ], dt Φ(τσ ) = 0, σ − ε ≤ τσ < σ ; Φ(σ ) = I2 , where σ ∈ [0, tc ],  A0 (ε) =

0 1 0 − cε1



 ,

A1 (ε) =

 0 0 , − cε2 0

and  Φ(t, σ, ε) =

 Φ1 (t, σ, ε) Φ2 (t, σ, ε) . Φ3 (t, σ, ε) Φ4 (t, σ, ε)

Due to Lemma 2.2, the scalar blocks Φi (t, σ, ε) (i = 1, . . . , 4) of Φ(t, σ, ε) satisfy the following inequalities for all sufficiently small ε > 0:

48

2 Singularly Perturbed Linear Time Delay Systems

  Φp (t, σ, ε) ≤ a, p = 1, 3,

  Φ2 (t, σ, ε) ≤ aε,

   

Φ4 (t, σ, ε) ≤ a ε + exp − 0.5c1 (t − σ )/ε , where 0 ≤ σ ≤ t ≤ tc ; a > 0 is some constant independent of ε. Now we consider the problems (2.20), (2.22), and (2.23) associated with the system (2.63). The first problem has the form dΨ (σ, ε) = −AT0 (ε)Ψ (σ, ε) − AT1 (ε)Ψ (σ + ε, ε), σ ∈ [0, tc ), dσ Ψ (tc , ε) = I2 , Ψ (σ, ε) = 0, σ > tc ,   Ψ1 (σ, ε) Ψ2 (σ, ε) . Ψ (σ, ε) = Ψ3 (σ, ε) Ψ4 (σ, ε)

(2.65)

The second and third problems are, respectively, dΨ1s (σ ) c2 = Ψ1s (σ ), dσ c1

σ ∈ [0, tc ),

Ψ1s (tc ) = 1,

and dΨ4f (ξ ) = −c1 Ψ4f (ξ ), dξ

ξ > 0,

Ψ4f (0) = 1.

These two problems have the unique solutions 

 c2 Ψ1s (σ ) = exp (σ − tc ) , c1

σ ∈ [0, tc ],

Ψ4f (ξ ) = exp(−c1 ξ ),

ξ ≥ 0.

By virtue of Theorem 2.1, the scalar blocks of the solution to the problem (2.65) satisfy the following inequalities for all sufficiently small ε > 0 and all σ ∈ [0, tc ]:      Ψ1 (σ, ε) − exp c2 (σ − tc )  ≤ aε,   c 1

  Ψ2 (σ, ε) ≤ a,

        Ψ3 (σ, ε) − ε exp c2 (σ − tc )  ≤ aε ε + exp − 0.5c1 (tc − σ ) ,   c1 c1 ε  ! c " 1   Ψ4 (σ, ε) − exp − (tc − σ )  ≤ aε, ε where a > 0 is some constant independent of ε.

2.2 Singularly Perturbed Systems with Small Delays

49

2.2.9 Proof of Lemma 2.2 We prove the inequalities for Φ2 (t, σ, ε) and Φ4 (t, σ, ε). The other inequalities are proven similarly. Using (2.9) and (2.19), we obtain from (2.12) the initial-value problem for Φ2 (t, σ, ε) and Φ4 (t, σ, ε) N

dΦ2 (t, σ, ε)  = A1j (t, ε)Φ2 (t − εhj , σ, ε) + A2j (t, ε)Φ4 (t − εhj , σ, ε) dt

 +

j =0

0 −h



G1 (t, η, ε)Φ2 (t+εη, σ, ε)+G2 (t, η, ε)Φ4 (t+εη, σ, ε) dη, t ∈ (σ, tc ],

Φ2 (t, σ, ε) = 0,

ε

t ∈ [σ − εh, σ ],

(2.66)

N

dΦ4 (t, σ, ε)  = A3j (t, ε)Φ2 (t − εhj , σ, ε) + A4j (t, ε)Φ4 (t − εhj , σ, ε) dt

 +

j =0

0 −h



G3 (t, η, ε)Φ2 (t+εη, σ, ε)+G4 (t, η, ε)Φ4 (t+εη, σ, ε) dη, t ∈ (σ, tc ],

Φ4 (t, σ, ε) = 0, t ∈ [σ − εh, σ ),

Φ4 (σ, σ, ε) = Im .

(2.67)

Let, for any given ε ∈ (0, ε2 ] and σ ∈ [0, tc ], X(t, σ, ε) be n × n-matrix-valued function satisfying the initial-value problem dX(t, σ, ε)  = A1j (t, ε)X(t − εhj , σ, ε) dt N

j =0

 +

0 −h

G1 (t, η, ε)X(t + εη, σ, ε)dη, t ∈ (σ, tc ],

X(t, σ, ε) = 0,

t ∈ [σ − εh, σ ),

X(σ, σ, ε) = In .

Since the matrix-valued functions A1j (t, ε) (j = 0, 1, . . . , N) are bounded for (t, ε) ∈ [0, tc ] × [0, ε2 ], and the matrix-valued function G1 (t, η, ε) is bounded for (t, η, ε) ∈ [0, tc ] × [−h, 0] × [0, ε2 ], then for all ε ∈ (0, ε2 ]

X(t, σ, ε) ≤ a,

0 ≤ σ ≤ t ≤ tc ,

where a > 0 is some constant independent of ε.

(2.68)

50

2 Singularly Perturbed Linear Time Delay Systems

Applying the variation-of-constant formula (see, e.g., [4, 22]) to the problems (2.66) and (2.67), and using the matrix-valued functions X(t, σ, ε) and Y (t, σ, ε) (see Eq. (2.16)), we can rewrite these problems in the equivalent integral form 

t

Φ2 (t, σ, ε) =

 N X(t, ρ, ε) A2j (ρ, ε)Φ4 (ρ − εhj , σ, ε)

σ

 +

j =0

 G2 (ρ, η, ε)Φ4 (ρ + εη, σ, ε)dη dρ, t ∈ [σ, tc ],

0 −h

1 Φ4 (t, σ, ε) = Y (t, σ, ε) + ε  +

0 −h



t

Y (t, ρ, ε) σ

 N

(2.69)

A3j (ρ, ε)Φ2 (ρ − εhi , σ, ε)

j =1

 G3 (ρ, η, ε)Φ2 (ρ + εη, σ, ε)dη dρ, t ∈ [σ, tc ].

(2.70)

Substituting (2.70) into (2.69), changing the integration order, and taking into account the initial conditions in (2.16), we obtain the following integral equation with respect to Φ2 (t, σ, ε): Φ2 (t, σ, ε) = F1 (t, σ, ε) +

 t  N σ

 +

0 −h

F2j (t, ρ, ε)Φ2 (ρ − εhj , σ, ε)

j =0



F3 (t, ρ, η, ε)Φ2 (ρ + εη, σ, ε)dη dρ, t ∈ [σ, tc ],

(2.71)

where  F1 (t, σ, ε) =

t

 N X(t, ρ1 , ε) A2j (ρ1 , ε)Y (ρ1 − εhj , σ, ε)

σ

 +

0 −h

j =0

 G2 (ρ1 , η1 , ε)Y (ρ1 + εη1 , σ, ε)dη1 dρ1 ,

F2j (t, ρ, ε) =

1 F1 (t, ρ, ε)A3j (ρ, ε), ε

F3 (t, ρ, η, ε) =

j = 0, 1, . . . , N,

1 F1 (t, ρ, ε)G3 (ρ, η, ε). ε

The estimates (2.18) and (2.68) yield the following inequality for all ε ∈ (0, ε2 ]:

2.2 Singularly Perturbed Systems with Small Delays

F1 (t, σ, ε) ≤ aε,

51

0 ≤ σ ≤ t ≤ tc ,

where a > 0 is some constant independent of ε. Now, applying the method of successive approximations with zero initial guess to the integral equation (2.71) and using the derived above estimate of F1 (t, σ, ε), we directly obtain the existence of a positive number ε3 (ε3 ≤ ε2 ) such that, for all ε ∈ (0, ε3 ], the following inequality is valid:

Φ2 (t, σ, ε) ≤ aε,

0 ≤ σ ≤ t ≤ tc ,

where a > 0 is some constant independent of ε. Finally, using Eq. (2.70), the estimate (2.18), and the estimate of Φ2 (t, σ, ε), we have immediately  

Φ4 (t, σ, ε) ≤ a ε + exp − β(t − σ )/ε ,

∀ε ∈ (0, ε3 ], 0 ≤ σ ≤ t ≤ tc ,

where a > 0 is some constant independent of ε. Thus, the lemma is proven.

2.2.10 Proof of Theorem 2.1 Let us set Ψps (σ ) = Ψps (tc ) ∀σ > tc , p = 1, 3.

(2.72)

The further proof of the theorem is based on one technical proposition.

2.2.10.1

Technical Proposition

Consider the following (n + m) × (n + m)-matrix-valued functions:  Θ(σ, ε) =

Γ (σ, ε) =  +

 Ψ1s (σ ) 0  , σ ∈ [0, tc ],  εΨ3s (σ ) Ψ4f (tc − σ )/ε

(2.73)

N T dΘ(σ, ε)   + Aj (σ + εhj , ε) Θ(σ + εhj , ε) dσ j =1

0 −h

 T G(σ − εη, η, ε) Θ(σ − εη, ε)dη, σ ∈ [0, tc ].

Let us partition the matrix Γ (σ, ε) into blocks as

(2.74)

52

2 Singularly Perturbed Linear Time Delay Systems

 Γ (σ, ε) =

Γ1 (σ, ε) Γ3 (σ, ε)

 Γ2 (σ, ε) , Γ4 (σ, ε)

(2.75)

where the matrices Γ1 (σ, ε), Γ2 (σ, ε), Γ3 (σ, ε), and Γ4 (σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Proposition 2.6 Let the assumptions (AI-1)–(AIII-1) be valid. Then, for all ε ∈ (0, ε3 ], the following inequalities are satisfied:  

Γp (σ, ε) ≤ aε, p = 1, 3, Γ2 (σ, ε) ≤ (a/ε) exp − β(tc − σ )/ε ,  

Γ4 (σ, ε) ≤ a exp − β(tc − σ )/ε , σ ∈ [0, tc ], (2.76) where ε3 is introduced in Lemma 2.2.9; a > 0 is some constant independent of ε. Proof We prove here the inequalities for Γ1 (σ, ε) and Γ4 (σ, ε). The other inequalities are proven similarly. Let us start with the inequality for Γ1 (σ, ε). We substitute the block representations for the matrices Aj (σ + εhj , ε) (j = 0, 1, . . . , N ), G(σ − εη, η, ε), Θ(σ − εη, ε), and Γ (σ, ε) (see Eqs. (2.9), (2.73), and (2.75)) into (2.74). Then, calculating the products of the corresponding block matrices and equating the upper left-hand blocks in both sides of the resulting equation yield the expression for Γ1 (σ, ε) Γ1 (σ, ε) =

N  T dΨ1s (σ )   A1j (σ + εhj , ε) Ψ1s (σ + εhj ) + dσ j =0

T  + A3j (σ + εhj , ε) Ψ3s (σ + εhj )  +

0 −h



  T G1 (σ − εη, η, ε) Ψ1s (σ − εη)

  T + G3 (σ − εη, η, ε) Ψ3s (σ − εη) dη.

(2.77)

Now, we consider two cases: (i) 0 < σ + εh ≤ tc ; (ii) σ + εh > tc . In the first case, based on the assumptions (AI-1)–(AII-1), and Eqs. (2.22) and (2.25), we directly obtain Apj (σ + εhj , ε) = Apj (σ, 0) + ΔApj (σ, ε), Gp (σ − εη, η, ε) = Gp (σ, η, 0) + ΔGp (σ, η, ε), Ψps (σ + εhj ) = Ψps (σ ) + Δ1 Ψps (σ, ε), Ψps (σ −εη)=Ψps (σ )+Δ2 Ψps (σ, η, ε), p=1, 3, j =0, 1, . . . , N,

(2.78)

2.2 Singularly Perturbed Systems with Small Delays

53

where ΔApj (σ, ε), ΔGp (σ, η, ε), Δ1 Ψps (σ, ε), and Δ2 Ψps (σ, η, ε) are matrixvalued functions of corresponding dimensions, satisfying the following inequalities for all ε ∈ (0, ε3 ], σ ∈ [0, tc − εh], η ∈ [−h, 0]:

ΔApj (σ, ε) ≤ aε, ΔGp (σ, η, ε) ≤ aε,

Δ1 Ψps (σ, ε) ≤aε, Δ2 Ψps (σ, η, ε) ≤aε, p=1, 3, j =0, 1, . . . , N,(2.79) and a > 0 is some constant independent of ε. Substituting the representations (2.78) into (2.77), and using the inequalities (2.79), we obtain for all ε ∈ (0, ε3 ], σ ∈ [0, tc − εh]  N  T  T dΨ1s (σ )   A1j (σ, 0) Ψ1s (σ ) + A3j (σ, 0) Ψ3s (σ ) + Γ1 (σ, ε) = dσ  +

0 −h



j =0

  T  T G1 (σ, η, 0) Ψ1s (σ ) + G3 (σ, η, 0) Ψ3s (σ ) dη + ΔΓ1 (σ, ε), (2.80)

where the n × n-matrix-valued function ΔΓ1 (σ, ε) satisfies the inequality

ΔΓ1 (σ, ε) ≤ aε, ε ∈ [0, ε3 ], σ ∈ [0, tc − εh],

(2.81)

and a > 0 is some constant independent of ε. Due to (2.4), Eq. (2.80) can be rewritten as Γ1 (σ, ε) =

dΨ1s (σ ) + AT1s (σ )Ψ1s (σ ) + AT3s (σ )Ψ3s (σ ) + ΔΓ1 (σ, ε), dσ ε ∈ [0, ε3 ], σ ∈ [0, tc − εh].

Substitution of (2.25) into this equation yields Γ1 (σ, ε) =

"  T T dΨ1s (σ ) ! T + A1s (σ ) − AT3s (σ ) A−1 4s (σ ) A2s (σ ) Ψ1s (σ ) dσ +ΔΓ1 (σ, ε), ε ∈ [0, ε3 ], σ ∈ [0, tc − εh]. (2.82)

 T T ¯ Remember that AT1s (σ ) − AT3s (σ ) A−1 4s (σ ) A2s (σ ) = A Therefore, by virtue of (2.22), Eq. (2.82) becomes

T (σ ) s

(see Eq. (2.7)).

Γ1 (σ, ε) = ΔΓ1 (σ, ε), ε ∈ [0, ε3 ], σ ∈ [0, tc − εh]. The latter, along with the inequality (2.81), immediately yields that the matrixvalued function Γ1 (σ, ε) satisfies the inequality in (2.76) for all ε ∈ (0, ε3 ], σ ∈ [0, tc − εh].

54

2 Singularly Perturbed Linear Time Delay Systems

Consider the case (ii) where σ + εh > tc . In this case, using the assumptions (AI-1)–(AII-1) and Eqs. (2.21) and (2.72), we have the following inequalities for all ε ∈ (0, ε3 ], σ ∈ (tc − εh, tc ]:

Apj (σ + εhj , ε) − Apj (tc , 0) ≤ aε,

p = 1, 3, j = 0, 1, . . . , N,

Gp (σ − εη, η, ε) − Gp (tc , η, 0) ≤ aε,

p = 1, 3. (2.83)

Moreover, from Eq. (2.22), we obtain for all ε ∈ (0, ε3 ], σ ∈ (tc − εh, tc ] T dΨ1s (σ )  ¯ + As (tc ) Ψ1s (tc ) ≤ aε. dσ

(2.84)

In (2.83) and (2.84), a > 0 is some constant independent of ε. Now, based on Eq. (2.72) and the inequalities (2.83)–(2.84), we obtain (similarly to the case (i)) that the matrix-valued function Γ1 (σ, ε) satisfies the inequality in (2.76) for all ε ∈ (0, ε3 ] and σ ∈ (tc − εh, tc ]. Thus, the inequality for Γ1 (σ, ε) is completely proven. Proceed to the proof of the inequality for Γ4 (σ, ε). Substitution of (2.9), (2.73), and (2.75) into (2.74) yields after a routine algebra the expression for Γ4 (σ, ε) Γ4 (σ, ε) = +

N    T  A4j (σ + εhj , ε) Ψ4f (tc − σ − εhj )/ε j =0

 +

 dΨ4f (ξ )  1 −  ξ =(tc −σ )/ε ε dξ

0 −h

   T  G4 (σ − εη, η, ε) Ψ4f (tc − σ + εη)/ε dη .

In this expression, let us transform the independent variable σ as σ = tc − εξ,

(2.85)

where ξ ≥ 0 is a new independent variable. The transformation (2.85) yields Γ4 (tc − εξ, ε) = +  +

N    T A4j tc − ε(ξ − hj ), ε Ψ4f (ξ − hj ) j =0

0 −h

 dΨ4f (ξ ) 1 − ε dξ

   T G4 tc − ε(ξ + η), η, ε Ψ4f (ξ + η)dη .

2.2 Singularly Perturbed Systems with Small Delays

55

This equation can be rewritten in the form Γ4 (tc − εξ, ε) = +

 N 

T A4j (tc , 0) Ψ4f (ξ − hj ) + j =0

+  +

0 −h



G4 (tc , η, 0))

 dΨ4f (ξ ) 1 − ε dξ

T

Ψ4f (ξ + η)dη

N   

T A4j tc − ε(ξ − hj ), ε − A4j (tc , 0) Ψ4f (ξ − hj ) j =0

0 −h

  

T G4 tc − ε(ξ + η), η, ε − G4 (tc , η, 0) Ψ4f (ξ + η)dη ,

yielding, by virtue of (2.23), Γ4 (tc − εξ, ε) =  +

 N  

T 1  A4j tc − ε(ξ − hj ), ε − A4j (tc , 0) Ψ4f (ξ − hj ) ε j =0

0 −h

  

T G4 tc − ε(ξ + η), η, ε − G4 (tc , η, 0) Ψ4f (ξ + η)dη .

(2.86)

Using the assumptions (AI-1)–(AII-1), Eq. (2.21), and the estimate (2.24), we obtain the following inequalities for all ε ∈ (0, ε3 ], ξ ∈ [0, tc /ε], η ∈ [−h, 0]:  

T A4j tc − ε(ξ − hj ), ε − A4j (tc , 0) Ψ4f (ξ − hj ) ≤ aε exp(−βξ ),  

T G4 tc − ε(ξ + η), η, ε − G4 (tc , η, 0) Ψ4f (ξ + η) ≤ aε exp(−βξ ), (2.87) where a > 0 is some constant independent of ε. Equation (2.86), along with the inequalities (2.87), yields the inequality Γ4 (tc − εξ, ε) ≤ a exp(−βξ ) for all ε ∈ (0, ε3 ], ξ ∈ [0, tc /ε]. This inequality and Eq. (2.85) directly imply the inequality for Γ4 (σ, ε) in (2.76).  

2.2.10.2

Main Part of the Proof

Let us transform the matrix Ψ (σ, ε) in the problem (2.20) as Ψ (σ, ε) = Δ(σ, ε) + Θ(σ, ε),

(2.88)

where Δ(σ, ε) is a new unknown matrix-valued function; Θ(σ, ε) is given by (2.73). Due to (2.88), the problem (2.20) becomes

56

2 Singularly Perturbed Linear Time Delay Systems N  T  dΔ(σ, ε) =− Aj (σ + εhi , ε) Δ(σ + εhj , ε) dσ

 −

j =1

0 −h



T G(σ − εη, η, ε) Δ(σ − εη, ε)dη + Γ (σ, ε), σ ∈ [0, tc ),

Δ(tc , ε) =



  −Ψ1 (tc ) 0 0 , Δ(σ, ε) = −εΨ3 (tc ) −εΨ3s (tc ) 0

 0 , σ > tc , (2.89) 0

where Γ (σ, ε) is given by (2.74). By application of the results of [22] (Section 4.3), we can rewrite the problem (2.89) in the equivalent integral form  Δ(σ, ε) = Φ T (tc , σ, ε)Δ(tc , ε) +

tc +εh

Λ(σ, ρ, ε)Δ(ρ, ε)dρ tc



σ

+

Φ T (t, σ, ε)Γ (t, ε)dt, σ ∈ [0, tc ],

(2.90)

tc

where Φ(t, σ, ε), 0 ≤ σ ≤ t ≤ tc , is given by (2.12) Λ(σ, ρ, ε) =

N 

Φ T (ρ − εhj , σ, ε)ATj (ρ, ε)

j =1

 +

(tc −ρ)/ε −h

Φ T (ρ + εη, σ, ε)GT (ρ, η, ε)dη,

and Φ(ρ, σ, ε) = 0 for ρ > tc .

(2.91)

Let us partition the matrix Λ(σ, ρ, ε) into blocks as  Λ(σ, ρ, ε) =

Λ1 (σ, ρ, ε) Λ3 (σ, ρ, ε)

 Λ2 (σ, ρ, ε) , Λ4 (σ, ρ, ε)

where the matrices Λ1 (σ, ρ, ε), Λ2 (σ, ρ, ε), Λ3 (σ, ρ, ε), and Λ4 (σ, ρ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Using Lemma 2.2 and Eq. (2.91), we obtain the following inequalities for all ε ∈ (0, ε3 ], ρ ∈ [tc , tc + εh], σ ∈ [0, tc ]:    

Λ1 (σ, ρ, ε) ≤ a 1 + (ρ − tc )/ε , Λ2 (σ, ρ, ε) ≤ (a/ε) 1 + (ρ − tc )/ε ,  

Λ3 (σ, ρ, ε) ≤ a ε + (ρ − tc ) + exp − β(ρ − σ )/ε ,

2.3 Singularly Perturbed Systems with Delays of Two Scales

57

 

Λ4 (σ, ρ, ε) ≤ (a/ε) ε + (ρ − tc ) + exp − β(ρ − σ )/ε , (2.92) where a > 0 is some constant independent of ε. Now, let us partition the matrix Δ(σ, ε) into blocks as  Δ(σ, ε) =

Δ1 (σ, ε) Δ3 (σ, ε)

 Δ2 (σ, ε) , Δ4 (σ, ε)

where the matrices Δ1 (σ, ε), Δ2 (σ, ε), Δ3 (σ, ε), and Δ4 (σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Now, let us substitute this block representation for Δ(σ, ε), as well as the block representations for Φ(t, σ, ε), Γ (t, ε), Λ(σ, ρ, ε), Δ(tc , ε), and Δ(σ, ε), σ > tc , into Eq. (2.90). Then, let us calculate the blocks of the matrix in the right-hand side of the resulting matrix equality. Finally, using Lemma 2.2 and the inequalities (2.76), (2.92), let us estimate these blocks. Thus, we obtain the following inequalities for all ε ∈ (0, ε3 ] and σ ∈ [0, tc ]:

Δl (σ, ε) ≤ aε, l = 1, 4, Δ2 (σ, ε) ≤ a,  

Δ3 (σ, ε) ≤ aε ε + exp − β(tc − σ )/ε ,

(2.93)

where a > 0 is some constant independent of ε. Equations (2.73), (2.88), and the inequalities (2.93) directly yield the inequalities stated in the theorem. Thus, the theorem is proven.

2.3 Singularly Perturbed Systems with Delays of Two Scales 2.3.1 Original System In this section we consider another type of singularly perturbed time delay systems. Namely,  dx(t)  = A1i (t, ε)x(t − gi ) + A2j (t, ε)y(t − εhj ) dt M

N

j =0

i=0

 +

0 −g

 G1 (t, η, ε)x(t + η)dη +

ε

0

−h

G2 (t, ζ, ε)y(t + εζ )dζ + b1 (t, ε), (2.94)

 dy(t)  A3i (t, ε)x(t − gi ) + A4j (t, ε)y(t − εhj ) = dt M

N

i=0

j =0

58

2 Singularly Perturbed Linear Time Delay Systems

 +

0 −g

 G3 (t, η, ε)x(t + η)dη +

0

−h

G4 (t, ζ, ε)y(t + εζ )dζ + b2 (t, ε), (2.95)

where t ≥ 0, x(t) ∈ E n , y(t) ∈ E m ; M ≥ 1 and N ≥ 1 are integers; ε > 0 is a small parameter; 0 = g0 < g1 < g2 < . . . < gM = g and 0 = h0 < h1 < h2 < . . . < hN = h are given constants independent of ε; Aki (t, ε), Alj (t, ε), Gk (t, η, ε), Gl (t, ζ, ε) (i = 0, 1, . . . , M; j = 0, 1, . . . , N; k = 1, 3; l = 2, 4) are matrices of corresponding dimensions, given for t ≥ 0, η ∈ [−g, 0], ζ ∈ [−h, 0], ε ∈ [0, ε0 ]; bp (t, ε), (p = 1, 2), are vectors of corresponding dimensions, given for t ≥ 0, ε ∈ [0, ε0 ]. In what follows of this section, we assume that the positive number ε0 satisfies the inequality ε0 ≤ g1 / h.

(2.96)

Like (2.1)–(2.2), the system (2.94)–(2.95) is also functional-differential and  infinite-dimensional. The state variables of this system have the form x(t), x(t +    η) , η ∈ [−g, 0), y(t), y(t + τ ) , τ ∈ [−εh, 0). For any given t ≥ 0 and     ε ∈ (0, ε0 ], we consider the states x(t), x(t + η) and y(t), y(t + τ ) in the spaces M [−g, 0; n], and M [−εh, 0; m], respectively. Similarly to the previous section, the components x(t) and y(t) of the state variables are called their Euclidean parts, while the components x(t + η) and y(t + τ ) are called the functional parts of the respective state variables. Moreover, by the same reason as in the previous section, the system (2.94)–(2.95) is singularly perturbed. Equation (2.94) is a slow mode of this system. However, in contrast with  the system (2.1)–(2.2), in (2.94)–(2.95),  the entire state  variable x(t), x(t + η) , η ∈ [−g, 0), is slow. The state variable y(t), y(t +τ ) , τ ∈ [−εh, 0), and Eq. (2.95) are a fast state variable and a fast mode of (2.94)–(2.95), respectively. It is important to note that the system (2.94)–(2.95) has the delays of two scales, the nonsmall delays (of order of 1) in the slow state variable and the small delays (of order of ε) in the fast state variable.

2.3.2 Slow–Fast Decomposition of the Original System Now, like in Sect. 2.2, we are going to decompose asymptotically the system (2.94)– (2.95) into two much simpler ε-free subsystems, the slow and fast ones. The slow subsystem is obtained by setting formally ε = 0 in (2.94)–(2.95), which yields dxs (t)  = A1i (t, 0)xs (t − gi ) + A2s (t)ys (t) dt M

i=0

2.3 Singularly Perturbed Systems with Delays of Two Scales

 +

0

G1 (t, η, 0)xs (t + η)dη + b1 (t, 0), t ≥ 0,

−g

0=

M 

A3i (t, 0)xs (t − gi ) + A4s (t)ys (t)

i=0

 +

59

0 −g

G3 (t, η, 0)xs (t + η)dη + b2 (t, 0), t ≥ 0,

(2.97)

where xs (t) ∈ E n , ys (t) ∈ E m ; Als (t) =

N 

 Alj (t, 0) +

j =0

0

−h

Gl (t, ζ, 0)dζ, l = 2, 4.

(2.98)

The slow subsystem (2.97) is an ε-free differential-algebraic system. However, in contrast with (2.3), the slow subsystem (2.97) has delays in the state xs (·). In the case of the standard system (2.94)–(2.95), i.e., if det A4s (t) = 0, t ≥ 0, the slow subsystem (2.97) can be converted into the set consisting of the explicit expression for ys (t) ys (t) = −  −

M 

A−1 4s (t)A3i (t, 0)xs (t − gi )

i=0 0 −g

−1 A−1 4s (t)G3 (t, η, 0)xs (t + η)dη − A4s (t)b2 (t, 0), t ≥ 0,

and the differential equation with state delays dxs (t)  ¯ = Ai,s (t)xs (t − gi ) dt M

 +

i=0

0 −g

¯ s (t, η)xs (t + η)dη + b¯s (t), t ≥ 0, G

(2.99)

where A¯ i,s (t) = A1i (t, 0) − A2s (t)A−1 4s (t)A3i (t, 0),

i = 0, . . . , M,

¯ s (t, η) = G1 (t, η, 0) − A2s (t)A−1 (t)G3 (t, η, 0), G 4s b¯s (t) = b1 (t, 0) − A2s (t)A−1 4s (t)b2 (t, 0).

(2.100)

Equation (2.99) is also called the slow subsystem, associated with the system (2.94)– (2.95).

60

2 Singularly Perturbed Linear Time Delay Systems

The fast subsystem, associated with (2.94)–(2.95), is obtained similarly to Sect. 2.2. Thus, we have dyf (ξ )  = A4j (t, 0)yf (ξ − hj ) dξ N

 +

j =0

0 −h

G4 (t, ζ, 0)yf (ξ + ζ )dζ + b2 (t, 0), ξ ≥ 0,

(2.101)

where t is a parameter; yf (ξ ) ∈ E m . Like in the previous section, the fast subsystem, associated with (2.94)–(2.95), is a differential equation with state delays. It is of a lower Euclidean dimension than the original system, and it is ε-free.

2.3.3 Fundamental Matrix Solution   For any given ε ∈ (0, ε0 ], consider the block vectors " z(t) = col x(t), y(t) , t ∈ ! [−g, +∞), and b(t, ε) = col b1 (t, ε), 1ε b2 (t, ε) , t ∈ [0, +∞), and the block matrices of the dimension (n + m) × (n + m)  Ax,i (t, ε) =  Ay,j (t, ε) =  Gx (t, η, ε)=

 0 , 0

i = 0, 1, . . . , M,

t ≥ 0,

(2.102)

 0 A2j (t, ε) , 0 1ε A4j (t, ε)

j = 0, 1, . . . , N,

t ≥ 0,

(2.103)

A1i (t, ε) 1 ε A3i (t, ε)

G1 (t, η, ε) 1 ε G3 (t, η, ε)

  0 0 , Gy (t, ζ, ε)= 0 0

 G2 (t, ζ, ε) , t≥0, 1 ε G4 (t, ζ, ε) (2.104)

Gx,y (t, η, ε) =

η ∈ [−g, −εh), Gx (t, η, ε), Gx (t, η, ε) + 1ε Gy (t, η/ε, ε), η ∈ [−εh, 0],

t ≥ 0. (2.105)

Using the above introduced vectors and matrices, we can rewrite the system (2.94)–(2.95) in the following equivalent form for all ε ∈ (0, ε0 ] and t ≥ 0:  dz(t)  = Ax,i (t, ε)z(t − gi ) + Ay,j (t, ε)z(t − εhj ) dt M

N

i=0

j =0

2.3 Singularly Perturbed Systems with Delays of Two Scales

 +

0 −g

61

Gx,y (t, η, ε)z(t + η)dη + b(t, ε).

(2.106)

For this system, let us give the initial conditions z(η) = φz (η),

η ∈ [−g, 0);

z(0) = z0 .

(2.107)

For any given σ ≥ 0 and ε ∈ (0, ε0 ], we consider the initial-value problem with respect to the (n + m) × (n + m)-matrix-valued function F (t):  dF (t)  = Ax,i (t, ε)F (t − gi ) + Ay,j (t, ε)F (t − εhj ) dt M

N

j =0

i=0

 +

0 −g

Gx,y (t, η, ε)F (t + η)dη,

F (t) = 0, σ − g ≤ t < σ,

t > σ,

F (σ ) = In+m .

(2.108)

Let tc > 0 be a given time instant independent of ε. Let ε ∈ (0, ε0 ] be given. Similarly to Propositions 2.1 and 2.2, we have the following assertions. Proposition 2.7 Let the matrix-valued functions Ax,i (t, ε) (i = 0, 1, . . . , M) and Ay,j (t, ε) (j = 0, 1, . . . , N ) be continuous in t ∈ [0, tc ]. Let the matrix-valued function Gx,y (t, η, ε) be piecewise continuous with respect to η ∈ [−g, 0] for each t ∈ [0, tc ] and be continuous with respect to t ∈ [0, tc ] uniformly in η ∈ [−g, 0]. Then, for any given σ ∈ [0, tc ], the initial-value problem (2.108) has the unique solution F (t) = F (t, σ, ε), 0 ≤ σ ≤ t ≤ tc . Proposition 2.8 Let the conditions of Proposition 2.7 be valid. Let b(·, ε) ∈ L2 [0, tc ; E n+m ] and φz (·) ∈ L2 [−g, 0; E n+m ]. Then, the initial-value problem (2.106)–(2.107) has the unique absolutely continuous solution z(t) = z(t, ε), t ∈ [0, tc ], and this solution has the form  z(t, ε) = F (t, 0, ε)z0 +

0 −g



t

+

(t, 0, η, ε)φz (η)dη F F (t, σ, ε)b(σ, ε)dσ,

(2.109)

0

(t, σ, η, ε) has the form where the (n + m) × (n + m)-matrix-valued function F A,y (t, σ, η, ε) + F G,x,y (t, σ, η, ε), (t, σ, η, ε) = F A,x (t, σ, η, ε) + F F and

62

2 Singularly Perturbed Linear Time Delay Systems A,x (t, σ, η, ε) = F

 M  F (t, σ + η + gi , ε)Ax,i (σ + η + gi , ε), σ + η − t ≤ −gi ≤ η , 0, otherwise i=1

(2.110)

A,y (t, σ, η, ε) = F

 N  F (t, σ + η + εhj , ε)Ay,j (σ + η + εhj , ε), σ + η − t ≤ −εhj ≤ η , 0, otherwise j =1

(2.111)

G,x,y (t, σ, η, ε) = F



η

−g

F (t, σ + η − κ, ε)Gx,y (σ + η − κ, κ, ε)dκ. (2.112)

The matrix F (t, σ, ε) is the fundamental matrix of the system (2.106) (and the system (2.94)–(2.95)). Let us partition the vector φz (η) and the matrix F (t, σ, ε) into blocks as  φz (η) =

 φx (η) , φy (η)

φx (η) ∈ E n ,

φy (η) ∈ E m ,

and  F (t, σ, ε) =

 F1 (t, σ, ε) F2 (t, σ, ε) , F3 (t, σ, ε) F4 (t, σ, ε)

(2.113)

where the matrices F1 (t, σ, ε), F2 (t, σ, ε), F3 (t, σ, ε), and F4 (t, σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Corollary 2.1 For any given ε ∈ (0, ε0 ], the solution z(t, ε) (see Eq. (2.109)) of the initial-value problem (2.106)–(2.107) depends on the function φx (η) given in the interval [−g, 0] and on the function φy (η) given in the interval [−εh, 0], while z(t, ε) is independent of φy (η) given in the interval [−g, −εh). Proof First of all let us note that, due to the inequality (2.96), we have −g < −εh A,y (t, σ, η, ε) = 0 for all for all ε ∈ (0, ε0 ]. Also, due to (2.111), we obtain that F η < −εh. Based on these observations and Eq. (2.105), we can represent the second addend in (2.109) as 

0 −g

(t, 0, η, ε)φz (η)dη = F

2.3 Singularly Perturbed Systems with Delays of Two Scales



63

−εh

A,x (t, 0, η, ε) + F G,x (t, 0, η, ε) φz (η)dη F

−g

 +

0 −εh

(t, 0, η, ε)φz (η)dη, F

(2.114)

where G,x (t, σ, η, ε) = F



η −g

F (t, σ + η − κ, ε)Gx (σ + η − κ, κ, ε)dκ.

(2.115)

Using the block representations of the matrices Ax,i (t, ε) (i = 1, . . . , M), Gx (t, η, ε), and F (t, σ, ε) (see Eqs. (2.102), (2.104), and (2.113)), we obtain for 0 ≤ σ ≤ t ≤ tc , η ∈ [−g, 0], ⎛ F (t, σ, ε)Ax,i (σ, ε) = ⎝

F1 (t, σ, ε) F2 (t, σ, ε)

⎞⎛ ⎠⎝

F3 (t, σ, ε) F4 (t, σ, ε)

A1i (σ, ε)

0

⎞ ⎠

1 ε A3i (σ, ε)

0 ⎞ F1 (t, σ, ε)A1i (σ, ε) + 1ε F2 (t, σ, ε)A3i (σ, ε) 0 ⎠, =⎝ ⎛

F3 (t, σ, ε)A1i (σ, ε) + 1ε F4 (t, σ, ε)A3i (σ, ε) 0 ⎛ F (t, σ, ε)Gx (σ, η, ε) = ⎝

F1 (t, σ, ε) F2 (t, σ, ε)

⎞⎛ ⎠⎝

F3 (t, σ, ε) F4 (t, σ, ε)

G1 (σ, η, ε)

0

⎞ ⎠

1 ε G3 (σ, η, ε)

0 ⎞ F1 (t, σ, ε)G1 (σ, η, ε) + 1ε F2 (t, σ, ε)G3 (σ, η, ε) 0 ⎠. =⎝ ⎛

F3 (t, σ, ε)G1 (σ, η, ε) + 1ε F4 (t, σ, ε)G3 (σ, η, ε) 0 These two equations, along with (2.110), (2.113), and (2.115), yield A,x (t, 0, η, ε)φz (η) = F $ ⎛# ⎞ F1 (t, η + gi , ε)A1i (η + gi , ε) + 1ε F2 (t, η + gi , ε)A3i (η + gi , ε) φx (η) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟, ⎜ ⎟ ⎝# ⎠ $ F3 (t, η + gi , ε)A1i (η + gi , ε) + 1ε F4 (t, η + gi , ε)A3i (η + gi , ε) φx (η) η − t ≤ −gi ≤ η,

i = 1, . . . , M, (2.116)

64

2 Singularly Perturbed Linear Time Delay Systems

G,x (t, 0, η, ε)φz (η) = F



 G,1 (t, η, ε)φx (η) F G,2 (t, η, ε)φx (η) , F

(2.117)

where, for η ∈ [−g, −εh), G,1 (t, η, ε) = F   η 1 F1 (t, η − κ, ε)G1 (η − κ, κ, ε) + F2 (t, η − κ, ε)G3 (η − κ, κ, ε) dκ, ε −g

G,2 (t, η, ε) = F   η 1 F3 (t, η − κ, ε)G1 (η − κ, κ, ε) + F4 (t, η − κ, ε)G3 (η − κ, κ, ε) dκ. ε −g

Now, Eqs. (2.109), (2.114), (2.116), and (2.117) directly yield the statements of the corollary.   Along with the fundamental matrix F (t, σ, ε) of the system (2.94)–(2.95), we consider the fundamental matrices Fs (t, σ ) and Ff (ξ, t) of the slow (2.99) and fast (2.101) subsystems, respectively. For a given σ ≥ 0, the n × n-matrix-valued function Fs (t, σ ) is the unique solution of the initial-value problem dFs (t, σ )  ¯ = Ai,s (t)Fs (t − gi , σ ) + dt M

i=0

Fs (t, σ ) = 0,



0

−g

¯ s (t, η)Fs (t + η, σ )dη, t > σ, G

σ − g ≤ t < σ,

Fs (σ, σ ) = In , (2.118)

while, for any t ≥ 0, the m × m-matrix-valued function Ff (ξ, t) is the unique solution of the initial-value problem dFf (ξ, t)  = A4j (t, 0)Ff (ξ − hj , t) dξ N

 +

j =0

0 −h

G4 (t, ζ, 0)Ff (ξ + ζ, t)dζ,

Ff (ξ, t) = 0,

ξ < 0,

ξ > 0,

Ff (0, t) = Im .

(2.119)

Remark 2.6 Comparing the problems (2.14) and (2.119), one can conclude that Ff (ξ, t) = Φf (ξ, t), ξ ∈ (−∞, +∞), t ≥ 0. Similarly to Proposition 2.8, we have the following two assertions.

2.3 Singularly Perturbed Systems with Delays of Two Scales

65

Proposition 2.9 Let the inequality det A4s (t) = 0 be valid for all t ∈ [0, tc ]. Let the matrix-valued functions A¯ i,s (t) (i = 0, 1, . . . , M) be continuous in the interval [0, tc ]. Let b¯s (·) ∈ L2 [0, tc ; E n ]. Then, for any given vector-valued function φs0 (·) ∈ L2 [−g, 0; E n ] and vector xs0 ∈ E n , the system (2.99) subject to the initial conditions xs (η) = φs0 (η), η ∈ [−g, 0), xs (0) = xs0 , has the unique absolutely continuous solution  xs (t) = Fs (t, 0)xs0 + 

t

+

0 −g

s (t, 0, η)φs0 (η)dη F

Fs (t, σ )b¯s (σ )dσ,

t ∈ [0, tc ],

0

where s (t, σ, η) = F  M  Fs (t, σ + η + gi )A¯ i,s (σ + η + gi ), σ + η − t ≤ −gi ≤ η 0, otherwise i=1  η ¯ s (σ + η − κ, κ)dκ. Fs (t, σ + η − κ)G + −g

Proposition 2.10 Let t ∈ [0, tc ] be any given value. Let, for this t, the matrixvalued function G4 (t, ζ, 0), ζ ∈ [−h, 0], be piecewise continuous. Then, for any given vector-valued function φf 0 (·) ∈ L2 [−h, 0; E m ] and vector yf 0 ∈ E m , the system (2.101) subject to the initial conditions yf (κ) = φf 0 (κ), κ ∈ [−h, 0), yf (0) = yf 0 has the unique absolutely continuous solution  yf (ξ, t) = Ff (ξ, t)yf 0 + 

ξ

+

0 −h

f (ξ, κ, t)φf 0 (κ)dκ F

Ff (ξ − ω, t)b2 (t, 0)dω,

ξ ≥ 0,

0

where f (ξ, κ, t) = F  N  Ff (ξ − κ − hj , t)A4j (κ + hj , 0), κ − ξ ≤ −hj ≤ κ 0, otherwise j =1

 +

κ −h

Ff (ξ − κ + η, t)G4 (t, η, 0)dη.

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2 Singularly Perturbed Linear Time Delay Systems

2.3.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Delays of Two Scales Here, in contrast with Sect. 2.2.4, we consider the case where singularly perturbed matrix differential systems have the delays of two different scales. Namely, they have nonsmall delays (of order of 1) in the slow state variables and small delays (proportional to the small parameter ε of singular perturbations) in the fast state variables. We consider the system in (2.108) and some other systems with such a feature in a given time interval [0, tc ], where tc is independent of ε and satisfies the inequality tc ≥ g. In what follows in this subsection, we assume that: (AI-2)

(AII-2)

(AIII-2)

(AIV-2)

The matrix-valued functions Aki (t, ε), Alj (t, ε) (i = 0, 1, . . . , M; j = 0, 1, . . . , N ; k = 1, 3; l = 2, 4) are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ]. The matrix-valued functions Gk (t, η, ε) (k = 1, 3) are piecewise continuous with respect to η ∈ [−g, 0] for each (t, ε) ∈ [0, tc ] × [0, ε0 ], and they are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in η ∈ [−g, 0]. The matrix-valued functions Gl (t, ζ, ε) (l = 2, 4) are piecewise continuous with respect to ζ ∈ [−h, 0] for each (t, ε) ∈ [0, tc ]×[0, ε0 ], and they are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in ζ ∈ [−h, 0]. All roots λ(t) of the equation ⎡ det ⎣λIm −

N 

 A4j (t, 0) exp(−λhj )−

j =0

⎤ 0

−h

G4 (t, ζ, 0) exp(λζ )dζ ⎦ =0

satisfy the inequality Reλ(t) < −2β for all t ∈ [0, tc ], where β > 0 is some constant. Similarly to Lemma 2.2, we obtain the following assertion. Lemma 2.3 Let the assumptions (AI-2)–(AIV-2) be valid. Then, there exists a positive number ε¯ 1 ≤ ε0 such that, for all ε ∈ (0, ε¯ 1 ], the following inequalities are satisfied:

Fk (t, σ, ε) ≤ a, k = 1, 3,

F2 (t, σ, ε) ≤ aε,

 

F4 (t, σ, ε) ≤ a ε + exp − β(t − σ )/ε , 0 ≤ σ ≤ t ≤ tc , where Fq (t, σ, ε) (q = 1, . . . , 4) are the corresponding blocks of the matrix F (t, σ, ε) (see Eq. (2.113)), which is the solution of the initial-value problem (2.108); a > 0 is some constant independent of ε.

2.3 Singularly Perturbed Systems with Delays of Two Scales

67

Now, for any given ε ∈ (0, ε0 ], we consider the following terminal-value problem for (n + m) × (n + m)-matrix-valued function K (σ ): M   T dK (σ ) =− Ax,i (σ + gi , ε) K (σ + gi ) dσ i=0

−  −

N 

 T Ay,j (σ + εhj , ε) K (σ + εhj )

j =0 0 −g

 T Gx,y (σ − χ , χ , ε) K (σ − χ )dχ , K (tc ) = In+m ,

σ ∈ [0, tc ),

K (σ ) = 0, σ > tc .

(2.120)

In this problem, it is assumed that the blocks of the matrices Ax,i (t, ε) (i = 0, 1, . . . , M), Ay,j (t, ε) (j = 0, 1, . . . , N ), and Gx,y (t, χ , ε) (see Eqs. (2.102)– (2.105)) satisfy the following equalities: Aki (t, ε) = Aki (tc , ε),

Alj (t, ε) = Alj (tc , ε), t > tc , ε ∈ [0, ε0 ],

Gk (t, χ , ε) = Gk (tc , χ , ε), t > tc , χ ∈ [−g, 0], ε ∈ [0, ε0 ], Gl (t, ζ, ε) = Gl (tc , ζ, ε), t > tc , ζ ∈ [−h, 0], ε ∈ [0, ε0 ], k=1, 3, l=2, 4, i=0, 1, . . . , M, j =0, 1, . . . , N. (2.121) Remark 2.7 By virtue of the results of [22] (Section 4.3), we obtain that the problem (2.120) has the unique solution K (σ ) = K (σ, ε). Moreover, for any given ε ∈ (0, ε0 ], F (tc , σ, ε) = K T (σ, ε),

σ ∈ [0, tc ].

Let us partition the matrix K (σ, ε) into blocks as follows:  K (σ, ε) =

 K1 (σ, ε) K2 (σ, ε) , K3 (σ, ε) K4 (σ, ε)

(2.122)

where the block K1 (σ, ε) is of the dimension n × n, the block K2 (σ, ε) is of the dimension n × m, the block K3 (σ, ε) is of the dimension m × n, and the block K4 (σ, ε) is of the dimension m × m. Along with the problem (2.120), we consider two more problems. The first one is the following terminal-value problem:  T dK1s (σ ) =− A¯ i,s (σ + gi ) K1s (σ + gi ) dσ M

i=0

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2 Singularly Perturbed Linear Time Delay Systems

 −

0 −g

  ¯ s (σ − χ , χ ) T K1s (σ − χ )dχ , G K1s (tc ) = In ,

σ ∈ [0, tc ),

K1s (σ ) = 0, σ > tc ,

(2.123)

where, taking into account (2.121), the matrix-valued functions A¯ i,s (t) (i = ¯ s (t, χ ) are given in (2.100). 0, 1, . . . , M) and G The problem (2.123) has the unique solution K1s (σ ), σ ∈ [0, tc ]. The second of the abovementioned problems is the initial-value problem T dK4f (ξ )   A4j (tc , 0) K4f (ξ − hj ) = dξ N

 +

j =0

0 −h

 T G4 (tc , ζ, 0) K4f (ξ + ζ )dζ, K4f (ξ ) = 0, ξ < 0;

ξ > 0,

K4f (0) = Im .

(2.124)

This problem has the unique solution K4f (ξ ), ξ ≥ 0. Due to the assumption (AIV2) and the results of [23], this solution satisfies the inequality K4f (ξ ) ≤ a exp(−2βξ ), ξ ≥ 0,

(2.125)

where a > 0 is some constant. Remark 2.8 By virtue of the results of [22] (Section 4.3), we have K1s (σ ) = FsT (tc , σ ), σ ∈ [0, tc ]; K4f (ξ ) = FfT (ξ, tc ), ξ ∈ (−∞, +∞), where Fs (t, σ ) is defined by the initial-value problem (2.118); Ff (ξ, t) is defined by the initial-value problem (2.119). Theorem 2.2 Let the assumptions (AI-2)–(AVI-2) be valid. Then, there exists a positive number ε¯ 2 ≤ ε¯ 1 such that, for all ε ∈ (0, ε¯ 2 ], the following inequalities are satisfied:

K1 (σ, ε) − K1s (σ ) ≤ aε,

σ ∈ [0, tc ],

 

K3 (σ, ε) − εK3s (σ ) ≤ aε ε + exp − β(tc − σ )/ε , σ ∈ [0, tc ], where Kk (σ, ε) (k = 1, 3) are the corresponding blocks of the solution to the terminal-value problem (2.120), %  T  T − A−1 (σ ) A2s (σ ) K1s (σ ), 4s K3s (σ ) = 0,

σ ∈ [0, tc ], σ > tc ;

(2.126)

2.3 Singularly Perturbed Systems with Delays of Two Scales

69

Als (t) (l = 2, 4) are given by (2.98); a > 0 is some constant independent of ε. Proof of the theorem is presented in Sect. 2.3.8. Similarly to Theorem 2.2, we have the following theorem. Theorem 2.3 Let the assumptions (AI-2)–(AVI-2) be valid. Then, there exists a positive number ε¯ 3 ≤ ε¯ 1 such that, for all ε ∈ (0, ε¯ 3 ], the following inequalities are satisfied:  K2 (σ, ε) ≤ a,

K4 (σ, ε) − K4f

tc − σ ε

 ≤ aε,

σ ∈ [0, tc ],

where Kl (σ, ε) (l = 2, 4) are the corresponding blocks of the solution to the terminal-value problem (2.120); a > 0 is some constant independent of ε.

2.3.5 Example 5 Consider the following system, a particular case of (2.94)–(2.95), dx(t) = (t + 1)x(t) − (2t + 5)x(t − 2) + 5y(t) − ty(t − ε) dt  0 +2 (t − ζ )y(t + εζ )dζ − (t + 1), t ≥ 0, −1

ε

dy(t) = (2 − t)x(t) + 2tx(t − 2) dt −6y(t) − ty(t − ε) + t, t ≥ 0,

(2.127)

where x(t) and y(t) are scalars; g = 2, h = 1; ε ∈ (0, ε0 ] (0 < ε0 ≤ 2). In what follows, we consider this system in the interval [0, tc ] = [0, 3]. Setting formally ε = 0 in this system, we obtain its slow subsystem in the differential-algebraic form dxs (t) = (t + 1)xs (t) − (2t + 5)xs (t − 2) + (t + 6)ys (t) − (t + 1), dt 0 = (2 − t)xs (t) + 2txs (t − 2) − (t + 6)ys (t) + t,

t ∈ [0, 3],

t ∈ [0, 3].

The second equation of this subsystem can be resolved with respect to ys (t), yielding ys (t) =

1 (2 − t)xs (t) + 2txs (t − 2) + t , t +6

t ∈ [0, 3].

70

2 Singularly Perturbed Linear Time Delay Systems

Substituting this expression for ys (t) into the first equation of the slow subsystem in the differential-algebraic form, we obtain the slow subsystem of (2.127) in the pure differential form dxs (t) = 3xs (t) − 5xs (t − 2) − 1, dt

t ∈ [0, 3].

The fast subsystem, associated with the original system (2.127), is dyf (ξ ) = −6yf (ξ ) − tyf (ξ − 1) + t, dξ

ξ ≥ 0,

(2.128)

where t ∈ [0, 3] is a parameter. The fundamental matrix F (t, σ, ε) of the system (2.127) is the unique solution of the initial-value problem dF (t) = Ax,0 (t, ε)F (t) + Ax,1 (t, ε)F (t − 2) + Ay,0 (t, ε)F (t) dt  +Ay,1 (t, ε)F (t − ε) + F (t) = 0,

0 −2

Gx,y (t, η, ε)F (t + η)dη,

σ − 2 ≤ t < σ,

t ∈ (σ, 3],

F (σ ) = I2 ,

where σ ∈ [0, 3],  Ax,0 (t, ε) =

 t +1 0 , 2−t 0 ε 

Ay,0 (t, ε) =

Gx,y (t, η, ε) =

Ax,1 (t, ε) =

 −(2t + 5) 0 , 2t 0 ε



0 5 0 − 6ε

 Gx (t, η, ε) =



 ,

 0 0 , 0 0

Ay,1 (t, ε) = 

Gy (t, ζ, ε) =

0 −t 0 − εt

 ,

 0 2(t − ζ ) , 0 0

η ∈ [−2, −ε), Gx (t, η, ε), 1 Gx (t, η, ε) + ε Gy (t, η/ε, ε), η ∈ [−ε, 0],

and  F (t, σ, ε) =

 F1 (t, σ, ε) F2 (t, σ, ε) . F3 (t, σ, ε) F4 (t, σ, ε)

Since the matrix-valued function Gx (t, η, ε) is identically zeroth, the differential equation in this initial-value problem can be rewritten as

2.3 Singularly Perturbed Systems with Delays of Two Scales

71

dF (t) = Ax,0 (t, ε)F (t) + Ax,1 (t, ε)F (t − 2) + Ay,0 (t, ε)F (t) dt  +Ay,1 (t, ε)F (t − ε) +

0

−1

Gy (t, ζ, ε)F (t + εζ )dζ,

t ∈ (σ, 3].

The assumptions (AI-2), (AII-2), and (AIII-2) are fulfilled for the coefficients of this differential equation. Let us check up the fulfillment of the assumption (AIV-2). For any t ∈ [0, 3], the characteristic equation of the fast subsystem (2.128) has the form λ + 6 + t exp(−λ) = 0. Similarly to the analysis of the quasi-polynomial equation (2.30) in Example 1 (see Sect. 2.2.5), one can show that all roots λ(t) of the characteristic equation of the fast subsystem (2.128) satisfy the inequality Reλ(t) < −2β, t ∈ [0, 3], β = 0.25, i.e., the assumption (AIV-2) is fulfilled in this example. Thus, for all sufficiently small ε > 0, the blocks of the fundamental matrix F (t, σ, ε) satisfy the inequalities   Fk (t, σ, ε) ≤ a, k = 1, 3,

  F2 (t, σ, ε) ≤ aε,

   

F4 (t, σ, ε) ≤ a ε + exp − 0.25(t − σ )/ε , where 0 ≤ σ ≤ t ≤ 3; a > 0 is some constant independent of ε. Let us proceed to the problems (2.120) and (2.123). In the present example, the first problem becomes  T  T dK (σ, ε) = − Ax,0 (σ, ε) K (σ, ε) − Ax,1 (σ + 2, ε) K (σ + 2, ε) dσ  T  T − Ay,0 (σ, ε) K (σ, ε) − Ay,1 (σ + ε, ε) K (σ + ε, ε)  −

0 −1

 T Gy (σ − εζ, ζ, ε) K (σ − εζ, ε)dζ, σ ∈ [0, 3), K (3, ε) = I2 ,

K (σ, ε) = 0, σ > 3,

where Ax,1 (t, ε) = Ax,1 (3, ε), Ay,1 (t, ε) = Ay,1 (3, ε), Gy (t, ζ, ε) = Gy (3, ζ, ε), t > 3,

ζ ∈ [−1, 0], 

K (σ, ε) =

ε ∈ (0, ε0 ];

 K1 (σ, ε) K2 (σ, ε) . K3 (σ, ε) K4 (σ, ε)

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2 Singularly Perturbed Linear Time Delay Systems

The second of the abovementioned problems becomes as dK1s (σ ) = −3K1s (σ ) + 5K1s (σ + 2), σ ∈ [0, 3), dσ K (3) = 1, K (σ ) = 0, σ > 3. Solving this terminal-value problem by the method of steps (see, e.g., [10]), we obtain  

exp − 3(σ − 3) , σ ∈ (1, 3],  

K1s (σ ) = exp − 3(σ − 1) exp(6) + 5(σ − 1) , σ ∈ [0, 1]. Thus, by virtue of Theorem 2.2, we obtain the following inequalities for all sufficiently small ε > 0 and all σ ∈ [0, 3]:   K1 (σ, ε) − K1s (σ ) ≤ aε,    

K3 (σ, ε) − εK3s (σ ) ≤ aε ε + exp − 0.25(3 − σ )/ε , where

K3s (σ ) =

K1s (σ ), 0,

σ ∈ [0, 3], σ > 3;

a > 0 is some constant independent of ε. Now, let us proceed to the problem (2.124), which in this example becomes as dK4f (ξ ) = −6K4f (ξ ) − 3K4f (ξ − 1), dξ K4f (ξ ) = 0,

ξ < 0;

ξ > 0,

K4f (0) = 1.

Using the mathematical induction method, we obtain the solution of this problem K4f (ξ ) =

k  i=0

  (ξ − i)i , (−3)i exp − 6(ξ − i) i!

ξ ∈ [k, k + 1), k = 0, 1, 2, . . . .

Moreover, the characteristic equation of the differential equation in this problem is λ + 6 + 3 exp(−λ) = 0. All roots of this quasi-polynomial equation satisfy the inequality Reλ < −0.5 because this equation can be obtained from the above considered characteristic

2.3 Singularly Perturbed Systems with Delays of Two Scales

73

equation of the fast subsystem (2.128) by setting there t = 3. Therefore, due to the inequality (2.125), we have   K4f (ξ ) ≤ a exp(−0.5ξ ),

ξ ≥ 0,

where a > 0 is some constant. Finally, by virtue of Theorem 2.3, we obtain the following inequalities for all sufficiently small ε > 0 and all σ ∈ [0, 3]:   K2 (σ, ε) ≤ a,

     K4 (σ, ε) − K4f 3 − σ  ≤ aε,   ε

where a > 0 is some constant independent of ε.

2.3.6 Example 6: Dynamics of Nuclear Reactor In this example, we study the systems (1.11) and (1.12), which are singularly perturbed versions of the normalized linearized model of a nuclear reactor dynamics [1] (for details, see Sect. 1.1.3). Let us start with the first of these systems dx(θ ) = y(θ ) − x(θ − g), dθ dy(θ ) ε = −δx(θ ) − y(θ − εh), dθ

(2.129)

where θ ∈ [0, θc ] is an independent variable (the nondimensional time); ε > 0 is a small parameter; the nondimensional time instant θc > 0 is independent of ε; δ ∈ (−1, 0], g > 0, and h > 0 are constants independent of ε. In what follows of this example, we assume that θc ≥ g. The system (2.129) is a particular case of the system (2.94)–(2.95). The slow subsystem in the differential-algebraic form of (2.129) is dxs (θ ) = ys (θ ) − xs (θ − g), dθ 0 = −δxs (θ ) − ys (θ ),

θ ∈ [0, θc ], θ ∈ [0, θc ],

and in the pure differential form is dxs (θ ) = −δxs (θ ) − xs (θ − g), dθ

θ ∈ [0, θc ].

The fast subsystem of the system (2.129) has the form

74

2 Singularly Perturbed Linear Time Delay Systems

dyf (ξ ) = −yf (θ − h), dξ

ξ ≥ 0.

The characteristic equation of the latter is λ + exp(−λh) = 0. Assuming that h < π2 , we can show similarly to Example 2 (see Sect. 2.2.6) the existence of a positive number β such that all roots λ of this quasi-polynomial equation satisfy the inequality Reλ < −2β. For all ε ≤ g/ h, the fundamental matrix F (θ, σ, ε) of the system (2.129) is the unique solution of the initial-value problem dF (θ ) = Ax,0 (ε)F (θ ) + Ax,1 F (θ − g) + Ay,0 F (θ ) dθ +Ay,1 (ε)F (θ − εh), F (θ ) = 0,

θ ∈ (σ, θc ],

σ − g ≤ θ < σ,

F (σ ) = I2 ,

where σ ∈ [0, θc ],  Ax,0 (ε) =  Ay,0 =

 0 0 , − εδ 0

 0 1 , 0 0

 Ax,1 = 

Ay,1 (ε) =

 −1 0 , 0 0

0 0 0 − 1ε

 ,

and  F (θ, σ, ε) =

 F1 (θ, σ, ε) F2 (θ, σ, ε) . F3 (θ, σ, ε) F4 (θ, σ, ε)

By virtue of Lemma 2.3, we obtain that, for all sufficiently small ε > 0, the scalar blocks of the fundamental matrix F (θ, σ, ε) satisfy the inequalities   Fk (θ, σ, ε) ≤ a, k = 1, 3,

  F2 (θ, σ, ε) ≤ aε,

   

F4 (θ, σ, ε) ≤ a ε + exp − β(t − σ )/ε , where 0 ≤ σ ≤ θ ≤ θc ; a > 0 is some constant independent of ε. Proceed to the problems (2.120) and (2.123). In the present example, the first problem becomes

2.3 Singularly Perturbed Systems with Delays of Two Scales

75

dK (σ, ε) = −ATx,0 (ε)K (σ, ε) − ATx,1 K (σ + g, ε) dσ −ATy,0 K (σ, ε) − ATy,1 (ε)K (σ + εh, ε), σ ∈ [0, θc ), K (θc , ε) = I2 ,

K (σ, ε) = 0,

σ > θc ,

where  K (σ, ε) =

 K1 (σ, ε) K2 (σ, ε) . K3 (σ, ε) K4 (σ, ε)

The second of the abovementioned problems becomes as dK1s (σ ) = δK1s (σ ) + K1s (σ + g), σ ∈ [0, θc ), dσ K1s (θc ) = 1, K1s (σ ) = 0, σ > θc . Solving this terminal-value problem by the method of steps (see, e.g., [10]), we obtain k   (σ + jg − θc )j , K1s (σ ) = exp δ(σ − θc ) exp(jg) j! j =0

# " σ ∈ θc − kg, max{0, θc − (k + 1)g} , where (k = 0, 1, . . . , K); K is the maximal nonnegative integer such that θc −Kg > 0. Thus, due to Theorem 2.2, we have the following inequalities for all sufficiently small ε > 0 and all σ ∈ [0, θc ]:   K1 (σ, ε) − K1s (σ ) ≤ aε,    

K3 (σ, ε) − εK3s (σ ) ≤ aε ε + exp − β(θc − σ )/ε , where

K3s (σ ) =

K1s (σ ), 0,

σ ∈ [0, θc ], σ > θc ;

a > 0 is some constant independent of ε. Now, we proceed to the problem (2.124), which in this example becomes as

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2 Singularly Perturbed Linear Time Delay Systems

dK4f (ξ ) = −K4f (ξ − h), dξ K4f (ξ ) = 0,

ξ < 0;

ξ > 0,

K4f (0) = 1.

Similarly to Example 2, using the method of the mathematical induction, we obtain the solution to this problem K4f (ξ ) =

k  (ξ − j h)j , (−1)j j!

 ξ ∈ kh, (k + 1)h , k = 0, 1, 2, . . . ,

j =0

and this solution satisfies the inequality   K4f (ξ ) ≤ a exp(−2βξ ),

ξ ≥ 0,

where a > 0 is some constant. Finally, using Theorem 2.3, we obtain the following inequalities for all sufficiently small ε > 0 and all σ ∈ [0, θc ]:   K2 (σ, ε) ≤ a,

     K4 (σ, ε) − K4f θc − σ  ≤ aε,   ε

where a > 0 is some constant independent of ε. Like the system (2.129), the system (1.12) is also a particular case of the system (2.94)–(2.95). The system (1.12) is analyzed quite similarly to the system (2.129), yielding the similar results. We leave this analysis to a reader as an exercise.

2.3.7 Example 7: Analysis of Car-Following Model in a Simple Closed Lane In this example we deal with the systems (1.30) and (1.31), which are singularly perturbed versions of the three-cars-following model in one lane of the shape of a simple closed curve (for details, see Sect. 1.1.6). Let us start with the second of the abovementioned systems dx(θ ) = −x(θ − g) + y2 (θ − εh2 ), dθ dy1 (θ ) = x(θ − g) − y1 (θ − εh1 ), ε dθ

2.3 Singularly Perturbed Systems with Delays of Two Scales

ε

77

  dy2 (θ ) = τ y1 (θ − εh1 ) − y2 (θ − εh2 ) , dθ

(2.130)

where θ ∈ [0, θc ] is an independent variable; ε > 0 is a small parameter; the value θc > 0 is independent of ε; g > 0, h1 > 0, h2 > 0, and τ > 0 are constants independent of ε. In what follows of this example, we assume that θc ≥ g. The system (2.130) is a singularly perturbed system with the delays of two scales. Thus, it is a particular case of the system (2.94)–(2.95). The slow subsystem in the differential-algebraic form of (2.130) is dxs (θ ) = −xs (θ − g) + ys2 (θ ), θ ∈ [0, θc ], dθ 0 = xs (θ − g) − ys1 (θ ), θ ∈ [0, θc ],   0 = τ ys1 (θ ) − ys2 (θ ) , θ ∈ [0, θc ], and in the pure differential form is dxs (θ ) = 0, dθ

θ ∈ [0, θc ].

The fast subsystem of the system (2.130) has the form dyf 1 (ξ ) = −yf 1 (ξ − h1 ), dξ   dyf 2 (ξ ) = τ yf 1 (ξ − h1 ) − yf 2 (ξ − h2 ) , dξ

ξ ≥ 0, ξ ≥ 0.

The characteristic equation of the fast subsystem is    λ + exp(−λh1 ) λ + τ exp(−λh2 ) = 0.

(2.131)

If λ is a root of this equation, then it is a root of one of the following equations: λ + exp(−λh1 ) = 0,

λ + τ exp(−λh2 ) = 0.

(2.132)

Vice versa, if λ is a root of one of the equations in (2.132), then it is a root of (2.131). Let us assume that h1
0 is some constant independent of ε. Now, let us treat the problems (2.120) and (2.123). In this example, the first problem becomes as dK (σ, ε) = −ATx,1 (ε)K (σ + g, ε) − ATy,1 (ε)K (σ + εh1 , ε) dσ −ATy,2 (ε)K (σ + εh2 , ε), K (θc , ε) = I3 , where

σ ∈ [0, θc ),

K (σ, ε) = 0,

σ > θc ,

2.3 Singularly Perturbed Systems with Delays of Two Scales

 K (σ, ε) =

79

 K1 (σ, ε) K2 (σ, ε) , K3 (σ, ε) K4 (σ, ε)

the block K1 (σ, ε) is scalar, the block K2 (σ, ε) is of the dimension 1 × 2, the block K3 (σ, ε) is of the dimension 2×1, and the block K4 (σ, ε) is of the dimension 2×2. The problem (2.123) becomes in this example as K1s (σ ) = 0, dσ

K1s (θc ) = 1;

σ ∈ [0, θc );

K1s (σ ) = 0,

σ > θc ,

yielding the solution K1s (σ ) = 1, σ ∈ [0, θc ]. Hence, by virtue of Theorem 2.2, the blocks K1 (σ, ε) and K3 (σ, ε) satisfy the following inequalities for all sufficiently small ε > 0 and all σ ∈ [0, θc ]:   K1 (σ, ε) − 1 ≤ aε,  

K3 (σ, ε) − εK3s (σ ) ≤ aε ε + exp − β(θc − σ )/ε , where

K3s (σ ) =

⎧  1 ⎪ ⎪ ⎨ 1 , ⎪ ⎪ ⎩

σ ∈ [0, θc ],

τ

σ > θc ;

0,

a > 0 is some constant independent of ε. Proceed with the problem (2.124), which in this example becomes as dK4f (ξ ) = AT41 K4f (ξ − h1 ) + AT42 K4f (ξ − h2 ), dξ K4f (ξ ) = 0,

ξ < 0;

ξ > 0,

K4f (0) = I2 ,

(2.133)

where  A41 =

 −1 0 , τ 0

 A42 =

0 0 0 −τ

 .

To solve this initial-value problem, we represent the 2 × 2-matrix K4f (ξ ) as  K4f (ξ ) =

 K4f,1 (ξ ) K4f,2 (ξ ) , K4f,3 (ξ ) K4f,4 (ξ )

where K4f,j (ξ ) (j = 1, . . . , 4) are scalars.

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2 Singularly Perturbed Linear Time Delay Systems

Substitution of this representation of K4f (ξ ) into the * problem (2.133) yields + the following two initial-value problem, with respect to K4f,1 (ξ ), K4f,3 (ξ ) and * + K4f,2 (ξ ), K4f,4 (ξ ) : dK4f,1 (ξ ) = −K4f,1 (ξ − h1 ) + τ K4f,3 (ξ − h1 ), dξ

ξ > 0,

dK4f,3 (ξ ) = −τ K4f,3 (ξ − h2 ), dξ

ξ > 0,

K4f,1 (ξ ) = 0, ξ < 0;

K4f,1 (0) = 1;

K4f,3 (ξ ) = 0, ξ ≤ 0, (2.134)

and dK4f,2 (ξ ) = −K4f,2 (ξ − h1 ) + τ K4f,4 (ξ − h1 ), dξ

ξ > 0,

dK4f,4 (ξ ) = −τ K4f,4 (ξ − h2 ), dξ

ξ > 0,

K4f,2 (ξ ) = 0, ξ ≤ 0;

K4f,4 (ξ ) = 0, ξ < 0;

K4f,4 (0) = 1. (2.135)

It is seen that each of these two problems can be solved consecutively, beginning from the second equation with the corresponding initial condition. Using this observation and the method of the mathematical induction, we obtain the solution to the problem (2.134) K4f,3 (ξ ) = 0, K4f,1 (ξ ) =

k  (ξ − j h1 )j (−1)j , j!

ξ ≥ 0,

 ξ ∈ kh1 , (k + 1)h1 , k = 0, 1, 2, . . . .

j =0

Note that K4f,1 (ξ − χ ), 0 ≤ χ ≤ ξ ≤ +∞, is the fundamental solution of the homogeneous equation corresponding to the first differential equation in (2.135). Based on this observation and using the variation-of-constant formula [23], we obtain the solution to the problem (2.135) K4f,4 (ξ ) =

k  j =0

(−1)j τ j

(ξ − j h2 )j , j! 

K4f,2 (ξ ) = τ

ξ

 ξ ∈ kh2 , (k + 1)h2 , k = 0, 1, 2, . . . ,

K4f,1 (ξ − χ )K4f,4 (χ )dχ ,

ξ ≥ 0.

0

Since all roots λ of the quasi-polynomial equation (2.131) satisfy the inequality Reλ < −2β, then the solution K4f (ξ ) of the initial-value problem (2.133) satisfies

2.3 Singularly Perturbed Systems with Delays of Two Scales

81

the inequality K4f (ξ ) ≤ a exp(−2βξ ),

ξ ≥ 0,

where a > 0 is some constant. Now, by virtue of Theorem 2.3, we have the following inequalities for all sufficiently small ε > 0 and all σ ∈ [0, θc ]:  K4 (σ, ε) − K4f

K2 (σ, ε) ≤ a,

θc − σ ε

 ≤ aε,

where a > 0 is some constant independent of ε. Like the system (2.130), the system (1.30) is also a particular case of the system (2.94)–(2.95). The system (1.30) can be analyzed similarly to the system (2.130), yielding the similar results. We leave this analysis to a reader as an exercise.

2.3.8 Proof of Theorem 2.2 First of all let us note that, due to the assumption (AVI-2), the inequality det A4s (t) = 0 is valid for all t ∈ [0, tc ]. Therefore, the problem (2.123) is feasible. Hence, the matrix-valued functions K1s (σ ) and K3s (σ ) exist and are unique for all σ ≥ 0. Let us denote     K1 (σ, ε) K1s (σ ) K13 (σ, ε) = , K13,s (σ, ε) = , (2.136) K3 (σ, ε) εK3s (σ )

Δ(σ, ε) = K13 (σ, ε) − K13,s (σ, ε).

(2.137)

Comparing the block representation (2.122) of the solution K (σ, ε) to the terminal-value problem (2.120) and the matrix K13 (σ, ε), we can conclude that the latter is the left-hand block of the former of the dimension (n + m) × n. Therefore, for any ε ∈ (0, ε0 ], the matrix-valued function K13 (σ, ε) satisfies the following terminal-value problem: M   T dK13 (σ, ε) =− Ax,i (σ + gi , ε) K13 (σ + gi , ε) dσ i=0

−

N   T Ay,j (σ + εhj , ε) K13 (σ + εhj , ε) j =0

82

2 Singularly Perturbed Linear Time Delay Systems

 −

0 −g

 T Gx,y (σ − χ , χ , ε) K13 (σ − χ , ε)dχ ,  K13 (tc , ε) =

In 0

σ ∈ [0, tc ),

 ,

K13 (σ, ε) = 0, σ > tc .

(2.138)

From (2.137), we have K13 (σ, ε) = Δ(σ, ε) + K13,s (σ, ε). Substitution of this expression for K13 (σ, ε) into the problem (2.138), and use of the expressions for K13,s (σ, ε) and K3s (σ ) (see Eqs. (2.136) and (2.126)) and the terminal-value problem (2.123) for K1s (σ ) yield the terminal-value problem for the matrix-valued function Δ(σ, ε) M   T dΔ(σ, ε) =− Ax,i (σ + gi , ε) Δ(σ + gi , ε) dσ i=0

−  −

N   T Ay,j (σ + εhj , ε) Δ(σ + εhj , ε) j =0

0 −g

 T Gx,y (σ − χ , χ , ε) Δ(σ − χ , ε)dχ + Γ (σ, ε), σ ∈ [0, tc ),  Δ(tc , ε) =

 0 , −εK3s (tc )

Δ(σ, ε) = 0, σ > tc , (2.139)

where the (n + m) × n-matrix-valued function Γ (σ, ε) has the block form  Γ (σ, ε) =

 Γ1 (σ, ε) , Γ2 (σ, ε)

(2.140)

and the upper and lower blocks of this matrix are of the dimensions n×n and m×n, respectively. The matrix-valued function Γ1 (σ, ε) has the form Γ1 (σ, ε) = −

M #   T A1i (σ + gi , ε) − A1i (σ + gi , 0) K1s (σ + gi ) i=0

$  T + A3i (σ + gi , ε) − A3i (σ + gi , 0) K3s (σ + gi )

 −

0 −g

#

T

G1 (σ − χ , χ , ε) − G1 (σ − χ , χ , 0)

K1s (σ − χ )

$  T + G3 (σ − χ , χ , ε) − G3 (σ − χ , χ , 0) K3s (σ − χ ) dχ .

2.3 Singularly Perturbed Systems with Delays of Two Scales

83

Note that K1s (σ ) and K3s (σ ) are bounded for σ ∈ [0, tc ]. Therefore, by virtue of the assumptions (AI-2) and (AII-2), Γ1 (σ, ε) ≤ c1 ε

∀σ ∈ [0, tc ], ε ∈ [0, ε0 ],

(2.141)

where c1 > 0 is some constant independent of ε. The matrix-valued function Γ2 (σ, ε) has the form Γ2 (σ, ε) = Γ21 (σ, ε) + Γ22 (σ, ε), where m × n-matrix-valued functions Γ21 (σ, ε) and Γ22 (σ, ε) are Γ21 (σ, ε) = −ε

Γ22 (σ, ε) = −

N  

dK3s (σ ) , dσ

T A2j (σ + εhj , ε) K1s (σ + εhj )

j =0

 −

 T

+ A4j (σ + εhj , ε) K3s (σ + εhj ) 0 −h

 T G2 (σ − εζ, ζ, ε) K1s (σ − εζ )

T

 + G4 (σ − εζ, ζ, ε) K3s (σ − εζ ) dζ.

(2.142)

Let us estimate Γ2 (σ, ε). We have for all σ ∈ [0, tc ], ε ∈ [0, ε0 ] Γ2 (σ, ε) ≤ Γ21 (σ, ε) + Γ22 (σ, ε) . Since K1s (σ ) is bounded for σ ∈ [0, tc ], then dK1s (σ )/dσ and dK3s (σ )/dσ are bounded for σ ∈ [0, tc ]. Moreover, due to the assumptions (AI-2), (AIII-2), the matrix-valued functions A2j (t, ε), A4j (t, ε) (j = 0, 1, . . . , N), G2 (t, ζ, ε), and G4 (t, ζ, ε) are bounded for t ∈ [0, tc ], ζ ∈ [−h, 0], ε ∈ [0, ε0 ]. These observations yield the inequalities Γ21 (σ, ε) ≤ c21 ε, σ ∈ [0, tc ], ε ∈ [0, ε0 ], Γ22 (σ, ε) ≤ c22 , σ ∈ [tc − εh, tc ], ε ∈ 0, ε1 ], where ε1 = min{ε0 , (tc − g)/ h}; c21 > 0 and c22 > 0 are some constants independent of ε. Now, let us estimate the matrix-valued function Γ22 (σ, ε) for all σ ∈ [0, tc − εh), and all sufficiently small ε > 0. Remember that the derivatives dK1s (σ )/dσ

84

2 Singularly Perturbed Linear Time Delay Systems

and dK3s (σ )/dσ are bounded for σ ∈ [0, tc ]. Moreover, these derivatives have a finite number of breakpoints in the interval σ ∈ [0, tc ]. Due to these properties of dK1s (σ )/dσ and dK3s (σ )/dσ , and the assumptions (AI-2), (AIII-2), there exists a positive number ε2 ≤ ε1 such that, for all ε ∈ [0, ε2 ], the matrix-valued function Γ22 (σ, ε) (see Eq. (2.142)) can be rewritten as  T  T Γ22 (σ, ε) = − A2s (σ ) K1s (σ ) − A4s (σ ) K3s (σ ) + RΓ (σ, ε), σ ∈ [0, tc − εh), (2.143) where Als (σ ) (l = 2, 4) are given by (2.98); RΓ (σ, ε) is a known matrix-valued function satisfying the inequality RΓ (σ, ε) ≤ c23 ε; c23 > 0 is some constant independent of ε. Substituting (2.126) into (2.143), we obtain Γ22 (σ, ε) = RΓ (σ, ε). Therefore, Γ22 (σ, ε) ≤ c23 ε for all σ ∈ [0, tc − εh), ε ∈ [0, ε2 ]. Summarizing all the above obtained estimates for Γ2 (σ, ε), Γ21 (σ, ε), and Γ22 (σ, ε), we obtain that, for all ε ∈ [0, ε2 ], the following inequality is valid:

Γ2 (σ, ε) ≤ c2

ε, 1,

σ ∈ [0, tc − εh), σ ∈ [tc − εh, tc ],

(2.144)

where c2 > 0 is some constant independent of ε. By application of the results of [22] (Section 4.3), we can rewrite the problem (2.139) in the equivalent integral form  +

Δ(σ, ε) = F T (tc , σ, ε)Δ(tc , ε) + σ

F T (t, σ, ε)Γ (t, ε)dt,

σ ∈ [0, tc ],

(2.145)

tc

where F (t, σ, ε), 0 ≤ σ ≤ t ≤ tc is given by (2.108). Let us partition the matrix Δ(σ, ε) into blocks as  Δ(σ, ε) =

 Δ1 (σ, ε) , Δ2 (σ, ε)

(2.146)

where the blocks Δ1 (σ, ε) and Δ2 (σ, ε) are of the dimensions n × n and m × n, respectively. Substitution of Eqs. (2.113), (2.140), and (2.146) into Eq. (2.145) and use of the block representation for Δ(tc , ε) (see Eq. (2.139)) yield after a routine matrix algebra the following two equations:

2.3 Singularly Perturbed Systems with Delays of Two Scales

85

 T Δ1 (σ, ε) = −ε F3 (tc , σ, ε) K3s (tc )  σ#  T F1 (t, σ, ε) Γ1 (t, ε) + tc

$  T + F3 (t, σ, ε) Γ2 (t, ε) dt,

σ ∈ [0, tc ],

(2.147)

 T Δ2 (σ, ε) = −ε F4 (tc , σ, ε) K3s (tc )  σ#  T F2 (t, σ, ε) Γ1 (t, ε) + tc

$  T + F4 (t, σ, ε) Γ2 (t, ε) dt,

σ ∈ [0, tc ].

(2.148)

Let us estimate Δ1 (σ, ε) and Δ2 (σ, ε). We start with the first matrix-valued function. Using Lemma 2.3 and the fact that K3s (σ ) is bounded, we obtain  T F3 (tc , σ, ε) K3s (tc ) ≤ a11 ∀σ ∈ [0, tc ], ε ∈ (0, ε¯ 1 ], where a11 > 0 is some constant independent of ε. Now, let us estimate the second addend in the right-hand side of (2.147). For this addend, two cases can be distinguished: (i) σ ∈ [tc − εh, tc ]; (ii) σ ∈ [0, tc − εh). In both cases, due to Lemma 2.3 and the inequalities (2.141) and (2.144), we have 

σ

# $ T  T F1 (t, σ, ε) Γ1 (t, ε) + F3 (t, σ, ε) Γ2 (t, ε) dt ≤ a12 ε

∀ε ∈ (0, ε¯ 2 ],

tc

where ε¯ 2 = min{ε2 , ε¯ 1 }; a12 > 0 is some constant independent of ε. Equation (2.147), along with the inequalities for both addends in its right-hand side, yields immediately Δ1 (σ, ε) ≤ a1 ε

∀σ ∈ [0, tc ], ε ∈ (0, ε¯ 2 ],

where a1 > 0 is some constant independent of ε. Proceed to the estimation of Δ2 (σ, ε). The estimate of this matrix-valued function is derived similarly to the estimate of Δ1 (σ, ε), yielding for all σ ∈ [0, tc ], ε ∈ (0, ε¯ 2 ]  

Δ2 (σ, ε) ≤ a2 ε ε + exp − β(tc − σ )/ε , where a2 > 0 is some constant independent of ε. The above obtained estimates for Δ1 (σ, ε) and Δ2 (σ, ε), along with the notations (2.136) and (2.137), imply the validity of the inequalities stated in the theorem.

86

2 Singularly Perturbed Linear Time Delay Systems

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays 2.4.1 Original System In this section we consider one class of singularly perturbed time delay systems where the delays are nonsmall. Namely,

dx(t)  A1i (t, ε)x(t − hi ) + A2i (t, ε)y(t − hi ) = dt 1

 +

i=0

0



−h

G1 (t, η, ε)x(t + η) + G2 (t, η, ε)y(t + η) dη + b1 (t, ε),

(2.149)

dy(t)  = A3i (t, ε)x(t − hi ) + A4i (t, ε)y(t − hi ) dt 1

ε  +

i=0

0 −h



G3 (t, η, ε)x(t + η) + G4 (t, η, ε)y(t + η) dη + b2 (t, ε),

(2.150)

where t ≥ 0, x(t) ∈ E n , y(t) ∈ E m ; ε > 0 is a small parameter; h0 = 0, h1 = h > 0 is a given constant independent of ε; Aki (t, ε), Gk (t, η, ε) (i = 0, 1; k = 1, . . . , 4) are matrices of corresponding dimensions, given for t ≥ 0, η ∈ [−h, 0], ε ∈ [0, ε0 ]; bp (t, ε), (p = 1, 2), are vectors of corresponding dimensions, given for t ≥ 0, ε ∈ [0, ε0 ]. Like (2.1)–(2.2) and (2.94)–(2.95), the system (2.149)–(2.150) is also functionaldifferential and infinite-dimensional. The     state variables of this system have the form x(t), x(t + η) , y(t), y(t + η) , η ∈ [−h, 0). For any given t ≥ 0, we consider the states x(t), x(t + η) and y(t), y(t + η) in the spaces M [−h, 0; n] and M [−h, 0; m], respectively. Similarly to the previous sections, the components x(t) and y(t) of the state variables are called their Euclidean parts, while the components x(t +η) and y(t +η) are called the functional parts of the respective state variables. Moreover, by the same reason as in the previous sections, the system (2.149)–(2.150) is singularly perturbed. is a slow mode of this system, and the entire state  Equation (2.149)   variable x(t), x(t + η) , η ∈ [−h, 0) is slow. The state variable y(t), y(t + η) , η ∈ [−h, 0), and Eq. (2.150) are a fast state variable and a fast mode of (2.149)– (2.150), respectively. It is important to note that in the system (2.149)–(2.150) the delays in both, slow and fast, state variables are nonsmall (of order of 1). Remark 2.9 In what follows of this section, we study the case where A31 (t, 0) ≡ 0, A41 (t, 0) ≡ 0, G3 (t, η, 0) ≡ 0, G4 (t, η, 0) ≡ 0,

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

87

t ≥ 0, η ∈ [−h, 0].

(2.151)

Moreover, we assume that the matrix-valued functions A30 (t, ε), A40 (t, ε) and the vector-valued function b2 (t, ε) are given in a larger domain, namely, for t ≥ −h and ε ∈ [0, ε0 ].

2.4.2 Slow–Fast Decomposition of the Original System Like in the previous sections, let us decompose asymptotically the original system (2.149)–(2.150) into the slow and fast subsystems. Setting formally ε = 0 in (2.149)–(2.150), we obtain the slow subsystem

dxs (t)  = A1i (t, 0)xs (t − hi ) + A2i (t, 0)ys (t − hi ) dt 1

 +

i=0

0 −h



G1 (t, η, 0)xs (t + η) + G2 (t, η, 0)ys (t + η) dη + b1 (t, 0), 0 = A30 (t, 0)xs (t) + A40 (t, 0)ys (t) + b2 (t, 0), (2.152)

where xs (t) ∈ E n , ys (t) ∈ E m . The slow subsystem (2.152) is an ε-free differential-algebraic system with pointwise and distributed delays in both states xs (·) and ys (·). If the system (2.149)–(2.150) is standard, i.e., if det A40 (t, 0) = 0,

t ≥ −h,

(2.153)

the slow subsystem (2.152) can be converted into the set consisting of the explicit expression for ys (t)   ys (t) = −A−1 40 (t, 0) A30 (t, 0)xs (t) + b2 (t, 0) ,

t ≥ −h,

and the time delay differential equation with respect to xs (·) dxs (t)  ¯ = Ai,s (t)xs (t − hi ) dt 1

 + where

i=0

0 −h

¯ s (t, η)xs (t + η)dη + b¯s (t), G

t ≥ 0,

(2.154)

88

2 Singularly Perturbed Linear Time Delay Systems

A¯ i,s (t) = A1i (t, 0) − A2i (t, 0)A−1 40 (t − hi , 0)A30 (t − hi , 0), ¯ s (t, η) = G1 (t, η, 0) − G2 (t, η, 0)A−1 (t + η, 0)A30 (t + η, 0), G 40

b¯s (t) = b1 (t, 0) −  −

1 

(2.155)

A20 (t, 0)A−1 40 (t − hi , 0)b2 (t − hi , 0)

i=0 0 −h

G2 (t, η, 0)A−1 40 (t + η, 0)b2 (t + η, 0)dη.

The fast subsystem, associated with (2.149)–(2.150), is obtainedin the following  way: (a) the terms containing the state variable x(t), x(t + η) and the terms becoming zero for ε = 0 are removed from (2.150); (b) the transformations of

the variables t = t1 + εξ , y(t1 + εξ ) = yf (ξ ), are made in the resulting system, where t1 ≥ 0 is any fixed time instant. Thus, we obtain the system dyf (ξ ) = A40 (t1 + εξ, ε)yf (ξ ) + b2 (t1 + εξ, ε). dξ Finally, setting formally ε = 0 in this system and replacing t1 with t yield the fast subsystem dyf (ξ ) = A40 (t, 0)yf (ξ ) + b2 (t, 0), ξ ≥ 0, dξ

(2.156)

where t ≥ 0 is a parameter; yf (ξ ) ∈ E m is a state variable. The fast subsystem (2.156) is of a lower Euclidean dimension than the original system (2.149)–(2.150), and it is ε-free. However, in contrast with the previous sections, this fast subsystem is an undelayed differential equation.

2.4.3 Fundamental Matrix Solution   For a given ε! ∈ (0, ε0 ], consider the block vectors z(t) = col x(t), y(t) and " 1 b(t, ε) = col b1 (t, ε), ε b2 (t, ε) . Also, we consider the block matrices  Ai (t, ε) =

 A1i (t, ε) A2i (t, ε) , i = 0, 1, 1 1 ε A3i (t, ε) ε A4i (t, ε)

(2.157)

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

 G(t, η, ε) =

89

 G1 (t, η, ε) G2 (t, η, ε) . 1 1 ε G3 (t, η, ε) ε G4 (t, η, ε)

(2.158)

Using the above introduced vectors and matrices, we can rewrite the system (2.149)–(2.150) in the following equivalent form for all ε ∈ (0, ε0 ] and t ≥ 0: dz(t)  = Ai (t, ε)z(t − hi ) + dt 1



0 −h

i=0

G(t, η, ε)z(t + η)dη + b(t, ε). (2.159)

For this system, let us give the initial conditions z(η) = ψz (η),

η ∈ [−h, 0);

z(0) = z0 .

(2.160)

For any given σ ≥ 0 and ε ∈ (0, ε0 ], let us consider the initial-value problem with respect to the (n + m) × (n + m)-matrix-valued function H (t) dH (t)  = Ai (t, ε)H (t − hi ) + dt 1



0

−h

i=0

G(t, η, ε)H (t + η)dη,

H (t) = 0, σ − h ≤ t < σ,

t > σ,

H (σ ) = In+m .

(2.161)

Let tc > 0 be a given time instant independent of ε. Let ε ∈ (0, ε0 ] be a given number. Similarly to Propositions 2.1, 2.2 and 2.7, 2.8, we have the following assertions. Proposition 2.11 Let the matrix-valued functions Ai (t, ε) (i = 0, 1) be continuous in t ∈ [0, tc ]. Let the matrix-valued function G(t, η, ε) be piecewise continuous with respect to η ∈ [−h, 0] for each t ∈ [0, tc ] and be continuous with respect to t ∈ [0, tc ] uniformly in η ∈ [−h, 0]. Then, for any given σ ∈ [0, tc ], the initial-value problem (2.161) has the unique solution H (t) = H (t, σ, ε), 0 ≤ σ ≤ t ≤ tc . Proposition 2.12 Let the conditions of Proposition 2.11 be valid. Let b(·, ε) ∈ L2 [0, tc ; E n+m ] and ψz (·) ∈ L2 [−h, 0; E n+m ]. Then, the initial-value problem (2.159)–(2.160) has the unique absolutely continuous solution z(t) = z(t, ε), t ∈ [0, tc ], and this solution has the form  z(t, ε) = H (t, 0, ε)z0 +

0 −h



t

+

,(t, 0, η, ε)ψz (η)dη H H (t, σ, ε)b(σ, ε)dσ,

0

,(t, σ, η, ε) has the form where the (n + m) × (n + m)-matrix-valued function H , , ,(t, σ, η, ε) = H H A (t, σ, η, ε) + HG (t, σ, η, ε),

90

2 Singularly Perturbed Linear Time Delay Systems

, H A (t, σ, η, ε) =

H (t, σ + η + h, ε)A1 (σ + η + h, ε), σ + η − t ≤ −h ≤ η, 0, otherwise,

, H G (t, σ, η, ε) =



η

−h

H (t, σ + η − κ, ε)G(σ + η − κ, κ, ε)dκ.

The matrix H (t, σ, ε) is the fundamental matrix of the system (2.159) (and the system (2.149)–(2.150)). Along with the fundamental matrix H (t, σ, ε) of the system (2.149)–(2.150), we consider the fundamental matrices Hs (t, σ ) and Hf (ξ, t) of the slow (2.154) and fast (2.156) subsystems, respectively. For a given σ ≥ 0, the n × n-matrix-valued function Hs (t, σ ) is the unique solution of the initial-value problem dHs (t, σ )  ¯ = Ai,s (t)Hs (t − hi , σ ) + dt 1



−h

i=0

Hs (t, σ ) = 0,

0

¯ s (t, η)Hs (t + η, σ )dη, t > σ, G

σ − h ≤ t < σ,

Hs (σ, σ ) = In , (2.162)

while, for any t ≥ 0, the m × m-matrix-valued function Hf (ξ, t) is the unique solution of the initial-value problem dHf (ξ, t) = A40 (t, 0)Hf (ξ, t), ξ > 0, dξ

Hf (0, t) = Im ,

meaning that   Hf (ξ, t) = exp A40 (t, 0)ξ ,

ξ ≥ 0.

(2.163)

Similarly to Proposition 2.12, we have the following two assertions. Proposition 2.13 Let the inequality (2.153) be valid for all t ∈ [−h, tc ]. Let the matrix-valued functions A¯ i,s (t) (i = 0, 1) be continuous in the interval [0, tc ]. Let b¯s (·) ∈ L2 [0, tc ; E n ]. Then, for any given vector-valued function ψs0 (·) ∈ L2 [−h, 0; E n ] and vector xs0 ∈ E n , the system (2.154) subject to the initial conditions xs (η) = ψs0 (η) and η ∈ [−h, 0), xs (0) = xs0 , has the unique absolutely continuous solution  xs (t) = Hs (t, 0)xs0 + 

t

+ 0

0 −h

, H s (t, 0, η)ψs0 (η)dη

Hs (t, σ )b¯s (σ )dσ,

t ∈ [0, tc ],

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

91

where

, H s (t, σ, η) =

 Hs (t, σ + η + h)A¯ i,s (σ + η + h), σ + η − t ≤ −h ≤ η 0, otherwise  η ¯ s (σ + η − κ, κ)dκ. + Hs (t, σ + η − κ)G −h

Proposition 2.14 Let t ∈ [0, tc ] be any given value. Then, for any given vector yf 0 ∈ E m , the system (2.156) subject to the initial condition yf (0) = yf 0 has the unique solution  yf (ξ, t) = Hf (ξ, t)yf 0 +

ξ

Hf (ξ − ω, t)b2 (t, 0)dω

0

   = exp A40 (t, 0)ξ yf 0 + A−1 40 (t, 0)b2 (t, 0) −A−1 40 (t, 0)b2 (t, 0),

ξ ≥ 0.

2.4.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Nonsmall Delays Here, in contrast with Sects. 2.2.4 and 2.3.4, we consider the case where singularly perturbed matrix differential systems have only nonsmall delays in both, slow and fast, state variables. We consider the system (2.161) and some other systems with such a feature in a given time interval [0, tc ], where tc is independent of ε and satisfies the inequality tc > h. The analysis of these systems is subject to the assumptions formulated in Remark 2.9. In addition to the assumptions of this remark, we assume that: (AI-3)

(AII-3) (AIII-3)

(AIV-3)

The matrix-valued functions Aki (t, ε) (i = 0, 1; k = 1, 2) and A31 (t, ε), A41 (t, ε) are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ]. The matrix-valued functions A30 (t, ε) and A40 (t, ε) are continuously differentiable with respect to (t, ε) ∈ [−h, tc ] × [0, ε0 ]. The matrix-valued functions Gk (t, η, ε) (k = 1, . . . , 4) are piecewise continuous with respect to η ∈ [−h, 0] for each (t, ε) ∈ [0, tc ] × [0, ε0 ], and they are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in η ∈ [−h, 0]. All eigenvalues λ(t) of the matrix A40 (t, 0) satisfy the inequality Reλ(t) < −2β for all t ∈ [−h, tc ], where β > 0 is some constant.

92

2 Singularly Perturbed Linear Time Delay Systems

Remark 2.10 Due to the assumptions (AII-3) and (AIV-3) and Lemma 2.1, there exists a positive value ε1 ≤ ε0 such that all eigenvalues ν(t, ε) of the matrix A40 (t, ε) satisfy the inequality Reν(t, ε) < −2β for all t ∈ [−h, tc ] and ε ∈ (0, ε1 ]. Let for any given ε ∈ (0, ε0 ] and σ ∈ [0, tc ], P(t, σ, ε) be the unique solution of the problem dP(t, σ, ε) = A0 (t, ε)P(t, σ, ε), dt

t ∈ (σ, tc ],

P(σ, σ, ε) = In+m . (2.164)

Now, we partition the matrix P(t, σ, ε) into blocks as  P(t, σ, ε) =

P1 (t, σ, ε) P3 (t, σ, ε)

 P2 (t, σ, ε) , P4 (t, σ, ε)

(2.165)

where the matrices P1 (t, σ, ε), P2 (t, σ, ε), P3 (t, σ, ε), and P4 (t, σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. The following proposition is a direct consequence of the Remark 2.10 and the results of [12] on the estimate of the fundamental matrix solution of a singularly perturbed differential system without delays. Proposition 2.15 Let the assumptions (AI-3)–(AIV-3) be valid. Then, there exists a positive number ε2 ≤ ε1 such that, for all ε ∈ (0, ε2 ], the following inequalities are satisfied: Pp (t, σ, ε) ≤ a, p = 1, 3,

P2 (t, σ, ε) ≤ aε,

 

P4 (t, σ, ε) ≤ a ε + exp − β(t − σ )/ε ,

0 ≤ σ ≤ t ≤ tc ,

where a > 0 is some constant independent of ε. Let us partition the solution H (t, σ, ε) of the initial-value problem (2.161) into blocks as   H1 (t, σ, ε) H2 (t, σ, ε) H (t, σ, ε) = , (2.166) H3 (t, σ, ε) H4 (t, σ, ε) where the matrices H1 (t, σ, ε), H2 (t, σ, ε), H3 (t, σ, ε), and H4 (t, σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Lemma 2.4 Let the assumptions (AI-3)–(AIV-3) be valid. Then, there exists a positive number ε3 ≤ ε2 such that, for all ε ∈ (0, ε3 ], the following inequalities are satisfied: Hp (t, σ, ε) ≤ a, p = 1, 3,

H2 (t, σ, ε) ≤ aε, 0 ≤ σ ≤ t ≤ tc ,

 

H4 (t, σ, ε) ≤ a ε + exp − β(t − σ )/ε ,

0 ≤ σ ≤ t < min{σ + h, tc },

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

93

and H4 (t, σ, ε) ≤ aε,

σ + h ≤ t ≤ tc ,

where a > 0 is some constant independent of ε. Proof of the lemma is presented in Sect. 2.4.6. Let, for any given ε ∈ (0, ε3 ], the (n + m) × (n + m)-matrix-valued function Υ (σ, ε) be the solution of the terminal-value problem 1   T dΥ (σ, ε) =− Ai (σ + hi , ε) Υ (σ + hi , ε) dσ

 −

i=0

0



−h

G(σ − η, η, ε)

T

Υ (σ − η, ε)dη,

Υ (tc , ε) = In+m ,

σ ∈ [0, tc ),

Υ (σ, ε) = 0,

σ > tc .

(2.167)

In this problem, it is assumed that for all ε ∈ [0, ε3 ] A1 (t, ε) = A1 (tc , ε), G(t, η, ε) = G(tc , η, ε), t > tc , η ∈ [−h, 0]. Remark 2.11 By virtue of the results of [22] (Section 4.3), we have the equality H (tc , σ, ε) = Υ T (σ, ε), σ ∈ [0, tc ], ε ∈ (0, ε3 ]. Let us partition the matrix Υ (σ, ε) into blocks as  Υ (σ, ε) =

Υ1 (σ, ε) Υ3 (σ, ε)

 Υ2 (σ, ε) , Υ4 (σ, ε)

where the matrices Υ1 (σ, ε), Υ2 (σ, ε), Υ3 (σ, ε), and Υ4 (σ, ε) are of the dimensions n × n, n × m, m × n and m × m, respectively. Along with the matrix-valued function Υ (σ, ε), we consider the matrix-valued functions Υ1s (σ ) and Υ4f (ξ ), which are the solutions of the problems  T dΥ1s (σ ) =− A¯ i,s (σ + hi ) Υ1s (σ + hi ) dt 1

 −

i=0

0 −h

  ¯ s (σ − η, η) T Υ1s (σ − η)dη, σ ∈ [0, tc ), G Υ1s (tc ) = In ,

Υ1s (σ ) = 0,

σ > tc ,

(2.168)

94

2 Singularly Perturbed Linear Time Delay Systems

dΥ4f (ξ ) = AT40 (tc , 0)Υ4f (ξ ), dξ

ξ > 0,

Υ4f (0) = Im ,

ξ < 0,

Υ4f (ξ ) = 0,

(2.169)

¯ s (t, η) are defined in (2.155); and where A¯ i,s (t) (i = 0, 1) and G A¯ 1,s (t) = A¯ 1,s (tc ),

¯ s (t, η) = G ¯ s (tc , η), t > tc , η ∈ [−h, 0]. G

From (2.163) and (2.169), we directly have   Υ4f (ξ ) = exp AT40 (tc , 0)ξ = HfT (ξ, tc ),

ξ ≥ 0.

Remark 2.12 By virtue of the results of [22] (Section 4.3), we have Υ1s (σ ) = HsT (tc , σ ),

σ ∈ [0, tc ],

where Hs (t, σ ) is defined by the initial-value problem (2.162). Denote  ⎧  −1 T -1  T ⎪ ⎪ − A40 (σ, 0) ⎪ i=0 A2i (σ + hi , 0) Υ1s (σ + hi ) ⎪ ⎨  .0  T Υ3s (σ ) = σ ∈ [0, tc ], ⎪ ⎪ + −h G2 (σ − η, η, 0) Υ1s (σ − η)dη , ⎪ ⎪ ⎩ 0, σ > tc . (2.170) In this equation, it is assumed that A21 (t, 0) = A21 (tc , 0),

G2 (t, η, 0) = G2 (tc , η, 0),

t > tc , η ∈ [−h, 0].

Theorem 2.4 Let the assumptions (AI-3)–(AIV-3) be valid. Then, for all ε ∈ (0, ε3 ], the following inequalities are satisfied: Υ1 (σ, ε) − Υ1s (σ ) ≤ aε,

Υ3 (σ, ε)−εΥ3s (σ ) ≤aε

σ ∈ [0, tc ],

 

⎧ ⎨ ε+ exp − β(tc − σ )/ε ,

if tc − h < σ ≤ tc ,

 

⎩ ε+ exp − β(tc − h − σ )/ε , if 0 ≤ σ ≤ tc − h,

where Υp (σ, ε) (p = 1, 3) are the corresponding blocks of the solution to the terminal-value problem (2.167); a > 0 is some constant independent of ε. Proof of the theorem is presented in Sect. 2.4.7. Similarly to Theorem 2.4, we have the following theorem.

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

95

Theorem 2.5 Let the assumptions (AI-3)–(AIV-3) be valid. Then, for all ε ∈ (0, ε3 ], the following inequalities are satisfied:  Υ2 (σ, ε) ≤ a,

Υ4 (σ, ε) − Υ4f

tc − σ ε

 ≤ aε, σ ∈ [0, tc ],

where Υl (σ, ε) (l = 2, 4) are the corresponding blocks of the solution to the terminal-value problem (2.167); a > 0 is some constant independent of ε.

2.4.5 Example 8 Consider the following system, a particular case of (2.149)–(2.150), dx(t) = tx(t − 1) − y(t − 1) − t 2 , dt ε

dy(t) = x(t) − ε(t + 1)x(t − 1) − y(t) − εy(t − 1) + t, dt

t ≥ 0, t ≥ 0, (2.171)

where x(t) and y(t) are scalars. In what follows, we consider this system in the interval [0, tc ] = [0, 2]. Setting formally ε = 0 in this system, we obtain its slow subsystem in the differential-algebraic form dxs (t) = txs (t − 1) − ys (t − 1) − t 2 , dt 0 = xs (t) − ys (t) + t,

t ∈ [0, 2], t ∈ [0, 2].

Resolving the second equation of this system with respect to ys (t), we obtain ys (t) = xs (t) + t,

t ∈ [0, 2].

Now, we substitute this expression for ys (t) into the first equation of the slow subsystem in the differential-algebraic form. Thus, we obtain the slow subsystem of (2.171) in the pure differential form dxs (t) = (t − 1)xs (t − 1) − t 2 − t, dt

t ∈ [0, 2].

The fast subsystem, associated with the singularly perturbed system (2.171), is dyf (ξ ) = −yf (ξ ) + t, dξ where t ∈ [0, 2] is a parameter.

ξ ≥ 0,

96

2 Singularly Perturbed Linear Time Delay Systems

The fundamental matrix H (t, σ, ε) of the system (2.171) is the unique solution to the initial-value problem dH (t, σ, ε) = A0 (t, ε)H (t, σ, ε) + A1 (t, ε)H (t − 1, σ, ε), t ∈ (σ, 2], dt H (t, σ, ε) = 0, σ − 1 ≤ t < σ, H (σ, σ, ε) = I2 , (2.172) where σ ∈ [0, 2],  A0 (t, ε) =

0

0

1 ε

− 1ε



 ,

A1 (t, ε) =

 t −1 . −(t + 1) −1

The assumptions (AI-3)–(AIV-3) are fulfilled for the coefficients of the differential equation in (2.172). In the assumption (AIV-3), we can choose β = 0.45. Let us solve the problem (2.172). For this purpose, we use the block representation (2.166) of the matrix H (t, σ, ε) and the method of steps (see, e.g., [10]). In this example, the blocks Hk (t, σ, ε) (k = 1, . . . , 4) are scalars. We consider the following two cases: σ ∈ [0, 1) and σ ∈ [1, 2]. Case σ ∈ [0, 1) In this case the blocks of H (t, σ, ε) have the form % H1 (t, σ, ε) =

1, t2 2

−t −

σ2 2

+

#

!

t−σ −1 ε

"$ t ∈ [σ, σ + 1), , t ∈ [σ + 1, 2],

+ ε 1 − exp − % 0,# ! " $ t ∈ [σ, σ + 1), H2 (t, σ, ε) = t−σ −1 ε exp − ε − 1 , t ∈ [σ + 1, 2], ⎧ ! " ⎪ 1 − exp − t−σ , t ∈ [σ, σ + 1), ⎪ ⎪ ε ⎨ 2 # 2 H3 (t, σ, ε) = t2 − t − σ2 + 32 − 2ε(t − ε) + 2ε(σ + 1) ⎪ " ! ⎪ $  ⎪ ⎩ −2ε2 − exp − 1 exp − t−σ −1 , t ∈ [σ + 1, 2], ε ε ⎧ " ! ⎨ exp − t−σ , t ∈ [σ, σ + 1), ε ! " ! " H4 (t, σ, ε) = ⎩ exp − t−σ − ε + ε exp − t−σ −1 , t ∈ [σ + 1, 2]. ε ε 3 2

Case σ ∈ [1, 2] In this case the blocks of H (t, σ, ε) have the form H1 (t, σ, ε) = 1,

t ∈ [σ, 2],

H2 (t, σ, ε) = 0,   t −σ , H3 (t, σ, ε) = 1 − exp − ε

t ∈ [σ, 2], t ∈ [σ, 2],

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays



t −σ H4 (t, σ, ε) = exp − ε

97

 ,

t ∈ [σ, 2].

Estimating Hk (t, σ, ε) (k = 1, . . . , 4) in both cases for all ε ∈ (0, 0.5], we obtain       H1 (t, σ, ε) ≤ 2, H2 (t, σ, ε) ≤ ε, H3 (t, σ, ε) ≤ 3, 0 ≤ σ ≤ t ≤ 2,     H4 (t, σ, ε) ≤ exp − t − σ , 0 ≤ σ ≤ t < min{σ + 1, 2}, ε   H4 (t, σ, ε) ≤ 2ε, σ + 1 ≤ t ≤ 2. These estimates clearly illustrate Lemma 2.4. Proceed to the terminal-value problem (2.167). In this example, it becomes as  T  T dΥ (σ, ε) = − A0 (σ, ε) Υ (σ, ε)− A1 (σ +1, ε) Υ (σ +1, ε), σ ∈ [0, 2), dσ Υ (2, ε)=I2 , Υ (σ, ε)=0, σ >2. (2.173) In this problem, A1 (t, ε) = A1 (2, ε), t > 2. Solving (2.173) by the method of steps (see, e.g., [10]), we obtain % Υ1 (σ, ε) =

3 2

−

σ2 2

"$ # ! , σ ∈ [0, 1], + ε 1 − exp σ −1 ε

1, σ ∈ (1, 2], ⎧ 2 3 ⎪ − σ − 2ε(2 − ε) ⎪ "$ ! " ! ⎨2# 2 1 σ −1 2 Υ2 (σ, ε) = + 2ε(σ + 1) − 2ε − exp − ε exp ε , σ ∈ [0, 1], ! " ⎪ ⎪ ⎩ 1 − exp σ −2 , σ ∈ (1, 2], ε ! " $ % # ε exp σ −1 − 1 , σ ∈ [0, 1], ε Υ3 (σ, ε) = 0, σ ∈ (1, 2], ⎧ ! " # ! "$ ⎨ exp σ −2 − ε 1 − exp σ −1 , σ ∈ [0, 1], ε ! ε " Υ4 (σ, ε) = ⎩ exp σ −2 , σ ∈ (1, 2], ε where Υ1 (σ, ε), Υ2 (σ, ε), Υ3 (σ, ε), and Υ4 (σ, ε) are the upper left-hand entry, the upper right-hand entry, the lower left-hand entry, and the lower right-hand entry, respectively, of the 2 × 2-matrix Υ (σ, ε)—the solution of the terminal-value problem (2.173). The comparison of Υk (σ, ε) (k = 1, . . . , 4) with the above obtained Hk (t, σ, ε) (k = 1, . . . , 4) directly yields

98

2 Singularly Perturbed Linear Time Delay Systems

Υ1 (σ, ε) = H1 (2, σ, ε),

Υ2 (σ, ε) = H3 (2, σ, ε),

Υ3 (σ, ε) = H2 (2, σ, ε),

Υ4 (σ, ε) = H4 (2, σ, ε),

meaning that  T Υ (σ, ε) = H (2, σ, ε) . Now, let us consider the problem (2.168), the expression (2.170), and the problem (2.169) in this example. The problem (2.168) and the expression (2.170) become  dΥ1s (σ )  = 1 − σ + 1 Υ1s (σ + 1), σ ∈ [0, 2), dσ Υ (2) = 1, Υ1s (σ ) = 0, σ > 2,

Υ3s (σ ) =

−Υ1s (σ + 1), σ ∈ [0, 2], 0, σ > 2.

(2.174)

(2.175)

In (2.174),

σ + 1 =

σ + 1, σ + 1 ≤ 2, 2, σ + 1 > 2.

The problem (2.169) becomes dΥ4f (ξ ) = −Υ4f (ξ ), dξ Υ4f (0) = 1,

Υ4f (ξ ) = 0,

ξ > 0, ξ < 0.

The problem (2.174) and the expression (2.175) yield % Υ1s (σ ) =

3 2

−

1,

Υ3s (σ ) =

σ2 2 ,

σ ∈ [0, 1], σ ∈ (1, 2],

−1, σ ∈ [0, 1], 0, σ > 1,

while the problem (2.176) yields Υ4f (ξ ) = exp(−ξ ),

ξ ≥ 0.

(2.176)

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

99

Using the above derived Υk (σ, ε) (k = 1, . . . , 4), and Υ1s (σ ), Υ3s (σ ), and Υ4f (ξ ), we obtain the following inequalities for all ε ∈ (0, 0.5]:   Υ1 (σ, ε) − Υ1s (σ ) ≤ ε,   Υ3 (σ, ε) − εΥ3s (σ ) ≤   Υ2 (σ, ε) ≤ 3,

%

0,

!

ε exp

σ ∈ [0, 2],

σ −1 ε

"

σ ∈ (1, 2], , σ ∈ [0, 1],

     Υ4 (σ, ε) − Υ4f − 2 − σ  ≤ ε, σ ∈ [0, 2].   ε

These inequalities clearly illustrate Theorems 2.4 and 2.5.

2.4.6 Proof of Lemma 2.4

First, consider the problem (2.161) in the interval t ∈ σ, min{σ + h, tc } . Due to the initial conditions of (2.161), it can be rewritten in this interval as follows: dH (t, σ, ε) = A0 (t, ε)H (t, σ, ε) + dt



t

G(t, χ − t, ε)H (χ , σ, ε)dχ ,

σ

H (σ, σ, ε) = In+m . Using (2.164), this problem can be rewritten in the equivalent integral form as 

t

+



χ1

P(t, χ1 , ε)

σ

H (t, σ, ε) = P(t, σ, ε)  G(χ1 , χ − χ1 , ε)H (χ , σ, ε)dχ dχ1 ,

(2.177)

σ

or as 

t

H (t, σ, ε) = P(t, σ, ε) +

Q(t, χ , ε)H (χ , σ, ε)dχ ,

(2.178)

σ

where  Q(t, χ , ε) =

t

P(t, χ1 , ε)G(χ1 , χ − χ1 , ε)dχ1 .

χ

Equation (2.178) is a matrix Volterra integral equation of the second kind with the kernel Q(t, χ , ε). Substituting the block representations of the matrices G(t, η, ε) and P(t, σ, ε) (see Eqs. (2.158) and (2.165)) into the integral expression of

100

2 Singularly Perturbed Linear Time Delay Systems

Q(t, χ , ε) and calculating the product of the corresponding block matrices appearing in the integrand, we can express this kernel in the block form as  Q(t, χ , ε) =

 Q1 (t, χ , ε) Q2 (t, χ , ε) , Q3 (t, χ , ε) Q4 (t, χ , ε)

where 

t

Q1 (t, χ , ε) =

P1 (t, χ1 , ε)G1 (χ1 , χ − χ1 , ε)

χ

1 + P2 (t, χ1 , ε)G3 (χ1 , χ − χ1 , ε)dχ1 dχ1 , ε  t P1 (t, χ1 , ε)G2 (χ1 , χ − χ1 , ε) Q2 (t, χ , ε) = χ

1 + P2 (t, χ1 , ε)G4 (χ1 , χ − χ1 , ε)dχ1 dχ1 , ε  t P3 (t, χ1 , ε)G1 (χ1 , χ − χ1 , ε) Q3 (t, χ , ε) = χ

1 + P4 (t, χ1 , ε)G3 (χ1 , χ − χ1 , ε)dχ1 dχ1 , ε  t P3 (t, χ1 , ε)G2 (χ1 , χ − χ1 , ε) Q4 (t, χ , ε) = χ

1 + P4 (t, χ1 , ε)G4 (χ1 , χ − χ1 , ε)dχ1 dχ1 . ε Using the assumption (AIII-3) and Proposition 2.15, one obtains the existence of a positive number ε˜ ≤ ε2 such that, for all ε ∈ (0, ε˜ ], the matrix-valued functions Qk (t, χ , ε) (k = 1, . . . , 4) are bounded, i.e., Q(t, χ , ε) ≤ a, t ∈ [σ, tc ], χ ∈ [σ, t], where a > 0 is some constant independent of ε. Since the kernel Q(t, χ , ε) of the integral equation (2.178) is bounded, then the resolvent R(t, χ , ε) of this equation is also bounded. Thus, the solution of (2.178) is  t R(t, χ , ε)P(χ , σ, ε)dχ . (2.179) H (t, σ, ε) = P(t, σ, ε) + σ

Let us partition the matrix-valued resolvent R(t, χ , ε) into blocks as  R(t, χ , ε) =

R1 (t, χ , ε) R3 (t, χ , ε)

 R2 (t, χ , ε) , R4 (t, χ , ε)

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

101

where the matrices R1 (t, σ, ε), R2 (t, σ, ε), R3 (t, σ, ε), and R4 (t, σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Using this block representation, and Eqs. (2.158) and (2.165), we can rewrite the equality (2.179) in the following equivalent form: H1 (t, σ, ε) = P1 (t, σ, ε)





R1 (t, χ , ε)P1 (χ , σ, ε) + R2 (t, χ , ε)P3 (χ , σ, ε) dχ ,

t

+ σ

H2 (t, σ, ε) = P2 (t, σ, ε)





R1 (t, χ , ε)P2 (χ , σ, ε) + R2 (t, χ , ε)P4 (χ , σ, ε) dχ ,

t

+ σ

H3 (t, σ, ε) = P3 (t, σ, ε)





R3 (t, χ , ε)P1 (χ , σ, ε) + R4 (t, χ , ε)P3 (χ , σ, ε) dχ ,

t

+ σ

H4 (t, σ, ε) = P4 (t, σ, ε)



t

+



R3 (t, χ , ε)P2 (χ , σ, ε) + R4 (t, χ , ε)P4 (χ , σ, ε) dχ .

σ

These four equalities, along with Proposition 2.15 and the fact that the resolvent R(t, χ , ε) is bounded, directly yield the validity of the following inequalities in the interval t ∈ σ, min{σ + h, tc } and for all ε ∈ (0, ε˜ ]: H2 (t, σ, ε) ≤ aε, Hp (t, σ, ε) ≤ a, p = 1, 3,  

H4 (t, σ, ε) ≤ a ε + exp − β(t − σ )/ε .

(2.180)

Moreover, if tc ≥ σ + h, then there exists a positive number ε˜ 1 ≤ ε˜ such that, for all ε ∈ (0, ε˜ 1 ], the following inequalities hold: Hl (σ + h, σ, ε) ≤ aε, l = 2, 4.

(2.181) The inequalities (2.180) prove the lemma for t ∈ σ, min{σ + h, tc } . Let us assume that tc > σ + h, and consider the problem (2.161) in the interval t ∈ σ + h, min{σ +2h, tc } . Denoting the solution of (2.161) in the interval t ∈ [σ, σ +h] as

H (t, σ, ε), we can rewrite this problem in the interval t ∈ σ + h, min{σ + 2h, tc } as  t dH (t, σ, ε) =A0 (t, ε)H (t, σ, ε)+ G(t, χ −t, ε)H (χ , σ, ε)dχ dt σ +h Hp (σ + h, σ, ε) ≤ a, p = 1, 3,

+Ω(t, σ, ε), H (σ +h, σ, ε)=H (σ +h, σ, ε), (2.182) where

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2 Singularly Perturbed Linear Time Delay Systems

 +

Ω(t, σ, ε) = A1 (t, ε)H (t − h, σ, ε) σ +h−t −h



G(t, η, ε)H (t + η, σ, ε)dη, t ∈ σ + h, min{σ + 2h, tc } .

Using the condition (2.151) and the inequalities (2.180), we obtain the existence of a positive number ε˜ 2 ≤ ε˜ 1 such that, for all ε ∈ (0, ε˜ 2 ], the following inequalities are satisfied in the interval t ∈ σ + h, min{σ + 2h, tc } : Ωp (t, σ, ε) ≤ a, p = 1, 3, 

Ωl (t, σ, ε) ≤ a ε + exp − β(t − h − σ )/ε , l = 2, 4,



(2.183)

where Ω1 (t, σ, ε), Ω2 (t, σ, ε), Ω3 (t, σ, ε), and Ω4 (t, σ, ε) are the upper left-hand block, upper right-hand block, lower left-hand block, and lower right-hand block of the dimensions n × n, n × m, m × n, and m × m, respectively, of the matrix Ω(t, σ, ε); a > 0 is some constant independent of ε. Using (2.164), the problem (2.182) can be rewritten in the equivalent integral form as  +

t σ +h

 P(t, χ1 , ε)

χ1

σ +h

H (t, σ, ε) = P(t, σ, ε)  G(χ1 , χ − χ1 , ε)H (χ , σ, ε)dχ dχ1

+Ω(t, σ, ε),



t ∈ σ + h, min{σ + 2h, tc } , (2.184)

where P(t, σ, ε) = P(t, σ + h, ε)H (σ + h, σ, ε),  Ω(t, σ, ε) =

t σ +h

P(t, χ1 , ε)Ω(χ1 , σ, ε)dχ1 .

Using the block representations of the matrices P(t, σ, ε), H (t, σ, ε), and Ω(t, σ, ε), as well as Proposition 2.15 and the inequalities (2.181), (2.183), we

obtain the following inequalities for all ε ∈ (0, ε˜ 2 ] and t ∈ σ +h, min{σ +2h, tc } : P p (t, σ, ε) ≤ a, p = 1, 3, Ω p (t, σ, ε) ≤ a, p = 1, 3,

P l (t, σ, ε) ≤ aε, l = 2, 4, Ω l (t, σ, ε) ≤ aε, l = 2, 4, (2.185)

where P 1 (t, σ, ε), P 2 (t, σ, ε), P 3 (t, σ, ε), and P 4 (t, σ, ε) are the upper lefthand block, upper right-hand block, lower left-hand block, and lower right-hand block of the dimensions n × n, n × m, m × n, and m × m, respectively, of the matrix P(t, σ, ε); Ω 1 (t, σ, ε), Ω 2 (t, σ, ε), Ω 3 (t, σ, ε), and Ω 4 (t, σ, ε) are the upper left-

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

103

hand block, upper right-hand block, lower left-hand block, and lower right-hand block of the dimensions n × n, n × m, m × n, and m × m, respectively, of the matrix Ω(t, σ, ε); a > 0 is some constant independent of ε. Now, treating Eq. (2.184) similarly to Eq. (2.177), and using the inequalities (2.185), we obtain the existence of a positive number ε˜ 3 ≤ ε˜ 2 such that, for all ε ∈ (0, ε˜ 3 ] and t ∈ σ + h, min{σ + 2h, tc } , the following inequalities are satisfied: Hp (t, σ, ε) ≤ a, p = 1, 3,

Hl (t, σ, ε) ≤ aε, l = 2, 4,

(2.186)

where a > 0 is some constant independent of ε. These inequalities prove the lemma

for t ∈ σ + h, min{σ + 2h, tc } . If tc > σ + 2h, we can analyze the initial-value problem (2.161) in the interval t ∈ [σ + 2h, min{σ + 3h, tc }] like we did this in the two previous intervals. Such an analysis yields the validity of the inequalities similar to (2.186) for all sufficiently small ε > 0. This procedure is stopped when tc ≤ σ +qmin h, where 0 < qmin < +∞ is the smallest integer satisfying this inequality. Since qmin is independent of ε, and the inequalities similar to (2.186) hold in each interval t ∈ [σ + qh, min{σ + (q + 1)h, tc }] (q = 1, . . . , qmin − 1), then the statement of the lemma is valid for all 0 ≤ σ ≤ t ≤ tc and all sufficiently small ε > 0.

2.4.7 Proof of Theorem 2.4 Let us denote

Υ13 (σ, ε) =



 Υ1 (σ, ε) , Υ3 (σ, ε)



Υ13,s (σ, ε) =



 Υ1s (σ ) , εΥ3s (σ )



Δ(σ, ε) = Υ13 (σ, ε) − Υ13,s (σ, ε).

(2.187)

Using the notations in (2.187), the terminal-value problems (2.167) and (2.168) and Eq. (2.170), we obtain (quite similarly to the problem (2.139)) the terminalvalue problem for the matrix-valued function Δ(σ, ε) 1   T dΔ(σ, ε) =− Ai (σ + hi , ε) Δ(σ + hi , ε) dσ

 −

i=0

0 −h

 T G(σ − η, η, ε) Δ(σ − η, ε)dη + Γ (σ, ε),  Δ(tc , ε) =

 0 , −εΥ3s (tc )

σ ∈ [0, tc ),

Δ(σ, ε) = 0, σ > tc , (2.188)

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2 Singularly Perturbed Linear Time Delay Systems

where the (n + m) × n-matrix-valued function Γ (σ, ε) has the block form  Γ (σ, ε) =

 Γ1 (σ, ε) , Γ2 (σ, ε)

(2.189)

and the n × n-block Γ1 (σ, ε) and m × n-block Γ2 (σ, ε) are, respectively,

 1 #  T  T A1i (σ + hi , ε) Υ1s (σ + hi )+ A3i (σ + hi , ε) Υ3s (σ + hi ) Γ1 (σ, ε)=− i=0

$  T − A¯ i,s (σ + hi ) Υ1s (σ + hi ) +



0

−h

# T G1 (σ − η, η, ε) Υ1s (σ − η)

$   T   ¯ s (σ − η, η) T Υ1s (σ − η) dη , + G3 (σ − η, η, ε) Υ3s (σ − η) − G (2.190)

Γ2 (σ, ε) = −ε

dΥ3s (σ ) dσ

 1 # $  T  T A2i (σ + hi , ε) Υ1s (σ + hi ) + A4i (σ + hi , ε) Υ3s (σ + hi ) −  +

i=0 0

−h

# $  T  T G2 (σ − η, η, ε) Υ1s (σ − η) + G4 (σ − η, η, ε) Υ3s (σ − η) dη . (2.191)

In Eqs. (2.190)–(2.191), like in (2.167), it is assumed that for all ε ∈ [0, ε3 ] Ak1 (t, ε)=Ak1 (tc , ε), Gk (t, η, ε)=Gk (tc , η, ε), k=1, . . . , 4, t>tc , η ∈ [−h, 0]. Let us estimate Γ1 (σ, ε) and Γ2 (σ, ε). We start with the first matrix. Using the assumptions (AI-3)–(AIII-3), we can represent the matrix-valued functions A1i (σ + hi , ε), A2i (σ + hi , ε) (i = 0, 1), A30 (σ, ε), A40 (σ, ε), G1 (σ − η, η, ε), and G2 (σ − η, η, ε) as A1i (σ + hi , ε) = A1i (σ + hi , 0) + ΔA,1i (σ + hi , ε), i = 0, 1, σ ∈ [0, tc ], A2i (σ + hi , ε) = A2i (σ + hi , 0) + ΔA,2i (σ + hi , ε), i = 0, 1, σ ∈ [0, tc ], A30 (σ, ε) = A30 (σ, 0) + ΔA,30 (σ, ε),

σ ∈ [0, tc ],

A40 (σ, ε) = A40 (σ, 0) + ΔA,40 (σ, ε),

σ ∈ [0, tc ],

G1 (σ − η, η, ε) = G1 (σ − η, η, 0) + ΔG,1 (σ − η, η, ε),

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

105

σ ∈ [0, tc ], η ∈ [−h, 0], G2 (σ − η, η, ε) = G2 (σ − η, η, 0) + ΔG,2 (σ − η, η, ε), σ ∈ [0, tc ], η ∈ [−h, 0],

(2.192)

where ΔA,1i (σ + hi , ε), ΔA,2i (σ + hi , ε), ΔA,30 (σ + hi , ε), ΔA,40 (σ + hi , ε), ΔG,1 (σ − η, η, ε), and ΔG,2 (σ − η, η, ε) are some matrix-valued functions of the corresponding dimensions satisfying the following inequalities for all σ ∈ [0, tc ], η ∈ [−h, 0], ε ∈ [0, ε3 ]: ΔA,1i (σ + hi , ε) ≤ aε,

ΔA,2i (σ + hi , ε) ≤ aε, i = 0, 1,

ΔA,30 (σ, ε) ≤ aε, ΔG,1 (σ − η, η, ε) ≤ aε,

ΔA,40 (σ, ε) ≤ aε,

ΔG,2 (σ − η, η, ε) ≤ aε. (2.193)

Also, due to the assumptions (2.151) and (AI-3)–(AIII-3), the matrices A31 (σ + h, ε), A41 (σ + h, ε), G3 (σ − η, η, ε), and G4 (σ − η, η, ε) satisfy the following inequalities for all σ ∈ [0, tc ], η ∈ [−h, 0], ε ∈ [0, ε3 ]: A31 (σ + h, ε) ≤ aε,

A41 (σ + h, ε) ≤ aε,

G3 (σ − η, η, ε) ≤ aε,

G4 (σ − η, η, ε) ≤ aε.

(2.194)

In (2.193) and (2.194), a > 0 is some constant independent of ε. Substitution of (2.155) and (2.192) into (2.190), and use of (2.170) yield after a routine algebra

 1  T  T Γ1 (σ, ε) = − ΔA,1i (σ + hi , ε) Υ1s (σ + hi ) + ΔA,30 (σ, ε) Υ3s (σ ) i=0

 T + A31 (σ + h, ε) Υ3s (σ + h) +



0

−h

# T ΔG,1 (σ − η, η, ε) Υ1s (σ − η)

$   T + G3 (σ − η, η, ε) Υ3s (σ − η) dη ,

σ ∈ [0, tc ].

Note that Υ1s (σ ) is bounded in the interval [0, +∞). Therefore, Υ3s (σ ) is also bounded in this interval. This observation and the inequalities (2.193) and (2.194) yield the estimate Γ1 (σ, ε) ≤ aε,

σ ∈ [0, tc ],

ε ∈ (0, ε3 ],

where a > 0 is some constant independent of ε. Proceed to Γ2 (σ, ε). We can rewrite this matrix as

(2.195)

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2 Singularly Perturbed Linear Time Delay Systems

Γ2 (σ, ε) = −ε

dΥ3s (σ ) + Γ (σ, ε), dσ

(2.196)

where

 1 # $  T  T Γ (σ, ε)=− A2i (σ +hi , ε) Υ1s (σ +hi )+ A4i (σ +hi , ε) Υ3s (σ +hi )  +

i=0

0 −h

# $  T  T G2 (σ −η, η, ε) Υ1s (σ − η)+ G4 (σ −η, η, ε) Υ3s (σ −η) dη .

For Γ (σ, ε), similarly to (2.195), we have the estimate Γ (σ, ε) ≤ aε,

σ ∈ [0, tc ],

ε ∈ (0, ε3 ],

(2.197)

where a > 0 is some constant independent of ε. Now, let us deal with the derivative dΥ3s (σ )/dσ , σ ∈ [0, tc ]. Due to (2.170), we can represent Υ3s (σ ) in the interval [0, tc ] as Υ3s (σ ) = Υ3s1 (σ ) + Υ3s2 (σ ) + Υ3s3 (σ ),

σ ∈ [0, tc ],

(2.198)

where  T  T A20 (σ, 0) Υ1s (σ ), Υ3s1 (σ ) = − A−1 40 (σ, 0)  T  T Υ3s2 (σ ) = − A−1 A21 (σ + h, 0) Υ1s (σ + h), 40 (σ, 0)    −1 T 0 T 3 Υ3s (σ ) = − A40 (σ, 0) G2 (σ − η, η, 0) Υ1s (σ − η)dη. max{−h,σ −tc }

Due to (2.168) and the assumptions (AI-3) and (AIII-3), the derivatives dΥ3s1 (σ )/dσ and dΥ3s3 (σ )/dσ are piecewise continuous functions in the interval [0, tc ] with a single finite break at σ = tc − h. Therefore, dΥ3s1 (σ ) ≤ a, dσ

dΥ3s3 (σ ) ≤ a, dσ

σ ∈ [0, tc ].

(2.199)

In contrast with dΥ3s1 (σ )/dσ and dΥ3s3 (σ )/dσ , the derivative dΥ3s2 (σ )/dσ is a generalized function because the function Υ3s2 (σ ) has a break at σ = tc − h. Let us represent this function and its derivative in a form convenient for the further analysis. Let F (σ ) be an m × n-matrix-valued function, differentiable in the interval (−∞, +∞), and such that F (σ ) = Υ3s2 (σ ) in the interval [0, tc − h]. Thus, we can represent the matrix-valued function Υ3s2 (σ ) in the form

2.4 One Class of Singularly Perturbed Systems with Nonsmall Delays

  Υ3s2 (σ ) = F (σ ) 1 − θsf (σ − tc + h) ,

107

σ ∈ [0, tc ],

where θsf (σ − tc + h) is the Heaviside step function with the break point σ = tc − h (see, e.g., [35]). Using this representation, we obtain the representation for the generalized derivative dΥ3s2 (σ )/dσ  dΥ3s2 (σ ) dF (σ )  1 − θsf (σ − tc + h) − F (σ )δ(σ − tc + h), = dσ dσ

σ ∈ [0, tc ], (2.200) where δ(σ − tc + h) is the Dirac delta-function with the impulse at σ = tc − h (see, e.g., [35]). Quite similarly to the results of [22] (Section 4.3), we can rewrite the problem (2.188) in the equivalent integral form for σ ∈ [0, tc ]  T Δ(σ, ε) = H (tc , σ, ε) Δ(tc , ε) +



σ

 T H (t, σ, ε) Γ (t, ε)dt, (2.201)

tc

where H (t, σ, ε), 0 ≤ σ ≤ t ≤ tc , is the solution of the initial-value problem (2.161). Equation (2.201), along with Lemma 2.4, the expression for Δ(tc , ε) (see Eq. (2.188)), the block representation of Γ (σ, ε) (see Eq. (2.189)), and the estimate for Γ1 (σ, ε) (see Eq. (2.195)), directly yields the following inequality: Δ1 (σ, ε) ≤ aε,

σ ∈ [0, tc ],

ε ∈ (0, ε3 ],

(2.202)

where Δ1 (σ, ε) is the upper block of the matrix Δ(σ, ε) of the dimension n × n; a > 0 is some constant independent of ε. Similarly, using Eqs. (2.196), (2.198), (2.200), and the inequalities (2.197), (2.199), we obtain for all ε ∈ (0, ε3 ] Δ2 (σ, ε) ≤ aε

 

⎧ ⎨ ε + exp − β(tc − σ )/ε ,

if tc − h < σ ≤ tc ,

 

⎩ ε + exp − β(tc − h − σ )/ε , if 0 ≤ σ ≤ tc − h, (2.203) where Δ2 (σ, ε) is the lower block of the matrix Δ(σ, ε) of the dimension m × n; a > 0 is some constants independent of ε. The inequalities (2.202) and (2.203), along with the notations (2.187), imply the validity of the inequalities stated in the theorem.

108

2 Singularly Perturbed Linear Time Delay Systems

2.5 Concluding Remarks and Literature Review In this chapter, three classes of singularly perturbed linear time-dependent differential systems with time delays (point-wise and distributed) were studied. The first class represents the systems, where the delays are small of order of the small positive multiplier ε for a part of the derivatives in the differential equations. The systems of the second class contain delays of two scales (small and nonsmall). Namely, the slow state variable is with the nonsmall delays (of order of 1), while the fast state variable is with the small delays. In the third class, the systems with only the nonsmall delays are considered. The asymptotic decomposition of these systems into two ε-free subsystems, the slow and fast ones, was carried out. For the original system with only the small delays, the slow subsystem is a differential-algebraic delay-free system, which can be converted (subject to a proper assumption) into a pure differential equation of a lower Euclidean dimension than the original system. For the original system with the delays of two scales, the slow subsystem is a differential-algebraic time delay system, which can be converted (subject to a proper assumption) into a pure differential time delay equation of a lower Euclidean dimension than the original system. The slow subsystem, associated with the original system containing only the nonsmall delays, is of the same type as the slow subsystem corresponding to the system with the delays of two scales. For the original systems with the small delays and the delays of two scales, the fast subsystem is a differential time delay equation of a lower Euclidean dimension than the original system. For the original system with the nonsmall delays, the fast subsystem is an undelayed differential equation of a lower Euclidean dimension than the original system. The slow–fast decomposition of a singularly perturbed system is one of the main approaches to its analysis. This approach is widely used in the literature. There are two types of the slow–fast decomposition of a singularly perturbed system, asymptotic and exact. The asymptotic slow–fast decomposition and its applications to analysis of singularly perturbed systems can be found, for instance, in [6, 11, 17, 20, 24, 27, 33, 34, 36, 37] (see also references therein). For the exact slow–fast decomposition and its applications to analysis of singularly perturbed systems, one can see the papers [2, 8, 9, 19, 21, 24, 25, 28, 30–32]. The fundamental matrix solutions of the original singularly perturbed systems and some other matrix-valued functions, associated with these fundamental matrices, were analyzed in this chapter. Their block-wise estimates were derived. Thus, Lemma 2.2 and Theorem 2.1 present estimates of the fundamental matrix solution and its adjoint matrix of the original singularly perturbed system with the small delays. These lemma and theorem were formulated and proven in [14]. In this chapter, the much more detailed proofs of these assertions are done. Particular cases of Lemma 2.2 were considered in [13] (a time-dependent system with a single point-wise delay and a distributed delay), and in [19] (a time independent system with a single point-wise delay and a distributed delay). In [15] the result of this lemma was extended to the case of the general delay in the form of the Stieltjes

References

109

integral. A particular case of Theorem 2.1 was considered in [13]. Lemma 2.3 and Theorems 2.2 and 2.3 establish estimates of the fundamental matrix solution and its adjoint matrix of the original singularly perturbed system with the delays of two scales. Estimates of the fundamental matrix solution and its adjoint matrix of the original singularly perturbed system with the nonsmall delays are obtained in Lemma 2.4, and Theorems 2.4 and 2.5. Lemma 2.4 has been formulated for the first time in [16] with a brief proof. Theorems 2.4 and 2.5 have been obtained in [18]. In this chapter, Lemma 2.4 and Theorem 2.4 are proven in details. It is also important to note that in this chapter (in contrast with the papers [13–16, 18]) we consider the case where all the matrix-valued coefficients of the original systems depend not only on the independent variables, but also on the small parameter of singular perturbation. Completing this section, we would like to emphasize the following. The subsequent chapters are essentially based on the results of the present chapter. Namely, the results of Chap. 3 are based on Lemmas 2.2 and 2.3, and Theorems 2.1–2.3. The results of Chap. 4 are based on the same lemmas and theorems. The results of Chap. 6 (Sect. 6.2) are based on Lemma 2.4, and Theorems 2.4 and 2.5. The results of Chap. 6 (Sect. 6.3) are based on Theorem 2.1.

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Chapter 3

Euclidean Space Output Controllability of Linear Systems with State Delays

3.1 Introduction In this chapter, two kinds of singularly perturbed linear time-dependent controlled systems with multiple point-wise delays and a distributed delay in the state variables are considered. In the system of the first kind, the delays are small of order of a small positive multiplier ε for a part of its derivatives. This multiplier is a parameter of the singular perturbation. In the system of the second kind, the delays are of two scales. Namely, the delays in the slow state variable are nonsmall (of order of 1), while the delays in the fast state variable are small (of order of ε). Along with the dynamic system of each kind, a linear algebraic delay-free output equation is considered. Two types of each considered singularly perturbed system, standard and nonstandard, are analyzed. The analysis of the controllability of each considered system, robust with respect to ε, is based on its asymptotic slow-fast decomposition. Namely, two much simpler parameter-free subsystems (the slow and fast ones) are associated with the original singularly perturbed system. Using this decomposition, it is established in the chapter that proper kinds of controllability of the slow and fast subsystems yield the Euclidean space output controllability of the original system robust with respect to the parameter of singular perturbation for all its sufficiently small values. The following main notations are applied in this chapter: 1. E n is the n-dimensional real Euclidean space. 2. The Euclidean norm of either a matrix or a vector is denoted by · . 3. The upper index T denotes the transposition either of a vector x (x T ) or of a matrix A (AT ). 4. In denotes the identity matrix of dimension n. 5. L2 [t1 , t2 ; E n ] denotes the linear space of all functions x(·) : [t1 , t2 ] → E n square integrable in the interval [t1 , t2 ].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Y. Glizer, Controllability of Singularly Perturbed Linear Time Delay Systems, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-65951-6_3

111

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

6. L2loc [t¯, +∞; E n ] denotes the linear space of all functions x(·) : [t¯, +∞) → E n square integrable in any subinterval [t1 , t2 ] ⊂ [t¯, +∞). 7. col(x, y), where x ∈ E n , y ∈ E m , denotes the column block vector of the dimension n + m with the upper block x and the lower block y, i.e., col(x, y) = (x T , y T )T . 8. Reλ denotes the real part of a complex number λ.

3.2 Systems with Small Delays: Main Notions and Definitions 3.2.1 Original System Consider the controlled system

dx(t)  = A1j (t, ε)x(t − εhj ) + A2j (t, ε)y(t − εhj ) dt N

 +

j =0

0 −h



G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)y(t + εη) dη + B1 (t, ε)u(t), t ≥ 0, (3.1)

dy(t)  = A3j (t, ε)x(t − εhj ) + A4j (t, ε)y(t − εhj ) dt N

ε  +

j =0

0 −h



G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη + B2 (t, ε)u(t), t ≥ 0, (3.2)

where x(t) ∈ E n , y(t) ∈ E m , u(t) ∈ E r (u(t) is a control); ε > 0 is a small parameter; N ≥ 1 is an integer; 0 = h0 < h1 < h2 < . . . < hN = h are some given constants independent of ε; Aij (t, ε), Gi (t, η, ε) and Bk (t, ε), (i = 1, . . . , 4; j = 0, . . . , N; k = 1, 2) are matrix-valued functions of corresponding dimensions, given for t ≥ 0, η ∈ [−h, 0] and ε ∈ [0, ε0 ], (ε0 > 0); Aij (t, ε) and Bk (t, ε), (i = 1, . . . , 4; j = 0, . . . , N ; k = 1, 2) are continuous in (t, ε) ∈ [0, +∞)×[0, ε0 ]; the functions Gi (t, η, ε), (i = 1, . . . , 4) are piecewise continuous in η ∈ [−h, 0] for any (t, ε) ∈ [0, +∞)×[0, ε0 ], and these functions are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0]. Due to the results of Sect. 2.2, for a given u(·) ∈ L2loc [0, +∞; E r ], the system (3.1)–(3.2) is a linear time-dependent nonhomogeneous functional-differential  system. It is infinite-dimensional with the state variables x(t), x(t + εη) and   y(t), y(t + εη) , η ∈ [−h, 0). Moreover, (3.1)–(3.2) is a singularly perturbed system.

3.2 Systems with Small Delays: Main Notions and Definitions

113

Along with the dynamic system (3.1)–(3.2), consider the algebraic delay-free output equation ζ (t) = X (t, ε)x(t) + Y (t, ε)y(t),

t ≥ 0,

(3.3)

where ζ (t) ∈ E q , (q ≤ n + m), is an output; X (t, ε) and Y (t, ε) are matrix-valued functions of corresponding dimensions, given for t ≥ 0 and ε ∈ [0, ε0 ]; X (t, ε) and Y (t, ε) are continuous in (t, ε) ∈ [0, +∞) × [0, ε0 ]. Definition 3.1 For a given ε ∈ (0, ε0 ], the system (3.1)–(3.2), (3.3) is said to be Euclidean space output controllable at a given time instant tc > 0 if for any x0 ∈ E n , y0 ∈ E m , ϕx (·) ∈ L2 [−εh, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ] and ζc ∈ E q there exists a control function u(·) ∈ L2 [0, tc ; E r ], for which the system (3.1)–(3.2), (3.3) with the initial and terminal conditions x(τ ) = ϕx (τ ), y(τ ) = ϕy (τ ), τ ∈ [−εh, 0), x(0) = x0 ,

y(0) = y0 ,

ζ (tc ) = ζc ,

(3.4) (3.5) (3.6)

has a solution. Remark 3.1 In the particular case where q = n + m, and the q × n-matrix X (t, ε) and the q × m-matrix Y (t, ε) have the block-form  X (t, ε) =

In 0



 ,

Y (t, ε) =

0 Im

 , t ∈ [0, tc ], ε ∈ [0, ε0 ],

(3.7)

  the output becomes ζ (t) = col x(t), y(t) , i.e., the output coincides with the Euclidean part of the state variable of (3.1)–(3.2), (see Sect. 2.2 for the details on the state variable structure). In this case, the Euclidean space output controllability of the system (3.1)–(3.2), (3.3) at the time instant tc is called the complete Euclidean space controllability of the system (3.1)–(3.2) at this time instant.

3.2.2 Asymptotic Decomposition of the Original System Let us decompose the original system, consisting of the singularly perturbed dynamic system (3.1)–(3.2) and the algebraic output Eq. (3.3), into two much simpler ε-free subsystems. The first subsystem is called a slow subsystem. The dynamic part of the slow subsystem is obtained from (3.1)–(3.2) similarly to the results of Sect. 2.2, i.e., by setting formally ε = 0 in these functional-differential equations. The output part of the slow subsystem is obtained from (3.3) by setting

114

3 Euclidean Space Output Controllability of Linear Systems with State Delays

formally ε = 0 in this algebraic equation and dropping formally the term with  the Euclidean part y(t) of the fast state variable y(t), y(t + εη) , η ∈ [−h, 0) (see Sect. 2.2 for the details on the slow and fast state variables). Thus, the slow subsystem has the form dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) + B1 (t, 0)us (t), t ≥ 0, dt

(3.8)

0 = A3s (t)xs (t) + A4s (t)ys (t) + B2 (t, 0)us (t), t ≥ 0,

(3.9)

ζs (t) = X (t, 0)xs (t), t ≥ 0,

(3.10)

where xs (t) ∈ E n and ys (t) ∈ E m are state variables; us (t) ∈ E r is a control; ζs (t) ∈ E q is an output; Ais (t) =

N  j =0

 Aij (t, 0) +

0 −h

Gi (t, η, 0)dη,

i = 1, . . . , 4.

(3.11)

The slow subsystem (3.8)–(3.10) consists of the descriptor (differentialalgebraic) system (3.8)–(3.9) and the algebraic output Eq. (3.10). The latter depends only on the state xs (t). The slow subsystem is ε-free. If det A4s (t) = 0,

t ≥ 0,

(3.12)

the differential-algebraic system (3.8)–(3.9) can be converted to an equivalent system. This new system consists of the explicit expression for ys (t)   ys (t) = −A−1 4s (t) A3s (t)xs (t) + B2 (t, 0)us (t) ,

t ≥ 0,

and the differential equation with respect to xs (t) dxs (t) = A¯ s (t)xs (t) + B¯ s (t)us (t), t ≥ 0, dt

(3.13)

where −1 ¯ A¯ s (t) = A1s (t) − A2s (t)A−1 4s (t)A3s (t), Bs (t) = B1 (t, 0) − A2s (t)A4s (t)B2 (t, 0). (3.14)

The second subsystem, obtained by the asymptotic decomposition of (3.1)–(3.2), (3.3), is called a fast subsystem. It is derived from  (3.2) and (3.3)  in the following way: (a) the terms containing the state variable x(t), x(t + εη) , η ∈ [−h, 0] are removed from (3.2) and (3.3); (b) the transformations of the variables t = t1 + εξ ,

3.2 Systems with Small Delays: Main Notions and Definitions



115

y(t1 + εξ ) = yf (ξ ), u(t1 + εξ ) = uf (ξ ), ζ (t1 + εξ ) = ζf (ξ ) are made in the resulting system, where t1 ≥ 0 is any fixed time instant. Thus, we obtain the system dyf (ξ )  = A4j (t1 + εξ, ε)yf (ξ − hj ) dξ N

j =0

 +

0 −h

G4 (t1 + εξ, η, ε)yf (ξ + η)dη + B2 (t1 + εξ, ε)uf (ξ ), ζf (ξ ) = Y (t1 + εξ, ε)yf (ξ ).

Finally, setting formally ε = 0 in this system and replacing t1 with t yield the fast subsystem dyf (ξ )  = A4j (t, 0)yf (ξ − hj ) dξ N

 +

j =0

0 −h

G4 (t, η, 0)yf (ξ + η)dη + B2 (t, 0)uf (ξ ), ξ ≥ 0, ζf (ξ ) = Y (t, 0)yf (ξ ), ξ ≥ 0,

(3.15) (3.16)

  where t ≥ 0 is a parameter; yf (ξ ) ∈ E m ; yf (ξ ), yf (ξ + η) , η ∈ [−h, 0) is a state variable; uf (ξ ) ∈ E r (uf (ξ ) is a control); ζf (ξ ) ∈ E q (ζf (ξ ) is an output). Like in Sect. 2.2, the new independent variable ξ is called the stretched time, and it is expressed by the original time t in the form ξ = (t − t1 )/ε. Thus, for any t > t1 , ξ → +∞ as ε → +0. The fast subsystem consists of the differential equation with state delays (3.15) and the algebraic output Eq. (3.16). It is of a lower Euclidean dimension than the original system (3.1)–(3.3), and it is ε-free. Definition 3.2 Subject to (3.12), the system (3.13), (3.10) is said to be output controllable at a given time instant tc > 0 if for any x0 ∈ E n and ζc ∈ E q there exists a control function us (·) ∈ L2 [0, tc ; E r ], for which (3.13) has a solution xs (t), t ∈ [0, tc ], satisfying the initial and terminal conditions xs (0) = x0 ,

(3.17)

X (tc , 0)xs (tc ) = ζc .

(3.18)

Remark 3.2 In the particular case of the original system (3.1)–(3.2), (3.3) where q = n and X (t, 0) ≡ In , the output controllability of the system (3.13), (3.10) at the time instant tc becomes the complete controllability of the system (3.13) at this time instant.

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

Definition 3.3 The system (3.8)–(3.10) is said to be impulse-free output controllable at a given time instant tc > 0 if for any x0 ∈ E n and ζc ∈ E q there exists a control function us (·) ∈ L2 [0, tc ; E r ], for which (3.8)–(3.9) has a solution col(xs (t), ys (t)) ∈ L2 [0, tc ; E n+m ], satisfying the initial condition (3.17) and the terminal condition (3.18). Remark 3.3 In the particular case of (3.1)–(3.2), (3.3) where q = n and X (t, 0) ≡ In , the impulse-free output controllability of the system (3.8)–(3.10) at the time instant tc becomes the impulse-free controllability with respect to xs (t) of the system (3.8)–(3.9) at this time instant. Definition 3.4 For a given t ≥ 0, the system (3.15)–(3.16) is said to be Euclidean space output controllable if for any y0 ∈ E m , ϕyf (·) ∈ L2 [−h, 0; E m ], and ζc ∈ E q there exist a number of ξc > 0, independent of y0 , ϕyf (·) and ζc , and a control function uf (·) ∈ L2 [0, ξc ; E r ], for which the system (3.15)–(3.16) with the initial and terminal conditions yf (η) = ϕyf (η),

η ∈ [−h, 0);

yf (0) = y0 ,

ζf (ξc ) = ζc ,

(3.19) (3.20)

has a solution. Remark 3.4 In the particular case of the original system (3.1)–(3.2), (3.3) where q = m and Y (t, 0) ≡ Im , the Euclidean space output controllability of the system (3.15)–(3.16) becomes the complete Euclidean space controllability of the system (3.15).

3.3 Auxiliary Results In this section, some properties of controlled time delay systems are studied. Also, some smoothness properties of solution to a hybrid set of Riccati-type equations are analyzed. Based on these results, in the next section different parameter-free conditions for the Euclidean space output controllability of the original singularly perturbed system (3.1)–(3.2) are derived.

3.3.1 Output Controllability of a System with State Delays: Necessary and Sufficient Conditions Consider the controlled system N

 d x(t) ˜ = Aj (t)x(t ˜ − gj ) + dt j =0



0

−g

G (t, τ )x(t ˜ + τ )dτ + B(t)u(t), ˜ t ≥ 0, (3.21)

3.3 Auxiliary Results

117

ζ˜ (t) = X (t)x(t), ˜

t ≥ 0,

(3.22)

where x(t) ˜ ∈ E n , u(t) ˜ ∈ E r (u(t) ˜ is a control), ζ˜ (t) ∈ E q , (q ≤ n), (ζ˜ (t) is an output); N ≥ 1 is an integer; 0 = g0 < g1 < g2 < . . . < gN = g are given time delays; Aj (t), (j = 0, . . . , N ), G (t, τ ), B(t) and X (t) are matrix-valued functions of corresponding dimensions, given for t ≥ 0, τ ∈ [−g, 0]; Aj (t), (j = 0, . . . , N ), B(t) and X (t) are continuous in t ∈ [0, +∞); for any t ∈ [0, +∞), G (t, τ ) is piecewise continuous in τ ∈ [−g, 0], and this function is continuous in t ∈ [0, +∞) uniformly with respect to τ ∈ [−g, 0]. Definition 3.5 The system (3.21)–(3.22) is said to be Euclidean space output controllable at a given time instant tc > 0 if for any x˜0 ∈ E n , ϕ˜x (·) ∈ L2 [−g, 0; E n ], and ζ˜c ∈ E q there exists a control function u(·) ˜ ∈ L2 [0, tc ; E r ], for which the system (3.21)–(3.22) with the initial and terminal conditions x(τ ˜ ) = ϕ˜x (τ ), τ ∈ [−g, 0);

x(0) ˜ = x˜0 ,

ζ˜ (tc ) = ζ˜c ,

(3.23) (3.24)

has a solution. Definition 3.6 The system (3.21) is said to be completely Euclidean space controllable at a given time instant tc > 0 if for any x˜0 ∈ E n , ϕ˜x (·) ∈ L2 [−g, 0; E n ], and x˜c ∈ E n there exists a control function u(·) ˜ ∈ L2 [0, tc ; E r ], for which the system (3.21) with the initial conditions (3.23) and the terminal condition x(t ˜ c ) = x˜c has a solution. Let for any σ ∈ [0, tc ], the n × n-matrix-valued function V (t, σ ) be the fundamental matrix of Eq. (3.21), i.e., V (t, σ ) is the unique solution to the following initial-value problem (see Sects. 2.2.3, 2.3.3 and 2.4.3 for the details): N

dV (t, σ )  = Aj (t)V (t − gj , σ ) dt j =0

 +

0 −g

G (t, τ )V (t + τ, σ )dτ,

V (t, σ ) = 0,

σ − g ≤ t < σ,

σ < t ≤ tc , V (σ, σ ) = In .

(3.25)

Consider the following q × q-matrix: 

tc

W (tc ) = X (tc ) 0

V (tc , σ )B(σ )B T (σ )V T (tc , σ )dσ X T (tc ).

(3.26)

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

Lemma 3.1 The system (3.21)–(3.22) is Euclidean space output controllable at a given time instant tc > 0 if and only if the matrix W (tc ) is nonsingular, i.e., det W (tc ) = 0. The proof of the lemma is presented in Sect. 3.3.4. Let the n × n-matrix-valued function Ξ (σ ) be the adjoint matrix to the fundamental matrix V (t, σ ) at t = tc , i.e., Ξ (σ ) is the unique solution to the following terminal-value problem (see Sects. 2.2.4, 2.3.4 and 2.4.4 for the details): N

 T dΞ (σ ) =− Aj (σ + gj ) Ξ (σ + gj ) dσ  −

j =0

 T G (σ − τ, τ ) Ξ (σ − τ )dτ,

0 −g

Ξ (tc ) = In ,

σ ∈ [0, tc ),

Ξ (σ ) = 0,

σ > tc .

In this problem, it is assumed that the matrix-valued functions Aj (t), (j = 0, 1, . . . , N ), and G (t, τ ) satisfy the following equalities: Aj (t) = Aj (tc ), G (t, τ ) = G (tc , τ ),

t > tc , τ ∈ [−g, 0].

Remark 3.5 By virtue of the results of [18] (Section 4.3), we have V (tc , σ ) = Ξ T (σ ),

σ ∈ [0, tc ],

(see also Remarks 2.3, 2.7 and 2.11 in Chap. 2). Remark 3.5 and Lemma 3.1 yield immediately the following corollary. Corollary 3.1 The system (3.21)–(3.22) is Euclidean space output controllable at a given time instant tc > 0 if and only if   det X (tc )

tc

 Ξ (σ )B(σ )B (σ )Ξ (σ )dσ X (tc ) = 0. T

T

T

(3.27)

0

Setting q = n and X (t) ≡ In , t ≥ 0 in Corollary 3.1, we directly obtain the following proposition. Proposition 3.1 The system (3.21) is completely Euclidean space controllable at a given time instant tc > 0 if and only if 

tc

det

 Ξ T (σ )B(σ )B T (σ )Ξ (σ )dσ

= 0.

(3.28)

0

Remark 3.6 Note that the Euclidean space output controllability and the complete Euclidean space controllability do not follow, in general, from each other. Thus,

3.3 Auxiliary Results

119

the system (3.21) is completely Euclidean space controllable at t = tc subject to (3.28), while the system (3.21)–(3.22) is not Euclidean space output controllable   at this time instant if rank X (tc ) < q. Similarly, the system (3.21)–(3.22) is Euclidean space output controllable at t = tc subject to (3.27), while the system!(3.21) is not completely Euclidean " space controllable at this time instant .t if rank 0c Ξ T (σ )B(σ )B T (σ )Ξ (σ )dσ < n. Remark 3.7 From the proof of Lemma 3.1 (see Sect. 3.3.4), we directly have the following. The statement of this lemma, as well as the statements of Corollary 3.1 and Proposition 3.1, are valid not only for the class of controls from L2 [0, tc ; E r ] but also for the classes of piecewise continuous and even continuous controls in the interval [0, tc ]. Moreover, the statements of the abovementioned assertions are valid not only for the class of initial functions from L2 [−g, 0; E n ] but also for the classes of piecewise continuous and even continuous initial functions in the interval [−g, 0].

3.3.2 Linear Control Transformation in Systems with Small Delays 3.3.2.1

Control Transformation in the Original System

Let us transform the control in the dynamic system (3.1)–(3.2) as:  u(t) = K1 (t, ε)y(t) +

0 −h

K2 (t, η, ε)y(t + εη)dη + w(t),

(3.29)

where w(t) is a new control; K1 (t, ε) and K2 (t, η, ε) are any specified matrixvalued functions of corresponding dimensions, given for t ≥ 0, η ∈ [−h, 0] and ε ∈ [0, ε0 ]; K1 (t, ε) is continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ]; K2 (t, η, ε) is continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0]; for any (t, ε) ∈ [0, +∞) × [0, ε0 ], K2 (t, η, ε) is piecewise continuous in η ∈ [−h, 0]. The control transformation (3.29) is invertible. Due to this transformation, for all (t, ε) ∈ [0, +∞) × (0, ε0 ], the system (3.1)–(3.2) becomes as: $ dx(t)  # = A1j (t, ε)x(t − εhj ) + AK 2j (t, ε)y(t − εhj ) dt N

 +

j =0

0 −h



G1 (t, η, ε)x(t + εη) + GK 2 (t, η, ε)y(t + εη) dη + B1 (t, ε)w(t), (3.30)

120

3 Euclidean Space Output Controllability of Linear Systems with State Delays

$ dy(t)  # = A3j (t, ε)x(t − εhj ) + AK 4j (t, ε)y(t − εhj ) dt N

ε  +

j =0

0 −h

G3 (t, η, ε)x(t + εη) + GK 4 (t, η, ε)y(t + εη) dη + B2 (t, ε)w(t), (3.31)

where K AK 20 (t, ε) = A20 (t, ε) + B1 (t, ε)K1 (t, ε), A2j (t, ε) = A2j (t, ε), j = 1, . . . , N, (3.32) K AK 40 (t, ε) = A40 (t, ε) + B2 (t, ε)K1 (t, ε), A4j (t, ε) = A4j (t, ε), j = 1, . . . , N, (3.33) GK (3.34) 2 (t, η, ε) = G2 (t, η, ε) + B1 (t, ε)K2 (t, η, ε),

GK 4 (t, η, ε) = G4 (t, η, ε) + B2 (t, ε)K2 (t, η, ε).

(3.35)

Lemma 3.2 For a given ε ∈ (0, ε0 ], the system (3.1)–(3.2), (3.3) is Euclidean space output controllable at a given time instant tc > 0, if and only if the system (3.30)– (3.31), (3.3) is Euclidean space output controllable at this time instant. Proof Necessity: Suppose that for some ε ∈ (0, ε0 ], the system (3.1)–(3.2), (3.3) is Euclidean space output controllable at tc > 0. Let x0 ∈ E n , y0 ∈ E m , ϕx (·) ∈ L2 [−εh, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ] and ζc ∈ E q be arbitrary given. Due to Definition 3.1, there exists a control u(·) ∈ L2 [0, tc ; E r ], for which the system (3.1)–(3.2), (3.3) with the (3.4)–(3.5) and the terminal  initial conditions  condition (3.6) has a solution col x(t), y(t) , t ∈ [0, tc ]. By virtue of Proposition 2.2 (see Sect. 2.2.3), the functions x(t) and y(t) are absolutely continuous in the interval [0, tc ]. For the abovementioned control u(t), t ∈ [0, tc ], the system (3.1)–(3.2) can be rewritten in the equivalent form $ dx(t)  # = A1j (t, ε)x(t − εhj ) + AK (t, ε)y(t − εh ) j 2j dt N

 + 

j =0

0 −h

# $ G1 (t, η, ε)x(t + εη) + GK 2 (t, η, ε)y(t + εη) dη

+B1 (t, ε) u(t) − K1 (t, ε)y(t) −



0 −h

 K2 (t, η, ε)y(t + εη)dη , t ≥ 0, (3.36)

3.3 Auxiliary Results

121

$ dy(t)  # = A3j (t, ε)x(t − εhj ) + AK 4j (t, ε)y(t − εhj ) dt N

ε

 + 

j =0

0 −h

# $ G3 (t, η, ε)x(t + εη) + GK (t, η, ε)y(t + εη) dη 4 

+B2 (t, ε) u(t) − K1 (t, ε)y(t) −

0 −h

 K2 (t, η, ε)y(t + εη)dη , t ≥ 0. (3.37)

Denote  w(t) = u(t) − K1 (t, ε)y(t) −

0

−h

K2 (t, η, ε)y(t + εη)dη, t ∈ [0, tc ].

Since u(·) ∈ L2 [0, tc ; E r ], then w(·) ∈ L2 [0, tc ; E r ]. Now, using the equivalent form (3.36)–(3.37) of the system (3.1)–(3.2), one directly has that the control function w(t) provides the existence of solution to the system (3.30)–(3.31), (3.3) subject to the conditions (3.4)–(3.5) and (3.6). The latter, along with Definition 3.1, implies the Euclidean space output controllability of the system (3.30)–(3.31), (3.3) at the time instant tc . Thus, the necessity is proven. Sufficiency The sufficiency is proven similarly to the necessity.

 

Corollary 3.2 For a given ε ∈ the system (3.1)–(3.2) is completely Euclidean space controllable at a given time instant tc > 0, if and only if the system (3.30)–(3.31) is completely Euclidean space controllable at this time instant. (0, ε0 ],

Proof The corollary directly follows from Lemma 3.2.

3.3.2.2

 

Asymptotic Decomposition of the Transformed System (3.30)–(3.31), (3.3)

Let us decompose asymptotically the singularly perturbed system (3.30)–(3.31), (3.3) into the slow and fast subsystems. This decomposition is carried out similarly to that for the system (3.1)–(3.2), (3.3). Thus, the slow subsystem, associated with (3.30)–(3.31), (3.3), consists of the differential-algebraic system dxs (t) = A1s (t)xs (t) + AK 2s (t)ys (t) + B1 (t, 0)ws (t), t ≥ 0, dt

(3.38)

0 = A3s (t)xs (t) + AK 4s (t)ys (t) + B2 (t, 0)ws (t), t ≥ 0,

(3.39)

and the algebraic output Eq. (3.10). In (3.38)–(3.39): xs (t) ∈ E n , ys (t) ∈ E m , ws (t) ∈ E r , (ws (t) is a control); the matrices A1s (t) and A3s (t) are given in (3.11);

122

3 Euclidean Space Output Controllability of Linear Systems with State Delays

and AK ls (t)

=

N 

 AK lj (t, 0) +

j =0

0 −h

GK l (t, η, 0)dη,

l = 2, 4.

(3.40)

If det AK 4s (t) = 0,

t ≥ 0,

(3.41)

the differential-algebraic system (3.38)–(3.39) can be converted to an equivalent system. This new system consists of the explicit expression for ys (t)  −1   A3s (t)xs (t) + B2 (t, 0)ws (t) , ys (t) = − AK 4s (t)

t ≥ 0,

(3.42)

and the differential equation with respect to xs (t) dxs (t) ¯ = A¯ K s (t)xs (t) + B dt

K s (t)ws (t),

t ≥ 0,

(3.43)

where  K −1 K A3s (t), A¯ K s (t) = A1s (t) − A2s (t) A4s (t) B¯

K s (t)

 K −1 = B1 (t, 0) − AK B2 (t, 0). 2s (t) A4s (t)

(3.44) (3.45)

The fast subsystem, associated with (3.30)–(3.31), (3.3), consists of the differential equation with state delays dyf (ξ )  K = A4j (t, 0)yf (ξ − hj ) dξ N

 +

j =0

0 −h

GK 4 (t, η, 0)yf (ξ + η)dη + B2 (t, 0)wf (ξ ), ξ ≥ 0,

(3.46)

and the algebraic output Eq. (3.16). Note that in (3.46), t ≥ 0 is a parameter, while ξ  m ; y (ξ ), y (ξ + (ξ ) ∈ E is an independent variable. Moreover, in this equation, y f f f  η) , η ∈ [−h, 0) is a state variable; wf (ξ ) ∈ E r , (wf (ξ ) is a control). Lemma 3.3 The system (3.8)–(3.10) is impulse-free output controllable at a given time instant tc > 0, if and only if the system (3.38)–(3.39), (3.10) is impulse-free output controllable at this time instant. Proof Let us make the following control transformation in the differential-algebraic system (3.8)–(3.9):

3.3 Auxiliary Results

123

us (t) = Ks (t)ys (t) + ws (t),

(3.47)

where ws (t) is a new control, and  Ks (t) = K1 (t, 0) +

0

−h

K2 (t, η, 0)dη.

(3.48)

Substitution of (3.47) into (3.8) yields the following equation for t ≥ 0:   dxs (t) = A1s (t)xs (t) + A2s (t) + B1 (t, 0)Ks (t) ys (t) + B1 (t, 0)ws (t). dt (3.49)  Let us treat the matrix-valued coefficient (A2s (t) + B1 (t, 0)Ks (t) for ys (t) in the right-hand side of this equation. Substitution of (3.48) into this coefficient yields  A2s (t) + B1 (t, 0)Ks (t) = A2s (t) + B1 (t, 0)K1 (t, 0) +

0

−h

B1 (t, 0)K2 (t, η, 0)dη.

Further, replacing A2s (t) in the right-hand side of this equation with its expression from (3.11), we obtain A2s (t) + B1 (t, 0)Ks (t) =

N 

 A2j (t, 0) +

j =0

0 −h

G2 (t, η, 0)dη + B1 (t, 0)K1 (t, 0)  +

0 −h

B1 (t, 0)K2 (t, η, 0)dη.

This equation can be rewritten as: N    A2s (t) + B1 (t, 0)Ks (t) = A20 (t, 0) + B1 (t, 0)K1 (t, 0) + A2j (t)

 +

j =1 0 −h

  G2 (t, η, 0) + B1 (t, 0)K2 (t, η, 0) dη.

 Finally, using Eqs. (3.32), (3.34) and (3.40), we obtain the expression for A2s (t) + B1 (t, 0)Ks (t) in the form A2s (t) + B1 (t, 0)Ks (t) =

N  j =0

 AK 2j (t) +

0 −h

K GK 2 (t, η, 0)dη = A2s (t).

124

3 Euclidean Space Output Controllability of Linear Systems with State Delays

The latter, along with (3.49), means that the control transformation (3.47) converts Eq. (3.8) to Eq. (3.38). In the same way, it is shown that the transformation (3.47) converts Eq. (3.9) to Eq. (3.39). Thus, we have shown that the invertible control transformation (3.47) converts the differential-algebraic system (3.8)–(3.9) to the differential-algebraic system (3.38)–(3.39). The rest of the proof is similar to the proof of Lemma 3.2.   Lemma 3.4 Let the condition (3.41) be satisfied. Then, the system (3.38)– (3.39), (3.10) is impulse-free output controllable at a given time instant tc > 0, if and only if the system (3.43), (3.10) is output controllable at this time instant. Proof Necessity: Let the system (3.38)–(3.39), (3.10) be impulse-free output controllable at a given time instant tc > 0. Then, due to Definition 3.3, for any x0 ∈ E n and ζc ∈ E q there exists a control function ws (·) ∈ L2 [0, tc ; E r ], for which (3.38)–(3.39) has a solution col(xs (t), ys (t)) ∈ L2 [0, tc ; E n+m ] satisfying the initial condition (3.17) and the terminal condition (3.18). Since the condition (3.41) is satisfied, the differential-algebraic system (3.38)– (3.39) can be converted to the equivalent system consisting of the explicit expression (3.42) for the state variable ys (t) and the differential equation (3.43) with respect to xs (t). It is clear that the abovementioned solution of the system (3.38)– (3.39) also is a solution of the system (3.42)–(3.43). The latter means immediately the output controllability of the system (3.43), (3.10) at the time instant tc . Sufficiency: The sufficiency is proven similarly to the necessity.   Lemma 3.5 Let the conditions (3.12) and (3.41) be satisfied. Then, the system (3.13), (3.10) is output controllable at a given time instant tc > 0, if and only if the system (3.43), (3.10) is output controllable at this time instant. Proof of the lemma is presented in Sect. 3.3.5. Lemma 3.6 For a given t ≥ 0, the system (3.15)–(3.16) is Euclidean space output controllable if and only if the system (3.46), (3.16) is Euclidean space output controllable. Proof Using the control transformation  uf (ξ ) = K1 (t, 0)yf (ξ ) +

0 −h

K2 (t, η, 0)yf (ξ + η)dη + wf (ξ ),

the lemma is proven similarly to Lemmas 3.3 and 3.2.

 

The following four corollaries are direct consequences of Lemmas 3.3, 3.4, 3.5 and 3.6, respectively. Corollary 3.3 The system (3.8)–(3.9) is impulse-free controllable with respect to xs (t) at a given time instant tc > 0, if and only if the system (3.38)–(3.39) is impulsefree controllable with respect to xs (t) at this time instant.

3.3 Auxiliary Results

125

Corollary 3.4 Let the condition (3.41) be satisfied. Then, the system (3.38)–(3.39) is impulse-free controllable with respect to xs (t) at a given time instant tc > 0, if and only if the system (3.43) is completely controllable at this time instant. Corollary 3.5 Let the conditions (3.12) and (3.41) be satisfied. Then, the system (3.13) is completely controllable at a given time instant tc > 0, if and only if the system (3.43) is completely controllable at this time instant. Corollary 3.6 For a given t ≥ 0, the system (3.15) is completely Euclidean space controllable if and only if the system (3.46) is completely Euclidean space controllable.

3.3.3 Hybrid Set of Riccati-Type Matrix Equations Let us denote

S22 (t, ε) = B2 (t, ε)B2T (t, ε),

t ≥ 0, ε ∈ [0, ε0 ].

(3.50)

Consider the following set, consisting of one algebraic and two differential equations (ordinary and partial) for matrices P , Q, and R: P (t)A40 (t, 0) + AT40 (t, 0)P (t) − P (t)S22 (t, 0)P (t) + Q(t, 0) + QT (t, 0) + Im = 0, (3.51) " dQ(t, η) ! T = A40 (t, 0) − P (t)S22 (t, 0) Q(t, η) + P (t)G4 (t, η, 0) dη +

N −1 

P (t)A4j (t, 0)δ(η + hj ) + R(t, 0, η),

(3.52)

j =1



 ∂ ∂ + R(t, η, χ ) = GT4 (t, η, 0)Q(t, χ ) ∂η ∂χ

+QT (t, η)G4 (t, χ , 0) +

N −1 

AT4j (t, 0)Q(t, χ )δ(η + hj )

j =1

+

N −1 

QT (t, η)A4j (t, 0)δ(χ + hj ) − QT (t, η)S22 (t, 0)Q(t, χ ),

(3.53)

j =1

where t ≥ 0 is a parameter; η ∈ [−h, 0] and χ ∈ [−h, 0] are independent variables; δ(·) is the Dirac delta-function (see, e.g., [32]).

126

3 Euclidean Space Output Controllability of Linear Systems with State Delays

The set of Eqs. (3.51)–(3.53) is subject to the boundary conditions Q(t, −h) = P (t)A4N (t, 0), R(t, −h, η) = AT4N (t, 0)Q(t, η),

R(t, η, −h) = QT (t, η)A4N (t, 0).

(3.54)

Let tc > 0 be a given time instant. In what follows of this subsection, we assume: (I) The matrix-valued functions A4j (t, 0), (j = 0, 1, . . . , N ), and S22 (t, 0) are continuously differentiable in the interval [0, tc ]. (II) The matrix-valued function G4 (t, η, 0) is piecewise continuous with respect to η ∈ [−h, 0] for each t ∈ [0, tc ]. (III) The matrix-valued function G4 (t, η, 0) is continuously differentiable with respect to t ∈ [0, tc ] uniformly in η ∈ [−h, 0]. For the sake of the further analysis of the set (3.51)–(3.54), we introduce the following definition. For a given t ∈ [0, tc ], consider the state-feedback control in the fast subsystem (3.15)

1f (t)yf (ξ ) + u˜ f yf (·) (ξ ) = K



0

−h

2f (t, η)yf (ξ + η)dη, K

(3.55)

1f (t) and K 2f (t, η) are an r × m-matrix and an r × m-matrix-valued where K 2f (t, η) is piecewise continuous in the interval [−h, 0]. function of η, respectively; K Definition 3.7 For a given t ∈ [0, tc ], the fast subsystem (3.15) is called L2 stabilizable if there exists the state-feedback control (3.55) such that for any given y0 ∈ E m , ϕyf (·) ∈ L2 [−h, 0; E m ], the solution y˜f (ξ ) of (3.15) with uf (ξ ) = u˜ f yf (·) (ξ ) and subject to the initial conditions (3.19) satisfies the inclusion y˜f (ξ ) ∈ L2 [0, +∞; E m ]. The following proposition is a direct consequence of the results of [4]. Proposition 3.2 Let the assumption (II) be valid. Let, for any t ∈ [0, tc ], the fast subsystem (3.15) be L2 -stabilizable. Then, for any t ∈ [0, tc ], the set of *Eqs. (3.51)–(3.53) subject to the boundary conditions (3.54) + has the unique solution P (t), Q(t, η), R(t, η, χ ), (η, χ ) ∈ [−h, 0] × [−h, 0] such that: (a) P T (t) = P (t); (b) the matrix-valued function Q(t, η) is piecewise absolutely continuous in η ∈ [−h, 0] with the bounded jumps at η = −hj , (j = 1, . . . , N − 1); (c) the matrix-valued function R(t, η, χ ) is piecewise absolutely continuous in η ∈ [−h, 0] and in χ ∈ [−h, 0] with the bounded jumps at η = −hj1 and χ = −hj2 , (j1 = 1, . . . , N − 1; j2 = 1, . . . , N − 1), moreover, R T (t, η, χ ) = R(t, χ , η); (d) all roots λ(t) of the equation

3.3 Auxiliary Results

127

 N ! "  det λIm − A40 (t, 0) − S22 (t, 0)P (t) − A4j (t, 0) exp(−λhj ) j =1

 −

0 −h

 ! " G4 (t, η, 0) − S22 (t, 0)Q(t, η) exp(λη)dη = 0 (3.56)

satisfy the inequality Reλ(t) < −2γ (t),

t ∈ [0, tc ],

(3.57)

where γ (t) > 0 is some function of t. Lemma 3.7 Let the assumptions (I)–(III) be valid. Let, for any t ∈ [0, tc ], the fast subsystem (3.15) be L2 -stabilizable. Then, the matrices P (t), Q(t, η), R(t, η, χ ) are continuous functions of t ∈ [0, tc ] uniformly in (η, χ ) ∈ [−h, 0] × [−h, 0]. Proof of the lemma is presented in Sect. 3.3.6. Lemma 3.8 Let the assumptions (I)–(III) be valid. Let, for any t ∈ [0, tc ], the fast subsystem (3.15) be L2 -stabilizable. Then, the derivatives dP (t)/dt, ∂Q(t, η)/∂t, ∂R(t, η, χ )/∂t exist and are continuous functions of t ∈ [0, tc ] uniformly in (η, χ ) ∈ [−h, 0] × [−h, 0]. Proof of the lemma is presented in Sect. 3.3.7. Lemma 3.9 Let the assumptions (I)–(III) be valid. Let, for any t ∈ [0, tc ], the fast subsystem (3.15) be L2 -stabilizable. Then, there exists a positive number γ¯ such that all roots λ(t) of Eq. (3.56) satisfy the inequality λ(t) < −2γ¯ , t ∈ [0, tc ]. Proof of the lemma is presented in Sect. 3.3.8.

3.3.4 Proof of Lemma 3.1 3.3.4.1

Sufficiency

Let the matrix W (tc ), given by (3.26), be nonsingular. Let x˜0 ∈ E n , ϕ˜x (·) ∈ L2 [−g, 0; E n ] and ξ˜c ∈ E q be any fixed. Consider the following vector-valued function of t ≥ −g: ψ˜ x (t) =

ϕ˜x (t), t ∈ [−g, 0), 0, t ≥ 0.

(3.58)

Since ϕ˜x (·) ∈ L2 [−g, 0; E n ], then, for any given number t˜ > 0, ψ˜ x (·) ∈ L2 [−g, t˜; E n ].

128

3 Euclidean Space Output Controllability of Linear Systems with State Delays

Using this function, let us transform the state variable in the initial-value problem (3.21), (3.23) as:       x(t), ˜ x(t ˜ + τ ) = z˜ (t), z˜ (t + τ ) + ψ˜ x (t), ψ˜ x (t + τ ) ,   where z˜ (t), z˜ (t + τ ) is a new state variable. Due to (3.58) and (3.59), x(t) ˜ = z˜ (t),

t ≥ 0, τ ∈ [−g, 0], (3.59)

t ≥ 0,

(3.60)

and the transformation (3.59) converts the problem (3.21), (3.23) to the equivalent initial-value problem N

d z˜ (t)  = Aj (t)˜z(t − gj )+ dt j =0



0

−g

G (t, τ )˜z(t + τ )dτ +f˜(t) + B(t)u(t), ˜ t ≥ 0, z˜ (τ ) = 0,

τ ∈ [−g, 0);

z˜ (0) = x˜0 , (3.61)

where f˜(t) =

N 

Aj (t)ψ˜ x (t − gj ) +

j =0



0

G (t, τ )ψ˜ x (t + τ )dτ, t ≥ 0.

−g

Applying the variation-of-constant formula (see, e.g., [3, 18]) to the problem (3.61), and using the initial-value problem (3.25) and the equality (3.60), we obtain for any given number t˜ > 0 and function u(·) ˜ ∈ L2 [0, t˜; E r ]: 

t

x(t) ˜ = V (t, 0)x˜0 +

V (t, σ )f˜(σ )dσ +



0

t

V (t, σ )B(σ )u(σ ˜ )dσ, t ∈ [0, t˜].

0

The latter, along with (3.22), yields the output of the system (3.21)–(3.22) subject to the initial conditions (3.23) ζ˜ (t) = X (t)V (t, 0)x˜0 + X (t)  +X (t)



t

V (t, σ )f˜(σ )dσ

0 t

V (t, σ )B(σ )u(σ ˜ )dσ, t ∈ [0, t˜].

(3.62)

0

Let us choose the control u(t) ˜ in the form ! "−1  u(t) ˜ = B (t)V (tc , t)X (tc ) W (tc ) ζ˜c − X (tc )V (tc , 0)x˜0 T

T

T



tc

−X (tc ) 0

 ˜ V (tc , σ )f (σ )dσ , t ∈ [0, tc ].

(3.63)

3.3 Auxiliary Results

129

Now, the substitution of this control into (3.62) and the use of Eq. (3.26) yield after a routine algebra that ζ˜ (tc ) = ζ˜c , meaning the Euclidean space output controllability of the system (3.21)–(3.22) at the time instant tc .

3.3.4.2

Necessity

We prove the necessity by contradiction. Namely, we assume that the system (3.21)– (3.22) is Euclidean space output controllable at the time instant tc , while the matrix W (tc ) is singular. The latter means the existence of a vector ζ¯ ∈ E q such that ζ¯ = 0,

ζ¯ T W (tc )ζ¯ = 0.

(3.64)

Consider the vector-valued function ϑ(t) = B T (t)V T (tc , t)X T (tc )ζ¯ ,

t ∈ [0, tc ].

(3.65)

By virtue of (3.26), (3.64) and (3.65), we have 

tc

ϑ T (t)ϑ(t)dt = ζ¯ T W (tc )ζ¯ = 0.

0

The latter, along with the continuity of ϑ(t) for t ∈ [0, tc ], yields ϑ(t) = 0 ∀t ∈ [0, tc ].

(3.66)

Now, let x˜0 ∈ E n , ϕ˜x (·) ∈ L2 [−g, 0; E n ] be any fixed, and ζ˜c = ζ¯ + X (tc )V (tc , 0)x˜0 + X (tc )



tc

V (tc , σ )f˜(σ )dσ.

(3.67)

0

Using the Euclidean space output controllability of the system (3.21)–(3.22) ˜ ∈ at the time instant tc and Eq. (3.22), we obtain the existence of a control u(·) L2 [0, tc ; E r ] such that ζ˜c = X (tc )V (tc , 0)x˜0 + X (tc )



tc

V (tc , σ )f˜(σ )dσ

0



tc

+X (tc )

V (tc , σ )B(σ )u(σ ˜ )dσ.

0

This equality, along with (3.67), yields ζ¯ = X (tc )



tc 0

V (tc , σ )B(σ )u(σ ˜ )dσ.

130

3 Euclidean Space Output Controllability of Linear Systems with State Delays

Using the latter, as well as Eqs. (3.65) and (3.66), we obtain 

tc

0=

ϑ T (σ )u(σ ˜ )dσ = ζ¯ T ζ¯ ,

0

implying ζ¯ = 0. This equality contradicts the inequality in (3.64), meaning that the assumption on the singularity of the matrix W (tc ) is wrong. This completes the proof of the necessity. Thus, the lemma is proven.

3.3.5 Proof of Lemma 3.5 Consider the following control transformation in the differential equation (3.13): ! "−1 (t) A3s (t)xs (t) + Ks (t)ws (t), us (t) = −Ks (t) AK 4s

(3.68)

where ws (t) is a new control, and ! "−1 Ks (t) = Ir − Ks (t) AK B2 (t, 0). 4s (t)

(3.69)

−1  Let us show that this transformation is invertible, i.e., the matrix Ks (t) exists. First, let us treat the product of the matrices B2 (t, 0)Ks (t). Substitution of the expression (3.48) for Ks (t) into this product yields  B2 (t, 0)Ks (t) = B2 (t, 0)K1 (t, 0) + B2 (t, 0)

0 −h

K2 (t, η, 0)dη.

From Eqs. (3.33) and (3.35), we obtain  B2 (t, 0)

0 −h

B2 (t, 0)K1 (t, 0) = AK 40 (t, 0) − A40 (t, 0),  0  0 K K2 (t, η, 0)dη = G4 (t, η, 0)dη − G4 (t, η, 0)dη. −h

−h

Thus,  B2 (t, 0)Ks (t) =

AK 40 (t, 0) − A40 (t, 0) +

0 −h

 GK 4 (t, η, 0)dη

−

0 −h

G4 (t, η, 0)dη.

Taking into account that AK 4j (t, 0) = A4j (t, 0), (j = 1, . . . , N ) (see Eq. (3.33)), we can rewrite this expression for B2 (t, 0)Ks (t) as:

3.3 Auxiliary Results

131

  N  K K B2 (t, 0)Ks (t) = A40 (t, 0) + A4j (t, 0) + j =1

−h

  N  A4j (t, 0) + − A40 (t, 0) + j =1

0

 GK 4 (t, η, 0)dη 

0 −h

G4 (t, η, 0)dη .

The latter, along with (3.11) and (3.40), yields B2 (t, 0)Ks (t) = AK 4s (t) − A4s (t). Now, let us calculate the following product of the matrices # ! "−1 $ B2 (t, 0) . Ks (t) Ir + Ks (t) A4s (t) Substitution of (3.69) into this product yields # ! "−1 $ Ks (t) Ir + Ks (t) A4s (t) B2 (t, 0) ! "−1 $# ! "−1 $ # B2 (t, 0) Ir + Ks (t) A4s (t) B2 (t, 0) = Ir − Ks (t) AK 4s (t) ! "−1 ! "−1 B2 (t, 0) − Ks (t) AK (t) B2 (t, 0) = Ir + Ks (t) A4s (t) 4s ! "−1 ! "−1 (t) B (t, 0)K (t) A (t) B2 (t, 0). −Ks (t) AK 2 s 4s 4s Using the equality (3.70), we have # ! "−1 $ Ks (t) Ir + Ks (t) A4s (t) B2 (t, 0) ! "−1 ! "−1 B2 (t, 0) − Ks (t) AK B2 (t, 0) = Ir + Ks (t) A4s (t) 4s (t) ! "−1 ! "! "−1 K A (t) (t) − A (t) A (t) B2 (t, 0) = Ir , −Ks (t) AK 4s 4s 4s 4s i.e., # ! "−1 $ Ks (t) Ir + Ks (t) A4s (t) B2 (t, 0) = Ir . Similarly, we obtain #

! "−1 $ Ir + Ks (t) A4s (t) B2 (t, 0) Ks (t) = Ir ,

(3.70)

132

3 Euclidean Space Output Controllability of Linear Systems with State Delays

 −1 which, along with the previous equality, means that the matrix Ks (t) exists and has form ! "−1  −1 Ks (t) = Ir + Ks (t) A4s (t) B2 (t, 0). Let us show that the invertible transformation (3.68) converts the differential equation (3.13) to the differential equation (3.43). Indeed, the substitution of (3.68) into (3.13) yields after a simple rearrangement dxs (t) = As (t)xs (t) + Bs (t)ws (t), dt

(3.71)

where ! "−1 A3s (t), As (t) = A¯ s (t) − B¯ s (t)Ks (t) AK 4s (t)

(3.72)

Bs (t) = B¯ s (t)Ks (t).

(3.73)

For the further treatment of the coefficients As (t) and Bs (t) we need in the equality (3.70) and the equality AK 2s (t) = A2s (t) + B1 (t, 0)Ks (t),

(3.74)

which is derived similarly to (3.70). Substituting (3.14) into Eq. (3.72), we obtain ! "−1 ! "−1 A3s (t) − B1 (t, 0)Ks (t) AK (t) A3s (t) As (t) = A1s (t) − A2s (t) A4s (t) 4s ! "−1 ! "−1 B2 (t, 0)Ks (t) AK (t) A3s (t). +A2s (t) A4s (t) 4s This equation can be rewritten as: # ! "−1 AK As (t) = A1s (t) − A2s (t) A4s (t) 4s (t) + B1 (t, 0)Ks (t) ! "−1 $! "−1 B2 (t, 0)Ks (t) AK (t) A3s (t). −A2s (t) A4s (t) 4s Due to Eq. (3.70), this expression for As (t) can be converted to the form # ! "−1 ! " A4s (t) + B2 (t, 0)Ks (t) As (t) = A1s (t) − A2s (t) A4s (t) ! "−1 $! "−1 B2 (t, 0)Ks (t) AK A3s (t) +B1 (t, 0)Ks (t) − A2s (t) A4s (t) 4s (t) # $! "−1 A3s (t). = A1s (t) − A2s (t) + B1 (t, 0)Ks (t) AK 4s (t)

3.3 Auxiliary Results

133

Finally, using Eqs. (3.74) and (3.44) yields ! "−1 K (t) A (t) A3s (t) = A¯ K As (t) = A1s (t) − AK s (t). 2s 4s

(3.75)

Quite similarly to Eq. (3.75), using Eqs. (3.14), (3.45), (3.70), (3.73) and (3.74), we obtain ! "−1 Bs (t) = B1 (t, 0) − A2s (t) A4s (t) B2 (t, 0) ! "−1 (t) B2 (t, 0) −B1 (t, 0)Ks (t) AK 4s ! "−1 ! "−1 B2 (t, 0)Ks (t) AK B2 (t, 0) +A2s (t) A4s (t) 4s (t) # ! "−1 AK = B1 (t, 0) − A2s (t) A4s (t) 4s (t) + B1 (t, 0)Ks (t) ! "−1 $! "−1 B2 (t, 0)Ks (t) AK B2 (t, 0) −A2s (t) A4s (t) 4s (t) # ! "−1 ! " A4s (t) + B2 (t, 0)Ks (t) = B1 (t, 0) − A2s (t) A4s (t) ! "−1 $! "−1 B2 (t, 0)Ks (t) AK (t) B2 (t, 0) +B1 (t, 0)Ks (t) − A2s (t) A4s (t) 4s # $! "−1 B2 (t, 0) = B1 (t, 0) − A2s (t) + B1 (t, 0)Ks (t) AK 4s (t) ! "−1 K B2 (t, 0) = B¯ = B1 (t, 0) − AK 2s (t) A4s (t)

K s (t).

(3.76) Thus, due to (3.75) and (3.76), the differential equation (3.71) coincides with the differential equation (3.43), meaning that the invertible control transformation (3.68) converts the differential equation (3.13) to the differential equation (3.43). The rest of the proof is similar to the proof of Lemma 3.2.

3.3.6 Proof of Lemma 3.7 Let t0 be an arbitrary but fixed point in the interval [0, tc ] and Δt = 0 be an arbitrary number such that t0 + Δt ∈ [0, tc ]. Let us denote ΔA4j = A4j (t0 + Δt, 0) − A4j (t0 , 0), j = 0, 1, . . . , N, ΔS22 = S2 (t0 + Δt, 0) − S22 (t0 , 0), ΔG4 (η) = G4 (t0 + Δt, η, 0) − G4 (t0 , η, 0), (3.77)

134

3 Euclidean Space Output Controllability of Linear Systems with State Delays

ΔP = P (t0 + Δt) − P (t0 ), ΔQ(η) = Q(t0 + Δt, η) − Q(t0 , η) − P (t0 )ΔA4N , ΔR(η, χ ) = R(t0 + Δt, η, χ ) − R(t0 , η, χ ) − QT (t0 , η)ΔA4N T T   − ΔA4N Q(t0 , χ ) − ΔA4N P (t0 )ΔA4N . (3.78) Based on these notations, we are going to derive a boundary-value problem for the matrices ΔP , ΔQ(η), ΔR(η, χ ). Let us start with the boundary conditions. Subtracting the first condition in (3.54) at t = t0 from this condition at t = t0 + Δt, as well as the second condition at t = t0 from this condition at t = t0 + Δt and the third condition at t = t0 from this condition at t = t0 + Δt, we obtain Q(t0 + Δt, −h) − Q(t0 , −h) = P (t0 + Δt)A4N (t0 + Δt, 0) − P (t0 )A4N (t0 , 0), R(t0 + Δt, −h, η) − R(t0 , −h, η) = AT4N (t0 + Δt, 0)Q(t0 + Δt, η) −AT4N (t0 , 0)Q(t0 , η), R(t0 + Δt, η, −h) − R(t0 , η, −h) = QT (t0 + Δt, η)A4N (t0 + Δt, 0) −QT (t0 , η)A4N (t0 , 0). From (3.77) and (3.78), we have A4N (t0 + Δt, 0) = A4N (t0 , 0) + ΔA4N ,

P (t0 + Δt) = P (t0 ) + ΔP

Q(t0 + Δt, η) − Q(t0 , η) = ΔQ(η) + P (t0 )ΔA4N , R(t0 + Δt, η, χ ) − R(t0 , η, χ ) = ΔR(η, χ ) + QT (t0 , η)ΔA4N   T T + ΔA4N Q(t0 , χ ) + ΔA4N P (t0 )ΔA4N . Substitution of these expressions into the previous set of equalities yields after a routine algebra the boundary conditions for the matrices ΔP , ΔQ(η), ΔR(η, χ )   ΔQ(−h) = ΔP A4N (t0 , 0) + ΔA4N ,  T ΔR(−h, η) = A4N (t0 , 0) + ΔA4N ΔQ(η),  T   ΔR(η, −h) = ΔQ(η) A4N (t0 , 0) + ΔA4N .

(3.79)

It is seen that these conditions have the form similar to the form of the conditions (3.54). Proceed to the derivation of the equations for ΔP , ΔQ(η), ΔR(η, χ ). Subtracting Eq. (3.51) at t = t0 from this equation at t = t0 + Δt, as well as Eq. (3.52) at t = t0 from this equation at t = t0 +Δt and Eq. (3.53) at t = t0 from this equation at t = t0 + Δt, and using Eqs. (3.77) and (3.78), we obtain the following set of matrix equations with respect to ΔP , ΔQ(η), ΔR(η, χ ):

3.3 Auxiliary Results

135

 T ΔP α(t0 ) + α T (t0 )ΔP + ΔQ(0) + ΔQ(0) + ΥP (ΔP ) = 0,

(3.80)

dΔQ(η) = α T (t0 )ΔQ(η) + ΔP θ (t0 , η) dη +

N −1 

  ΔP A4j (t0 )δ(η + hj ) + ΔR(0, η) + ΥQ ΔP , ΔQ(η) ,

(3.81)

j =1



  T ∂ ∂ + ΔR(η, χ ) = ΔQ(η) θ (t0 , χ ) ∂η ∂χ

+θ T (t0 , η)ΔQ(χ ) +

N −1 

AT4j (t0 , 0)ΔQ(χ )δ(η + hj )

j =1

+

N −1 



ΔQ(η)

T

  A4j (t0 , 0)δ(χ + hj ) + ΥR ΔQ(η), ΔQ(χ ) .

(3.82)

j =1

In this set of the equations only the terms, linearly dependent on the unknown matrices ΔP , ΔQ(η), ΔR(η, χ ), are presented explicitly. The coefficients α(t0 ) and θ (t0 , η), appearing in some of these terms, have the form (for the sake of further analysis, we replace t0 with t in the expressions of these coefficients) α(t) = A40 (t, 0) − S22 (t, 0)P (t), θ (t, η) = G4 (t, η, 0) − S22 (t, 0)Q(t, η).

(3.83)

Due to the assumption (II) and Proposition 3.2, for any t ∈ [0, tc ], the matrix-valued function θ (t, η) is piecewise continuous inη ∈ [−h, 0].    The terms ΥP (ΔP ), ΥQ ΔP , ΔQ(η) , and ΥR ΔQ(η), ΔQ(χ ) , appearing in Eqs. (3.80)–(3.82), are nonlinear matrix-valued functions of ΔP , ΔQ(η), and ΔR(η, χ ). These functions have the form T      ΥP (ΔP ) = P (t0 ) + ΔP ΔA40 + ΔA40 P (t0 ) + ΔP     − P (t0 ) + ΔP ΔS22 P (t0 ) + ΔP − ΔP S22 (t0 , 0)ΔP T  −P (t0 )ΔA4N − ΔA4N P (t0 ),

(3.84)

    T   ΥQ ΔP , ΔQ(η) = ΔA40 − ΔS22 P (t0 ) + ΔP Q(t0 , η) + ΔQ(η)   −ΔP S22 (t0 , 0)ΔQ(η) + P (t0 ) + ΔP ΔG4 (η) +

N −1 

  P (t0 ) + ΔP ΔA4j δ(η + hj ) + α T (t0 )P (t0 )ΔA4N ,

j =1

(3.85)

136

3 Euclidean Space Output Controllability of Linear Systems with State Delays

T      ΥR ΔQ(η), ΔQ(χ ) = ΔG4 (η) Q(t0 , χ ) + ΔQ(χ ) N −1    T  T  + Q(t0 , η) + ΔQ(η) ΔG4 (χ ) + ΔA4j Q(t0 , χ ) + ΔQ(χ ) δ(η + hj ) j =1

+

N −1 

 T  T Q(t0 , η) + ΔQ(η) ΔA4j δ(χ + hj ) − ΔQ(η) S22 (t0 , 0)ΔQ(χ )

j =1

  T  − Q(t0 , η) + ΔQ(η) ΔS22 Q(t0 , χ ) + ΔQ(χ )   T dQ(t0 , χ )  dQ(t0 , η) T ΔA4N + ΔA4N + dη dχ T  + ΔA4N P (t0 )θ (t0 , χ ) + θ T (t0 , η)P (t0 )ΔA4N +

N −1 

AT4j (t0 , 0)P (t0 )ΔA4N δ(η + hj ) +

j =1

N −1 

 T ΔA4N P (t0 )A4j (t0 , 0)δ(χ + hj ).

j =1

(3.86) Consider the following initial-value problem for the m × m-matrix-valued function L (σ ): N −1  dL (σ ) = α(t0 )L (σ ) + A4j (t0 , 0)L (σ − hj ) dσ

 +A4N (t0 + Δt, 0)L (σ − hN ) +

j =1

0

−h

θ (t0 , η)L (σ + η)dη, σ > 0,

L (σ ) = 0 ∀σ < 0,

L (0) = Im .

(3.87)

By virtue of Proposition 2.1 (see Sect. 2.2.3), the problem (3.87) has the unique solution L (σ ) = L (σ, t0 , Δt), σ ∈ (−∞, +∞). Based on this solution, let us construct the following m × m-matrix-valued function: L (σ, η, t0 , Δt) =

N −1  j =1

+  +

h −η

L (σ − η − hj , t0 , Δt)A4j (t0 , 0), η − σ < −hj ≤ η 0, otherwise

L (σ − η − h, t0 , Δt)A4N (t0 + Δt, 0), η − σ < −h ≤ η 0, otherwise L (σ − η − χ , t0 , Δt)θ (t0 , −χ )dχ ,

 

σ ≥ 0, η ∈ [−h, 0]. (3.88)

3.3 Auxiliary Results

137

Now, using the results of [4] and the matrix-valued functions L (σ, t0 , Δt), L (σ, η, t0 , Δt), we can rewrite the problem (3.80)–(3.82), (3.79) in the equivalent integral form +∞ 



L T (σ, t0 , Δt)ΥP (ΔP )L (σ, t0 , Δt)

ΔP =  +  +  +

0



0

−h −h

0

  L T (σ, t0 , Δt)ΥQ ΔP , ΔQ(η) L (σ + η, t0 , Δt)dη

0 −h

  L T (σ + η, t0 , Δt)ΥQT ΔP , ΔQ(η) L (σ, t0 , Δt)dη

0 −h

   L (σ + η, t0 , Δt)ΥR ΔQ(η), ΔQ(χ ) L (σ + χ , t0 , Δt)dηdχ dσ, T

(3.89)

ΔQ(η) = + + +

 0  0 −h −h

 0

 +∞ 

L T (σ, t0 , Δt)ΥP (ΔP )L (σ, η, t0 , Δt)

0

  L T (σ, t0 , Δt)ΥQ ΔP , ΔQ(χ ) L (σ + χ , η, t0 , Δt)dχ

−h

 0

T ΔP , ΔQ(χ )L (σ, η, t , Δt)dχ L T (σ + χ , t0 , Δt)ΥQ 0

−h

   L T (σ + χ , t0 , Δt)ΥR ΔQ(χ ), ΔQ(χ1 ) L (σ + χ1 , η, t0 , Δt)dχ dχ1 dσ + +

 0 −h

 η+h  0

  L T (σ, t0 , Δt)ΥQ ΔP , ΔQ(η − σ )

   L T (σ + χ , t0 , Δt)ΥR ΔQ(χ ), ΔQ(η − σ ) dχ dσ, (3.90)

+∞ 



L T (σ, η, t0 , Δt)ΥP (ΔP )L (σ, χ , t0 , Δt)

ΔR(η, χ ) =  +  +

0 0 −h 0

−h

  L T (σ, η, t0 , Δt)ΥQ ΔP , ΔQ(χ1 ) L (σ + χ1 , χ , t0 , Δt)dχ1   L T (σ + χ1 , η, t0 , Δt)ΥQT ΔP , ΔQ(χ1 ) L (σ, χ , t0 , Δt)dχ1  +

0



0

−h −h

  L T (σ + χ1 , η, t0 , Δt)ΥR ΔQ(χ1 ), ΔQ(χ2 )

138

3 Euclidean Space Output Controllability of Linear Systems with State Delays

 ×L (σ + χ2 , χ , t0 , Δt)dχ1 dχ2 dσ η+h 

 + 0

 +

0 −h

   ΥR ΔQ(χ1 ), ΔQ(η − σ ) L (σ + χ1 , χ , t0 , Δt)dχ1 dσ χ +h 



  L T (σ, η, t0 , Δt)ΥQ ΔP , ΔQ(χ − σ )

+  +

 ΥQT ΔP , ΔQ(η − σ ))L (σ, χ , t0 , Δt)

0

0 −h

  T L (σ + χ1 , η, t0 , Δt)ΥR ΔQ(χ1 ), ΔQ(χ − σ ))dχ1 dσ 

min(η+h,χ +h)

+

  ΥR ΔQ(η − σ ), ΔQ(χ − σ ) dσ.

(3.91)

0

Using Proposition 3.2, Eq. (3.83) and the continuity of A4N (t, 0), one can show (similarly to Lemma 2.1 in Sect. 2.2) that, for all sufficiently small |Δt|, all roots λ of the equation  N −1  det λIm − α(t0 ) − A4j (t0 , 0) exp(−λhj ) j =1

 −A4N (t0 + Δt, 0) exp(−λh) −

0

−h

 θ (t0 , η) exp(λη)dη = 0

satisfy the inequality Reλ < −2γ (t0 ). The latter, along with the results of [4], and Eqs. (3.87) and (3.88), yields the following inequalities for all sufficiently small |Δt|, and all σ ≥ 0, η ∈ [−h, 0]:   L (σ, t0 , Δt) ≤ a exp − γ (t0 )σ ,   L (σ, η, t0 , Δt) ≤ a exp − γ (t0 )σ ,

(3.92)

where a > 0 is some constant independent of Δt. Due to the assumptions (I)–(III), we have the following inequalities for all sufficiently small |Δt|:

ΔS22

  ≤ a Δt ,

  ΔA4j ≤ a Δt ,

j = 0, 1, . . . , N,   ΔG4 (η) ≤ a Δt , η ∈ [−h, 0],

where a > 0 is some constant independent of Δt.

(3.93)

3.3 Auxiliary Results

139

Now, applying the procedure of successive approximations to the set (3.89)– (3.91) with the initial guess equals zero, and using Eqs. (3.83)–(3.86) and the inequalities (3.92)–(3.93), one can show after a routine algebra that for all suffi* + ciently small |Δt| there exists the unique solution ΔP , ΔQ(η), ΔR(η, χ ) of this set, such that:  T (i) ΔP = ΔP ; (ii) the matrix-valued function ΔQ(η) is piecewise absolutely continuous in η ∈ [−h, 0] with the bounded jumps at η = −hj , (j = 1, . . . , N − 1); (iii) the matrix-valued function ΔR(η, χ ) is piecewise absolutely continuous in η ∈ [−h, 0] and in χ ∈ [−h, 0] with the bounded jumps at η = −hj1 and χ =  T −hj2 , (j1 = 1, . . . , N − 1; j2 = 1, . . . , N − 1), moreover, ΔR(η, χ ) = ΔR(χ , η); (iv) the following inequality is satisfied: $

# max

ΔP , ΔQ(η) , ΔR(η, χ )

≤ aΔt,

(η, χ ) ∈ [−h, 0] × [−h, 0],

(3.94)

where a > 0 is some constant independent of Δt. Since (3.89)–(3.91) is a set of nonlinear equations and may have multiple solutions, we must show that * its solution, satisfying the items (i)–(iv), indeed + satisfies Eq. (3.78), where P (t * + 0 + Δt), Q(t0 + Δt, η), R(t0 + Δt, η, χ ) and P (t0 ), Q(t0 , η), R(t0 , η, χ ) are the solutions of the set (3.51)–(3.54) at the parameter values t = t0 + Δt and t = t0 , satisfying Proposition 3.2. First of all let us observe that, by virtue of Proposition 3.2, the expressions in (3.78) do satisfy the items (i)–(iii). To show that these expressions satisfy the item (iv), we consider the matrices = P (t0 ) + ΔP , P

Q(η) = Q(t0 , η) + ΔQ(η) + P (t0 )ΔA4N ,

χ) = R(t0 , η, χ ) + ΔR(η, χ ) + QT (t0 , η)ΔA4N R(η, T T   + ΔA4N Q(t0 , χ ) + ΔA4N P (t0 )ΔA4N .

* + , Q(η), χ ) satisfies the set (3.51)–(3.53), (3.54) It is clear that the triplet P R(η, for t = t0 + Δt, i.e., − P S22 (t0 +Δt, 0)P + Q(0)+ T (0)+Im = 0, A40 (t0 +Δt, 0)+AT40 (t0 +Δt, 0)P Q P (3.95) ! " d Q(η) S22 (t0 + Δt, 0) Q(η) +P G4 (t0 + Δt, η, 0) = AT40 (t0 + Δt, 0) − P dη

140

3 Euclidean Space Output Controllability of Linear Systems with State Delays

+

N −1 

η), A4j (t0 + Δt, 0)δ(η + hj ) + R(0, P

j =1

(3.96) 

 ∂ ∂ ) + R(η, χ ) = GT4 (t0 + Δt, η, 0)Q(χ ∂η ∂χ

T (η)G4 (t0 + Δt, χ , 0) + +Q

N −1 

)δ(η + hj ) AT4j (t0 + Δt, 0)Q(χ

j =1

+

N −1 

T (η)S22 (t0 + Δt, 0)Q(χ ), T (η)A4j (t0 + Δt, 0)δ(χ + hj ) − Q Q

j =1

(3.97) A4N (t0 + Δt, 0), Q(−h) =P −h) = Q T (η)A4N (t0 + Δt, 0), R(−h, η) = AT4N (t0 + Δt, 0)Q(η), R(η, (3.98) where η ∈ [−h, 0], χ ∈ [−h, 0]. Consider the following quasi-polynomial equation with respect to λ˜ : N #  det λ˜ Im − α˜ − A4j (t0 + Δt, 0) exp(−λ˜ hj ) j =1

−



0 −h

$ θ˜ (η) exp(λ˜ η)dη = 0,

(3.99)

where , α˜ = A40 (t0 + Δt, 0) − S22 (t0 + Δt, 0)P θ˜ (η) = G4 (t0 + Δt, η, 0) − S22 (t0 + Δt, 0)Q(η). Due to the inequality (3.94), this quasi-polynomial equation for |Δt| = 0 becomes Eq. (3.56) with t = t0 . All roots λ(t0 ) of the latter satisfy the inequality (3.57) for t = t0 (see Proposition 3.2). Now, using Lemma 2.1, we obtain that, for all sufficiently small |Δt|, all roots λ˜ of Eq. (3.99) satisfy the inequality Reλ˜ < −2γ (t0 + Δt),

3.3 Auxiliary Results

141

where γ (t0 + Δt) ≡ * γ (t0 ) for all such |Δt|. + , Q(η), χ ) satisfies the item (d) of Proposition 3.2 for Thus, the triplet P R(η, * + , Q(η), χ ) , being the solution to the (3.95)– t = t0 + Δt. Moreover, P R(η, (a)–(c) of this proposition. This means that the triplet *(3.98), satisfies the items + , Q(η), χ ) is the unique solution of (3.95)–(3.98), which satisfies the P R(η, items (a)–(d) of Proposition 3.2 for t = t0 + Δt. However, the set (3.95)– (3.97), (3.98) coincides with the set (3.51)–(3.53), * + (3.54) for t = t0 + Δt, and P (t0 + Δt), Q(t0 + Δt, η), R(t0 + Δt, η, χ ) , (η, χ ) ∈ [−h, 0] × [−h, 0] is the unique solution of the latter satisfying the items (a)–(d) of Proposition 3.2 for t = t0 + Δt. Therefore, for all sufficiently small |Δt|, = P (t0 + Δt), Q(η) χ ) = R(t0 + Δt, η, χ ), P = Q(t0 + Δt, η), R(η, * + meaning that the solution ΔP , ΔQ(η), ΔR(η, χ ) of the set (3.89)–(3.91) does satisfy Eq. (3.78). This observation, along with the inequality (3.94), implies the continuity of the matrix-valued functions P (t), Q(t, η), R(t, η, χ ) with respect to t at t = t0 uniformly in (η, χ ) ∈ [−h, 0] × [−h, 0]. Since t0 is any point of the interval [0, tc ], then P (t), Q(t, η), R(t, η, χ ) are continuous functions of t ∈ [0, tc ] uniformly in (η, χ ) ∈ [−h, 0] × [−h, 0]. This completes the proof of the lemma.

3.3.7 Proof of Lemma 3.8 In the proof of this lemma, we use the notations introduced in the proof of Lemma 3.7. Let us note the following. The problem (3.87) is the initial-value problem for the linear autonomous time delay equation where the coefficient for L (σ − hN ) continuously depends on the parameter Δt for all its sufficiently small values such that (t0 + Δt) ∈ [0, tc ]. Therefore, by virtue of the results of [1], the solution L (σ, t0 , Δt) of this problem continuously depends on Δt at Δt = 0. Let us denote,

Llim (σ, t0 ) = lim L (σ, t0 , Δt). Δt→0

Then, Llim (σ, t0 ) is the unique solution of the following initial-value problem:  dLlim (σ, t) = α(t0 )Llim (σ, t0 ) + A4j (t0 , 0)Llim (σ − hj , t0 ) dσ N

 +

j =1

0 −h

θ (t0 , η)Llim (σ + η, t0 )dη, σ > 0,

Llim (σ, t) = 0 ∀σ < 0,

Llim (0, t) = Im .

(3.100)

142

3 Euclidean Space Output Controllability of Linear Systems with State Delays

Also, let us denote L lim (σ, η, t0 ) = lim L (σ, η, t0 , Δt). Δt→0

Then, using (3.88), we have L lim (σ, η, t0 ) =

 +

 N  Llim (σ − η − hj , t)A4j (t0 , 0), η − σ < −hj ≤ η 0, otherwise j =1

h −η

Llim (σ − η − χ , t0 )θ (t0 , −χ )dχ ,

σ ≥ 0, η ∈ [−h, 0]. (3.101)

Let us calculate the limits ΥP (ΔP ) lim , Δt→0 Δt

  ΥQ ΔP , ΔQ(η) lim , Δt→0 Δt

  ΥR ΔQ(η), ΔQ(χ ) lim . Δt→0 Δt

We start with the first one. Using Eq. (3.84), we have  T   ΔA40  ΔA40  ΥP (ΔP ) = lim P (t0 ) + ΔP + lim P (t0 ) + ΔP lim Δt→0 Δt→0 Δt→0 Δt Δt Δt  ΔS22    ΔP P (t0 ) + ΔP − lim S22 (t0 , 0)ΔP − lim P (t0 ) + ΔP Δt→0 Δt→0 Δt Δt  T ΔA4N ΔA4N − lim P (t0 ). − lim P (t0 ) Δt→0 Δt→0 Δt Δt Calculating each limit in the right-hand side of this equality, and using the assumption (I) and the inequality (3.94), we obtain   dA40 (t0 , 0) T ΥP (ΔP ) dA40 (t0 , 0) = P (t0 ) + P (t0 ) Δt→0 Δt dt dt   dA4N (t0 , 0) T dS22 (t0 , 0) dA4N (t0 , 0) P (t0 ) − P (t0 ) − P (t0 ). −P (t0 ) dt dt dt

ΠP (t0 ) = lim

Similarly, using Eqs. (3.85), (3.86), the assumptions (II)–(III) and the results of [32], we have    T ΥQ ΔP , ΔQ(η) dA40 (t0 , 0) dS22 (t0 , 0) ΠQ (η, t0 ) = lim = − P (t0 ) Q(t0 , η) Δt→0 Δt dt dt

3.3 Auxiliary Results

143 N −1

+P (t0 )

dA4j (t0 , 0) ∂G4 (t0 , η, 0)  dA4N (t0 , 0) + δ(η+hj )+α T (t0 )P (t0 ) , P (t0 ) ∂t dt dt j =1

    ΥR ΔQ(η), ΔQ(χ ) ∂G4 (t0 , η, 0) T ΠR (η, χ , t0 ) = lim = Q(t0 , χ ) Δt→0 Δt ∂t  N −1  ∂G4 (t0 , χ , 0)  dA4j (t0 , 0) T T + +Q (t0 , η) Q(t0 , χ )δ(η + hj ) ∂t dt

j =1

+

N −1 

QT (t0 , η)

j =1

dA4j (t0 , 0) dS22 (t0 , 0) δ(χ + hj ) − QT (t0 , η) Q(t0 , χ ) dt dt

   dA4N (t0 , 0) T dQ(t0 , χ ) dQ(t0 , η) T dA4N (t0 , 0) + + dη dt dt dχ  T dA4N (t0 , 0) dA4N (t0 , 0) + P (t0 )θ (t0 , χ ) + θ T (t0 , η)P (t0 ) dt dt 

+

N −1 

AT4j (t0 , 0)P (t0 )

j =1

+

N −1   j =1

dA4N (t0 , 0) dt

dA4N (t0 , 0) δ(η + hj ) dt

T P (t)A4j (t0 , 0)δ(χ + hj ).

Now, using the above derived limit equalities, we are going to calculate the following limits: lim

Δt→0

ΔP , Δt

lim

Δt→0

ΔQ(η) + P (t0 )ΔA4N Δt

and   T T ΔR(η, χ ) + QT (t0 , η)ΔA4N + ΔA4N Q(t0 , χ ) + ΔA4N P (t0 )ΔA4N . Δt→0 Δt lim

By virtue of Eq. (3.78), these limits (if they exist) equal to the derivatives dP (t0 )/dt, ∂Q(t0 , η)/∂t, and ∂R(t0 , η, χ )/∂t, respectively. Let us start with the calculation of the first limit. Dividing the equality (3.89) by Δt yields

144

3 Euclidean Space Output Controllability of Linear Systems with State Delays

ΔP = Δt



+∞ 

ΥP (ΔP ) L (σ, t0 , Δt) Δt 0    0 ΥQ ΔP , ΔQ(η) T L (σ + η, t0 , Δt)dη + L (σ, t0 , Δt) Δt −h    0 ΥQT ΔP , ΔQ(η) T L (σ, t0 , Δt)dη + L (σ + η, t0 , Δt) Δt −h     0 0 ΥR ΔQ(η), ΔQ(χ ) T L (σ + χ , t0 , Δt)dηdχ dσ. + L (σ + η, t0 , Δt) Δt −h −h L T (σ, t0 , Δt)

Calculating the limit of this equation for Δt → 0, we obtain that the derivative dP (t0 )/dt exists and has the form  +∞ # dP (t0 ) ΔP T = lim = Llim (σ, t0 )ΠP (t0 )Llim (σ, t0 ) Δt→0 Δt dt 0  0 T + Llim (σ, t0 )ΠQ (η, t0 )Llim (σ + η, t0 )dη  +  +

0



0

−h −h

−h 0

−h

T T Llim (σ + η, t0 )ΠQ (η, t0 )Llim (σ, t0 )dη

$ T Llim (σ + η, t0 )ΠR (η, χ , t0 )Llim (σ + χ , t0 )dηdχ dσ.

(3.102)

Similarly, using the equalities (3.90), (3.91) and the assumption (I), and calculating the two other limits, we obtain that the derivatives ∂Q(t0 , η)/∂t, ∂R(t0 , η, χ )/∂t exist for any pair (η, χ ) ∈ [−h, 0] × [−h, 0], and these derivatives have the form ∂Q(t0 , η) ΔQ(η) dA4N (t0 , 0) = lim + P (t0 ) Δt→0 ∂t Δt dt  +∞ # T = Llim (σ, t0 )ΠP (t0 )L lim (σ, η, t0 )  +  +  +

0



0

−h −h

0

0 −h 0

−h

T Llim (σ, t0 )ΠQ (χ , t0 )L lim (σ + χ , η, t0 )dχ T T Llim (σ + χ , t0 )ΠQ (χ , t0 )L lim (σ, η, t0 )dχ

$ T Llim (σ + χ , t0 )ΠR (χ , χ1 , t0 )L lim (σ + χ1 , η, t0 )dχ dχ1 dσ 

η+h #

+ 0

T Llim (σ, t0 )ΠQ (η − σ, t0 )

3.3 Auxiliary Results

 +

145

$  dA4N (t0 , 0) T , Llim (σ + χ , t0 )ΠR χ , η − σ, t0 )dχ dσ + P (t0 ) dt −h (3.103) 0

∂R(t0 , η, χ ) ΔR(η, χ ) dA4N (t0 , 0) = lim + QT (t0 , η) Δt→0 ∂t Δt dt  T  +∞ # dA4N (t0 , 0) + L Tlim (σ, η, t0 )ΠP (t0 )L lim (σ, χ , t0 ) Q(t0 , χ ) = dt 0  0 + L Tlim (σ, η, t0 )ΠQ (χ1 , t0 )L lim (σ + χ1 , χ , t0 )dχ1  +

−h 0

T (χ1 , t0 )L lim (σ, χ , t0 )dχ1 L Tlim (σ + χ1 , η, t0 )ΠQ

−h

 +

0



0

−h −h

 L Tlim (σ + χ1 , η, t0 )ΠR χ1 , χ2 , t0 ) $ ×L lim (σ + χ2 , χ , t0 )dχ1 dχ2 dσ



η+h #

+  +

0 −h

$ ΠR (χ1 , η − σ, t0 )L lim (σ + χ1 , χ , t0 )dχ1 dσ 

χ +h #

+ 0

 +

T ΠQ (η − σ, t0 )L lim (σ, χ , t0 )

0

0 −h

 L Tlim (σ, η, t0 )ΠQ χ − σ, t0 )

$ L Tlim (σ + χ1 , η, t0 )ΠR (χ1 , χ − σ, t0 )dχ1 dσ 

min(η+h,χ +h)

+ 0

dA4N (t0 , 0) +Q (t0 , η) + dt T



ΠR (η − σ, χ − σ, t0 )dσ dA4N (t0 , 0) dt

T Q(t0 , χ ). (3.104)

Since t0 is any given point of the interval [0, tc ], then the derivatives dP (t)/dt, ∂Q(t, η)/∂t, ∂R(t, η, χ )/∂t exist for all t ∈ [0, tc ] and any pair (η, χ ) ∈ [−h, 0] × [−h, 0]. Now, let us show that these derivatives are continuous in t ∈ [0, tc ] uniformly with respect to (η, χ ) ∈ [−h, 0] × [−h, 0].

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

Let us denote ∂Q(t, η) dA4N (t, 0) − P (t) , ∂t dt   dA4N (t, 0) dA4N (t, 0) T ∂R(t, η, χ ) R(t, η, χ ) = Q(t, χ ). − QT (t, η) − ∂t dt dt (3.105)

P(t) =

dP (t) , dt



Q(t, η) =

Using these notations and Eqs. (3.102)–(3.104), we have for any t ∈ [0, tc ] 

+∞ #

P(t) = 0

 +  +  +

0



0

−h −h

0 −h 0

−h

T Llim (σ, t)ΠQ (η, t)Llim (σ + η, t)dη T T Llim (σ + η, t)ΠQ (η, t)Llim (σ, t)dη

$ T Llim (σ + η, t)ΠR (η, χ , t)Llim (σ + χ , t)dηdχ dσ,  Q(t, η) =

 + 

0



0

−h −h

+∞ #

0

 +

+

T Llim (σ, t)ΠP (t)Llim (σ, t)

0

T Llim (σ, t)ΠP (t)L lim (σ, η, t)

T Llim (σ, t)ΠQ (χ , t)L lim (σ + χ , η, t)dχ

−h 0

T T Llim (σ + χ , t)ΠQ (χ , t)L lim (σ, η, t)dχ

−h

$ T Llim (σ + χ , t)ΠR (χ , χ1 , t)L lim (σ + χ1 , η, t)dχ dχ1 dσ 

η+h #

+  +  

0 0 −h

0 0 −h

T Llim (σ, t)ΠQ (η − σ, t)

$  T Llim (σ + χ , t)ΠR χ , η − σ, t)dχ dσ,

+∞ #

R(t, η, χ ) = +

(3.106)

L Tlim (σ, η, t)ΠP (t)L lim (σ, χ , t)

L Tlim (σ, η, t)ΠQ (χ1 , t)L lim (σ + χ1 , χ , t)dχ1

(3.107)

3.3 Auxiliary Results

 +

0 −h

147

T (χ1 , t)L lim (σ, χ , t)dχ1 L Tlim (σ + χ1 , η, t)ΠQ

 +

0



0

−h −h

 T (σ + χ1 , η, t)ΠR χ1 , χ2 , t) L lim

$ ×L lim (σ + χ2 , χ , t)dχ1 dχ2 dσ



η+h #

+ 0

 +

0 −h

$ ΠR (χ1 , η − σ, t)L lim (σ + χ1 , χ , t)dχ1 dσ 

χ +h #

+  +

T ΠQ (η − σ, t)L lim (σ, χ , t)

 L Tlim (σ, η, t)ΠQ χ − σ, t)

0 0 −h

$ L Tlim (σ + χ1 , η, t)ΠR (χ1 , χ − σ, t)dχ1 dσ 

min(η+h,χ +h)

+

ΠR (η − σ, χ − σ, t)dσ,

(3.108)

0

where Llim (σ, t) and L lim (σ, η, t) are defined in (3.100) and (3.101), respectively. Remember that in the proof of Lemma 3.7, we transformed equivalently the problem (3.80)–(3.82), (3.79) to the set of integral equations (3.89)–(3.91). In the present proof, we apply the inverse transformation of the set (3.106)–(3.108). Due to this transformation, we obtain the following set of equations, equivalent to (3.106)– (3.108): P(t)α(t) + α T (t)P(t) + Q(t, 0) + Q T (t, 0) + ΠP (t) = 0,

(3.109)

dQ(t, η) = α T (t)Q(t, η) + P(t)θ (t, η) dη +

N −1 

P(t)A4j (t)δ(η + hj ) + R(t, 0, η) + ΠQ (η, t),

(3.110)

j =1



 ∂ ∂ + R(t, η, χ ) = Q T (t, η)θ (t, χ ) ∂η ∂χ

+θ T (t, η)Q(t, χ ) +

N −1 

AT4j (t, 0)Q(t, χ )δ(η + hj )

j =1

+

N −1  j =1

Q T (t, η)A4j (t, 0)δ(χ + hj ) + ΠR (η, χ , t),

(3.111)

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

Q(t, −h) = P(t)A4N (t, 0), R(t, −h, η) = AT4N (t, 0)Q(t, η), R(t, η, −h) = Q T (t, η)A4N (t, 0), (3.112) where t ∈ [0, tc ] is a parameter. Now, based on the set (3.109)–(3.112) and using the statement of Lemma 3.7, one can show (similarly to the proof of this lemma) the continuity of P(t), Q(t, η), R(t, η, χ ) in t ∈ [0, tc ] uniformly with respect to (η, χ ) ∈ [−h, 0] × [−h, 0]. The latter, along with Eq. (3.105), implies the continuity of dP (t)/dt, ∂Q(t, η)/∂t, ∂R(t, η, χ )/∂t in t ∈ [0, tc ] uniformly with respect to (η, χ ) ∈ [−h, 0] × [−h, 0]. This completes the proof of the lemma.

3.3.8 Proof of Lemma 3.9 We prove the lemma by contradiction, i.e., we assume that the statement of the lemma is wrong. Due to the inequality (3.57), this means the existence of two sequences {tl } and {λl } satisfying the properties: (a) tl ∈ [0, tc ], (l = 1, 2, . . .); (b) Reλl < −2γ (tl ) < 0, (l = 1, 2, . . .) and liml→+∞ Reλl = 0; (c) Eq. (3.56) is satisfied for any pair t, λ = (tl , λl ), (l = 1, 2, . . .). Due to the property (a), there exists a convergent subsequence of {tl }. For the sake of simplicity (but without loss of generality), we assume that the sequence {tl } itself is convergent, and t¯ = liml→+∞ tl . Hence, t¯ ∈ [0, tc ]. The following two cases can be distinguished with respect to the sequence {λl }: (i) {λl } is bounded; (ii) {λl } is unbounded. We start with the first case. In this case, there exists a convergent subsequence of {λl }. For the sake of simplicity (but without loss of generality), we assume that the sequence {λl } itself is such a subsequence. Let λ¯ = liml→+∞ λl . Due to the abovementioned property (b), Reλ¯ = 0. Also, let us note that, by virtue of Lemma 3.7, the matrix P (t) is a continuous function of t ∈ [0, tc ], and the matrix Q(t, η) is a continuous function of t ∈ [0, tc ] uniformly in η ∈ [−h, 0].   Now, the substitution of t, λ = (tl , λl ) into (3.56) and the calculation of the limit of the resulting equality for l → +∞ yield  N    ¯ j) ¯ m − A40 (t¯, 0) − S22 (t¯, 0)P (t¯) − det λI A4j (t¯, 0) exp(−λh  −

j =1

0 −h

   ¯ ¯ ¯ ¯ G4 (t , η, 0) − S22 (t , 0)Q(t , η) exp(λη)dη = 0.

3.4 Parameter-Free Controllability Conditions for Systems with Small Delays

149

The latter means that λ¯ is a root of Eq. (3.56) for t = t¯. Thus, due to the inequality (3.57), Reλ¯ < −2γ (t¯) < 0, which contradicts the above obtained equality Reλ¯ = 0. Proceed to the case (ii) where the sequence {λl } is unbounded. In this case, there exists a subsequence of {λl }, modules of elements of which tend to infinity. Similarly to the case (i), we assume   that {λl } itself is such a subsequence, i.e., liml→+∞ |λl | = +∞. Substituting t, λ = (tl , λl ) into (3.56), dividing the resulting equality by λm l and, then, calculating the limit of the last equality for l → +∞, one obtains the contradiction 1 = 0. The contradictions, obtained in the cases (i) and (ii), prove the lemma.

3.4 Parameter-Free Controllability Conditions for Systems with Small Delays In this section, based on the auxiliary results, various ε-independent conditions, providing the Euclidean space output controllability of the original system (3.1)– (3.2), (3.3) for all sufficiently small ε > 0, are derived. Let tc > 0 be a given time instant independent of ε.

3.4.1 Case of the Standard System (3.1)–(3.2) In this subsection, we assume that the condition (3.12) holds for all t ∈ [0, tc ]. In the literature, singularly perturbed systems with such a feature are called standard (see, e.g., [5, 22]). Let us assume: (AI)

(AII)

(AIII)

The matrix-valued functions Aij (t, ε), Bk (t, ε), (i = 1, . . . , 4; j = 0, 1, . . . , N ; k = 1, 2), are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ]. The matrix-valued functions Gi (t, η, ε), (i = 1, . . . , 4) are piecewise continuous with respect to η ∈ [−h, 0] for each (t, ε) ∈ [0, tc ]×[0, ε0 ], and they are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in η ∈ [−h, 0]. All roots λ(t) of the equation ⎡ ⎤  0 N  det ⎣λIm − A4j (t, 0) exp(−λhj ) − G4 (t, η, 0) exp(λη)dη⎦ = 0 j =0

−h

(3.113) satisfy the inequality Reλ(t) < −2β for all t ∈ [0, tc ], where β > 0 is some constant.

150

3 Euclidean Space Output Controllability of Linear Systems with State Delays

Remark 3.8 If the assumption (AIII) is valid, then λ(t) ≡ 0 cannot be a root of Eq. (3.113). Therefore, for all t ∈ [0, tc ]: ⎡ det ⎣λIm −

N 

 A4j (t, 0) exp(−λhj ) −

j =0

⎤

0 −h

  ⎦ G4 (t, η, 0) exp(λη)dη 

= 0,

λ=0

which means the fulfilment of the condition (3.12) for all t ∈ [0, tc ]. Consider the block matrix ! " Z (tc , 0) = X (tc , 0), Y (tc , 0) . The following three theorems present different ε-free sufficient conditions for the Euclidean space output controllability of the singularly perturbed system (3.1)–(3.2), (3.3). These conditions depend considerably on relations between the Euclidean space dimensions of the slow state variable, the fast state variable, and the output of the system. Theorem 3.1 Let the assumptions (AI)–(AIII) be valid. Let q ≤ m. Let, for t = tc , the system (3.15)–(3.16) be Euclidean space output controllable. Then, there exists a positive number εc,1 , (εc,1 ≤ ε0 ), such that for all ε ∈ (0, εc,1 ], the singularly perturbed system (3.1)–(3.2), (3.3) is Euclidean space output controllable at the time instant tc . Theorem 3.2 Let the assumptions (AI)–(AIII) be valid. Let m < q ≤ n. Let rankY (tc , 0) = m. Let the system (3.13), (3.10) be output controllable at the time instant tc . Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number εc,2 , (εc,2 ≤ ε0 ), such that for all ε ∈ (0, εc,2 ], the singularly perturbed system (3.1)–(3.2), (3.3) is Euclidean space output controllable at the time instant tc . Theorem 3.3 Let the assumptions (AI)–(AIII) be valid. Let rankZ (tc , 0) = q. Let the system (3.13) be completely controllable at the time instant tc . Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number εc,3 , (εc,3 ≤ ε0 ), such that for all ε ∈ (0, εc,3 ], the singularly perturbed system (3.1)–(3.2), (3.3) is Euclidean space output controllable at the time instant tc . Proofs of Theorems 3.1–3.3 are presented in Sect. 3.4.3. Now, we consider the case where the assumption (AIII) is violated. The following three theorems are based on the presented below less restrictive and easier verified assumption. However, the proofs of these theorems are essentially based on the three above considered theorems. Thus, we assume: (AIV) For all t ∈ [0, tc ] and any complex number λ with Reλ ≥ 0, the following equality is valid:

3.4 Parameter-Free Controllability Conditions for Systems with Small Delays

151

 N    rank Wf (t, λ) = rank λIm − A4j (t, 0) exp(−λhj )  −

j =0

0 −h

 G4 (t, η, 0) exp(λη)dη , B2 (t, 0) = m.

As a direct consequence of the results of [31], we obtain the following assertion. Proposition 3.3 Let the assumption (AIV) be valid. Then, the fast subsystem (3.15) is L2 -stabilizable for each value of the parameter t ∈ [0, tc ]. Theorem 3.4 Let the assumptions (AI)–(AII), (AIV) be valid. Let q ≤ m. Let, for t = tc , the system (3.15)–(3.16) be Euclidean space output controllable. Then, there exists a positive number ε¯ c,1 , (¯εc,1 ≤ ε0 ), such that for all ε ∈ (0, ε¯ c,1 ], the singularly perturbed system (3.1)–(3.2), (3.3) is Euclidean space output controllable at the time instant tc . Proof Due to Proposition 3.3, for any t ∈ [0, tc ], the fast subsystem (3.15) is L2 stabilizable. Therefore, Proposition 3.2 is valid. The latter means that, for any t ∈ [0, tc ], the set of*Eqs. (3.51)–(3.53) subject to the boundary conditions (3.54) + has the unique solution P (t), Q(t, η), R(t, η, χ ), (η, χ ) ∈ [−h, 0] × [−h, 0] satisfying the items (a)–(d) of this proposition. Now, in the system (3.1)–(3.2), let us make the control transformation (3.29), where K1 (t, ε) = −B2T (t, 0)P (t),

K2 (t, η, ε) = −B2T (t, 0)Q(t, η),

t ∈ [0, tc ], η ∈ [−h, 0], ε ∈ [0, ε0 ].

(3.114)

Due to this transformation, we obtain the system (3.30)–(3.31). By virtue of Proposition 3.2 (item (b)) and Lemma 3.8, the coefficients (3.32)–(3.35) of this system have the properties, similar to the assumptions (AI) and (AII) on the coefficients of the system (3.1)–(3.2). The slow subsystem, associated with (3.30)–(3.31), has the form (3.38)–(3.39), where, due to (3.50) and (3.114), N    (t) = A (t, 0) − S (t, 0)P (t) + A4j (t, 0) AK 40 22 4s

 +

j =1 0 −h

  G4 (t, η, 0) − S22 (t, 0)Q(t, η) dη.

By virtue of Proposition 3.2 (item (d)), the inequality (3.41) is valid. Thus, the slow subsystem (3.38)–(3.39) is reduced to the differential equation (3.43).

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

The fast subsystem, associated with (3.30)–(3.31), has the form (3.46), where due to (3.50) and (3.114), AK 40 (t, 0) = A40 (t, 0) − S22 (t, 0)P (t),

AK 4j (t, 0) = A4j (t, 0), j = 1, . . . , N,

GK 4 (t, 0) = G4 (t, η, 0) − S22 (t, 0)Q(t, η). Thus, by virtue of Lemma 3.9, these coefficients satisfy the stability property similar to the assumption (AIII) on the coefficients of the system (3.15). Moreover, due to Lemma 3.6, the system (3.46), (3.16) for t = tc is Euclidean space output controllable. Thus, the transformed system (3.30)–(3.31), (3.3) satisfies all the conditions of Theorem 3.1, yielding the existence of a positive number ε¯ c,1 , (¯εc,1 ≤ ε0 ), such that for all ε ∈ (0, ε¯ c,1 ], this system is Euclidean space output controllable at the time instant tc . The latter, along with Lemma 3.2, means the Euclidean space output controllability at the time instant tc of the system (3.1)–   (3.2), (3.3) for all ε ∈ (0, ε¯ c,1 ]. This completes the proof of the theorem. Theorem 3.5 Let the assumptions (AI)–(AII), (AIV) be valid. Let m < q ≤ n. Let rankY (tc , 0) = m. Let the system (3.13), (3.10) be output controllable at the time instant tc . Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number ε¯ c,2 , (¯εc,2 ≤ ε0 ), such that for all ε ∈ (0, ε¯ c,2 ], the singularly perturbed system (3.1)–(3.2), (3.3) is Euclidean space output controllable at the time instant tc . Theorem 3.6 Let the assumptions (AI)–(AII), (AIV) be valid. Let rankZ (tc , 0) = q. Let the system (3.13) be completely controllable at the time instant tc . Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number ε¯ c,3 , (¯εc,3 ≤ ε0 ), such that for all ε ∈ (0, ε¯ c,3 ], the singularly perturbed system (3.1)–(3.2), (3.3) is Euclidean space output controllable at the time instant tc . Based on Lemmas 3.2, 3.5, Corollaries 3.5, 3.6 and Theorems 3.2–3.3, Theorems 3.5–3.6 are proven similarly to Theorem 3.4.

3.4.2 Case of the Nonstandard System (3.1)–(3.2) In this subsection, in contrast with the previous one, we consider the case where the condition (3.12) does not hold at least for one value of t ∈ [0, tc ]. In the literature, singularly perturbed systems with such a feature are called nonstandard (see, e.g., [5, 22]). Remark 3.9 It follows directly from the proof of Theorem 3.4 that this theorem (where q ≤ m) is also valid for the nonstandard system (3.1)–(3.2), (3.3).

3.4 Parameter-Free Controllability Conditions for Systems with Small Delays

153

The following two theorems present ε-free sufficient conditions for the Euclidean space output controllability of the singularly perturbed nonstandard system (3.1)– (3.2), (3.3) subject to the other relations between n, m, and q. Theorem 3.7 Let the assumptions (AI)–(AII), (AIV) be valid. Let m < q ≤ n. Let rankY (tc , 0) = m. Let the system (3.8)–(3.10) be impulse-free output controllable at the time instant tc . Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number ε˜ c,2 , (˜εc,2 ≤ ε0 ), such that for all ε ∈ (0, ε˜ c,2 ], the singularly perturbed system (3.1)–(3.2), (3.3) is Euclidean space output controllable at the time instant tc . Proof For a given ε ∈ (0, ε0 ] in the system (3.1)–(3.2), let us make the control transformation (3.29), where the matrices K1 (t, ε) and K2 (t, η, ε) are given by (3.114). As a result of this transformation, we obtain the system (3.30)– (3.31). The slow and fast subsystems, associated with the latter, are (3.38)–(3.39) and (3.46), respectively. Since, for t = tc , the system (3.15) is completely Euclidean space controllable, then due to Corollary 3.6, the system (3.46) for t = tc is completely Euclidean space controllable. Furthermore, since the system (3.8)–(3.10) is impulse-free output controllable at the time instant tc , then due to Lemma 3.3, the system (3.38)–(3.39), (3.10) is impulse-free output controllable at the time instant tc . By virtue of Proposition 3.2 (item d), the inequality (3.41) is valid. Thus, the slow subsystem (3.38)–(3.39) is reduced to the differential equation (3.43). Therefore, the abovementioned impulse-free output controllability of the system (3.38)–(3.39), (3.10) yields the output controllability of the system (3.43), (3.10) at the time instant tc . Now, by application of Theorem 3.5 to the system (3.30)–(3.31), (3.3), we directly obtain the existence of a positive number ε˜ c,2 , (˜εc,2 ≤ ε0 ), such that for all ε ∈ (0, ε˜ c,2 ], this system is Euclidean space output controllable at the time instant tc . Finally, using Lemma 3.2 yields the Euclidean space output controllability of the system (3.1)–(3.2), (3.3) at the time instant tc for all ε ∈ (0, ε˜ c,2 ], which completes the proof of the theorem.   Theorem 3.8 Let the assumptions (AI)–(AII), (AIV) be valid. Let rankZ (tc , 0) = q. Let the system (3.8)–(3.9) be impulse-free controllable with respect to xs (t) at the time instant tc . Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number ε˜ c,3 , (˜εc,3 ≤ ε0 ), such that for all ε ∈ (0, ε˜ c,3 ], the singularly perturbed system (3.1)–(3.2), (3.3) is Euclidean space output controllable at the time instant tc . Based on Lemmas 3.2, 3.3, 3.6 and Theorem 3.6, Theorem 3.8 is proven similarly to Theorem 3.7.

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

3.4.3 Proofs of Theorems 3.1, 3.2, and 3.3 3.4.3.1

Proof of Theorem 3.1

Consider the block matrices:   A (t, ε) A2j (t, ε) , j = 0, 1, . . . , N, Aj (t, ε) = 1 1j 1 ε A3j (t, ε) ε A4j (t, ε)   G (t, η, ε) G2 (t, η, ε) . G(t, η, ε) = 1 1 1 ε G3 (t, η, ε) ε G4 (t, η, ε)   ! " B1 (t, ε) B(t, ε) = 1 , Z (tc , ε) = X (tc , ε), Y (tc , ε) . ε B2 (t, ε)

(3.115)

Let for any given ε ∈ (0, ε0 ], the (n + m) × (n + m)-matrix-valued function Φ(t, σ, ε) be the fundamental matrix of the system (3.1)–(3.2), while the (n + m) × (n + m)-matrix-valued function Ψ (σ, ε) be the adjoint matrix of the fundamental matrix at the time tc . Due to results of Sects. 2.2.3 and 2.2.4, for any given σ ∈ [0, tc ], the matrix Φ(t, σ, ε) is the unique solution of the initial-value problem dΦ(t, σ, ε)  = Aj (t, ε)Φ(t − εhj , σ, ε) dt N

 +

j =0

0 −h

G(t, η, ε)Φ(t + εη, σ, ε)dη, t ∈ (σ, tc ],

Φ(τσ , σ, ε) = 0, σ − εh ≤ τσ < σ ;

Φ(σ, σ, ε) = In+m ,

while the matrix Ψ (σ, ε) is the unique solution of the terminal-value problem  T dΨ (σ, ε) Aj (σ + εhj , ε) Ψ (σ + εhj , ε) =− dσ N

 −

j =0

0 −h

 T G(σ − εη, η, ε) Ψ (σ − εη, ε)dη, σ ∈ [0, tc ), Ψ (tc , ε) = In+m ,

Ψ (σ, ε) = 0, σ > tc .

In the latter problem, it is assumed that the blocks of the matrices Aj (t, ε), (j = 0, 1, . . . , N), and G(t, η, ε) satisfy the following equalities for all ε ∈ [0, ε0 ]: Aij (t, ε) = Aij (tc , ε), Gi (t, η, ε) = Gi (tc , η, ε), t > tc , η ∈ [−h, 0], i = 1, . . . , 4.

3.4 Parameter-Free Controllability Conditions for Systems with Small Delays

155

Also, let us consider the matrices  M(ε) =

tc

Ψ T (σ, ε)B(σ, ε)B T (σ, ε)Ψ (σ, ε)dσ,

(3.116)

0

and MZ (ε) = Z (tc , ε)M(ε)Z T (tc , ε).

(3.117)

Due to Corollary 3.1, in order to prove the theorem, it is necessary and sufficient to show the existence of a positive number εc,1 such that det MZ (ε) = 0 ∀ε ∈ (0, εc,1 ].

(3.118)

Let, for a given ε ∈ (0, ε0 ], matrices M1 (ε), M2 (ε) and M3 (ε) be the upper left-hand, upper right-hand, and lower right-hand blocks of the matrix M(ε) of dimensions n × n, n × m, and m × m, respectively, i.e., ⎛ M(ε) = ⎝

M1 (ε)

M2 (ε)

M2T (ε)

M3 (ε)

⎞ ⎠.

(3.119)

Let us partition the matrix Ψ (σ, ε) into blocks as:  Ψ (σ, ε) =

Ψ1 (σ, ε) Ψ3 (σ, ε)

 Ψ2 (σ, ε) , Ψ4 (σ, ε)

where the matrices Ψ1 (σ, ε), Ψ2 (σ, ε), Ψ3 (σ, ε), and Ψ4 (σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Also, let us denote

Skl (σ, ε) = Bk (σ, ε)BlT (σ, ε), k = 1, 2, l = 1, 2. Now, substitution of the block representation for the matrix B(t, ε) (see Eq. (3.115)), as well as the block representations for the matrices M(ε) and Ψ (σ, ε), into Eq. (3.116) yields 

 M1 (ε) M2 (ε) = M2T (ε) M3 (ε)     tc  T S11 (σ, ε) 1ε S12 (σ, ε) Ψ1 (σ, ε) Ψ2 (σ, ε) Ψ1 (σ, ε) Ψ3T (σ, ε) dσ. 1 T 1 Ψ2T (σ, ε) Ψ4T (σ, ε) Ψ3 (σ, ε) Ψ4 (σ, ε) 0 ε S12 (σ, ε) ε2 S22 (σ, ε)

156

3 Euclidean Space Output Controllability of Linear Systems with State Delays

Calculating the product of the block matrices in the integrand and equating the corresponding blocks of the matrices on the left-hand side and the right-hand side of the resulting equation, we obtain the following expressions for the matrices M1 (ε), M2 (ε), and M3 (ε):  tc # T Ψ1T (σ, ε)S11 (σ, ε)Ψ1 (σ, ε) + (1/ε)Ψ3T (σ, ε)S12 (σ, ε)Ψ1 (σ, ε) M1 (ε) = 0

$ +(1/ε)Ψ1T (σ, ε)S12 (σ, ε)Ψ3 (σ, ε) + (1/ε2 )Ψ3T (σ, ε)S22 (σ, ε)Ψ3 (σ, ε) dσ, (3.120) 

tc

M2 (ε) = 0

# T Ψ1T (σ, ε)S11 (σ, ε)Ψ2 (σ, ε) + (1/ε)Ψ3T (σ, ε)S12 (σ, ε)Ψ2 (σ, ε)

$ +(1/ε)Ψ1T (σ, ε)S12 (σ, ε)Ψ4 (σ, ε) + (1/ε2 )Ψ3T (σ, ε)S22 (σ, ε)Ψ4 (σ, ε) dσ, (3.121) 

tc

M3 (ε) = 0

# T Ψ2T (σ, ε)S11 (σ, ε)Ψ2 (σ, ε) + (1/ε)Ψ4T (σ, ε)S12 (σ, ε)Ψ2 (σ, ε)

$ +(1/ε)Ψ2T (σ, ε)S12 (σ, ε)Ψ4 (σ, ε) + (1/ε2 )Ψ4T (σ, ε)S22 (σ, ε)Ψ4 (σ, ε) dσ. (3.122) Let us estimate the matrices M1 (ε), M2 (ε), and M3 (ε) for all sufficiently small ε > 0. For this purpose, we use the matrix-valued functions Ψ1s (σ ), Ψ4f (ξ ), and Ψ3s (σ ) defined as follows. The first two matrices are the unique solutions of the following terminal-value and initial-value problems:  T dΨ1s (σ ) = − A¯ s (σ ) Ψ1s (σ ), σ ∈ [0, tc ), Ψ1s (tc ) = In , dσ T dΨ4f (ξ )   = A4j (tc , 0) Ψ4f (ξ − hj ) dξ N

 +

j =0

0 −h

 T G4 (tc , η, 0) Ψ4f (ξ + η)dη, ξ > 0, Ψ4f (ξ ) = 0, ξ < 0,

Ψ4f (0) = Im ,

while the third matrix has the form  T T Ψ3s (σ ) = − A−1 4s (σ ) A2s (σ )Ψ1s (σ ).

(3.123)

3.4 Parameter-Free Controllability Conditions for Systems with Small Delays

157

Remember that the matrix A¯ s (t) is defined in (3.14), while the matrices A2s (t) and A4s (t) are defined in (3.11). From the assumption (AIII) and the results of [19], we directly obtain

Ψ4f (ξ ) ≤ a exp(−2βξ ), ξ ≥ 0,

(3.124)

where a > 0 is some constant. Now, we are ready to estimate the abovementioned matrices. We start with the matrix M1 (ε). Let us denote



ΔΨ1 (σ, ε) = Ψ1 (σ, ε) − Ψ1s (σ ), ΔΨ3 (σ, ε) = Ψ3 (σ, ε) − εΨ3s (σ ). Using these notations and Eq. (3.120), we can express the matrix M1 (ε) as:  M1 (ε) = 0

tc

# T T ΔΨ1 (σ, ε) S11 (σ, ε)ΔΨ1 (σ, ε) + Ψ1s (σ )S11 (σ, ε)ΔΨ1 (σ, ε)

T  T (σ )S11 (σ, ε)Ψ1s (σ ) + ΔΨ1 (σ, ε) S11 (σ, ε)Ψ1s (σ ) + Ψ1s  T T T T +(1/ε) ΔΨ3 (σ, ε) S12 (σ, ε)ΔΨ1 (σ, ε) + Ψ3s (σ )S12 (σ, ε)ΔΨ1 (σ, ε)  T T T T +(1/ε) ΔΨ3 (σ, ε) S12 (σ, ε)Ψ1s (σ ) + Ψ3s (σ )S12 (σ, ε)Ψ1s (σ )  T T +(1/ε) ΔΨ1 (σ, ε) S12 (σ, ε)ΔΨ3 (σ, ε) + (1/ε)Ψ1s (σ )S12 (σ, ε)ΔΨ3 (σ, ε)  T T + ΔΨ1 (σ, ε) S12 (σ, ε)Ψ3s (σ ) + Ψ1s (σ )S12 (σ, ε)Ψ3s (σ )  T T +(1/ε2 ) ΔΨ3 (σ, ε) S22 (σ, ε)ΔΨ3 (σ, ε) + (1/ε)Ψ3s (σ )S22 (σ, ε)ΔΨ3 (σ, ε) $ T  T (σ )S22 (σ, ε)Ψ3s (σ ) dσ. (1/ε) ΔΨ3 (σ, ε) S22 (σ, ε)Ψ3s (σ ) + Ψ3s (3.125) By virtue of Theorem 2.1 (see Sect. 2.2.4), there exists a positive number ε1 ≤ ε0 such that for all ε ∈ (0, ε1 ] the following inequalities are satisfied:

ΔΨ1 (σ, ε) ≤ aε, σ ∈ [0, tc ],  

ΔΨ3 (σ, ε) ≤ aε ε + exp − β(tc − σ )/ε , σ ∈ [0, tc ],

where a > 0 is some constant independent of ε. Applying these inequalities to the expression (3.125) of the matrix M1 (ε) and using the fact that the matrix-valued functions Skl (σ, ε), (k = 1, 2; l = 1, 2) are bounded for all (σ, ε) ∈ [0, tc ] × [0, ε0 ], we obtain 

tc

M1 (ε) − 0

# T T T Ψ1s (σ )S11 (σ, ε)Ψ1s (σ ) + Ψ3s (σ )S12 (σ, ε)Ψ1s (σ )

158

3 Euclidean Space Output Controllability of Linear Systems with State Delays

$ T T +Ψ1s (σ )S12 (σ, ε)Ψ3s (σ ) + Ψ3s (σ )S22 (σ, ε)Ψ3s (σ ) dσ ≤ aε,

ε ∈ (0, ε1 ], (3.126)

where a > 0 is some constant independent of ε. Due to the assumption (AI), there exists a positive number ε2 ≤ ε0 such that for all ε ∈ (0, ε2 ] the matrix-valued functions Skl (σ, ε), (k = 1, 2; l = 1, 2) can be expressed in the form: Skl (σ, ε) = Skl (σ, 0) + Okl (σ, ε), k = 1, 2, l = 1, 2,

σ ∈ [0, tc ],

(3.127)

where Okl (σ, ε), (k = 1, 2; l = 1, 2) are some matrices of the corresponding dimensions satisfying the inequalities

Okl (σ, ε) ≤ aε,

σ ∈ [0, tc ],

(3.128)

where a > 0 is some constant independent of ε. Substitution of the expressions (3.127) into (3.126) and use of the inequalities (3.128) yield the inequality 

tc

M1 (ε) − 0

# T T T Ψ1s (σ )S11 (σ, 0)Ψ1s (σ ) + Ψ3s (σ )S12 (σ, 0)Ψ1s (σ )

$ T T +Ψ1s (σ )S12 (σ, 0)Ψ3s (σ ) + Ψ3s (σ )S22 (σ, 0)Ψ3s (σ ) dσ ≤ aε,

ε ∈ (0, ε3 ], (3.129)

where ε3 = min{ε1 , ε2 }; a > 0 is some constant independent of ε. Let us transform equivalently the integral in the left-hand side of (3.129). Using the expression (3.123) for Ψ3s (σ ), the definitions of Skl (σ, ε), (k = 1, 2; l = 1, 2), and the expression for B¯ s (t) (see Eq. (3.14)), we obtain



tc

M1s = 0

# T T T Ψ1s (σ )S11 (σ, 0)Ψ1s (σ ) + Ψ3s (σ )S12 (σ, 0)Ψ1s (σ )

$ T T (σ )S12 (σ, 0)Ψ3s (σ ) + Ψ3s (σ )S22 (σ, 0)Ψ3s (σ ) dσ +Ψ1s 

tc

= 0

# ! T T Ψ1s (σ ) B1 (σ, 0)B1T (σ, 0) − A2s (σ )A−1 4s (σ )B2 (σ, 0)B1 (σ, 0)

 T T −B1 (σ, 0)B2T (σ, 0) A−1 4s (σ ) A2s (σ ) " $  −1 T T T (σ )B (σ, 0)B (σ, 0) A (σ ) A (σ ) Ψ (σ ) dσ +A2s (σ )A−1 2 1s 2 2s 4s 4s  tc  T T = Ψ1s (σ )B¯ s (σ ) B¯ s (σ ) Ψ1s (σ )dσ, 0

(3.130)

3.4 Parameter-Free Controllability Conditions for Systems with Small Delays

159

which, along with the inequality (3.129), yields

M1 (ε) − M1s ≤ aε,

ε ∈ (0, ε3 ],

(3.131)

where a > 0 is the constant introduced in (3.129). Using the inequality (3.124) and the assumption (AI), we directly obtain (similarly to the inequality (3.131)) the existence of a positive number ε4 , (ε4 ≤ ε0 ), such that for all ε ∈ (0, ε4 ] the following inequalities are satisfied:

M2 (ε) ≤ a,

εM3 (ε) − M3f ≤ aε,

(3.132)

T Ψ4f (ξ )S22 (tc , 0)Ψ4f (ξ )dξ,

(3.133)

where 

+∞

M3f = 0

a > 0 is some constant independent of ε. Let us observe that, by virtue of Remark 2.4 (see Sect. 2.2.4), the matrixvalued function Ψ4f (ξ ) is the adjoint matrix to the fundamental matrix of the fast subsystem (3.15) for t = tc . Due to this observation, as well as the Euclidean space output controllability of the system (3.15)–(3.16) for t = tc (see Definition 3.4), and Corollary 3.1, there exists a positive number ξc such that   det Y (tc , 0)

ξc 0

 T Ψ4f (ξ )S22 (tc , 0)Ψ4f (ξ )dξ Y T (tc , 0)

= 0.

Moreover, since S22 (tc , 0) is positive semi-definite, this determinant is positive. Therefore, using the expression (3.133) for M3f and the results of [2], we have " ! det Y (tc , 0)M3f Y T (tc , 0)   = det Y (tc , 0)  +Y (tc , 0)

0 +∞ ξc

  ≥ det Y (tc , 0)

ξc 0

ξc

T Ψ4f (ξ )S22 (tc , 0)Ψ4f (ξ )dξ Y T (tc , 0)

 T Ψ4f (ξ )S22 (tc , 0)Ψ4f (ξ )dξ Y T (tc , 0)

 T Ψ4f (ξ )S22 (tc , 0)Ψ4f (ξ )dξ Y T (tc , 0) ≥ a1 ,

(3.134)

where a1 > 0 is some constant. Now, let us proceed to analysis of the matrix MZ (ε). Using Eq. (3.117), and the block representations of the matrices B(t, ε) and M(ε) (see Eqs. (3.115) and (3.119)), we can represent this matrix in the form:

160

3 Euclidean Space Output Controllability of Linear Systems with State Delays

MZ (ε) = X (tc , ε)M1 (ε)X T (tc , ε) + Y (tc , ε)M2T (ε)X T (tc , ε) +X (tc , ε)M2 (ε)Y T (tc , ε) + Y (tc , ε)M3 (ε)Y T (tc , ε).

(3.135)

Using the notations

ΔM1 (ε) = M1 (ε) − M1s ,



ΔM3 (ε) = εM3 (ε) − M3f ,

(3.136)

we can rewrite this expression for MZ (ε) as: εMZ (ε) = Y (tc , ε)M3f Y T (tc , ε) + εM(ε),

(3.137)

where = X (tc , ε)ΔM1 (ε)X T (tc , ε) + X (tc , ε)M1s X T (tc , ε) M(ε)

+Y (tc , ε)M2T (ε)X T (tc , ε) + X (tc , ε)M2 (ε)Y T (tc , ε) 1 + Y (tc , ε)ΔM3 (ε)Y T (tc , ε). ε Using the continuity of X (tc , ε) and Y (tc , ε) for ε ∈ [0, ε0 ], as well as the inequalities (3.131)–(3.132) and the notations (3.136), we have ≤ cε, εM(ε)



ε ∈ 0, min{ε3 , ε4 } ,

where c > 0 is some constant independent of ε. This inequality means that     ≤ a2 (ε)q ,  det εM(ε)



ε ∈ 0, min{ε3 , ε4 } ,

(3.138)

where a2 > 0 is some constant independent of ε. From the other hand, using the inequality (3.134) and the continuity of Y (tc , ε) for ε ∈ [0, ε0 ], we obtain the existence of a positive number ε5 ≤ ε0 such that for all ε ∈ (0, ε5 ] the following inequality is satisfied " a ! 1 det Y (tc , ε)M3f Y T (tc , ε) ≥ . 2 Thus, Eq. (3.137), along with this inequality and the inequality (3.138), yields   det εMZ (ε) ≥ a,

ε ∈ (0, εc,1 ],

where εc,1 = min{ε3 , ε4 , ε5 }; a > 0 is some constant independent of ε. This inequality directly implies the inequality (3.118), which completes the proof of the theorem.

3.4 Parameter-Free Controllability Conditions for Systems with Small Delays

3.4.3.2

161

Proof of Theorem 3.2

Here, similarly to the proof of Theorem 3.1, we are going to show the existence of a positive number εc,2 such that det MZ (ε) = 0 for all ε ∈ (0, εc,2 ], which will prove the theorem. Since q > m and rankY (tc , 0) = m, then there exists a nonsingular q ×q-matrix Y such that   0 . Y Y (tc , 0) = Im Using the continuity of X (tc , ε) and Y (tc , ε) in ε ∈ [0, ε0 ], we obtain for all these ε: Y X (tc , ε) = Y X (tc , 0) + Y ΔX (ε),   0 + Y ΔY (ε), Y Y (tc , ε) = Y Y (tc , 0) + Y ΔY (ε) = Im

(3.139)

where ΔX (ε) and ΔY (ε) are some matrices of the corresponding dimensions, satisfying the limit equalities lim ΔX (ε) = 0,

ε→+0

lim ΔY (ε) = 0.

ε→+0

(3.140)

Also, let us partition the matrix Y X (tc , 0) into blocks as: Y X (tc , 0) =



 ,1 X ,2 , X

(3.141)

,1 and X ,2 are of the dimensions (q − m) × n and m × n, where the blocks X respectively.   Iq−m 0 √ . Let us consider the q × q-matrix L(ε) = εIm 0 Multiplication of Eq. (3.135) from the left and from the right by L(ε)Y and by Y T L(ε), respectively, and use of Eqs. (3.139) and (3.141) yield after a routine algebra: 

 2 (ε) 1 (ε) M M 3 (ε) 2 T (ε) M M   2 (ε) 1 (ε) ΔM ΔM  +  , 2 (ε) T ΔM 3 (ε) ΔM

L(ε)Y MZ (ε)Y T L(ε) =

(3.142)

162

3 Euclidean Space Output Controllability of Linear Systems with State Delays

where " √ ! ,1 M1 (ε)X ,T , M ,1 M1 (ε)X ,T + X ,1 M2 (ε) , 1 (ε) = X (ε) = ε X M 2 1 2 ! " ,2 M1 (ε)X , T + M2 (ε)X ,T + X ,2 M2 (ε) + εM3 (ε), 3 (ε) = ε X M 2 2 (3.143) 1 (ε), ΔM 2 (ε), and ΔM 3 (ε) are some matrices of the dimensions (q −m)×(q − ΔM m), (q − m) × m, and m × m, respectively. Due to the inequalities (3.131)–(3.132) and the limit equalities (3.140), l (ε) = 0, lim ΔM

l = 1, 2, 3.

ε→+0

(3.144)

Using Eqs. (3.143)–(3.144) and the inequalities (3.131)–(3.132), we obtain   ! " ,T 0 ,1 M1s X X 1 lim det L(ε)Y MZ (ε)Y T L(ε) = det ε→+0 0 M3f "  ,1 M1s X ,T det M3f . = det X 1

(3.145)

Let us observe that, by virtue of Remark 2.4 (see Sect. 2.2.4), the matrixvalued function Ψ1s (σ ) is the adjoint matrix to the fundamental matrix of the slow subsystem (3.13). Due to this observation, as well as the output controllability of the system (3.13), (3.10) at the time instant t = tc (see Definition 3.9) and Corollary 3.1, we obtain that ! " det X (tc , 0)M1s X T (tc , 0) = 0. Moreover, since the matrix M1s is at least positive semi-definite, then the matrix X (tc , 0)M1s X T (tc , 0) is positive definite. Multiplication of this matrix from the left and from the right by the nonsingular matrices Y and Y T , respectively, and use of Eq. (3.141) yield the following positive definite matrix:  ! " ,1 X ,T , X ,T M X 1s 1 2 ,2 X   ,T X ,M X ,T ,M X X = ,1 1s ,1T ,1 1s ,2T . X2 M1s X1 X2 M1s X2

Y X (tc , 0)M1s X T (tc , 0)Y T =



Since the block matrix in the right-hand side of this equation is positive definite, ,1 M1s X ,T also is positive definite. Thus, then its left-hand upper block X 1

3.4 Parameter-Free Controllability Conditions for Systems with Small Delays

163

"  ,1 M1s X ,T > 0. det X 1 Moreover, since the system (3.15) for t = tc is completely Euclidean space T (ξ ) is adjoint to the fundamental controllable and the matrix-valued function Ψ4f matrix of this system, then (by virtue of Definition 3.4 and Proposition 3.1) there exists a positive number ξc such that 

ξc

det 0

 T Ψ4f (ξ )S22 (tc , 0)Ψ4f (ξ )dξ

= 0.

Moreover, since S22 (tc , 0) is positive semi-definite, then the matrix  0

ξc

T Ψ4f (ξ )S22 (tc , 0)Ψ4f (ξ )dξ

is positive definite. This observation yields (similarly to (3.134)) that det M3f > 0. Since both determinants in the right-hand side !of (3.145) are positive, then " there 0 T exists a positive number εc,2 ≤ ε such that det L(ε)Y MZ (ε)Y L(ε) > 0 for all ε ∈ (0, εc,2 ]. The latter, along with the nonsingularity of the matrix L(ε) for all ε > 0 and the nonsingularity of the matrix Y , means that det MZ (ε) = 0 for all ε ∈ (0, εc,2 ]. Thus, the theorem is proven.

3.4.3.3

Proof of Theorem 3.3

Since the matrix Z (tc , ε) (see Eq. (3.115)) is continuous in ε ∈ [0, ε0 ], then the condition rankZ (tc , 0) = q yields the following equality for all sufficiently small ε > 0: rankZ (tc , ε) = q.

(3.146)

Consider the (n + m) × (n + m)-matrix ¯ L(ε) =



In 0 √ εIm 0

 .

Multiplying the block representation of the matrix M(ε) (see Eq. (3.119)) from the ¯ left and from the right by L(ε), we obtain ⎛ ¯ ¯ L(ε)M(ε) L(ε) =⎝

M1 (ε)

√

εM2 (ε)

√ T εM2 (ε) εM3 (ε)

⎞ ⎠.

164

3 Euclidean Space Output Controllability of Linear Systems with State Delays

Using this equality and the inequalities (3.131)–(3.132), we obtain  ! " M1s ¯ ¯ L(ε) = det lim det L(ε)M(ε) ε→+0 0

0 M3f

 = det M1s det M3f .

(3.147)

Due to the complete controllability of the system (3.13) at t = tc and the complete Euclidean space controllability of the system (3.15) for t = tc , we directly obtain (similarly to the proof of Theorem 3.2) that both determinants in the right-hand ¯ side of (3.147) are nonzero. This observation and the fact that det L(ε) = 0 for all ε > 0 mean that det M(ε) = 0 for all sufficiently small ε > 0. The latter, along Eqs. (3.117) and (3.146), means the existence of a positive number εc,3 , (εc,3 ≤ ε0 ), such that det MZ (ε) = 0 for all ε ∈ (0, εc,3 ]. Hence, by virtue of Corollary 3.1, the singularly perturbed system (3.1)–(3.2), (3.3) is Euclidean space output controllable at the time instant tc for all ε ∈ (0, εc,3 ].

3.5 Special Cases of Controllability for Systems with Small Delays In this section, we consider several important cases of the controllability for the system (3.1)–(3.2), (3.3).

3.5.1 Complete Euclidean Space Controllability Let q = n + m, and the condition (3.7) be valid. In this case, Theorems 3.3, 3.6, 3.8 directly yield the following theorems. Theorem 3.9 Let the assumptions (AI)–(AIII) be valid. Let the system (3.13) be completely controllable at the time instant tc . Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number εˆ c,1 , (ˆεc,1 ≤ ε0 ), such that for all ε ∈ (0, εˆ c,1 ], the singularly perturbed system (3.1)– (3.2) is completely Euclidean space controllable at the time instant tc . Theorem 3.10 Let the condition (3.12) hold for all t ∈ [0, tc ]. Let the assumptions (AI)–(AII), (AIV) be valid. Let the system (3.13) be completely controllable at the time instant tc . Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number εˆ c,2 , (ˆεc,2 ≤ ε0 ), such that for all ε ∈ (0, εˆ c,2 ], the singularly perturbed system (3.1)–(3.2) is completely Euclidean space controllable at the time instant tc . Theorem 3.11 Let the assumptions (AI)–(AII), (AIV) be valid. Let the system (3.8)– (3.9) be impulse-free controllable with respect to xs (t) at the time instant tc . Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then,

3.5 Special Cases of Controllability for Systems with Small Delays

165

there exists a positive number εˆ c,3 , (ˆεc,3 ≤ ε0 ), such that for all ε ∈ (0, εˆ c,3 ], the singularly perturbed system (3.1)–(3.2) is completely Euclidean space controllable at the time instant tc .

3.5.2 Controllability with Respect to x(t) Let q = n, and the matrices X (t, ε) and Y (t, ε) be X (t, ε) = In ,

Y (t, ε) = 0,

t ∈ [0, tc ], ε ∈ [0, ε0 ].

In this case, the Euclidean space output controllability of the system (3.1)–(3.2), (3.3) is called the controllability with respect to x(t). The following three theorems present different ε-free sufficient conditions for this type of controllability. Theorem 3.12 Let the assumptions (AI)–(AIII) be valid. Let the system (3.13) be completely controllable at the time instant tc . Then, there exists a positive number εˇ x,1 , (ˇεx,1 ≤ ε0 ), such that for all ε ∈ (0, εˇ x,1 ], the singularly perturbed system (3.1)–(3.2), (3.3) is controllable with respect to x(t) at the time instant tc . Proof Similarly to the proof of Theorem 3.1, we should show the existence of a positive number εˇ x,1 such that det MZ (ε) = 0 for all ε ∈ (0, εˇ x,1 ], which will prove the theorem. Remember that the matrix MZ (ε) is defined by (3.117) and then is represented in the form (3.135). Using this representation and the fact that X (tc , ε) = In , Y (tc , ε) = 0, we obtain MZ (ε) = M1 (ε). Note that the system (3.13) is completely controllable at the time instant tc . Then, similarly to the proof of Theorem 3.2, we have det M1s = 0, where M1s is given in (3.130). Therefore, due to the inequality (3.131), there exists a positive number εˇ x,1 such that det M1 (ε) = 0 for all ε ∈ (0, εˇ x,1 ]. The latter, along with the equality MZ (ε) = M1 (ε), proves the theorem.   Theorem 3.13 Let the condition (3.12) hold for all t ∈ [0, tc ]. Let the assumptions (AI)–(AII), (AIV) be valid. Let the system (3.13) be completely controllable at the time instant tc . Then, there exists a positive number εˇ x,2 , (ˇεx,2 ≤ ε0 ), such that for all ε ∈ (0, εˇ x,2 ], the singularly perturbed system (3.1)–(3.2), (3.3) is controllable with respect to x(t) at the time instant tc . Theorem 3.14 Let the assumptions (AI)–(AII), (AIV) be valid. Let the system (3.8)– (3.9) be impulse-free controllable with respect to xs (t) at the time instant tc . Then, there exists a positive number εˇ x,3 , (ˇεx,3 ≤ ε0 ), such that for all ε ∈ (0, εˇ x,3 ], the singularly perturbed system (3.1)–(3.2), (3.3) is controllable with respect to x(t) at the time instant tc . Based on Lemmas 3.2, 3.3 and Theorem 3.12, Theorems 3.13 and 3.14 are proven similarly to Theorems 3.4 and 3.7, respectively.

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

3.5.3 Controllability with Respect to y(t) Let q = m, and the matrices X (t, ε) and Y (t, ε) be X (t, ε) = 0,

Y (t, ε) = Im ,

t ∈ [0, tc ], ε ∈ [0, ε0 ].

In this case, the Euclidean space output controllability of the system (3.1)–(3.2), (3.3) is called the controllability with respect to y(t). The following two theorems present different ε-free sufficient conditions for this type of controllability. Theorem 3.15 Let the assumptions (AI)–(AIII) be valid. Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number εˇ y,1 , (ˇεy,1 ≤ ε0 ), such that for all ε ∈ (0, εˇ y,1 ], the singularly perturbed system (3.1)–(3.2), (3.3) is controllable with respect to y(t) at the time instant tc . Proof Here, similarly to the proof of Theorem 3.12, we should show the existence of a positive number εˇ y,1 such that det MZ (ε) = 0 for all ε ∈ (0, εˇ y,1 ]. Using (3.135) and the fact that X (tc , ε) = 0, Y (tc , ε) = Im , we obtain MZ (ε) = M3 (ε). Since the system (3.15) is completely Euclidean space controllable, then by virtue of (3.134), we have det M3f = 0, where M3f is given in (3.133). Therefore, due to the second inequality in (3.132), there exists a positive number εˇ y,1 such that det M3 (ε) = 0 for all ε ∈ (0, εˇ y,1 ]. The latter, along with the equality MZ (ε) = M3 (ε), proves the theorem.   Theorem 3.16 Let the assumptions (AI)–(AII), (AIV) be valid. Let, for t = tc , the system (3.15) be completely Euclidean space controllable. Then, there exists a positive number εˇ y,2 , (ˇεy,2 ≤ ε0 ), such that for all ε ∈ (0, εˇ y,2 ], the singularly perturbed system (3.1)–(3.2), (3.3) is controllable with respect to y(t) at the time instant tc . Based on Lemmas 3.2, 3.3 and Theorem 3.15, Theorem 3.16 is proven similarly to Theorem 3.4.

3.6 Examples: Systems with Small Delays In this section, we present several examples, illustrating the above obtained theoretic results for the systems with the small delays.

3.6 Examples: Systems with Small Delays

167

3.6.1 Example 1 Consider the following system, a particular case of (3.1)–(3.2), (3.3): dx(t) = 4(t − 1)x(t) + 4(1 + εt)y(t) − 2tx(t − ε) + y(t − ε) dt  0

2ηx(t + εη) − y(t + εη) dη + (1 − ε2 t)u(t), t ≥ 0, − −1

dy(t) = −4x(t) + y(t) + 2x(t − ε) − 2y(t − ε) dt

x(t + εη) + 3(η + ε)2 y(t + εη) dη + (t + 1)u(t), t ≥ 0, ε

 +

0 −1

ζ (t) = (2 − εt 2 )x(t) + (t − 3)y(t),

t ≥ 0,

(3.148)

where x(t), y(t), u(t), and ζ (t) are scalars, i.e., n = m = r = q = 1. In this example, we study the Euclidean space output controllability of the system (3.148) at the time instant tc = 2 for all sufficiently small ε > 0. The asymptotic decomposition of the system (3.148) in the time interval [0, 2] yields the slow and fast subsystems, respectively, dxs (t) = (2t − 3)xs (t) + 6ys (t) + us (t), dt

t ∈ [0, 2],

0 = −xs (t) + (t + 1)us (t),

t ∈ [0, 2],

ζs (t) = 2xs (t),

t ∈ [0, 2],

(3.149)

and dyf (ξ ) = yf (ξ ) − 2yf (ξ − 1) + dξ



0 −1

3η2 yf ((ξ + η)dη + (t + 1)uf (ξ ),

ξ ≥ 0,

ζf (ξ ) = (t − 3)yf (ξ ),

ξ ≥ 0. (3.150)

In the system (3.150), t ∈ [0, 2] is a parameter. It is seen that the assumptions (AI)–(AII) are satisfied for the system (3.148) with any given ε0 . The condition (3.12) is not satisfied for (3.148), i.e., this system is nonstandard, and it does not satisfy the assumption (AIII). Let us show the fulfilment of the assumption (AIV) for the system (3.148). Indeed, since m = 1 and the coefficient for the control uf (ξ ) in the fast subsystem (3.150) differs from zero for all t ∈ [0, 2], then this assumption is valid. Moreover, since q = m = 1 and, for t = 2, the coefficient t − 3 in the output equation of the fast subsystem (3.150) differs from zero, then this subsystem

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

for t = 2 is Euclidean space output controllable. Thus, due to Theorem 3.4 and Remark 3.9, the system (3.148) is Euclidean space output controllable at the time instant t = 2 for all sufficiently small ε > 0. Let us show that Theorem 3.8 also is applicable to the analysis of the Euclidean space output controllability of the system (3.148). For this purpose, we have to show that: (i) the differential-algebraic system in the slow subsystem (3.149) is impulsefree controllable with respect to xs (t) at the time instant tc = 2; (ii) the differential equation in the fast subsystem (3.150) for t = tc = 2 is completely Euclidean space controllable; (iii)   the rank of the matrix Z (2, 0) equals q = 1, where Z (t, ε) = 2 − εt 2 , t − 3 is the 1 × 2-matrix of the coefficients in the output equation of the system (3.148). We start with the item (i). Let us make the following invertible transformation of the control in the differential-algebraic system of (3.149): us (t) =

 1  ys (t) + ws (t) , t +1

t ∈ [0, 2],

where ws (t) is a new control. Due to this transformation, the differential-algebraic system becomes as:   dxs (t) 1 1 = (2t − 3)xs (t) + 6 + ys (t) + ws (t), t ∈ [0, 2], dt t +1 t +1 0 = −xs (t) + ys (t) + ws (t),

t ∈ [0, 2].

Eliminating the state variable ys (t) from this system, we obtain the differential equation   1 dxs (t) = 2t + 3 + xs (t) − 6ws (t), dt t +1

t ∈ [0, 2].

Note that the coefficient for the control ws (t) in the latter is nonzero. Therefore, due to Proposition 3.1, this differential equation is completely controllable at the time instant tc = 2. Thus, by virtue of Corollaries 3.3 and 3.4, the differential-algebraic system in the slow subsystem (3.149) is impulse-free controllable with respect to xs (t) at this time instant. Proceed to the item (ii). Since the coefficient for the control uf (ξ ) in the differential equation of the fast subsystem (3.150) differs from zero for t = 2, then, by virtue of Proposition 3.1, this differential equation for t = tc = 2 is completely Euclidean space controllable.  Finally, let us deal with the item (iii). We have that Z (2, 0) = 2, −1). Therefore, rankZ (2, 0) = 1 = q. Consequently, all the condition of Theorem 3.8 are satisfied for the system (3.148), implying its Euclidean space output controllability for all sufficiently small ε > 0. Thus, in this example two theoretical results (Theorem 3.4, along with Remark 3.9, and Theorem 3.8) are applicable to the analysis of the Euclidean space output controllability of the system (3.148).

3.6 Examples: Systems with Small Delays

169

3.6.2 Example 2 Consider the following system, a particular case of (3.1)–(3.2), (3.3): dx(t) = 4(t − 1)x(t) − 2y(t) − 2tx(t − ε) + y(t − ε) dt  0

− 2ηx(t + εη) − y(t + εη) dη + (t + 1)u(t), t ≥ 0, −1

dy(t) = −4x(t) + y(t) + 2x(t − ε) − 2y(t − ε) dt

x(t + εη) + 3η2 y(t + εη) dη + (2t − 1)u(t), t ≥ 0, ε

 +

0 −1

ζ (t) = (1 − ε)x(t) + (t + ε)y(t),

t ≥ 0,

(3.151)

where x(t), y(t), u(t), and ζ (t) are scalars, i.e., n = m = r = q = 1. In this example, we study the Euclidean space output controllability of the system (3.151) at the time instant tc = 1 for all sufficiently small ε > 0. The asymptotic decomposition of the system (3.151) in the time interval [0, 1] yields the slow and fast subsystems, respectively, dxs (t) = (2t − 3)xs (t) + (t + 1)us (t), dt

t ∈ [0, 1],

0 = −xs (t) + (2t − 1)us (t),

t ∈ [0, 1],

ζs (t) = xs (t),

t ∈ [0, 1],

(3.152)

and dyf (ξ ) = yf (ξ ) − 2yf (ξ − 1) + dξ



0 −1

3η2 yf (ξ + η)dη + (2t − 1)uf (ξ ),

ξ ≥ 0,

ζf (ξ ) = tyf (ξ ),

ξ ≥ 0, (3.153)

In the system (3.153), t ∈ [0, 1] is a parameter. Similarly to the previous example, one can see that the assumptions (AI)– (AII) are satisfied for the system (3.151) with any prechosen ε0 . However, the condition (3.12) is not satisfied for (3.151), i.e., like in the previous example, the original singularly perturbed system is nonstandard and it does not satisfy the assumption (AIII). This occurs because λ = 0 is a root of Eq. (3.113) in this example, which has the form  λ − 1 + 2 exp(−λ) −

0 −1

3η2 exp(λη)dη = 0.

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

Note that, similarly to the previous example, one can show the fulfilment of the assumption (AIV) for the system (3.151). Moreover, since q = m = 1 and the coefficient t in the output equation of the fast subsystem (3.153) differs from zero for t = 1, then this subsystem for t = 1 is Euclidean space output controllable. Thus, all the conditions of Theorem 3.4 are fulfilled for the system (3.151). Therefore, by virtue of this theorem and Remark 3.9, the system (3.151) is Euclidean space output controllable at the time instant tc = 1 for all sufficiently small ε > 0. Remark 3.10 Note that, in contrast with the previous example, Theorem 3.8 is not applicable to the analysis of the Euclidean space output controllability of the system (3.151) at tc = 1. This occurs because the differential-algebraic system in the slow subsystem (3.152) is not impulse-free controllable with respect to xs (t) at tc = 1. Indeed, there is not a control function us (t) ∈ L2 [0, 1; E 1 ] transferring this differential-algebraic system from the initial position xs (0) = 0 to any given terminal position xs (1) = x1 = 0.

3.6.3 Example 3 Consider the following particular case of the system (3.1)–(3.2), (3.3): dx1 (t) = tx1 (t) + x2 (t) − y(t) + (1 − t)x1 (t − ε) dt −2x2 (t − ε) + y(t − ε) + u1 (t),

t ≥ 0,

dx2 (t) = x1 (t) − 2tx2 (t) + 2y(t) − x1 (t − ε) dt +(t + 2)x2 (t − ε) − y(t − ε) − 2u2 (t), ε

t ≥ 0,

dy(t) = 2x1 (t) + x2 (t) − y(t) − x1 (t − ε) dt

+x2 (t − ε) + y(t − ε) + u1 (t) + u2 (t),

t ≥ 0,

ζ1 (t) = exp(−t)x1 (t) + t 2 x2 (t) + 4y(t),

t ≥ 0,

ζ2 (t) = −x1 (t) + tx2 (t) − y(t),

t ≥ 0,

(3.154)

where x1 (t), x2 (t), y(t), u1 (t), u2 (t), ζ1 (t), and ζ2 (t) are scalars, i.e., n = 2, m = 1, r = 2, q = 2. We study the Euclidean space output controllability of the system (3.154) at the time instant tc = 1. The slow subsystem, associated with (3.154), is dxs1 (t) = xs1 (t) − xs2 (t) + us1 (t), dt

t ∈ [0, 1],

3.6 Examples: Systems with Small Delays

171

dxs2 (t) = (2 − t)xs2 (t) + ys (t) − 2us2 (t), dt

t ∈ [0, 1],

0 = xs1 (t) + 2xs2 (t) + us1 (t) + us2 (t),

t ∈ [0, 1],

ζs1 (t) = exp(−t)xs1 (t) + t 2 xs2 (t),

t ∈ [0, 1],

ζs2 (t) = −xs1 (t) + txs2 (t),

t ∈ [0, 1],

(3.155)

(3.156)

while the fast subsystem, associated with the original system, is dyf (ξ ) = −yf (ξ ) + yf (ξ − 1) + uf 1 (ξ ) + uf 2 (ξ ), dξ

ξ ≥ 0,

ζf 1 (ξ ) = 4yf (ξ ),

ξ ≥ 0,

ζf 2 (ξ ) = −yf (ξ ),

ξ ≥ 0.

(3.157)

The assumptions (AI)–(AII) are satisfied for the system (3.154), while the condition (3.12) is not satisfied for this system. Thus, (3.154) is nonstandard, and it does not satisfy the assumption (AIII). However, the assumption (AIV) is fulfilled for this system by the same arguments as in the example of Sect. 3.6.1. Moreover, since r > m and the coefficients for the controls uf 1 (ξ ) and uf 2 (ξ ) in the differential equation of the fast subsystem (3.157) differ from zero, then this equation is completely Euclidean space controllable. Now, let us show that the slow subsystem (3.155)–(3.156) is impulse-free output controllable at the time instant tc = 1. For this purpose, we make in the differential-algebraic system (3.155) the control transformation usl (t) = −ys (t) + wsl ,

l = 1, 2,

t ∈ [0, 1],

where ws1 (t) and ws2 (t) are new controls. This transformation and the elimination of ys (t) from the resulting system yield the following differential system with respect to xs1 (t) and xs2 (t): dxs1 (t) 1 1 1 = xs1 (t) − 2xs2 (t) + ws1 (t) − ws2 (t), dt 2 2 2 3 dxs2 (t) 3 1 = xs1 (t) + (5 − t)xs2 (t) + ws1 (t) − ws2 (t), dt 2 2 2

t ∈ [0, 1], t ∈ [0, 1]. (3.158)

Due to Lemma 3.4, the output controllability of the system (3.158), (3.156) at tc = 1 yields the impulse-free output controllability of the slow subsystem (3.155)–(3.156) 2 2 at tc = 1. Let us show that for  any2given x02∈ E and ζc ∈ E there exists a control ws (t) = col ws1 (t), ws2 (t) ∈ L [0, 1; E ], such that the system (3.158), (3.156)

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

  has a solution col  xs1 (t), xs2  (t) , satisfying  the initial and terminal conditions col xs1 (0), xs2 (0) = x0 , col ζs1 (1), ζs2 (1) = ζc . Consider the matrices of the coefficients for the states and the controls in (3.158), and for the states in (3.156) ⎛1 As (t) = ⎝

2

−2

3 2

5−t

⎛1

⎞

⎠ , Bs = ⎝

2

−

1 2

3 2

−

1 2

⎞

⎛

⎠ , Xs (t) = ⎝

exp(−t) t 2 −1

⎞ ⎠.

t

Let Φs (t, σ ) be the 2 × 2-matrix-valued function, satisfying for any σ ∈ [0, 1] the initial-value problem dΦs (t, σ ) = As (t)Φs (t, σ ), t ∈ (σ, 1], dt Φs (σ, σ ) = I2 . Since the matrices Bs and Xs (1) are nonsingular, then the matrix 

1

Ws = Xs (1) 0

Φs (1, σ )Bs BsT ΦsT (1, σ )dσ XsT (1)

is nonsingular. Thus, using the proof of Lemma 3.1 (see Eq. (3.63) in Sect. 3.3.4), we obtain that the continuous, and therefore belonging to L2 [0, 1; E 2 ], control ! " ws (t) = BsT (t)Φs (1, t)XsT (1)Ws−1 ζc − Xs (1)Φs (1, 0)x0 , t ∈ [0, 1] transfers the system (3.158), (3.156) from the initial state position x0 at t = 0 to the terminal output position ζc at t = 1. Hence, this system is output controllable at tc = 1, and therefore, the slow subsystem (3.155)–(3.156) is impulse-free output controllable at tc = 1. Also, note that in this example the matrix of the coefficients for y(t) in the output equations of the original system (3.154) has the form  Y =

 4 . −1

The rank of this matrix equals to m = 1. Hence, all the conditions of Theorem 3.7 are fulfilled for the system (3.154), implying its Euclidean space output controllability at tc = 1 for all sufficiently small ε > 0. Remark 3.11 Note that in this example, not only the above introduced matrix Ws is nonsingular but the matrix

3.6 Examples: Systems with Small Delays

, W s =



1 0

173

Φs (1, σ )Bs BsT ΦsT (1, σ )dσ

also is nonsingular. The latter means that the system (3.158) is completely consystem trollable at tc = 1, and therefore, the differential-algebraic   (3.155) is impulse-free controllable with respect to xs (t) = col xs1 (t), xs2 (t) at tc = 1. Moreover, the matrix of the coefficients in the output equation of the original system (3.154) has the form  Z (t) =

 exp(−t) t 2 4 . −1 t −1

The rank of this matrix at t = tc = 1 equals to q = 2. Therefore, along with Theorem 3.7, Theorem 3.8 also is applicable to the analysis of the Euclidean space output controllability of the system (3.154) at tc = 1. In the next section, a more complicated example, excluding such a situation, is considered.

3.6.4 Example 4 Consider the following particular case of the system (3.1)–(3.2), (3.3): dx1 (t) = tx1 (t) + x2 (t) − 2x3 (t) + (1 − t)x1 (t − ε) dt −x2 (t − ε) + 2x3 (t − ε) + 2y(t − ε) + u1 (t) − εu2 (t),

t ≥ 0,

dx2 (t) = x1 (t) − 2tx2 (t) + x3 (t) + 2y(t) − x1 (t − ε) dt +2(t + 2)x2 (t − ε) − x3 (t − ε) − y(t − ε) − εu1 (t) − 2u2 (t),

t ≥ 0,

dx3 (t) = −x1 (t) + x2 (t) − 2x3 (t) + 2y(t) + x1 (t − ε) dt −x2 (t − ε) + x3 (t − ε) − y(t − ε) − u1 (t) − u2 (t), ε

t ≥ 0,

dy(t) = x1 (t) − x2 (t) − 2x3 (t) − 4y(t) − x1 (t − ε) dt

+x2 (t − ε) + 2x3 (t − ε) + y(t − ε) + 3u1 (t) + 3u2 (t),

t ≥ 0,

ζ1 (t) = exp(−t + ε)x1 (t) + t 2 x2 (t) − x3 (t) + 4(t − 1)y(t),

t ≥ 0,

ζ2 (t) = −x1 (t) + tx2 (t) + 2x3 (t) − (t − 2)y(t),

t ≥ 0, (3.159)

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

where x1 (t), x2 (t), x3 (t), y(t), u1 (t), u2 (t), ζ1 (t), and ζ2 (t) are scalars, i.e., n = 3, m = 1, r = 2, q = 2. We study the Euclidean space output controllability of the system (3.159) at the time instant tc = 1 for all sufficiently small ε > 0. For this system, the coefficient A4s (t) (see Eq. (3.11)) is A4s (t) ≡ −3 = 0. Therefore, the system (3.159) is standard. The slow subsystem, associated with (3.159), is a particular case of (3.13), (3.10), and it has the form dxs1 (t) = xs1 (t) + 3us1 (t) + 2us2 (t), dt dxs2 (t) = 4xs2 (t) + us1 (t) − us2 (t), dt dxs3 (t) = −xs3 (t), dt

t ∈ [0, 1], t ∈ [0, 1], t ∈ [0, 1],

(3.160)

ζs1 (t) = exp(−t)xs1 (t) + t 2 xs2 (t) − xs3 (t),

t ∈ [0, 1],

ζs2 (t) = −xs1 (t) + txs2 (t) + 2xs3 (t),

t ∈ [0, 1].

(3.161)

The fast subsystem, associated with (3.159), is dyf (ξ ) = −4yf (ξ ) + yf (ξ − 1) + 3uf 1 (ξ ) + 3uf 2 (ξ ), dξ

ξ ≥ 0,

ζf 1 (ξ ) = 4(t − 1)yf (ξ ),

ξ ≥ 0,

ζf 2 (ξ ) = −(t − 2)yf (ξ ),

ξ ≥ 0.

(3.162)

In the system (3.162), t ∈ [0, 1] is a parameter. The assumptions (AI)–(AII) are satisfied for the system (3.159) with any given ε0 > 0. Let us show the fulfilment of the assumption (AIII). In this example, Eq. (3.113) becomes λ + 4 − exp(−λ) = 0.

(3.163)

For all complex numbers λ with Reλ ≥ −1, we have the inequality   Re λ + 4 − exp(−λ) > 0.28, meaning that such λ are not roots of Eq. (3.163). Therefore, all roots of (3.163) satisfy the inequality Reλ < −1. Thus, the assumption (AIII) is satisfied for β = 0.5.

3.6 Examples: Systems with Small Delays

175

By the same arguments, as in the previous example (see Sect. 3.6.3), the differential equation in the fast subsystem (3.162) is completely Euclidean space controllable. Now, let us show that the slow subsystem (3.160)–(3.161) is output controllable at the time instant tc = 1. Consider the matrices of coefficients for the states and the controls in (3.160), and for the states in (3.161) ⎛

⎞ ⎛ ⎞   1 0 0 3 2 exp(−t) t 2 − 1 ⎝ ⎠ ⎝ ⎠ As (t) = 0 4 . 0 , Bs = 1 − 1 , Xs (t) = −1 t 2 0 0 −1 0 0 Let Φs (t, σ ) be the 3 × 3-matrix-valued function, satisfying for any σ ∈ [0, 1] the initial-value problem dΦs (t, σ ) = As (t)Φs (t, σ ), t ∈ (σ, 1], dt Φs (σ, σ ) = I3 . Then, !    " Φs (t, σ ) = diag exp(t − σ ), exp 4(t − σ ) , exp − (t − σ ) . Thus, we have s = W



1 0

Φs (1, σ )Bs BsT ΦsT (1, σ )dσ

   ⎞ 6.5 exp(2) − 1 0.2 exp(5) − 1) 0    = ⎝ 0.2 exp(5) − 1) 0.25 exp(8) − 1 0 ⎠ . 0 0 0 ⎛

s is singular, and its rank equals 2. However, the matrix The matrix W s XsT (1) Ws = Xs (1)W is nonsingular. Therefore, due to Lemma 3.1, the slow subsystem (3.160)–(3.161) is output controllable at tc = 1. Also, note that in this example the matrix of coefficients for y(t) in the output equations of the original system (3.159) has the form   4(t − 1) Y (t) = . −(t − 2)

176

3 Euclidean Space Output Controllability of Linear Systems with State Delays

Thus, for t = tc = 1, the rank of this matrix equals to m = 1. Hence, all the conditions of Theorem 3.2 are fulfilled for the system (3.159). Therefore, this system is Euclidean space output controllable at tc = 1 for all sufficiently small ε > 0. s is singular, then the system (3.160) is not Remark 3.12 Since the matrix W completely controllable at t = tc = 1. This means that Theorem 3.3, which is analogous to Theorem 3.8 in the case of the standard original system, cannot be applied to the analysis of the Euclidean space output controllability of the system (3.159) at tc = 1.

3.6.5 Example 5 Consider the following particular case of (3.1)–(3.2), (3.3): dx1 (t) = (t + 1)x1 (t) + x2 (t) + y1 (t) − y2 (t) − 2x2 (t − ε) dt  0 +y1 (t − ε) + 2y2 (t − ε) − tx1 (t + εη)dη + tu1 (t) − u2 (t), t ≥ 0, −1

dx2 (t) = −x1 (t) − y1 (t) + 2y2 (t) − (t − 1)x2 (t − ε) + 2y1 (t − ε) − y2 (t − ε) dt  0  0 + x1 (t + εη)dη + 2tx2 (t + εη)dη − 2u1 (t) + (t − 1)u2 (t), t ≥ 0, −1

−1

ε

dy1 (t) = 2x1 (t) + x2 (t) − y1 (t) + y2 (t) − x1 (t − ε) dt

+x2 (t − ε) + 2y1 (t − ε) − 2y2 (t − ε) + 3u1 (t) − 2u2 (t), t ≥ 0, ε

dy2 (t) = x1 (t) + 2x2 (t) − 2y1 (t) + 2y2 (t) + 2x1 (t − ε) dt

−x2 (t − ε) + y1 (t − ε) − y2 (t − ε) − 2u1 (t) + 3u2 (t), t ≥ 0, ζ1 (t) = (t − 1)x1 (t) + (t + 1)x2 (t) + ty1 (t) − y2 (t), t ≥ 0, ζ2 (t) = −x1 (t) + tx2 (t) − y1 (t) + 2y2 (t), t ≥ 0, ζ3 (t) = x1 (t) + 2x2 (t) + exp(−t)y1 (t) + exp(−2t)y2 (t), t ≥ 0, (3.164) where x1 (t), x2 (t), y1 (t), y2 (t), u1 (t), u2 (t), ζ1 (t), ζ2 (t), and ζ3 (t) are scalars, i.e., n = 2, m = 2, r = 2, q = 3. We study the Euclidean space output controllability of the system (3.164) at the time instant tc = 2 for all sufficiently small ε > 0.

3.6 Examples: Systems with Small Delays

177

The slow and fast subsystems, associated with (3.164), are dxs1 (t) = xs1 (t) − xs2 (t) + 2ys1 (t) + ys2 (t) + tus1 (t) − us2 (t), dt dxs2 (t) = (t + 1)xs2 (t) + ys1 (t) + ys2 (t) − 2us1 (t) + (t − 1)us2 (t), dt

t ∈ [0, 2],

0 = xs1 (t) + 2xs2 (t) + ys1 (t) − ys2 (t) + 3us1 (t) − 2us2 (t),

t ∈ [0, 2],

0 = 3xs1 (t) + xs2 (t) − ys1 (t) + ys2 (t) − 2us1 (t) + 3us2 (t),

t ∈ [0, 2], (3.165)

ζs1 (t) = (t − 1)xs1 (t) + (t + 1)xs2 (t),

t ∈ [0, 2],

ζs2 (t) = −xs1 (t) + txs2 (t),

t ∈ [0, 2],

ζs3 (t) = xs1 (t) + 2xs2 (t),

t ∈ [0, 2],

t ∈ [0, 2],

(3.166)

and dyf 1 (ξ ) = −yf 1 (ξ ) + yf 2 (ξ ) + 2yf 1 (ξ − 1) − 2yf 2 (ξ − 1) dξ +3uf 1 (ξ ) − 2uf 2 (ξ ),

ξ ≥ 0,

dyf 2 (ξ ) = −2yf 1 (ξ ) + 2yf 2 (ξ ) + yf 1 (ξ − 1) − yf 2 (ξ − 1) dξ −2uf 1 (ξ ) + 3uf 2 (ξ ),

ξ ≥ 0,

ζf 1 (ξ ) = tyf 1 (ξ ) − yf 2 (ξ ),

ξ ≥ 0,

ζf 2 (ξ ) = −yf 1 (ξ ) + 2yf 2 (ξ ),

ξ ≥ 0,

ζf 3 (ξ ) = exp(−t)yf 1 (ξ ) + exp(−2t)yf 2 (ξ ),

ξ ≥ 0.

(3.167)

(3.168)

In the system (3.167), t ∈ [0, 2] is a parameter. The assumptions (AI)–(AII) are satisfied for the system (3.164), while the condition (3.12) is not satisfied for this system. Thus, (3.164) is nonstandard, and it does not satisfy the assumption (AIII). Let us show that this system satisfies the assumption (AIV). The matrix of coefficients for the controls u1 (t) and u2 (t) in the fast mode differential equations of (3.164) has the form  B2 =

 3 −2 . −2 3

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

The rank of this matrix equals 2. Therefore, the assumption (AIV) is fulfilled for the system (3.164). Moreover, since rankB2 = 2, then the differential system in (3.167) is completely Euclidean space controllable. Now, we proceed to the differentialalgebraic system in (3.165).  Our goal isto prove its impulse-free controllability with respect to xs (t) = col xs1 (t), xs2 (t) at tc = 2. Let us transform the controls in this system as usl (t) = −(1/2)ysl (t) + wsl (t),

l = 1, 2,

where ws1 (t) and ws2 (t) are new controls. This transformation and the elimination of ys1 (t) and ys2 (t) from the resulting system yield the following differential system with respect to xs1 (t) and xs2 (t): dxs1 (t) =(14 − t)xs1 (t)+(10 − 2t)xs2 (t)+(6 − 2t)ws1 (t)+2tws2 (t), t ∈ [0, 2], dt dxs2 (t) =(13 − 3t)xs1 (t)+12xs2 (t)+(4 + 2t)ws1 (t)−2tws2 (t), t ∈ [0, 2]. dt (3.169) Similarly to the example in Sect. 3.6.3, the complete controllability of this system at tc = 2 yields the impulse-free controllability with respect to xs (t) = col xs1 (t), xs2 (t) of the differential-algebraic system in (3.165) at tc = 2. Let us show that for any x0 ∈ E 2 and xc ∈ E 2 there exists a control ws (t) =  given 2 col ws1 (t),ws2 (t) ∈ L[0, 2; E 2 ], such that the system (3.169) has a solution xs (t) = col xs1 (t), xs2 (t) , satisfying the initial and terminal conditions xs (0) = x0 , xs (2) = xc . Consider the matrices of the coefficients for the states and the controls in (3.169)  As (t) =

14 − t 10 − 2t 13 − 3t 12



 ,

Bs (t) =

6 − 2t 4 + 2t

2t − 2t

 .

Let Φs (t, σ ) be the 2 × 2-matrix-valued function, satisfying for any σ ∈ [0, 2] the initial-value problem dΦs (t, σ ) = As (t)Φs (t, σ ), t ∈ (σ, 2] dt Φs (σ, σ ) = I2 . Note that the matrix Bs (t) is nonsingular for t ∈ (0, 2], while this matrix is singular for t = 0. Nevertheless, the matrix 

2

Ws = 0

Φs (2, σ )Bs (σ )BsT (σ )ΦsT (2, σ )dσ

3.6 Examples: Systems with Small Delays

179

is nonsingular. The latter yields (similarly to Sect. 3.6.3) the continuous control ! " ws (t) = BsT (t)Φs (2, t)Ws−1 xc − Φs (2, 0)x0 , t ∈ [0, 2] transferring the system (3.169) from the initial position x0 at t = 0 to the terminal position xc at t = 2. Hence, this system is completely controllable at tc = 2, and therefore, the system (3.165) is impulse-free controllable with respect to xs (t) =   col xs1 (t), xs2 (t) at tc = 2. Also, note that in this example the matrix Z (t) of the coefficients for col x(t), y(t) in the output equations of the original system (3.164), calculated at t = tc = 2, has the form ⎛

⎞ 1 3 2 −1 ⎠, Z (2) = ⎝ −1 2 − 1 2 1 2 exp(−2) exp(−4) and rankZ (2) = q = 3. Hence, all the conditions of Theorem 3.8 are fulfilled for the system (3.164), implying its Euclidean space output controllability at tc = 2 for all sufficiently small ε > 0.

3.6.6 Example 6: Pursuit-Evasion Engagement with Constant Speeds of Participants In this subsection, we consider a mathematical model of an engagement between two flying vehicles, a pursuer and an evader. In a vicinity of the collision course a nonlinear three-dimensional motion of the vehicles can be linearized and decoupled into two planar motions in perpendicular planes (see, e.g., [20, 26]). Therefore, in what follows, we consider the linear planar model of the engagement. In the linear model, the duration of the engagement is td =

R0 , Vc

where R0 is a known initial range of the vehicles and Vc is the closing speed of the engagement. For instance, in the head-on scenario of the engagement Vc = Vp +Ve , where Vp and Ve are constant magnitudes of the velocity vectors of the pursuer and the evader, respectively, (see, e.g., [28]). The state vector of this engagement is ⎛

⎞ ⎛ ⎞ x1 x1 (t) ⎜ x2 ⎟ ⎜ x2 (t) ⎟ ⎜ ⎟=⎜ ⎟ ⎝ x3 ⎠ ⎝ x3 (t) ⎠ , x4 x4 (t)

t ∈ [0, td ],

180

3 Euclidean Space Output Controllability of Linear Systems with State Delays

where x1 = x1 (t) is the relative separation of the vehicles, normal to the initial line of sight; x2 = x2 (t) is the relative normal velocity of the vehicles; x3 = x3 (t) and x4 = x4 (t) are the lateral accelerations of the evader and the pursuer, respectively, both normal to the initial line of sight. Thus, assuming the first-order dynamics of the pursuer and the evader, the set of linear differential equations, describing the engagement, has the following form (see, e.g., [26]): dx1 = x2 , dt dx2 = x3 − x4 , dt dx3 ue − x3 = , dt τe up − x4 dx4 = , dt τp

(3.170)

where τe , τp are the respective time constants; ue and up are controls of the evader and the pursuer, respectively. The objective of the pursuer is, starting from a given initial position of the engagement at t = 0, to approach the evader as close as possible, i.e., to minimize |x1 (td )|. The evader tries to avoid such an approach, i.e., its objective is to maximize |x1 (td )|. In the literature, four types of the pursuer’s state-feedback control are proposed: (I) the bang-bang control with a linear switch function (see, e.g., [26] and the references therein); (II) the saturated linear control (see [28]); (III) the linear control (see, e.g., [15, 29, 30] and the references therein); (IV) the sliding mode control (see [17]). In this example, we analyze the engagement with the linear pursuer’s control. This type of the control has some advantages in comparison with the others. Namely, it requires less control expenditure than the other controls and it is chattering-free. Thus, we choose up in the form 4    up = up x(t), t = kl (t)xl (t),

(3.171)

l=1

  where x(t) = col x1 (t), x2 (t), x3 (t), x4 (t) is the current value of the state vector; kl (t), (l = 1, 2, 3, 4) are known gain coefficients, sufficiently smooth in the interval [0, td ]. For the implementation of the control (3.171), the pursuer has to know the current values of all the coordinate of the state vector. Unfortunately, a realistic scenario is of imperfect information (see, e.g., [13, 14, 16]). In a real-life situation, the relative separation x1 (t) is obtained from noise corrupted measurements of the line-of-sight angle and the range, which are needed in a filtering. The relative velocity x2 (t) and

3.6 Examples: Systems with Small Delays

181

the evader’s lateral acceleration x3 (t) cannot be directly measured and have to be reconstructed by an estimator. The dynamics of the filtering and the estimation are not ideal and involve certain time delays. The delay in the estimated acceleration is larger than in the estimated relative velocity, and the latter is larger than the delay in the filtered relative position. However, these delays are small in comparison with the engagement duration (see, e.g., [25]). Therefore, the actual control of the pursuer is 3   up = up x(t, ˜ ε), t) = kl (t)xl (t − εhl ) + k4 (t)x4 (t),

(3.172)

l=1

  where x(t, ˜ ε) = col x1 (t − εh1 ), x2 (t − εh2 ), x3 (t − εh3 ), x4 (t) ; ε > 0 is a small parameter; 0 < h1 < h2 < h3 are known constants independent of ε. In (3.172) the state coordinate x4 is undelayed because of the assumption that the pursuer can measure perfectly its own lateral acceleration. If the evader knows that the pursuer gets the imperfect information on the current state of the engagement, it can use this circumstance to increase |x1 (td )| considerably and thus to win the engagement. The controllability of the system (3.170), (3.172) with respect to x1 at the time instant td by the evader’s control ue is a sufficient condition for such a result of the engagement. We analyze this controllability subject to the assumption that the evader is much more agile than the pursuer, i.e., τe /τp 0. It should be noted that the verification of the complete controllability of the slow subsystem (3.175) is a much simpler task than the direct verification of the Euclidean space output controllability of the original system (3.173)–(3.174). In order to check up the complete controllability of the slow subsystem (3.175), we use the results of [27]. For this purpose, let us consider the matrices of the coefficients for the state variables and for the control in this subsystem ⎛ ⎛ ⎞ ⎞ 0 1 0 0 ⎠ , Bs (t) = ⎝ 1 ⎠ , t ∈ [0, td ]. As (t) = ⎝ 0 0 −1 k1 (t) k2 (t) k3 (t) k4 (t) − 1 Due to the results of [27], we construct the controllability matrix ! " C (t) = D0 (t), D1 (t), D2 (t) ,

t ∈ [0, td ],

where D0 (t) = Bs (t),

Dp+1 (t) = −As (t)Dp (t) +

dDp (t) , dt

p = 0, 1.

Then, the slow subsystem (3.175) is completely controllable at the time instant td if and only if there exists a value t¯ ∈ [0, td ], such that rankC (t¯) = 3. Calculating the vector-valued functions D1 (t) and D2 (t), we obtain ⎛

⎞ −1 ⎠, D1 (t) = ⎝ k3 (t)   −k2 (t) − k4 (t) − 1 k3 (t) + dk3 (t)/dt

t ∈ [0, td ],

3.6 Examples: Systems with Small Delays

183

and the coordinates D21 (t), D22 (t), D23 (t) of D2 (t) have the form   D22 (t)=−k2 (t)− k4 (t)−1 k3 (t)+2dk3 (t)/dt,    2 D23 (t)=k1 (t)−k2 (t)k3 (t)+ k4 (t)−1 k2 (t)+ k4 (t)−1 k3 (t)

D21 (t)= − k3 (t),

  −2 k4 (t)−1 dk3 (t)/dt−dk2 (t)/dt−k3 (t)dk4 (t)/dt+d 2 k3 (t)/dt 2 ,

t ∈ [0, td ].

Using the vector-valued functions D0 (t), D1 (t), and D2 (t), we obtain    2 det C (t) = k1 (t) + k2 (t) k3 (t) + k4 (t) − 1 + k3 (t) k3 (t) + k4 (t) − 1   −dk2 (t)/dt − 3k3 (t) + 2k4 (t) − 2 dk3 (t)/dt −k3 (t)dk4 (t)/dt + d 2 k3 (t)/dt 2 ,

t ∈ [0, td ].

Thus, rankC (t¯) = 3, and therefore the slow subsystem is completely controllable at the time instant td if and only if det C (t¯) = 0. The latter means that the singularly perturbed system (3.173)–(3.174) is Euclidean space output controllable at the time instant td for all sufficiently small ε > 0 if there exists a value t¯ ∈ [0, td ], for which det C (t¯) = 0. If all the gain coefficients kl , (l = 1, 2, 3, 4) are constant, then the singularly perturbed system (3.173)–(3.174) is Euclidean space output controllable at the time instant td for all sufficiently small ε > 0 if k1 + k2 (k3 + k4 − 1) + k3 (k3 + k4 − 1)2 = 0.

3.6.7 Example 7: Pursuit-Evasion Engagement with Variable Speeds of Participants In this example, we analyze a more general model of the engagement between two flying vehicles (the pursuer and the evader) than the one studied in the previous example. Namely, we consider the case where the magnitudes of the velocities vectors of the pursuer and the evader are known sufficiently smooth, positive functions of time, i.e., Vp = Vp (t) > 0, Ve = Ve (t) > 0, t ≥ 0. In this case, the kinematics of the planar engagement is similar to the one of the previous example (for details, see Sect. 3.6.6). However, in contrast with the previous example, the linear model in the present example has the state vector of six coordinates. This vector contains the first four coordinates similar to those in Example 6. Namely, x1 is the relative separation of the vehicles normal to the initial line of sight; x2 is the relative normal velocity; x3 and x4 are the lateral accelerations of the evader and the pursuer, respectively, both normal to the initial line of sight. Along with these coordinates, two more coordinates appear in the model of this example. These coordinates are x5 = ϕe and x6 = ϕp , the respective angles between the velocity

184

3 Euclidean Space Output Controllability of Linear Systems with State Delays

vectors of the vehicles (the pursuer and the evader) and the initial line of sight. Thus, due to [24], the linear planar model of the engagement has the form dx1 = x2 , dt     dx2 = x3 − x4 + dVe (t)/dt x5 − dVp (t)/dt x6 , dt dx3 ue − x3 = , dt τe up − x4 dx4 = , dt τp dx5 x3 = , dt Ve (t) dx6 x4 = , dt Vp (t)

(3.177)

where t ∈ [0, td ]; τe and τp are the time constants of the evader and the pursuer, respectively; ue and up are controls of the evader and the pursuer, respectively. In this model, it is assumed that the pursuer and the evader know perfectly the functions Vp (t) and Ve (t), t ≥ 0. The time instant td is the smallest positive root of the equation 

t

Vc (ρ)dρ = R0 ,

0

where, like in the previous example, R0 is a known initial range of the vehicles and Vc (t) is the closing speed of the engagement. Like in Example 6 (Sect. 3.6.6), the objective of the pursuer is, starting from a given initial position of the engagement at t = 0, to minimize |x1 (td )|. The objective of evader is to maximize |x1 (td )|. Using the same arguments as in the previous example, we consider the pursuer’s feedback control in the form 3   up = up x(t, ˜ ε), t) = kl (t)xl (t − εhl ) + k4 (t)x4 (t) l=1

+k5 (t)x5 (t − εh5 ) + k6 (t)x6 (t),

(3.178)

  where x(t, ˜ ε) = col x1 (t −εh1 ), x2 (t −εh2 ), x3 (t −εh3 ), x4 (t), x5 (t −εh5 ), x6 (t) ; ε > 0 is a small parameter; 0 < h1 < h2 < h3 < h5 are known constants independent of ε; kl (t), (l = 1, . . . , 6) are known gain coefficients, sufficiently smooth in the interval [0, td ].

3.6 Examples: Systems with Small Delays

185

Remember that the state coordinates x4 and x6 are the pursuer’s lateral acceleration and the angle between the pursuer’s velocity vectors and the initial line of sight. Therefore, similarly to Example 6 (Sect. 3.6.6), the pursuer measures perfectly these coordinates. Now, we consider the case where the evader is aware of the imperfect information in the pursuer’s control. The evader tries to use this circumstance to reach a considerably large value |x1 (td )|, and thus to win the engagement. Moreover, in this example we assume that the evader wants to increase not only the value |x1 (td )| but also the values |x2 (td )| and |x3 (td ) − x4 (td )|. To find out whether such a result of the engagement is possible, it is sufficient to establish the controllability of the system (3.177), (3.178) with respect to the values x1 , x2 , and x3 − x4 at the time instant td by the evader’s control ue . Based on the same arguments as in the previous example, we analyze this controllability for the following time constants of the evader and the pursuer: τe = ε, τp = 1. Thus, we are going to analyze the Euclidean space output controllability at the time instant td for the following system: dx1 (t) = x2 (t), dt     dx2 (t) = x3 (t) − x4 (t) + dVe (t)/dt x5 (t) − dVp (t)/dt x6 (t), dt dx4 (t)  = kl (t)xl (t − εhl ) + (k4 (t) − 1)x4 (t) dt 3

l=1

+k5 (t)x5 (t − εh5 ) + k6 (t)x6 (t), dx5 (t) x3 (t) = , dt Ve (t) x4 (t) dx6 (t) = , dt Vp (t) ε

dx3 (t) = ue (t) − x3 (t), dt

  ζ (t) = X col x1 (t), x2 (t), x4 (t), x5 (t), x6 (t) + Y x3 (t), ⎛ ⎞ ⎛ ⎞ 1 0 0 0 0 0 X = ⎝0 1 0 0 0 ⎠, Y = ⎝1⎠. 0 0 −1 0 0 0

(3.179)

(3.180)

The system (3.179)–(3.180) is a particular case of the system (3.1)–(3.2), (3.3). In this particular case,  x1 (t), x2 (t), x4(t), x5 (t), and x6 (t) are the slow Euclidean state variables, while x3 (t), x3 (t + εη) , η ∈ [−h3 , 0) is the fast state variable. In this case, n = 5, m = 1, r = 1, q = 3.

186

3 Euclidean Space Output Controllability of Linear Systems with State Delays

The assumptions (AI)–(AIII) are satisfied for the system (3.179)–(3.180). Also, we have that m < q < n and rankY = 1 = m, meaning that the first three conditions of Theorem 3.2 are valid. Let us find out whether the other two conditions of this theorem are valid, namely, the output controllability at t = td for the slow subsystem of the system (3.179)–(3.180) and the complete controllability for the fast subsystem of (3.179). The slow subsystems, associated with (3.179)–(3.180), have the form dxs1 (t) = xs2 (t), dt     dxs2 (t) = −xs4 (t) + dVe (t)/dt xs5 (t) − dVp (t)/dt xs6 (t) + use (t), dt dxs4 (t)  = kl (t)xsl (t) + (k4 (t) − 1)xs4 (t) dt 2

l=1

+k5 (t)xs5 (t) + k6 (t)x6 (t) + k3 (t)use (t), dxs5 (t) use (t) = , dt Ve (t) dxs6 (t) xs4 (t) = , dt Vp (t)   ζs (t) = X col xs1 (t), xs2 (t), xs4 (t), xs5 (t), xs6 (t) , (3.181) where t ∈ [0, td ]; use (t) is a control. The fast subsystem, associated with (3.179), is the same as in Example 6 (see Eq. (3.176)), and it is completely controllable. Proceed to the verification of the output controllability for the system (3.181). Let As (t) and Bs (t), t ∈ [0, td ] be the matrices of the coefficients for the state coordinates and the control of the differential system in (3.181). Then, ⎛

⎛ ⎞ ⎞ 0 1 0 0 0 0 ⎜ 0 ⎜1 ⎟ 0 −1 dVe (t)/dt −dVp (t)/dt ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ As (t)= ⎜ k1 (t) k2 (t) k4 (t) − 1 k5 (t) k6 (t) ⎟ , Bs (t)= ⎜ k3 (t) ⎟ . ⎜ ⎜ ⎟ ⎟ ⎝ 0 ⎝ 1/Ve (t) ⎠ ⎠ 0 0 0 0 0 0 0 0 0 1/Vp (t) Based on the results of [27] and [23], we construct the output controllability matrix for the subsystem (3.181) ! " C (t) = X D0 (t), D1 (t), D2 (t), D3 (t), D4 (t) ,

t ∈ [0, td ],

3.6 Examples: Systems with Small Delays

187

where D0 (t) = Bs (t),

Dp+1 (t) = −As (t)Dp (t) +

dDp (t) , dt

p = 0, 1, 2, 3.

Then, the slow subsystem (3.181) is output controllable at the time instant td if and only if there exists a value t¯ ∈ [0, td ], such that rankC (t¯) = 3. Note that the matrix C (t) can be rewritten in the form ! " C (t) = X D0 (t), X D1 (t), X D2 (t), X D3 (t), X D4 (t) ,

t ∈ [0, td ],

where each block is a column with three entries. Thus, the condition rankC (t¯) = 3 is equivalent to the linear independence of any three vectors among the five ones X Dp (t¯), (p = 0, 1, 2, 3, 4). The latter allows to derive up to 10 conditions yielding rankC (t¯) = 3. Each of these conditions provides the output controllability at t = td of the slow subsystem (3.181) and, therefore, the Euclidean space output controllability at the time instant td of the system (3.179)– (3.180) for all sufficiently small ε > 0. To complete this example, let us analyze the following simple case: Ve = const, kl = const, (l = 1, . . . , 6). If Ve = const, then in the system (3.177) the state coordinate x5 does not influence the state coordinate x2 and, therefore, x1 . Hence, there is not a need for the pursuer to include x5 in its feedback control. Thus, we can set k5 = 0. Moreover, in this simple case, there is not a need to include the column X D4 (t) into the matrix C (t). Based on the above made simplification, we obtain ⎛

⎞ 0 X D0 = ⎝ 1 ⎠ , −k3

⎛

⎞ −1 ⎠, X D1 = ⎝ k 3 k2 + k3 (k4 − 1)

⎛

⎞ k3    ⎠. X D2 (t) = − ⎝ k2 + k3 (k4 − 1) + k3 1/Vp (t) dVp (t)/dt     k1 − k2 k3 + (k4 − 1) k2 + k3 (k4 − 1) + k3 k6 1/Vp (t) Thus, in this simple case, if there exists t¯ ∈ [0, td ] such that   det X D0 , X D1 , X D2 (t¯) = 0, then the system (3.179)–(3.180) is Euclidean space output controllable at t = td for all sufficiently small ε > 0. Additional more complicated ε-free sufficient conditions for the Euclidean space output controllability of (3.179)–(3.180) can be derived by the inclusion into the consideration the vector-valued function X D3 (t).

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

3.6.8 Example 8: Analysis of Controlled Coupled-Core Nuclear Reactor Model In this example we analyze the system (1.16), which is the linearized model of the controlled coupled-core nuclear reactor (for details, see Sect. 1.1.4). For the sake of the book’s reading convenience, we rewrite this system here as: dx1 (t) ∗ = a11 (ε)x1 (t) + a13 εx3 (t) + c15 X1d εx5 (t) + X1d y1 (t) + x2 (t − ε), dt dx2 (t) ∗ = a22 (ε)x2 (t) + a24 εx4 (t) + c26 X2d εx6 (t) + X2d y2 (t) + x1 (t − ε), dt   dx3 (t) = ε a31 x1 (t) + a33 x3 (t) , dt   dx4 (t) = ε a42 x2 (t) + a44 x4 (t) , dt   dx5 (t) = ε a51 x1 (t) + a55 x5 (t) , dt   dx6 (t) = ε a62 x2 (t) + a66 x6 (t) , dt dy1 (t) = −y1 (t) + u1 (t), ε dt dy2 (t) = −y2 (t) + u2 (t), ε dt (3.182) ∗ (ε) and a ∗ (ε) are smooth functions of ε ≥ 0, where ε > 0 is a small parameter; a11 22 (see Eq. (1.15) and (1.18)); a13 , a24 . a31 , a33 , a42 , a44 , a51 , a55 , a62 , a66 , c15 , c26 , X1d , and X2d are constants independent of ε (for details, see Sect. 1.1.4); u1 (t) and u2 (t) are control functions. Along with the singularly perturbed differential system (3.182), we consider the output system of four algebraic equations

ζ1 (t) = x1 (t),

t ≥ 0,

ζ2 (t) = x2 (t),

t ≥ 0,

ζ3 (t) = y1 (t),

t ≥ 0,

ζ4 (t) = y2 (t),

t ≥ 0.

(3.183)

The system (3.182), (3.183) is a particular case of the system (3.1)–(3.2), (3.3) with n = 6, m = 2, r = 2, q = 4 and

3.6 Examples: Systems with Small Delays

⎛

1 ⎜0 X (t, ε) = X = ⎜ ⎝0 0

0 1 0 0

0 0 0 0

0 0 0 0

189

0 0 0 0

⎛ ⎞ ⎞ 0 0 0 ⎜ ⎟ 0⎟ ⎟ , Y (t, ε) = Y = ⎜ 0 0 ⎟ . ⎝ ⎠ 1 0⎠ 0 0 1 0

We consider (3.182), (3.183) in the time interval [0, tc ], and we study its Euclidean space output controllability at t = tc for all sufficiently small ε > 0. Due to (1.17), such a kind of controllability means the controllability at t = tc of the differential system (3.182) with respect to the shifted levels of the power and the shifted values of the normalized external control reactivity in both cores. The system (3.182), (3.183) satisfies the assumptions (AI)–(AIII). Moreover, we have that m < q < n and rankY = 2 = m, meaning that the first three conditions of Theorem 3.2 are valid. Let us find out whether the other two conditions of this theorem are valid. These conditions are the output controllability at t = tc for the slow subsystem of the system (3.182), (3.183) and the complete controllability for the fast subsystem of (3.182). The slow subsystem in the pure differential form, associated with the system (3.182), (3.183), has the following form in the interval [0, tc ]: dxs1 (t) ∗ = a11 (0)xs1 (t) + xs2 (t) + X1d us1 (t), dt dxs2 (t) ∗ = xs1 (t) + a22 (0)xs2 (t) + X2d us2 (t), dt dxs3 (t) = 0, dt dxs4 (t) = 0, dt dxs5 (t) = 0, dt dxs6 (t) = 0, dt ζs1 (t) = xs1 (t), ζs2 (t) = xs2 (t), while the fast subsystem, associated with (3.182), is dyf 1 (ξ ) = −yf 1 (ξ ) + uf 1 (ξ ), dξ

ξ ≥ 0,

dyf 2 (ξ ) = −yf 2 (ξ ) + uf 2 (ξ ), dξ

ξ ≥ 0.

(3.184)

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

Let us start with the controllability analysis of the fast subsystem. It is seen that the differential equations of this subsystem are independent of each other and the coefficients for the controls in these equations are nonzero. Therefore, the fast subsystem is completely controllable. Proceed with the slow subsystem (3.184). We can see that the system of the first two differential equations in (3.184) is independent  of the other differential equations, while the output of (3.184) is the vector col xs1 (t), xs2 (t) . Therefore, the output controllability at t = tc of the slow subsystem (3.184) is equivalent to the complete controllability at t = tc of the system, consisting of the first two differential equations in (3.184). Since X1d > 0 and X2d > 0, then the matrix of the coefficients for the controls in this system of the two differential equations has the full rank equals 2. The latter means that the system of the first two differential equations in (3.184) is completely controllable at t = tc . Hence, the slow subsystem (3.184) is output controllable at t = tc . Thus, all the conditions of Theorem 3.2 are valid for the system (3.182), (3.183) meaning that, for all sufficiently small ε > 0, this system is Euclidean space output controllable at t = tc .

3.7 Systems with Delays of Two Scales: Main Notions and Definitions 3.7.1 Original System In this section we consider another type of singularly perturbed time delay controlled systems. Namely,  dx(t)  = A1i (t, ε)x(t − gi ) + A2j (t, ε)y(t − εhj ) dt M

N

j =0

i=0

 +

0 −g

 G1 (t, ρ, ε)x(t + ρ)dρ +

0 −h

G2 (t, η, ε)y(t + εη)dη + B1 (t, ε)u(t), (3.185)

 dy(t)  = A3i (t, ε)x(t − gi ) + A4j (t, ε)y(t − εhj ) dt M

ε

N

j =0

i=0

 +

0 −g

 G3 (t, ρ, ε)x(t + ρ)dρ +

0 −h

G4 (t, η, ε)y(t + εη)dη + B2 (t, ε)u(t), (3.186)

3.7 Systems with Delays of Two Scales: Main Notions and Definitions

191

where t ≥ 0; x(t) ∈ E n , y(t) ∈ E m , u(t) ∈ E r (u(t) is a control); M ≥ 1 and N ≥ 1 are integers; ε > 0 is a small parameter; 0 = g0 < g1 < g2 < . . . < gM = g and 0 = h0 < h1 < h2 < . . . < hN = h are given constants independent of ε; Aki (t, ε), Alj (t, ε), Gk (t, ρ, ε), Gl (t, η, ε) and Bp (t, ε), (i = 0, 1, . . . , M; j = 0, 1, . . . , N ; k = 1, 3; l = 2, 4; p = 1, 2) are matrices of corresponding dimensions, given for t ≥ 0, ρ ∈ [−g, 0], η ∈ [−h, 0], ε ∈ [0, ε0 ]; Aki (t, ε), Alj (t, ε) and Bp (t, ε), (i = 0, 1, . . . , M; j = 0, 1, . . . , N ; k = 1, 3; l = 2, 4; p = 1, 2) are continuous in (t, ε) ∈ [0, +∞) × [0, ε0 ]; the functions Gk (t, ρ, ε), (k = 1, 3) are piecewise continuous in ρ ∈ [−g, 0] for any (t, ε) ∈ [0, +∞) × [0, ε0 ], and these functions are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in ρ ∈ [−g, 0]; the functions Gl (t, η, ε), (l = 2, 4) are piecewise continuous in η ∈ [−h, 0] for any (t, ε) ∈ [0, +∞)×[0, ε0 ], and these functions are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0]. In what follows of this section, we assume that the positive number ε0 satisfies the inequality ε0 ≤ g1 / h. Similarly to the system (3.1)–(3.2), for a given u(·) ∈ L2loc [0, +∞; E r ], the system (3.185)–(3.186) is a linear time-dependent nonhomogeneous functionaldifferential (infinite-dimensional) system, and this system is singularly perturbed. However, in contrast with (3.1)–(3.2), the state variables of (3.185)–(3.186) are     x(t), x(t + ρ) , ρ ∈ [−g, 0) and y(t), y(t + εη) , η ∈ [−h, 0). Similarly to Sect. 3.2, along with the dynamic system (3.185)–(3.186), we consider the algebraic delay-free output equation ζ (t) = X (t, ε)x(t) + Y (t, ε)y(t),

t ≥ 0,

(3.187)

where ζ (t) ∈ E q , (q ≤ n + m), is an output; X (t, ε) and Y (t, ε) are matrix-valued functions of corresponding dimensions, given for t ≥ 0 and ε ∈ [0, ε0 ]; X (t, ε) and Y (t, ε) are continuous in (t, ε) ∈ [0, +∞) × [0, ε0 ]. Note that Eq. (3.187) has the same form as Eq. (3.3). Nevertheless, for the sake of the integrity of the input-output system with the delays of two scales, we write this equation once again in this section. Let tc ≥ g be a given time instant independent of ε. Definition 3.8 For a given ε ∈ (0, ε0 ], the system (3.185)–(3.186), (3.187) is said to be Euclidean space output controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , ϕx (·) ∈ L2 [−g, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ] and ζc ∈ E q there exists a control function u(·) ∈ L2 [0, tc ; E r ], for which the system (3.185)–(3.186), (3.187) with the initial and terminal conditions x(ρ)=ϕx (ρ), ρ ∈ [−g, 0); y(τ )=ϕy (τ ), τ ∈ [−εh, 0); x(0)=x0 , y(0)=y0 , ζ (tc ) = ζc , has a solution.

192

3 Euclidean Space Output Controllability of Linear Systems with State Delays



In 0



, Remark 3.13 Similarly to Remark 3.1, if q = n + m and X (t, ε) ≡     0 Y (t, ε) ≡ , the output becomes ζ (t) = col x(t), y(t) . In such a case, the Im Euclidean space output controllability of the system (3.185)–(3.186), (3.187) at the time instant tc becomes the complete Euclidean space controllability of the dynamic system (3.185)–(3.186) at this time instant. Now, let us derive a criterion of the Euclidean space output controllability for the system (3.187). To do this, we introduce the block vector z(t) =  (3.185)–(3.186),  col x(t), y(t) , t ≥ −g and, for a given ε ∈ (0, ε0 ], the block matrices  A1i (t, ε) 0 , i = 0, 1, . . . , M, t ≥ 0, Ax,i (t, ε) = 1 ε A3i (t, ε) 0   0 A2j (t, ε) , j = 0, 1, . . . , N, t ≥ 0, Ay,j (t, ε) = 0 1ε A4j (t, ε)   G (t, ρ, ε) 0 , t ≥ 0, ρ ∈ [−g, 0], Gx (t, ρ, ε) = 1 1 ε G3 (t, ρ, ε) 0   0 G2 (t, η, ε) , t ≥ 0, η ∈ [−h, 0], Gy (t, η, ε) = 0 1ε G4 (t, η, ε)

ρ ∈ [−g, −εh), Gx (t, ρ, ε), t ≥ 0, Gx,y (t, ρ, ε) = Gx (t, ρ, ε) + 1ε Gy (t, ρ/ε, ε), ρ ∈ [−εh, 0],   B1 (t, ε) , t ≥ 0, B(t, ε) = 1 ε B2 (t, ε) ! " Z (t, ε) = X (t, ε) , Y (t, ε) , t ≥ 0. 

Using the above introduced block vector and block matrices, we can rewrite the system (3.185)–(3.186), (3.187) in the following equivalent form for all ε ∈ (0, ε0 ] and t ≥ 0:  dz(t)  = Ax,i (t, ε)z(t − gi ) + Ay,j (t, ε)z(t − εhj ) dt M

N

j =0

i=0

 +

0 −g

Gx,y (t, ρ, ε)z(t + ρ)dρ + B(t, ε)u(t), ζ (t) = Z (t, ε)z(t).

(3.188)

3.7 Systems with Delays of Two Scales: Main Notions and Definitions

193

The Euclidean space output controllability of the system (3.188) is defined quite similarly to such a controllability of the system (3.185)–(3.186), (3.187). Moreover, the Euclidean space output controllability of the latter is equivalent to such a controllability of the former. For any given ε ∈ (0, ε0 ], we consider the following terminal-value problem with respect to (n + m) × (n + m)-matrix-valued function H (σ ):  T dH (σ ) =− Ax,i (σ + gi , ε) H (σ + gi ) dσ M

i=0

−  −

N   T Ay,j (σ + εhj , ε) H (σ + εhj ) j =0

0 −g

 T Gx,y (σ − ρ, ρ, ε) H (σ − ρ)dρ, H (tc ) = In+m ,

σ ∈ [0, tc ),

H (σ ) = 0, σ > tc .

In this problem and in the sequel, it is assumed that the blocks of the matrices Ax,i (t, ε), (i = 0, 1, . . . , M), Ay,j (t, ε), (j = 0, 1, . . . , N ) and Gx,y (t, ρ, ε) satisfy the following equalities: Aki (t, ε) = Aki (tc , ε),

Alj (t, ε) = Alj (tc , ε), t > tc , ε ∈ [0, ε0 ],

Gk (t, ρ, ε) = Gk (tc , ρ, ε), t > tc , ρ ∈ [−g, 0], ε ∈ [0, ε0 ], Gl (t, η, ε) = Gl (tc , η, ε), t > tc , η ∈ [−h, 0], ε ∈ [0, ε0 ], k = 1, 3, l = 2, 4, i = 0, 1, . . . , M, j = 0, 1, . . . , N. By virtue of the results of [18] (Section 4.3), this terminal-value problem has the unique solution H (σ ) = H (σ, ε). Proposition 3.4 For a given ε ∈ (0, ε0 ], the system (3.185)–(3.186), (3.187) is Euclidean space output controllable at the time instant tc if and only if   det Z (tc , ε)

tc

 H T (σ, ε)B(t, ε)B T (t, ε)H (σ, ε)dσ Z T (tc , ε) = 0.

0

Proof The proposition follows immediately from the results of Sect. 3.3.1 (see Corollary 3.1) and the equivalence of the Euclidean space output controllability of the systems (3.188) and (3.185)–(3.186), (3.187).  

194

3 Euclidean Space Output Controllability of Linear Systems with State Delays

3.7.2 Asymptotic Decomposition of the Original System Now, we are going to decompose the original system (3.185)–(3.186), (3.187) into the slow and fast subsystems. This decomposition is carried out quite similarly to such a decomposition of the system (3.1)–(3.2), (3.3) in Sect. 3.2.2. Thus, the slow subsystem, associated with the system (3.185)–(3.186), (3.187), has the form dxs (t)  = A1i (t, 0)xs (t − gi ) + dt M



0

−g

i=0

G1 (t, ρ, 0)xs (t + ρ)dρ

+A2s (t)ys (t) + B1 (t, 0)us (t), t ≥ 0,

0=

M 

 A3i (t, 0)xs (t − gi ) +

i=0

0 −g

(3.189)

G3 (t, ρ, 0)xs (t + ρ)dρ

+A4s (t)ys (t) + B2 (t, 0)us (t), t ≥ 0,

(3.190)

ζs (t) = X (t, 0)xs (t), t ≥ 0,

(3.191)

  where xs (t) ∈ E n and ys (t) ∈ E m ; xs (t), xs (t + ρ) , ρ ∈ [−g, 0) and ys (t) are state variables; us (t) ∈ E r is a control; ζs (t) ∈ E q is an output; Als (t) =

N 

 Alj (t, 0) +

j =0

0

−h

Gl (t, η, 0)dη,

l = 2, 4.

The slow subsystem (3.189)–(3.191) consists of the descriptor (differentialalgebraic) time delay system (3.189)–(3.190) and the algebraic output Eq. (3.191).   The latter depends only on the Euclidean part xs (t) of the state xs (t), xs (t + ρ) , ρ ∈ [−g, 0). The slow subsystem is ε-free. Note that the dynamic part of the slow subsystem (3.8)–(3.9), (3.10), associated with the small time delay system (3.1)– (3.2), (3.3), does not have delays in the state variables. If det A4s (t) = 0, t ≥ 0, the time delay differential-algebraic system (3.189)– (3.190) can be converted to an equivalent system. This new system consists of the explicit expression for ys (t) ys (t) = −A−1 4s (t)

 M i=0

 A3i (t, 0)xs (t − gi ) +

0 −g

G3 (t, ρ, 0)xs (t + ρ)dρ

 +B2 (t, 0)us (t) ,

t ≥ 0,

3.7 Systems with Delays of Two Scales: Main Notions and Definitions

195

and the time delay differential equation with respect to xs (t) dxs (t)  ¯ = Ai,s (t)xs (t − gi ) + dt M

i=0



0 −g

¯ s (t, ρ)xs (t + ρ)dρ G

+B¯ s (t)us (t),

t ≥ 0,

(3.192)

where A¯ i,s (t) = A1i (t, 0) − A2s (t)A−1 4s (t)A3i (t, 0), ¯ s (t, ρ) = G1 (t, ρ, 0) − A2s (t)A−1 (t)G3 (t, ρ, 0), G 4s B¯ s (t) = B1 (t, 0) − A2s (t)A−1 4s (t)B2 (t, 0). The fast subsystem, associated with the system (3.185)–(3.186), (3.187), has the form dyf (ξ )  A4j (t, 0)yf (ξ − hj ) = dξ N

 +

j =0

0 −h

G4 (t, η, 0)yf (ξ + η)dη + B2 (t, 0)uf (ξ ), ξ ≥ 0, ζf (ξ ) = Y (t, 0)yf (ξ ), ξ ≥ 0,

(3.193) (3.194)

  where t ≥ 0 is a parameter; yf (ξ ) ∈ E m ; yf (ξ ), yf (ξ + η) , η ∈ [−h, 0) is a state variable; uf (ξ ) ∈ E r (uf (ξ ) is a control); ζf (ξ ) ∈ E q (ζf (ξ ) is an output). The system (3.193)–(3.194) coincides with the fast subsystem, associated with the system (3.1)–(3.2), (3.3). Definition 3.9 Subject to the inequality det A4s (t) = 0, t ∈ [0, tc ], the system (3.192), (3.191) is said to be Euclidean space output controllable at the time instant tc if for any x0 ∈ E n , ϕx (·) ∈ L2 [−g, 0; E n ] and ζc ∈ E q there exists a control function us (·) ∈ L2 [0, tc ; E r ], for which (3.192) has a solution xs (t), t ∈ [0, tc ], satisfying the initial and terminal conditions x(ρ) = ϕx (ρ), ρ ∈ [−g, 0), xs (0) = x0 ;

X (tc , 0)xs (tc ) = ζc .

(3.195)

Remark 3.14 If q = n and X (t, ε) ≡ In , the Euclidean space output controllability of the system (3.192), (3.191) at the time instant tc becomes the complete Euclidean space controllability of the system (3.192) at this time instant. Definition 3.10 The system (3.189)–(3.191) is said to be impulse-free Euclidean space output controllable at the time instant tc if for any x0 ∈ E n , ϕx (·) ∈ L2 [−g, 0; E n ] and ζc ∈ E q there exists a control function us (·) ∈ L2 [0, tc ; E r ],

196

3 Euclidean Space Output Controllability of Linear Systems with State Delays

for which the differential-algebraic system (3.189)–(3.190) has a solution col(xs (t), ys (t)) ∈ L2 [0, tc ; E n+m ] satisfying the initial and terminal conditions (3.195). Remark 3.15 If q = n and X (t, ε) ≡ In , the impulse-free Euclidean space output controllability of the system (3.189)–(3.191) at the time instant tc becomes the impulse-free Euclidean space controllability with respect to xs (t) of the system (3.189)–(3.190) at this time instant. Now, let us proceed to a definition of controllability of the fast subsystem (3.193)–(3.194). Since this subsystem coincides with the fast subsystem, associated with the system (3.1)–(3.2), (3.3), then the definition of its Euclidean space output controllability is quite similar to Definition 3.4 for such a kind of the controllability of the system (3.15)–(3.16). Moreover, similarly to Remark 3.4, if q = m and Y (t, ε) ≡ Im , the Euclidean space output controllability of the system (3.193)–(3.194) becomes the complete Euclidean space controllability of the system (3.193). To complete this subsection, we will derive criterions of the Euclidean space output controllability for the systems (3.192), (3.191) and (3.193)–(3.194). We start with the system (3.192), (3.191). Let the n × n-matrix-valued function Hs (σ ), σ ∈ [0, tc ] be the unique solution of the terminal-value problem  T dHs (σ ) =− A¯ i,s (σ + gi ) Hs (σ + gi ) dσ M

 −

i=0

0 −g

  ¯ s (σ − ρ, ρ) T Hs (σ − ρ)dρ, G Hs (tc ) = In ;

σ ∈ [0, tc ),

Hs (σ ) = 0, σ > tc .

Applying the results of Sect. 3.3.1 (see Corollary 3.1) to the system (3.192), (3.191), we directly obtain the following proposition. Proposition 3.5 The system (3.192), (3.191) is Euclidean space output controllable at the time instant tc , if and only if   det X (tc , 0)

tc 0

 HsT (σ )B¯ s (σ )B¯ Ts (σ )Hs (σ )dσ X T (tc , 0) =  0.

Proceed to the system (3.193)–(3.194). Let, for any given t ≥ 0, the m × m-matrix-valued function Hf (ξ, t) be the unique solution of the following initial-value problem: T dHf (ξ )   = A4j (t, 0) Hf (ξ − hj ) dξ N

j =0

3.8 Linear Control Transformation in Systems with Delays of Two Scales

 +

0 −h



G4 (t, η, 0)

T

Hf (ξ + η)dη,

Hf (ξ ) = 0, ξ < 0;

197

ξ > 0,

Hf (0) = Im .

Application of the results of Sect. 3.3.1 (see Corollary 3.1) to the system (3.193)– (3.194) yields immediately the following proposition. Proposition 3.6 For a given t ≥ 0, the system (3.193)–(3.194) is Euclidean space output controllable at a prescribed instant ξ = ξc > 0 if and only if   det Y (t, 0)

ξc 0

 HfT (χ , t)B2 (t, 0)B2T (t, 0)Hf (χ , t)dχ Y T (t, 0) =  0.

3.8 Linear Control Transformation in Systems with Delays of Two Scales 3.8.1 Control Transformation in the Original System Let us make the control transformation (3.29) in the dynamic system (3.185)– (3.186). This control transformation converts the system (3.185)–(3.186) to the system  dx(t)  = A1i (t, ε)x(t − gi ) + AK 2j (t, ε)y(t − εhj ) dt M

N

j =0

i=0

 +

0 −g

 G1 (t, ρ, ε)x(t + ρ)dρ +

0

−h

GK 2 (t, η, ε)y(t + εη)dη + B1 (t, ε)w(t), (3.196)

 dy(t)  = A3i (t, ε)x(t − gi ) + AK 4j (t, ε)y(t − εhj ) dt M

ε

N

j =0

i=0

 +

0 −g

 G3 (t, ρ, ε)x(t + ρ)dρ +

0

−h

GK 4 (t, η, ε)y(t + εη)dη + B2 (t, ε)w(t), (3.197)

K where t ≥ 0; the matrix-valued functions AK 2j (t, ε), A4j (t, ε), (j = 0, 1, . . . , N) K K and G2 (t, η, ε), G4 (t, η, ε) are given by Eqs. (3.32)–(3.35).

198

3 Euclidean Space Output Controllability of Linear Systems with State Delays

Lemma 3.10 For a given ε ∈ (0, ε0 ], the system (3.185)–(3.186), (3.187) is Euclidean space output controllable at the time instant tc , if and only if the system (3.196)–(3.197), (3.187) is Euclidean space output controllable at this time instant.  

Proof The lemma is proven similarly to Lemma 3.2. As a direct consequence of Lemma 3.10, we obtain the following assertion.

Corollary 3.7 For a given ε ∈ (0, ε0 ], the system (3.185)–(3.186) is completely Euclidean space controllable at the time instant tc , if and only if the system (3.196)– (3.197) is completely Euclidean space controllable at this time instant.

3.8.2 Asymptotic Decomposition of the Transformed System (3.196)–(3.197), (3.187) Let us decompose asymptotically the singularly perturbed system (3.196)– (3.197), (3.187) into the slow and fast subsystems. The slow subsystem, associated with this system, consists of the differential-algebraic system dxs (t)  = A1i (t, 0)x(t − gi ) + dt M



0

−g

i=0

G1 (t, ρ, ε)x(t + ρ)dρ

+AK 2s (t)ys (t) + B1 (t, 0)ws (t), t ≥ 0,

0=

M 

 A3i (t, 0)x(t − gi ) +

i=0

0 −g

(3.198)

G3 (t, ρ, ε)x(t + ρ)dρ

+AK 4s (t)ys (t) + B2 (t, 0)ws (t), t ≥ 0,

(3.199)

 and the algebraic output Eq. (3.191), where xs (t) ∈ E n , ys (t) ∈ E m ; x(t), x(t + ρ) , ρ ∈ [−g, 0) and ys (t) are state variables; ws (t) ∈ E r , (ws (t) is a control); the matrix-valued functions AK ls (t), (l = 2, 4) are given by (3.40). If det AK (t) =  0, t ≥ 0, the differential-algebraic system (3.198)–(3.199) can 4s be converted to an equivalent system, consisting of the explicit expression for ys (t)  M   K −1  ys (t) = − A4s (t) A3i (t, 0)x(t − gi ) + i=0

0

−g

G3 (t, ρ, ε)x(t + ρ)dρ

 +B2 (t, 0)ws (t) ,

t ≥ 0,

3.8 Linear Control Transformation in Systems with Delays of Two Scales

199

and the time delay differential equation with respect to xs (t) dxs (t)  ¯ K = A i,s (t)xs (t − gi ) + dt M

i=0



0 −g

¯ K G s (t, ρ)xs (t + ρ)dρ

+B¯

K s (t)ws (t),

t ≥ 0,

(3.200)

where  K −1 K A3i (t, 0), A¯ K i,s (t) = A1i (t, 0) − A2s (t) A4s (t)   −1 K K ¯ K G3 (t, ρ, 0), G s (t, ρ) = G1 (t, ρ, 0) − A2s (t) A4s (t)  −1 K K B¯ K B2 (t, 0). s (t) = B1 (t, 0) − A2s (t) A4s (t) The fast subsystem, associated with (3.196)–(3.197), (3.187), consists of the differential equation with state delays dyf (ξ )  K = A4j (t, 0)yf (ξ − hj ) dξ N

 +

j =0

0 −h

GK 4 (t, η, 0)yf (ξ + η)dη + B2 (t, 0)wf (ξ ), ξ ≥ 0,

(3.201)

and Eq. (3.194), where t ≥ 0 is a parameter; yf (ξ ) ∈ E m ;  the algebraic output  yf (ξ ), yf (ξ + η) , η ∈ [−h, 0) is a state variable; wf (ξ ) ∈ E r , (wf (ξ ) is a control). The system (3.201), (3.194) coincides with the fast subsystem, associated with the system (3.30)–(3.31), (3.3). The following three Lemmas and three Corollaries are obtained quite similarly to Lemmas 3.3–3.5 and Corollaries 3.3–3.5, respectively. Lemma 3.11 The system (3.189)–(3.191) is impulse-free Euclidean space output controllable at the time instant tc , if and only if the system (3.198)–(3.199), (3.191) is impulse-free Euclidean space output controllable at this time instant. Lemma 3.12 Let the inequality det AK 4s (t) = 0, t ∈ [0, tc ] be valid. Then, the system (3.198)–(3.199), (3.191) is impulse-free Euclidean space output controllable at the time instant tc , if and only if the system (3.200), (3.191) is Euclidean space output controllable at this time instant. Lemma 3.13 Let the inequalities det A4s (t) = 0 and det AK 4s (t) = 0 be valid for all t ∈ [0, tc ]. Then, the system (3.192), (3.191) is Euclidean space output controllable at the time instant tc , if and only if the system (3.200), (3.191) is Euclidean space output controllable at this time instant.

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

Corollary 3.8 The system (3.189)–(3.190) is impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc , if and only if the system (3.198)– (3.199) is impulse-free Euclidean space controllable with respect to xs (t) at this time instant. Corollary 3.9 Let the inequality det AK 4s (t) = 0, t ∈ [0, tc ] be valid. Then, the system (3.198)–(3.199) is impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc , if and only if the system (3.200) is completely Euclidean space controllable at this time instant. Corollary 3.10 Let the inequalities det A4s (t) = 0 and det AK 4s (t) = 0 be valid for all t ∈ [0, tc ]. Then, the system (3.192) is completely Euclidean space controllable at the time instant tc , if and only if the system (3.200) is completely Euclidean space controllable at this time instant. Remark 3.16 Since the systems (3.193)–(3.194) and (3.201), (3.194) coincide with the systems (3.15)–(3.16) and (3.46), (3.16), respectively, then due to Lemma 3.6, the Euclidean space output controllability properties of the systems (3.193)–(3.194) and (3.201), (3.194) are equivalent to each other. Moreover, due to Corollary 3.6, the complete Euclidean space controllability properties of the systems (3.193) and (3.201) are equivalent to each other.

3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two Scales In this section, using the results of Sects. 3.7–3.8, various ε-independent conditions, providing the Euclidean space output controllability of the original system (3.185)– (3.186), (3.187) for all sufficiently small ε > 0, are derived. In what follows of this section, we assume: (AV)

The matrix-valued functions Aki (t, ε), Alj (t, ε), Bp (t, ε), (i = 0, 1, . . . , M; j = 0, 1, . . . , N; k = 1, 3; l = 2, 4; p = 1, 2) are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ].

(AVI)

The matrix-valued functions Gk (t, ρ, ε), (k = 1, 3) are piecewise continuous with respect to ρ ∈ [−g, 0] for each (t, ε) ∈ [0, tc ] × [0, ε0 ], and these functions are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in ρ ∈ [−g, 0].

(AVII)

The matrix-valued functions Gl (t, η, ε), (l = 2, 4) are piecewise continuous with respect to η ∈ [−h, 0] for each (t, ε) ∈ [0, tc ] × [0, ε0 ], and these functions are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in η ∈ [−h, 0].

3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

201

In what follows, we consider two cases: (i) the assumption (AIII) is valid for the original dynamic system (3.185)–(3.186); (ii) the assumption (AIII) is violated, while the assumption (AIV) is valid for the original dynamic system (3.185)– (3.186). Remember that the assumptions (AIII) and (AIV) are introduced in Sect. 3.4.1.

3.9.1 Case of the Validity of the Assumption (AIII) ! " Like in Sect. 3.4.1, we consider the block matrix Z (tc , 0) = X (tc , 0), Y (tc , 0) . The following three theorems present different ε-free sufficient conditions for the Euclidean space output controllability of the singularly perturbed system (3.185)– (3.186), (3.187). These conditions depend considerably on relations between the Euclidean space dimensions of the slow state variable, the fast state variable and the output of the system. Theorem 3.17 Let the assumptions (AV)–(AVII), (AIII) be valid. Let q ≤ m. Let, for t = tc , the system (3.193)–(3.194) be Euclidean space output controllable. Then, there exists a positive number εc,1 , (εc,1 ≤ ε0 ), such that for all ε ∈ (0, εc,1 ], the singularly perturbed system (3.185)–(3.186), (3.187) is Euclidean space output controllable at the time instant tc . Theorem 3.18 Let the assumptions (AV)–(AVII), (AIII) be valid. Let m < q ≤ n. Let rankY (tc , 0) = m. Let the system (3.192), (3.191) be Euclidean space output controllable at the time instant tc . Let, for t = tc , the system (3.193) be completely Euclidean space controllable. Then, there exists a positive number εc,2 , (εc,2 ≤ ε0 ), such that for all ε ∈ (0, εc,2 ], the singularly perturbed system (3.185)–(3.186), (3.187) is Euclidean space output controllable at the time instant tc . Theorem 3.19 Let the assumptions (AV)–(AVII), (AIII) be valid. Let rankZ (tc , 0) = q. Let the system (3.192) be completely Euclidean space controllable at the time instant tc . Let, for t = tc , the system (3.193) be completely Euclidean space controllable. Then, there exists a positive number εc,3 , (εc,3 ≤ ε0 ), such that for all ε ∈ (0, εc,3 ], the singularly perturbed system (3.185)–(3.186), (3.187) is Euclidean space output controllable at the time instant tc . Using the results of Chap. 2 (see Sect. 2.3) and Propositions 3.4–3.6 (see Sect. 3.7.1–3.7.2), Theorems 3.17, 3.18 and 3.19 are proven quite similarly to Theorems 3.1, 3.2 and 3.3, respectively.

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

3.9.2 Case of the Validity of the Assumption (AIV) Based on the results of the previous section and Sect. 3.8, the following three theorems are obtained quite similarly to Theorems 3.4, 3.7 and 3.8. Theorem 3.20 Let the assumptions (AV)–(AVII), (AIV) be valid. Let q ≤ m. Let, for t = tc , the system (3.193)–(3.194) be Euclidean space output controllable. Then, there exists a positive number ε˜ c,1 , (˜εc,1 ≤ ε0 ), such that for all ε ∈ (0, ε˜ c,1 ], the singularly perturbed system (3.185)–(3.186), (3.187) is Euclidean space output controllable at the time instant tc . Theorem 3.21 Let the assumptions (AV)–(AVII), (AIV) be valid. Let m < q ≤ n. Let rankY (tc , 0) = m. Let the system (3.189)–(3.191) be impulse-free Euclidean space output controllable at the time instant tc . Let, for t = tc , the system (3.193) be completely Euclidean space controllable. Then, there exists a positive number ε˜ c,2 , (˜εc,2 ≤ ε0 ), such that for all ε ∈ (0, ε˜ c,2 ], the singularly perturbed system (3.185)– (3.186), (3.187) is Euclidean space output controllable at the time instant tc . Theorem 3.22 Let the assumptions (AV)–(AVII), (AIV) be valid. Let rankZ (tc , 0) = q. Let the system (3.189)–(3.190) be impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc . Let, for t = tc , the system (3.193) be completely Euclidean space controllable. Then, there exists a positive number ε˜ c,3 , (˜εc,3 ≤ ε0 ), such that for all ε ∈ (0, ε˜ c,3 ], the singularly perturbed system (3.185)–(3.186), (3.187) is Euclidean space output controllable at the time instant tc .

3.9.3 Special Cases of Controllability In this section, we consider several important particular cases of the Euclidean space output controllability of the system (3.185)–(3.186), (3.187).

3.9.3.1

Complete Euclidean Space Controllability

Let q = n + m and Z (t, ε) ≡ In+m , meaning that the Euclidean space output controllability of the system (3.185)–(3.186), (3.187) becomes the complete Euclidean space controllability of its dynamic part (3.185)–(3.186). In this case, we obtain the following two theorems, which are direct consequences of Theorems 3.19 and 3.22, respectively. Theorem 3.23 Let the assumptions (AV)–(AVII), (AIII) be valid. Let the system (3.192) be completely Euclidean space controllable at the time instant tc . Let, for t = tc , the system (3.193) be completely Euclidean space controllable. Then, there exists a positive number εˆ c,1 , (ˆεc,1 ≤ ε0 ), such that for all ε ∈ (0, εˆ c,1 ], the singularly perturbed system (3.185)–(3.186) is completely Euclidean space controllable at the time instant tc .

3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

203

Theorem 3.24 Let the assumptions (AV)–(AVII), (AIV) be valid. Let the system (3.189)–(3.190) be impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc . Let, for t = tc , the system (3.193) be completely Euclidean space controllable. Then, there exists a positive number εˆ c,2 , (ˆεc,2 ≤ ε0 ), such that for all ε ∈ (0, εˆ c,2 ], the singularly perturbed system (3.185)–(3.186) is completely Euclidean space controllable at the time instant tc .

3.9.3.2

Controllability with Respect to x(t)

Here we assume that q = n and X (t, ε) ≡ In , Y (t, ε) ≡ 0. In this case, similarly to Sect. 3.5.2, the Euclidean space output controllability of the system (3.185)–(3.186), (3.187) becomes the controllability with respect to x(t). The following two theorems present different ε-free sufficient conditions for this type of controllability. Based on the results of Chap. 2 (Sect. 2.3) and the results of Sect. 3.8, these theorems are obtained quite similarly to Theorems 3.12 and 3.14, respectively. Theorem 3.25 Let the assumptions (AV)–(AVII), (AIII) be valid. Let the system (3.192) be completely Euclidean space controllable at the time instant tc . Then, there exists a positive number εˇ x,1 , (ˇεx,1 ≤ ε0 ), such that for all ε ∈ (0, εˇ x,1 ], the singularly perturbed system (3.185)–(3.186), (3.187) is controllable with respect to x(t) at the time instant tc . Theorem 3.26 Let the assumptions (AV)–(AVII), (AIV) be valid. Let the system (3.189)–(3.190) be impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc . Then, there exists a positive number εˇ x,2 , (ˇεx,2 ≤ ε0 ), such that for all ε ∈ (0, εˇ x,2 ], the singularly perturbed system (3.185)–(3.186), (3.187) is controllable with respect to x(t) at the time instant tc .

3.9.3.3

Controllability with Respect to y(t)

Here we assume that q = m and X (t, ε) ≡ 0, Y (t, ε) ≡ Im . In this case, similarly to Sect. 3.5.3, the Euclidean space output controllability of the system (3.185)– (3.186), (3.187) becomes the controllability with respect to y(t). The following two theorems present different ε-free sufficient conditions for this type of controllability. Based on the results of Chap. 2 (Sect. 2.3) and the results of Sect. 3.8, these theorems are obtained quite similarly to Theorems 3.15 and 3.16, respectively. Theorem 3.27 Let the assumptions (AV)–(AVII), (AIII) be valid. Let, for t = tc , the system (3.193) be completely Euclidean space controllable. Then, there exists a positive number εˇ y,1 , (ˇεy,1 ≤ ε0 ), such that for all ε ∈ (0, εˇ y,1 ], the singularly perturbed system (3.185)–(3.186), (3.187) is controllable with respect to y(t) at the time instant tc . Theorem 3.28 Let the assumptions (AV)–(AVII), (AIV) be valid. Let, for t = tc , the system (3.193) be completely Euclidean space controllable. Then, there exists

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

a positive number εˇ y,2 , (ˇεy,2 ≤ ε0 ), such that for all ε ∈ (0, εˇ y,2 ], the singularly perturbed system (3.185)–(3.186), (3.187) is controllable with respect to y(t) at the time instant tc .

3.9.4 Example 9 Consider the following particular case of the system (3.185)–(3.186), (3.187): dx(t) = tx(t) − 2y1 (t) + t 2 y2 (t) − (t − 2)x(t − 0.5) dt  0 −2x(t − 1.5) + y1 (t − ε) + 2y2 (t − ε) − ρ 2 x(t + ρ)dρ + tu(t), t ≥ 0, −1.5

dy1 (t) = 2x(t) − y1 (t) − y2 (t) + tx(t − 0.5) − x(t − 1.5) dt  0 +2y1 (t − ε) + 2y2 (t − ε) + (η + 0.5)y1 (t + εη)dη − (t + 2)u(t), t ≥ 0, ε

−1

ε

dy2 (t) = x(t) − 3y1 (t) − 3y2 (t) + 2x(t − 0.5) − tx(t − 1.5) dt +y1 (t − ε) + y2 (t − ε) + (t + 2)u(t), t ≥ 0, ζ (t) = (t − 1)x(t) + ty1 (t) − y2 (t), t ≥ 0, (3.202)

where x(t), y1 (t), y2 (t), u(t), and ζ (t) are scalars, i.e., n = 1, m = 2, r = 1, q = 1. Moreover, in this example M = 2, N = 1, g1 = 0.5, g2 = g = 1.5, h1 = h = 1, 0 < ε0 ≤ 0.5. We study the Euclidean space output controllability of the system (3.202) at the time instant tc = 2 robust with respect to ε ∈ (0, ε0 ] for all its sufficiently small values. The slow and fast subsystems, associated with (3.202), are dxs (t) = txs (t) − ys1 (t) + (t 2 + 2)ys2 (t) − (t − 2)xs (t − 0.5) dt  0 −2xs (t − 1.5) − ρ 2 xs (t + ρ)dρ + tus (t), t ∈ [0, 2], −1.5

0 = 2xs (t) + ys1 (t) + ys2 (t) + txs (t − 0.5) − xs (t − 1.5) − (t + 2)us (t), t ∈ [0, 2], 0 = xs (t) − 2ys1 (t) − 2ys2 (t) + 2xs (t − 0.5) − txs (t − 1.5) + (t + 2)us (t), t ∈ [0, 2], ζs (t) = (t − 1)xs (t), t ∈ [0, 2],

3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

205

and dyf 1 (ξ ) = −yf 1 (ξ ) − yf 2 (ξ ) + 2yf 1 (ξ − 1) + 2yf 2 (ξ − 1) dξ  0 (η + 0.5)yf 1 (ξ + η)dη − (t + 2)uf (t), ξ ≥ 0, + −1

dyf 2 (ξ ) = −3yf 1 (ξ ) − 3yf 2 (ξ ) + yf 1 (ξ − 1) + yf 2 (ξ − 1) dξ +(t + 2)uf (ξ ), ξ ≥ 0, ζf (ξ ) = tyf 1 (ξ ) − yf 2 (ξ ), ξ ≥ 0.

(3.203)

In the system (3.203), t ∈ [0, 2] is a parameter. The assumptions (AV)–(AVII) are satisfied for the system (3.202). The matrix A4s (t) in this example has the form  A4s (t) ≡

 1 1 . −2 − 2

This matrix is singular. Therefore, the system (3.202) is nonstandard, and it does not satisfy the assumption (AIII). Let us show that this system satisfies the assumption (AIV). The matrix, mentioned in this assumption, has the form  Wf (t, λ) =

 Wf,11 (λ) Wf,12 (λ) Wf,13 (t) , Wf,21 (λ) Wf,22 (λ) Wf,23 (t)

where  Wf,11 (λ) = λ + 1 − 2 exp(−λ) −

0

−1

(η + 0.5) exp(λη)dη,

Wf,12 (λ) = 1 − 2 exp(−λ), Wf,13 (t) = −(t + 2), Wf,21 (λ) = 3 − exp(−λ), Wf,22 (λ) = λ + 3 − exp(−λ), Wf,23 (t) = (t + 2).   To show the validity of the assumption (AIV), we should prove that rank Wf (t, λ) = m = 2 for all t ∈ [0, tc ] = [0, 2] and all complex λ : Reλ ≥ 0. For this purpose, let us consider the minor of Wf (t, λ), consisting of its first and third columns. This minor has the form Mf (t, λ) = Wf,11 (λ)Wf,23 (t) − Wf,21 (λ)Wf,13 (t)   0 (η + 0.5) exp(λη)dη , t ∈ [0, 2], Reλ ≥ 0. = (t + 2) λ + 4 − 3 exp(−λ) − 

−1

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

Since Reλ ≥ 0, then | exp(−λ)| ≤ 1, and    

0

−1

   (η + 0.5) exp(λη)dη ≤



0

−1

|η + 0.5|| exp(λη)|dη ≤

0

−1

|η + 0.5|dη = 0.25,

implying    3 exp(−λ) + 

0

−1

  (η + 0.5) exp(λη)dη ≤ 3.25.

Therefore,     Re 3 exp(−λ) + 

0

−1

  (η + 0.5) exp(λη)dη  ≤ 3.25.

This inequality, along with the expression for Mf (t, λ), directly yields ReMf (t, λ) ≥ 0.75(t + 2) ≥ 1.5 ∀ t ∈ [0, 2], Reλ ≥ 0. The latter means immediately that Mf (t, λ) = 0 for all t ∈ [0, 2] and Reλ ≥ 0.   Hence, rank Wf (t, λ) = m = 2 for all these t and λ, i.e., the assumption (AIV) is satisfied for the system (3.202). Now, we are going to show that, for t = tc = 2, the fast subsystem (3.203) is Euclidean space output controllable. We do this by contradiction, i.e., we assume that (3.203) is not Euclidean space output controllable. This assumption, along with Definition 3.4, Corollary 3.1 and the data of (3.203), means that for any given number ξc > 0 the following equality holds: 

ξc 0

 (2 , −1)ΨfT (σ )

   −4 2 (−4 , 4)Ψf (σ ) dσ = 0. 4 −1

(3.204)

In this equality, Ψf (σ ) is the 2 × 2-matrix-valued function satisfying the following terminal-value problem:     dΨf (σ ) 1 3 2 1 = Ψf (σ ) − Ψf (σ + 1) 1 3 2 1 dσ   0 η + 0.5 0 − Ψf (σ − η)dη, σ ∈ [0, ξc ), 0 −1 0 Ψf (ξc ) = I2 ,

Ψf (σ ) = 0, σ > ξc .

The function Ψf (σ ) is continuous for σ ∈ [0, ξc ]. Moreover, the integrand in (3.204) is a continuous and nonnegative function for σ ∈ [0, ξc ]. Therefore, due to this equality, the integrand equals zero for all σ ∈ [0, ξc ]. However, for

3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

207

σ = ξc , this integrand equals 144. Thus, we have obtained a contradiction. Due this contradiction, the assumption that the fast subsystem (3.203) is not Euclidean space output controllable is wrong. Summarizing the above obtained results on the original singularly perturbed system (3.202), and taking into account that q = 1 < m = 2, we can conclude that for this system all the conditions of Theorem 3.20 are fulfilled. Therefore, for all sufficiently small ε > 0, the system (3.202) is Euclidean space output controllable at t = tc = 2.

3.9.5 Example 10 Consider the following particular case of the system (3.185)–(3.186), (3.187): dx1 (t) = (1 − 3t)x1 (t) − 4x2 (t) + (2 − t)y(t) + 4x1 (t − 1) dt −(1 + 3t)x2 (t − 1) + (1 + t)y(t − 2ε) + (3t 2 − 2)u(t),

t ≥ 0,

dx2 (t) = (2 − t)x1 (t) + 2x2 (t) + ty(t) + 4x1 (t − 1) dt +(3 − t)x2 (t − 1) + (1 − t)y(t − 2ε) + (t 2 + 1)u(t), ε

t ≥ 0,

dy(t) = tx1 (t) + 2x2 (t) − 7y(t) − x1 (t − 1) + (t + 1)x2 (t − 1) dt  0 +2y(t − 2ε) − 2 ηy(t + εη)dη − t 2 u(t), t ≥ 0, −2

ζ1 (t) = tx1 (t) − x2 (t) + exp(−t)y(t),

t ≥ 0,

ζ2 (t) = (t − 1)x1 (t) + 2tx2 (t) − y(t),

t ≥ 0,

(3.205)

where x1 (t), x2 (t), y(t), u(t), ζ1 (t), and ζ2 (t) are scalars, i.e., n = 2, m = 1, r = 1, q = 2. Also, in this example M = 1, N = 1, g1 = g = 1, h1 = h = 2, 0 < ε0 ≤ 0.5. We study the Euclidean space output controllability of the system (3.205) at the time instant tc = 1.5. The slow subsystem, associated with (3.205), is dxs1 (t) = (1 − 3t)xs1 (t) − 4xs2 (t) + 3ys (t) + 4xs1 (t − 1) dt −(1 + 3t)xs2 (t − 1) + (3t 2 − 2)us (t),

t ∈ [0, 1.5],

dxs2 (t) = (2 − t)xs1 (t) + 2xs2 (t) + ys (t) + 4xs1 (t − 1) dt +(3 − t)xs2 (t − 1) + (t 2 + 1)us (t),

t ∈ [0, 1.5],

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

0 = txs1 (t) + 2xs2 (t) − ys (t) − xs1 (t − 1) + (t + 1)xs2 (t − 1) −t 2 us (t),

t ∈ [0, 1.5],

ζs1 (t) = txs1 (t) − xs2 (t),

t ∈ [0, 1.5],

ζs2 (t) = (t − 1)xs1 (t) + 2txs2 (t),

t ∈ [0, 1.5].

(3.206)

(3.207)

Eliminating ys (t) from (3.206), we can reduce this time delay differentialalgebraic system to the following time delay pure differential system with respect to xs1 and xs2 : dxs1 (t) = xs1 (t) + 2xs2 (t) + xs1 (t − 1) + 2xs2 (t − 1) − 2us (t), t ∈ [0, 1.5], dt dxs2 (t) = 2xs1 (t) + 4xs2 (t) + 3xs1 (t − 1) + 4xs2 (t − 1) + us (t), t ∈ [0, 1.5]. dt (3.208) The fast subsystem, associated with the original system (3.205), has the form dyf (ξ ) = −7yf (ξ ) + 2yf (ξ − 2) − 2 dξ



0 −2

ηyf (ξ + η)dη − t 2 uf (ξ ), ξ ≥ 0, ζf 1 (ξ ) = exp(−t)yf (ξ ),

ξ ≥ 0,

ζf 2 (ξ ) = −yf (ξ ),

ξ ≥ 0, (3.209)

where t ∈ [0, 1.5] is a parameter. The assumptions (AV)–(AVII) are satisfied for the system (3.205). Let us show the fulfilment of the assumption (AIII). In this example Eq. (3.113), appearing in the assumption (AIII), becomes as: 



Df (λ) = λ + 7 − 2 exp(−2λ) + 2

0 −2

η exp(λη)dη = 0.

First, we are going to show that the real part of any root of this equation is negative. Indeed, if Reλ ≥ 0, then | exp(−2λ)| ≤ 1 and | exp(λη)| ≤ 1, η ∈ [−2, 0], meaning that  0      ≤ 2. η exp(λη)dη   −2

  Therefore, Re Df (λ) ≥ 1 for all λ : Reλ ≥ 0. Hence, all roots λ of the equation Df (λ) = 0 have negative real parts. Moreover, due to the results of [19], this

3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

209

equation has no more than a finite number of roots in any strip −ω ≤ Reλ ≤ 0 (ω > 0 is a given number). This observation, along with the above proven inequality Reλ < 0 for all roots λ of the equation Df (λ) = 0, implies the existence of a positive number β such that these roots satisfy the inequality Reλ < −2β. Thus, the assumption (AIII) is satisfied. Since n = 2, m = 1, and q = 2, then m < q = n, which corresponds to the inequality between these values in Theorem 3.18. Moreover, rankY (tc ) = m = 1,   where Y (t) = col exp(−t) , −1 is the matrix of the coefficients for yf (ξ ) in the output equations of the fast subsystem (3.209). Due to these observations, to show the Euclidean space output controllability of the singularly perturbed system (3.205), we can try to use Theorem 3.18. For this purpose, we should prove that the system (3.208), (3.207) is Euclidean space output controllable at t = tc = 1.5, while for t = 1.5 the dynamic system in (3.209) is completely Euclidean space controllable. Let us start with the system (3.208), (3.207). First of all, we introduce ¯ ¯ ¯ into the consideration the   matrices A0s , A1s , and Bs of the coefficients for col xs1 (t) , xs2 (t) , col xs1 (t − 1) , xs2 (t − 1) , and us (t), respectively, in Eq. (3.208). These matrices are A¯ 0s =



 1 2 , 2 4

A¯ 1s =



 1 2 , 3 4

B¯ s =



 −2 . 1

  Also, we consider the matrix of the coefficients X (t) for col xs1 (t) , xs2 (t) in the output Eq. (3.207):  X (t) =

t −1 t −1 2t

 ,

t ∈ [0, 1.5].

Since the Euclidean dimensions of the systems (3.208) and (3.207) are n = 2 and q = 2, respectively, then due to the results of [33], the system (3.208), (3.207) is Euclidean space output controllable at t = tc = 1.5 if and only if the rank of the matrix !  3 " Fs = X0 B¯ s , X0 A¯ 0s B¯ s , X1 B˜ s , X1 A˜ s B˜ s , . . . , X1 A˜ s B˜ s equals q = 2, where  X0 = X (1.5) =

     0 0 1 0 1.5 − 1 0 0 1.5 − 1 = , X1 = X0 , 0 0 0 1 0.5 3 0 0 0.5 3

B˜ s =



 I2  B¯ s ,  exp A¯ 0s

A˜ s =



A¯ 0s O2×2 A¯ 1s A¯ 0s

 ,

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3 Euclidean Space Output Controllability of Linear Systems with State Delays

and   exp A¯ 0s =



   0.2 exp(5) + 0.8 0.4 exp(5) − 0.4 0 0 , O2×2 = . 0.4 exp(5) − 0.4 0.8 exp(5) + 0.2 0 0

Hence, ⎛

⎞ −2 ⎜ 1⎟ ⎟ B˜ s = ⎜ ⎝ −2 ⎠ , 1

⎛

1 ⎜2 A˜ s = ⎜ ⎝1 3

2 4 2 4

0 0 1 2

⎞ 0 0⎟ ⎟. 2⎠ 4

Note that each block  of the matrix Fs is a vector of the dimension 2. Therefore, to prove that rank Fs = 2, it is sufficient to find two linearly independent blocks of this matrix. Such blocks are     −4 2 X0 B¯ s = , X1 A˜ s B˜ s = . 2 −6 Thus, the system (3.208), (3.207) is Euclidean space output controllable at t = tc = 1.5. Proceed to the dynamic system in (3.209). Since for t = tc = 1.5 the coefficient for the control uf (ξ ) in this system is nonzero, then by virtue of Proposition 3.1, this system is completely Euclidean space controllable at any given ξc > 0. Based on the above presented analysis of the singularly perturbed system (3.205) and its slow (3.208), (3.207) and fast (3.209) subsystems, we can conclude that all the conditions of Theorem 3.18 are fulfilled in this example. Thus, due to Theorem 3.18, the original system (3.205) is Euclidean space output controllable at the time instant tc = 1.5 for all sufficiently small ε > 0.

3.9.6 Example 11: Controlled Car-Following Model in a Simple Open Lane In this example we are going to analyze the system (1.25), which is a controlled singularly perturbed system with state delays of two scales. This system models a car-following process of three vehicles, which follow each other in one lane in the shape of a simple open curve (for details, see Sect. 1.1.5). For the sake of the book’s reading convenience, we write this system here once again dx(θ ) = −x(θ − g) + y(θ − εh), dθ

3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

ε

dy(θ ) = −y(θ − εh) + u(θ ), dθ

211

(3.210)

where θ ∈ [0, θc ]: ε > 0 is a small parameter; the value θc > 0 is independent of ε; g > 0 and h > 0 are constants independent of ε; u(θ ) is a control function. In what follows of this example, we assume that θc ≥ g. The system (3.210) is a particular case of the system (3.185)–(3.186). We study the complete Euclidean space controllability, as well as the controllability with respect to x(t) and the controllability with respect to y(t), of the system (3.210) at the nondimensional time instant θc for all sufficiently small ε > 0. Let us start with the complete Euclidean space controllability. To study this kind of the controllability, we write down the slow and fast subsystems associated with (3.210). The slow subsystem in the differential-algebraic form is dxs (θ ) = −xs (θ − g) + ys (θ ), dθ

θ ∈ [0, θc ],

0 = −ys (θ ) + us (θ ),

θ ∈ [0, θc ],

(3.211)

θ ∈ [0, θc ].

(3.212)

and in the pure differential form is dxs (θ ) = −xs (θ − g) + us (θ ), dθ The fast subsystem has the form dyf (ξ ) = −yf (ξ − h) + uf (ξ ), dξ

ξ ≥ 0.

(3.213)

The system (3.210) satisfies the assumptions (AV)–(AVII). Let us check up whether this system satisfies the assumption (AIII). In this example Eq. (3.113), appearing in the assumption (AIII), becomes as: λ + exp(−λh) = 0.

(3.214)

First, we assume that h
0 such that all roots λ of this equation satisfy the inequality Reλ < −2β, implying the fulfilment of the assumption (AIII) for the system (3.210). Now, in order to apply Theorem 3.23 to the controllability analysis of the system (3.210), we should show the complete Euclidean space controllability of the slow (3.212) and fast (3.213) subsystems. Since the coefficient for the control in (3.212) is nonzero, this subsystem is completely Euclidean space controllable at θ = θc . By the same argument, the fast subsystem (3.213) is completely Euclidean space controllable. Thus, all the conditions of Theorem 3.23 are fulfilled for the system (3.210), meaning that this system is completely Euclidean space controllable at θ = θc for all sufficiently small ε > 0. Remark 3.17 It is important to note the following. Due to the proof of Lemma 3.1 (see Sect. 3.3.4), the control, transferring the system (3.210) from a given statespace initial position to a given Euclidean space terminal position, can be chosen as a continuous function in the interval [0, θc ]. Such a choice is feasible, because the control u(θ ) in this system is the speed of the leading vehicle (for details, see Sect. 1.1.5). Proceed to the controllability analysis of the system (3.210) with respect to x(t) and with respect to y(t) at θ = θc for all sufficiently small ε > 0. The first type of the controllability directly follows from Theorem 3.25, while the second type of the controllability is an immediate consequence of Theorem 3.27. Now, we assume that h ≥ π/2. Due to this inequality, Eq. (3.214) can have roots with nonnegative real parts. Indeed, if h = π/2, then λ = i (i is the imaginary unit) is a root of this equation. Therefore, if h ≥ π/2, the assumption (AIII) can be violated. However, since Eq. (3.213) is scalar and the coefficient for the control in this equation is nonzero, the assumption (AIV) is valid in the present example. Remember that the complete Euclidean space controllability of the fast subsystem (3.213) has already been shown above. Proceed to the analysis of the impulse-free Euclidean space controllability with respect to xs (t) of the slow subsystem (3.211). Since the slow subsystem in the pure differential form (3.212)

3.10 Concluding Remarks and Literature Review

213

is completely Euclidean space controllable at θ = θc , then due to Corollary 3.9, the slow subsystem in the differential-algebraic form (3.211) is impulse-free Euclidean space controllable with respect to xs (t) at θ = θc . Thus, by virtue of Theorem 3.24, the singularly perturbed system (3.210) is completely Euclidean space controllable at θ = θc robustly with respect to ε > 0 for all its sufficiently small values. Similarly to Remark 3.17, it can be shown that the corresponding transferring control can be chosen feasible. If h ≥ π/2, then the controllability of the system (3.210) with respect to x(t) and with respect to y(t) at θ = θc for all sufficiently small ε > 0 directly follows from Theorems 3.26 and 3.28, respectively.

3.10 Concluding Remarks and Literature Review In this chapter, a singularly perturbed linear time-dependent controlled system with time delays (point-wise and distributed) in the state variables was considered. This system consists of a set of time delay differential equations, describing its dynamics, and a set of delay-free algebraic equations, describing its output. Two cases for the delays were treated in the chapter. In the first case, all the delays are small of the order of the small positive multiplier ε for a part of the derivatives in the differential equations. In the second case, the delays in the slow state variable are nonsmall (of order of 1), while the delays in the fast state variable are small (of order of ε), i.e., in this case the delays are of two scales. For each case, the notion of the Euclidean space output controllability of the considered system was introduced. The asymptotic decomposition of this system into two ε-free subsystems, slow and fast ones, was carried out. In the first case, the dynamics of the slow subsystem is described by a differential-algebraic delay-free system, while the dynamics of the fast subsystem is described by a lower Euclidean dimension differential time delay system. In the second case the dynamics of both, slow and fast, subsystems are described by time delay systems. For the slow subsystem, the notion of the impulsefree output controllability at a given time instant was introduced. For the fast subsystem, the notion of the Euclidean space output controllability was presented. Several auxiliary results were derived. Namely, Lemma 3.1 establishes the necessary and sufficient condition for the Euclidean space output controllability at a given time instant of a linear time-dependent system with point-wise and distributed state delays. This result generalizes the result of [3] on the complete Euclidean space controllability of such a system. Lemma 3.1 also generalizes the result of [33] on the Euclidean space output controllability of differential-difference equations with constant coefficients. The proof of Lemma 3.1 is a generalization of the proof of the well-known Kalman’s result [21] on the complete controllability of systems without delays. Lemma 3.1 was obtained in [12]. Lemmas 3.2–3.6 establish the invariance of the controllability properties for the original system, as well as the slow and fast subsystems, with respect to a linear control transformation. Particular cases of these lemmas were obtained (formulated and proven) in [8, 9] where the

214

3 Euclidean Space Output Controllability of Linear Systems with State Delays

complete Euclidean space controllability of the original system was considered, and in [10] where the Euclidean space output controllability of the original system with a special output equation was treated. Note that Lemmas 3.2–3.4 and Lemma 3.6 were presented in [12], however, without proofs. Lemmas 3.7 and 3.8 state the continuity and smoothness of the solution to a hybrid set of Riccati-type matrix equations with respect to a parameter, varying in the closed bounded interval. A particular case of these lemmas was presented in [6]. More general case of these lemmas was considered in [11]. However, in this chapter the simpler proofs of Lemmas 3.7 and 3.8 are proposed than in [11]. Based on the abovementioned asymptotic decomposition of the original singularly perturbed system with small delays and on the auxiliary results, different ε-free sufficient conditions for the Euclidean space output controllability of this system were established in Theorems 3.1–3.8. One such a kind of conditions was obtained in [10], were a singularly perturbed time delay system with a special form of the output equation was analyzed. Theorems 3.1–3.3 and Theorem 3.7–3.8 were formulated and proven (briefly) in [12]. In the present chapter, these theorems are proven in much more detailed manner. Theorems 3.9–3.16 present the ε-free sufficient controllability conditions in three special cases of the original system with small delays. Namely, in the case where the output coincides with the Euclidean part of the entire state variable (the complete Euclidean space controllability), in the case where the output coincides with the slow Euclidean state variable, and in the case where the output coincides with the fast Euclidean state variable. The ε-free sufficient controllability conditions for the complete Euclidean space controllability of singularly perturbed systems with small state delays were derived in [7–9]. A brief analysis of the controllability with respect to the slow and fast Euclidean state variables for a singularly perturbed system with small state delays was carried out in [12]. In Sects. 3.7–3.9, the Euclidean space output controllability of singularly perturbed systems with the delays of two scales is studied. The results of these sections are a straightforward extension of the abovementioned results on singularly perturbed systems with small delays. Example 6 on the controllability in the engagement between two flying vehicles with constant speeds (Sect. 3.6.6) was considered for the first time in [12]. In the present chapter a more detailed analysis of this example is carried out. As it was mentioned above, in the present chapter the Euclidean space controllability (output and complete) was studied for the systems with only state delays. In the next chapter, we will study the Euclidean space controllability for a more general type of systems. Namely, we will study the complete Euclidean space controllability for the systems with state and control delays. The method, proposed in the next chapter for this study, connects the analysis of the complete Euclidean space controllability for the original system with the analysis of the Euclidean space output controllability for a new (auxiliary) system with only state delays. Moreover, this auxiliary system is of a new type, not studied in Theorems 3.1–3.8 of the present chapter.

References

215

References 1. Ait Dads, E.: Some general results and remarks on delay differential equations. In: Arino, O., Hbid, M.L., Ait Dads, E. (eds.) Delay Differential Equations and Applications, pp. 31–40, Springer, Dordrecht (2006) 2. Bellman, R.: Introduction to Matrix Analysis. SIAM, Philadelphia (1997) 3. Delfour, M.C., Mitter, S. K.: Controllability, observability and optimal feedback control of affine hereditary differential systems. SIAM J. Control 10, 298–328 (1972) 4. Delfour, M.C., McCalla, C., Mitter, S.K.: Stability and the infinite-time quadratic cost problem for linear hereditary differential systems. SIAM J. Control 13, 48–88 (1975) 5. Gajic, Z., Lim, M.T.: Optimal Control of Singularly Perturbed Linear Systems and Applications. High Accuracy Techniques. Marsel Dekker, New York (2001) 6. Glizer, V.Y.: Asymptotic solution of a singularly perturbed set of functional-differential equations of Riccati type encountered in the optimal control theory. NoDEA Nonlinear Differ. Equ. Appl. 5, 491–515 (1998) 7. Glizer, V.Y.: Euclidean space controllability of singularly perturbed linear systems with state delay. Syst. Control Lett. 43, 181–191 (2001) 8. Glizer, V.Y.: Controllability of singularly perturbed linear time-dependent systems with small state delay. Dyn. Control 11, 261–281 (2001) 9. Glizer, V.Y.: Controllability of nonstandard singularly perturbed systems with small state delay. IEEE Trans. Autom. Control 48, 1280–1285 (2003) 10. Glizer, V.Y.: Controllability conditions of linear singularly perturbed systems with small state and input delays. Math. Control Signals Syst. 28:1, 1–29 (2016) 11. Glizer, V.Y.: Dependence on parameter of the solution to an infinite horizon linear-quadratic optimal control problem for systems with state delays. Pure Appl. Funct. Anal. 2, 259–283 (2017) 12. Glizer, V.Y.: Euclidean space output controllability of singularly perturbed systems with small state delays. J. Appl. Math. Comput. 57, 1–38 (2018) 13. Glizer, V.Y., Shinar, J.: Optimal evasion from a pursuer with delayed information. J. Optim. Theory Appl. 111, 7–38 (2001) 14. Glizer, V.Y., Turetsky, V.: A linear differential game with bounded controls and two information delays. Optimal Control Appl. Methods 30, 135–161 (2009) 15. Glizer, V.Y., Turetsky, V.: Robust Controllability of Linear Systems. Nova Science Publishers, New York (2012) 16. Glizer, V.Y., Turetsky, V., Shinar, J.: Differential game with linear dynamics and multiple information delays. In: Proceedings of the 13th WSEAS International Conference on Systems, pp. 179–184, Rodos, Greece (2009) 17. Glizer, V.Y., Turetsky, V., Fridman, L., Shinar, J.: History-dependent modified sliding mode interception strategies with maximal capture zone. J. Franklin Inst. 349, 638–657 (2012) 18. Halanay, A.: Differential Equations : Stability, Oscillations, Time Lags. Academic Press, New York (1966) 19. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) 20. Isaacs, R.: Differential Games. Wiley, New York (1967) 21. Kalman, R. E.: Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5, 102–119 (1960) 22. Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic Press, London (1986) 23. Kopeikina, T.B.: The qualitative theory of control processes, Notas de Matematika, vol. 215, 74 p., Departamento de Matematika, Universidad de los Andes, Merida (2001) 24. Shima, T., Shinar, J.: Time-varying linear pursuit-evasion game models with bounded controls. J. Guid. Control Dyn. 25, 425–432 (2002)

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25. Shinar, J., Glizer, V.Y.: Solution of a delayed information linear pursuit-evasion game with bounded controls. Int. Game Theory Rev. 1, 197–217 (1999) 26. Shinar, J., Glizer, V.Y., Turetsky, V.: The effect of pursuer dynamics on the value of linear pursuit-evasion games with bounded controls. In Krivan, V., Zaccour, G. (eds.) Advances in Dynamic Games—Theory, Applications, and Numerical Methods, Annals of the International Society of Dynamic Games, vol. 13, pp. 313–350. Birkhauser, Basel (2013) 27. Silverman, L.M., Meadows, H. E.: Controllability and observability in time-variable linear systems. SIAM J. Control 5, 64–73 (1967) 28. Turetsky, V., Glizer, V.Y.: Continuous feedback control strategy with maximal capture zone in a class of pursuit games. Int. Game Theory Rev. 7, 1–24 (2005) 29. Turetsky, V., Glizer, V.Y.: Robust solution of a time-variable interception problem: a cheap control approach. Int. Game Theory Rev. 9, 637–655 (2007) 30. Turetsky, V., Shinar, J.: Missile guidance laws based on pursuit-evasion game formulations. Automatica J. IFAC 39, 607–618 (2003) 31. Vinter, R.B., Kwong, R.H.: The infinite time quadratic control problem for linear systems with state and control delays: an evolution equation approach. SIAM J. Control Optim. 19, 139–153 (1981) 32. Vladimirov, V.S.: Equations of Mathematical Physics. Marcel Dekker, New York (1971) 33. Zmood, R.B.: On Euclidean space and function space controllability of control systems with delay. Technical report. The University of Michigan, Ann Arbor, MI, p. 99 (1971)

Chapter 4

Complete Euclidean Space Controllability of Linear Systems with State and Control Delays

4.1 Introduction In this chapter, singularly perturbed linear time-dependent controlled systems with multiple point-wise delays and distributed delays in the state and control variables are considered. Two kinds of such systems are analyzed. In the systems of the first kind, the state delays are small of order of a small positive multiplier ε for a part of their derivatives. This multiplier is a parameter of the singular perturbation. In the systems of the second kind, the state delays are of two scales. Namely, the delays in the slow state variable are nonsmall (of order of 1), while the delays in the fast state variable are small (of order of ε). In both kinds of the systems, the control delays are small of order of ε. Two types of the considered singularly perturbed systems, standard and nonstandard, are studied. For each type, two much simpler parameterfree subsystems (the slow and fast ones) are associated with the original system. It is established in the chapter that proper kinds of controllability of the slow and fast subsystems yield the complete Euclidean space controllability of the original system robust with respect to the parameter of singular perturbation for all its sufficiently small values. The following main notations are applied in this chapter: E n is the n-dimensional real Euclidean space. E n1 ×n2 is the linear space of real matrices of the dimension n1 × n2 . The Euclidean norm of either a vector or a matrix is denoted by · . The upper index T denotes the transposition of either a vector x (x T ) or a matrix A (AT ). 5. In denotes the identity matrix of dimension n.

1. 2. 3. 4.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Y. Glizer, Controllability of Singularly Perturbed Linear Time Delay Systems, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-65951-6_4

217

218

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

6. The notation On1 ×n2 is used for the zero matrix of the dimension n1 × n2 , excepting the cases where the dimension of zero matrix is obvious. In such cases, we use the notation 0 for the zero matrix. 7. L2 [t1 , t2 ; E n ] denotes the linear space of all vector-valued functions x(·) : [t1 , t2 ] → E n square integrable in the interval [t1 , t2 ]; for any x(·) ∈ L2 [t1 , t2 ; E n ] and y(·) ∈ L2 [t1 , t2 ; E n ], the inner product in this space is defined as  t2 / 0 x T (t)y(t)dt; x(·), y(·) L2 = t1

the norm of any x(·) ∈ L2 [t1 , t2 ; E n ] is defined as 

x(·) L2 =

t2

1/2 T

x (t)x(t)dt

.

t1

8. L2loc [t¯, +∞; E n ] denotes the linear space of all vector-valued functions x(·) : [t¯, +∞) → E n square integrable in any subinterval [t1 , t2 ] ⊂ [t¯, +∞). 9. L2 [t1 , t2 ; E n1 ×n2 ] denotes the linear space of all matrix-valued functions X(·) : [t1 , t2 ] → E n1 ×n2 , each column of which belongs to L2 [t1 , t2 ; E n1 ]. 10. L2loc [t¯, +∞; E n1 ×n2 ] denotes the linear space of all matrix-valued functions X(·) : [t¯, +∞) → E n1 ×n2 , each column of which belongs to L2loc [t¯, +∞; E n1 ]. 11. W 1,2 [t1 , t2 ; E n ] denotes the corresponding Sobolev space, i.e., the linear space of all vector-valued functions x(·) : [t1 , t2 ] → E n square integrable in the interval [t1 , t2 ] with the first derivatives (generalized) square integrable in this interval. 1,2 ¯ 12. Wloc [t , +∞; E n ] denotes the linear space of all vector-valued functions x(·) : [t¯, +∞) → E n , and such that x(·) ∈ W 1,2 [t1 , t2 ; E n ] for any subinterval [t1 , t2 ] ⊂ [t¯, +∞). 13. W 1,2 [t1 , t2 ; E n1 ×n2 ] denotes the linear space of all matrix-valued functions X(·) : [t1 , t2 ] → E n1 ×n2 , each column of which belongs to W 1,2 [t1 , t2 ; E n1 ]. 1,2 ¯ [t , +∞; E n1 ×n2 ] denotes the linear space of all matrix-valued functions 14. Wloc 1,2 X(·) : [t¯, +∞) → E n1 ×n2 , each column of which belongs to Wloc [t1 , t2 ; E n1 ] for any subinterval [t1 , t2 ] ⊂ [t¯, +∞). 15. col(x, y), where x ∈ E n , y ∈ E m , denotes the column block vector of the dimension n + m with the upper block x and the lower block y, i.e., col(x, y) = (x T , y T )T . 16. Reλ denotes the real part of a complex number λ. 17. The inequality A < (≤) B, where A and B are quadratic symmetric matrices of the same dimension, means that the matrix A − B is negative definite (negative semi-definite).

4.2 System with Small State Delays: Main Notions and Definitions

219

4.2 System with Small State Delays: Main Notions and Definitions 4.2.1 Original System Consider the controlled system

dx(t)  = A1j (t, ε)x(t − εhj ) + A2j (t, ε)y(t − εhj ) dt N

 + +

N 

j =0 0

−h



G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)y(t + εη) dη 

B1j (t, ε)u(t − εhj ) +

0 −h

j =0

H1 (t, η, ε)u(t + εη)dη,

t ≥ 0,

(4.1)

dy(t)  = A3j (t, ε)x(t − εhj ) + A4j (t, ε)y(t − εhj ) dt N

ε

 + +

N  j =0

j =0 0

−h



G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη 

B2j (t, ε)u(t − εhj ) +

0 −h

H2 (t, η, ε)u(t + εη)dη,

t ≥ 0,

(4.2)

where x(t) ∈ E n , y(t) ∈ E m , u(t) ∈ E r (u(t) is a control); ε > 0 is a small parameter; N ≥ 1 is an integer; 0 = h0 < h1 < h2 < . . . < hN = h are some given constants independent of ε; Aij (t, ε), Gi (t, η, ε), Bkj (t, ε), and Hk (t, η, ε) (i = 1, . . . , 4; j = 0, . . . , N; k = 1, 2) are matrix-valued functions of corresponding dimensions, given for t ≥ 0, η ∈ [−h, 0], and ε ∈ [0, ε0 ] (ε0 > 0); Aij (t, ε) and Bkj (t, ε) (i = 1, . . . , 4; j = 0, . . . , N ; k = 1, 2) are continuous in (t, ε) ∈ [0, +∞) × [0, ε0 ]; and the functions Gi (t, η, ε) and Hk (t, η, ε) (i = 1, . . . , 4; k = 1, 2) are piecewise continuous in η ∈ [−h, 0] for any (t, ε) ∈ [0, +∞)×[0, ε0 ], and these functions are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0]. Due to the results of Sect. 2.2, for a given u(·) ∈ L2loc [−εh, +∞; E r ], the system (4.1)–(4.2) is a linear time-dependent nonhomogeneous functional-differential  system. It is infinite-dimensional with the state variables x(t), x(t + εη) and   y(t), y(t + εη) , η ∈ [−h, 0). Moreover, the system (4.1)–(4.2) is a singularly perturbed system.

220

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Let tc > 0 be a given time instant independent of ε. Definition 4.1 For a given ε ∈ (0, ε0 ], the system (4.1)–(4.2) is said to be completely Euclidean space controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , u0 ∈ E r , ϕx (·) ∈ L2 [−εh, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ], ϕu (·) ∈ L2 [−εh, 0; E r ], xc ∈ E n , and yc ∈ E m , there exists a control function u(·) ∈ W 1,2 [0, tc ; E r ] satisfying u(0) = u0 , for which the system (4.1)–(4.2) with the initial and terminal conditions x(τ ) = ϕx (τ ), y(τ ) = ϕy (τ ), u(τ ) = ϕu (τ ), τ ∈ [−εh, 0),

(4.3)

x(0) = x0 ,

y(0) = y0 ,

(4.4)

x(tc ) = xc ,

y(tc ) = yc

(4.5)

has a solution.

4.2.2 Asymptotic Decomposition of the Original System Let us decompose asymptotically the original singularly perturbed system (4.1)– (4.2) into two much simpler ε-free subsystems, the slow and fast ones. The slow subsystem is obtained from (4.1)–(4.2) by setting formally ε = 0 in these controlled functional-differential equations. Thus, the slow subsystem has the form dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) + B1s (t)us (t), t ≥ 0, dt

(4.6)

0 = A3s (t)xs (t) + A4s (t)ys (t) + B2s (t)us (t), t ≥ 0,

(4.7)

where xs (t) ∈ E n and ys (t) ∈ E m are state variables, us (t) ∈ E r is a control, and Ais (t) =

N 

 Aij (t, 0) +

j =0

Bks (t) =

N  j =0

0 −h

 Bkj (t, 0) +

Gi (t, η, 0)dη,

0 −h

Hk (t, η, 0)dη,

i = 1, . . . , 4,

(4.8)

k = 1, 2.

(4.9)

The slow subsystem (4.6)–(4.7) is a descriptor (differential-algebraic) system, and it is delay-free and ε-free. If det A4s (t) = 0,

t ≥ 0,

(4.10)

4.2 System with Small State Delays: Main Notions and Definitions

221

the slow subsystem (4.6)–(4.7) can be converted to an equivalent system, consisting of the explicit expression for ys (t)   ys (t) = −A−1 4s (t) A3s (t)xs (t) + B2s (t)us (t) ,

t ≥ 0,

and the differential equation with respect to xs (t) dxs (t) = A¯ s (t)xs (t) + B¯ s (t)us (t), t ≥ 0, dt

(4.11)

where −1 ¯ A¯ s (t) = A1s (t) − A2s (t)A−1 4s (t)A3s (t), Bs (t) = B1s (t) − A2s (t)A4s (t)B2s (t). (4.12)

The differential equation (4.11) is also called the slow subsystem, associated with the original system (4.1)–(4.2). The fast subsystem is derived from (4.2) in the following way: (a) the terms  containing the state variable x(t), x(t + εη) , η ∈ [−h, 0), are removed from (4.2)

and (b) the transformations of the variables t = t1 + εξ , y(t1 + εξ ) = yf (ξ ), and

u(t1 + εξ ) = uf (ξ ) are made in the resulting system, where t1 ≥ 0 is any fixed time instant. Thus, we obtain the system dyf (ξ )  = A4j (t1 + εξ, ε)yf (ξ − hj ) + dξ N



−h

j =0

+

N 

0

 B2j (t1 + εξ, ε)uf (ξ − hj ) +

0

−h

j =0

G4 (t1 + εξ, η, ε)yf (ξ + η)dη

H2 (t1 + εξ, η, ε)uf (ξ + η)dη.

Finally, setting formally ε = 0 in this system and replacing t1 with t yield the fast subsystem dyf (ξ )  = A4j (t, 0)yf (ξ − hj ) + dξ N

j =0

+

N  j =0

 B2j (t, 0)uf (ξ − hj ) +

0

−h



0 −h

G4 (t, η, 0)yf (ξ + η)dη

H2 (t, η, 0)uf (ξ + η)dη, ξ ≥ 0,

(4.13)

  where t ≥ 0 is a parameter; yf (ξ ) ∈ E m ; yf (ξ ), yf (ξ + η) , η ∈ [−h, 0) is a state variable; and uf (ξ ) ∈ E r (uf (ξ ) is a control).

222

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Like in Chap. 2, the new independent variable ξ is called the stretched time, and it is expressed by the original time t in the form ξ = (t − t1 )/ε. Thus, for any t > t1 , ξ → +∞ as ε → +0. The fast subsystem (4.13) is a differential equation with state and control delays. It is of a lower Euclidean dimension than the original system (4.1)–(4.2), and it is ε-free. Definition 4.2 Subject to (4.10), the system (4.11) is said to be completely controllable at the time instant tc if for any x0 ∈ E n and xc ∈ E n there exists a control function us (·) ∈ L2 [0, tc ; E r ], for which (4.11) has a solution xs (t), t ∈ [0, tc ], satisfying the initial and terminal conditions xs (0) = x0 ,

xs (tc ) = xc .

(4.14)

Definition 4.3 The system (4.6)–(4.7) is said to be impulse-free controllable with respect to xs (t) at the time instant tc if for any x0 ∈ E n and xc ∈ E n , there exists a control function us (·) ∈ L2 [0, tc ; E r ], for which the system (4.6)–(4.7) has a solution col(xs (t), ys (t)) ∈ L2 [0, tc ; E n+m ] satisfying the initial and terminal conditions (4.14). Definition 4.4 For a given t ≥ 0, the system (4.13) is said to be completely Euclidean space controllable if for any y0 ∈ E m , u0 ∈ E r , ϕyf (·) ∈ L2 [−h, 0; E m ], ϕuf (·) ∈ L2 [−h, 0; E r ], and yc ∈ E m , there exist a number ξc > 0, independent of y0 , u0 , ϕyf (·), ϕuf (·), and yc , and a control function uf (·) ∈ W 1,2 [0, ξc ; E r ] satisfying uf (0) = u0 , for which the system (4.13) with the initial and terminal conditions yf (η) = ϕyf (η),

uf (η) = ϕuf (η),

η ∈ [−h, 0);

yf (ξc ) = yc

yf (0) = y0 ,

(4.15) (4.16)

has a solution.

4.3 Preliminary Results In this section, some properties of systems with state and control delays are studied. Based on these results, in the next section different parameter-free conditions for the complete Euclidean space controllability of the original singularly perturbed system (4.1)–(4.2) are derived.

4.3 Preliminary Results

223

4.3.1 Auxiliary System with Small State Delays and Delay-Free Control Consider the dynamic system, consisting of Eqs. (4.1), (4.2), and the equation du(t) = −u(t) + v(t), t ≥ 0. (4.17) dt       In this new system, x(t), x(t + εη) , y(t), y(t + εη) , and u(t), u(t + εη) , η ∈ [−h, 0), are state variables, while v(t) ∈ E r is a control. Thus, in the system (4.1), (4.2), and (4.17), only the state variables have delays, while the control is delay-free. Moreover, in contrast with the original system (4.1)–(4.2), the new system contains two fast modes, Eqs. (4.2) and (4.17) (see Sect. 2.2 for the details on the slow and fast modes of a singularly perturbed differential system). For the new differential system (4.1), (4.2), and (4.17), we consider the algebraic output equation ε

  ζ (t) = Zcol x(t), y(t), u(t) ,

t ≥ 0,

(4.18)

where the (n + m) × (n + m + r)-matrix Z has the block form ! " Z = In+m , 0 .

(4.19)

  It is clear that ζ (t) ∈ E n+m and ζ (t) = col x(t), y(t) , t ≥ 0. Adapting Definition 3.1 (see Sect. 3.2.1) to the system (4.1), (4.2), (4.17), and (4.18), we have the following definition. Definition 4.5 For a given ε ∈ (0, ε0 ], the system (4.1), (4.2), (4.17), and (4.18) is said to be Euclidean space output controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , u0 ∈ E r , ϕx (·) ∈ L2 [−εh, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ], ϕu (·) ∈ L2 [−εh, 0; E r ], xc ∈ E n , and yc ∈ E m , there exists a control function v(·) ∈ L2 [0, tc ; E r ], for which the solution col x(t), y(t), u(t) , t ∈ [0, tc ], of the system (4.1), (4.2), and (4.17) with the initial conditions (4.3), (4.4), and u(0) = u0

(4.20)

satisfies the terminal condition (4.5). Lemma 4.1 For a given ε ∈ (0, ε0 ], the system (4.1)–(4.2) is completely Euclidean space controllable at the time instant tc , if and only if the system (4.1), (4.2), (4.17), and (4.18) is Euclidean space output controllable at this time instant. Proof (Necessity) Let us assume that, for some ε ∈ (0, ε0 ], the system (4.1)–(4.2) is completely Euclidean space controllable at tc . Let x0 ∈ E n , y0 ∈ E m , u0 ∈ E r , ϕx (·) ∈ L2 [−εh, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ], ϕu (·) ∈ L2 [−εh, 0; E r ], and

224

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

xc ∈ E n , yc ∈ E m be arbitrary given. Then, there exists a control function u(·) ∈ W 1,2 [0, tc ; E r ], satisfying u(0) = u0 , for  which the  boundary-value problem (4.1)– (4.2) and (4.3)–(4.5) has a solution col x(t), y(t) , t ∈ [0, tc ]. For the abovementioned control function u(t), consider the function v(t) = ε

du(t) + u(t), t ∈ [0, tc ]. dt

(4.21)

Since u(·) ∈ W 1,2 [0, tc ; E r ], then v(·) ∈ L2 [0, tc ; E r ]. Moreover,  for the control function (4.21), the vector-valued function col x(t), y(t), u(t) , t ∈ [0, tc ], is a solution of the system (4.1), (4.2), and (4.17) satisfying the initial conditions (4.3), (4.4), and (4.20) and the terminal conditions (4.5). The latter, along with the inclusion v(·) ∈ L2 [0, tc ; E r ] and Definition 4.5, means that the system (4.1), (4.2), (4.17), and (4.18) is Euclidean space output controllable at the time instant tc . Thus, the necessity is proven. Sufficiency The sufficiency is proven similarly to the necessity.  

Thus, the lemma is proven.

Now, let us decompose asymptotically the system (4.1), (4.2), (4.17), and (4.18) into the slow and fast subsystems. These subsystems are constructed similarly to the results of Sect. 3.2.2. Let us start with the slow subsystem. The dynamic part of this subsystem is obtained from (4.1), (4.2), and (4.17) by setting formally ε = 0 in these controlled functional-differential equations. The output part of the slow subsystem is obtained from (4.18) by removing formally the terms with the    Euclidean parts  y(t) and u(t) of the fast state variables y(t), y(t + εη) and u(t), u(t + εη) , η ∈ [−h, 0) (see Sect. 2.2 for the details on the slow and fast state variables). Thus, the slow subsystem has the form dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) + B1s (t)us (t), dt 0 = A3s (t)xs (t) + A4s (t)ys (t) + B2s (t)us (t), 0 = −us (t) + vs (t), ζs (t) = xs (t),

t ≥ 0, t ≥ 0,

t ≥ 0,

(4.22) (4.23) (4.24)

t ≥ 0.

(4.25)

In (4.22)–(4.24) and (4.25), xs (t) ∈ E n , ys (t) ∈ E m , and us (t) ∈ E r are state variables, vs (t) ∈ E r is a control, ζs (t) ∈ E n is an output, and Ais (t) and Bks (t) (i = 1, . . . , 4; k = 1, 2) are given in (4.8)–(4.9). Eliminating the state variable us (t) from Eqs. (4.22)–(4.24), we obtain us (t) = vs (t) and the system dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) + B1s (t)vs (t), dt

t ≥ 0,

(4.26)

4.3 Preliminary Results

225

0 = A3s (t)xs (t) + A4s (t)ys (t) + B2s (t)vs (t),

t ≥ 0.

(4.27)

This system, along with the output equation (4.25), constitutes the simplified slow subsystem associated with the system (4.1), (4.2), (4.17), and (4.18). Finally, subject to the condition (4.10), the slow subsystem (4.26)–(4.27) and (4.25) can be reduced to the system, consisting of the differential equation dxs (t) = A¯ s (t)xs (t) + B¯ s (t)vs (t), dt

t ≥ 0,

(4.28)

and the output equation (4.25). Note that the matrix-valued coefficients A¯ s (t) and B¯ s (t) are given in (4.12). Remark 4.1 Comparison of the differential equations (4.28) and (4.11) directly yields that the former can be obtained from the latter by replacing in it us (t) with vs (t) and vice versa. Similar conclusion is correct for the differential-algebraic systems (4.26)–(4.27) and (4.6)–(4.7). Moreover, the output in the systems (4.28), (4.25) and (4.26)–(4.27), (4.25) coincides with xs (t). Hence, the output controllability of the system (4.28) and (4.25) coincides with the complete controllability of the system (4.28) and, therefore, it is equivalent to the complete controllability of the system (4.11). Similarly, the impulse-free output controllability of the system (4.26)–(4.27) and (4.25) coincides with the impulse-free controllability with respect to xs (t) of the system (4.26)–(4.27) and, therefore, it is equivalent to the impulse-free controllability with respect to xs (t) of the system (4.6)–(4.7). Proceed to the fast subsystem, associated with the system (4.1), (4.2), (4.17), and (4.18). The dynamic part of this subsystem is constructed similarly to the fast subsystem (4.13), associated with the original system (4.1)–(4.2). The output part of the fast subsystem is obtained from (4.18)  by removingformally the term with the Euclidean part x(t) of the state variable x(t), x(t + εη) , η ∈ [−h, 0). Thus the fast subsystem, associated with the auxiliary system (4.1), (4.2), (4.17), and (4.18), consists of the differential equations dyf (ξ )  = A4j (t, 0)yf (ξ − hj ) + dξ N

j =0

+

N  j =0

 B2j (t, 0)uf (ξ − hj ) +

0

−h



0 −h

G4 (t, η, 0)yf (ξ + η)dη

H2 (t, η, 0)uf (ξ + η)dη, ξ ≥ 0,

duf (ξ ) = −uf (ξ ) + vf (ξ ), dξ

ξ ≥ 0,

(4.29)

(4.30)

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4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

and the output equation ζf (ξ ) = yf (ξ ),

ξ ≥ 0,

(4.31)

  where t ≥ 0 is a parameter; yf (ξ ) ∈ E m , uf (ξ ) ∈ E r ; yf (ξ ), yf (ξ + η) and   uf (ξ ), uf (ξ + η) , η ∈ [−h, 0), are state variables; vf (ξ ) ∈ E r is a control; and ζf (ξ ) ∈ E m is an output. Note that in contrast with the system (4.13), in the differential system (4.29)– (4.30) both variables yf (·) and uf (·) are time-delayed state variables, while vf (ξ ) is an undelayed control. Definition 4.6 For a given t ≥ 0, the system (4.29)–(4.30) and (4.31) is said to be Euclidean space output controllable if for any y0 ∈ E m , u0 ∈ E r , ϕyf (·) ∈ L2 [−h, 0; E m ], ϕuf (·) ∈ L2 [−h, 0; E r ], and yc ∈ E m , there exist a number ξc > 0, independent of y0 , u0 , ϕyf (·), ϕuf (·), and  yc , and a control function vf (·) ∈ L2 [0, ξc ; E r ], for which the solution col yf (ξ ), uf (ξ ) , ξ ∈ [0, ξc ], of the system (4.29)–(4.30) with the initial conditions yf (η) = ϕyf (η),

uf (η) = ϕuf (η),

η ∈ [−h, 0);

yf (0) = y0 ,

uf (0) = u0 (4.32)

satisfies the terminal condition (4.16). Lemma 4.2 For a given t ≥ 0, the system (4.13) is completely Euclidean space controllable if and only if the system (4.29)–(4.30) and (4.31) is Euclidean space output controllable. Proof (Sufficiency) Let us assume that, for some given t ≥ 0, the system (4.29)– (4.30) and (4.31) is Euclidean space output controllable. Let y0 ∈ E m , u0 ∈ E r , ϕyf (·) ∈ L2 [−h, 0; E m ], ϕuf (·) ∈ L2 [−h, 0; E r ], and yc ∈ E m be arbitrary given. Then, there exists a number ξc > 0, independent of y0 , u0 , ϕyf (·), ϕuf (·), and yc , and a control function vf (·) ∈ L2 [0, ξc ; E r ], for which the system (4.29)–(4.30) with the initial (4.32) and terminal (4.16) conditions has a  solution col yf (ξ ), uf (ξ ) , ξ ∈ [0, ξc ]. Note that the component uf (ξ ) of this solution satisfies the condition uf (0) = u0 . Moreover, since vf (·) ∈ L2 [0, ξc ; E r ], then uf (ξ ) ∈ W 1,2 [0, ξc ; E r ]. Thus, for the control function uf (ξ ), the vectorvalued function yf (ξ ), ξ ∈ [0, ξc ], is a solution of the system (4.13) satisfying the initial (4.15) and terminal (4.16) conditions. The latter, along with Definition 4.4, implies the complete Euclidean space controllability of the system (4.13) for the given t ≥ 0. Thus, the sufficiency is proven. Necessity The necessity is proven similarly to the Lemma 4.1. Thus, the lemma is proven.

 

4.3 Preliminary Results

227

4.3.2 Output Controllability of the Auxiliary System and Its Slow and Fast Subsystems: Necessary and Sufficient Conditions 4.3.2.1

Equivalent Forms of the Auxiliary System

For a given ε ∈ (0, ε0 ], let us introduce the block vector   ω(t) = col y(t), u(t) , t ≥ −εh, and the block matrices ! " A1j (t, ε)=A1j (t, ε), A2j (t, ε)= A2j (t, ε), B1j (t, ε) , j =0, 1, . . . , N, t ≥ 0, (4.33)   A3j (t, ε) , j = 0, 1, . . . , N, t ≥ 0, (4.34) A3j (t, ε) = Or×n  A40 (t, ε) =  A4j (t, ε) =

 A40 (t, ε) B20 (t, ε) , − Ir Or×m

t ≥ 0,

 A4j (t, ε) B2j (t, ε) , j = 1, . . . , N, t ≥ 0, Or×r Or×m

G1 (t, η, ε) = G1 (t, η, ε),

G3 (t, η, ε) =

(4.36)

! " G2 (t, η, ε) = G2 (t, η, ε), H1 (t, η, ε) , t ≥ 0,



(4.35)

 G3 (t, η, ε) , Or×n

 G4 (t, η, ε) =

η, ∈ [−h, 0],

 G4 (t, η, ε) H2 (t, η, ε) , Or×r Or×m t ≥ 0,

 B1 = On×r ,

B2 =

(4.37)

Om×r Ir

η ∈ [−h, 0], (4.38)

 .

(4.39)

Based on the above introduced vector and matrices, we can rewrite the auxiliary system (4.1), (4.2), (4.17), and (4.18) in the equivalent form

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4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

$ dx(t)  # = A1j (t, ε)x(t − εhj ) + A2j (t, ε)ω(t − εhj ) dt N

 +

j =0

0 −h

# $ G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)ω(t + εη) dη + B1 v(t), t ≥ 0, (4.40) $ dω(t)  # = A3j (t, ε)x(t − εhj ) + A4j (t, ε)ω(t − εhj ) dt N

ε  +

j =0

0 −h

# $ G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)ω(t + εη) dη + B2 v(t), t ≥ 0, (4.41)   ζ (t) = Zcol x(t), ω(t) , t ≥ 0.

(4.42)

Now, for a given ε ∈ (0, ε0 ], let us consider the block vector   z(t) = col x(t), ω(t) ,

t ≥ −εh,

and the block matrices   A2j (t, ε) A (t, ε) Aj (t, ε) = 1 1j , j = 0, 1, . . . , N, 1 ε A3j (t, ε) ε A4j (t, ε)  G (t, η, ε) =

 G2 (t, η, ε) G1 (t, η, ε) , 1 1 ε G3 (t, η, ε) ε G4 (t, η, ε)

 B(ε) =

B1 1 ε B2

(4.43)  .

(4.44)

Thus, the system (4.40)–(4.42) can be rewritten in the equivalent form dz(t)  = Aj (t, ε)z(t − εhj )+ dt N

j =0



0

−h

G (t, η, ε)z(t + εη)dη+B(ε)v(t), t ≥ 0, (4.45)

ζ (t) = Zz(t), t ≥ 0.

(4.46)

It is clear that the system (4.45)–(4.46) is equivalent to the auxiliary system (4.1), (4.2), (4.17), and (4.18).

4.3 Preliminary Results

4.3.2.2

229

Output Controllability of the Auxiliary System

Definition 4.7 For a given ε ∈ (0, ε0 ], the system (4.45)–(4.46) is said to be Euclidean space output controllable at the time instant tc if for any z0 ∈ E n+m+r , ϕz (·) ∈ L2 [−εh, 0; E n+m+r ], and ζc ∈ E n+m , there exists a control function v(·) ∈ L2 [0, tc ; E r ], for which the solution z(t), t ∈ [0, tc ], of the system (4.45) with the initial conditions z(τ ) = ϕz (τ ), τ ∈ [−h, 0), and z(0) = z0 satisfies the terminal condition Zz(tc ) = ζc . Let, for a given ε ∈ (0, ε0 ], the (n + m + r) × (n + m + r)-matrix-valued function Ψ (σ, ε), σ ∈ [0, tc ], be a solution of the terminal-value problem N   T dΨ (σ, ε) =− Aj (σ + εhj , ε) Ψ (σ + εhj , ε) dσ

 −

j =0

0 −h

 T G (t − εη, η, ε) Ψ (σ − εη, ε)dη, Ψ (tc , ε) = In+m+r ;

σ ∈ [0, tc ),

Ψ (σ, ε) = 0, σ > tc ,

(4.47) (4.48)

where it is assumed that Aij (t, ε) = Aij (tc , ε), Gi (t, η, ε) = Gi (tc , η, ε), t > tc , η ∈ [−h, 0], and ε ∈ [0, ε0 ], (i = 1, . . . , 4; j = 1, . . . , N ). Due to the results of Sect. 2.2.3 (see Proposition 2.1) and Sect. 2.2.4 (see Remark 2.3), Ψ (σ, ε) exists and is unique for σ ∈ [0, tc ], ε ∈ (0, ε0 ]. Consider the following two matrices of the dimensions (n + m + r) × (n + m + r) and (n + m) × (n + m), respectively:  W (tc , ε) =

tc

Ψ T (σ, ε)B(ε)B T (ε)Ψ (σ, ε)dσ

(4.49)

0

and WZ (tc , ε) = ZW (tc , ε)Z T .

(4.50)

Proposition 4.1 For a given ε ∈ (0, ε0 ], the auxiliary system (4.1), (4.2), (4.17), and (4.18) is Euclidean space output controllable at the time instant tc if and only if the matrix WZ (tc , ε) is nonsingular, i.e., det WZ (tc , ε) = 0. Proof By virtue of the results of Sect. 3.3.1 (see Definition 3.5, Lemma 3.1, Remark 3.5, and Corollary 3.1), the system (4.45)–(4.46) is Euclidean space output controllable at the time instant tc if and only if det WZ (tc , ε) = 0. Since this system is equivalent to the auxiliary system (4.1), (4.2), (4.17), and (4.18), then, due to Definitions 4.5 and 4.7, the auxiliary system is also Euclidean space output controllable at tc if and only if det WZ (tc , ε) = 0. This completes the proof of the proposition.  

230

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Remark 4.2 Since the system (4.40)–(4.42) is equivalent to the auxiliary system (4.1), (4.2), (4.17), and (4.18), the fulfillment of the inequality det WZ (tc , ε) = 0 is necessary and sufficient for the Euclidean space output controllability of (4.40)– (4.42) at the time instant tc .

4.3.2.3

Output Controllability of the Slow and Fast Subsystems Associated with the Auxiliary System

We start with the slow subsystem (4.28). Let, for a given tc > 0, the n × n-matrix-valued function Ψs (σ ), σ ∈ [0, tc ], be the unique solution of the terminal-value problem  T dΨs (σ ) = − A¯ s (σ ) Ψs (σ ), σ ∈ [0, tc ), Ψs (tc ) = In . dσ

(4.51)

Consider the n × n-matrix  Ws (tc ) =

tc

0

ΨsT (σ )B¯ s (σ )B¯ s T (σ )Ψs (σ )dσ.

Applying Remark 4.1, Definition 4.2, and the results of Sect. 3.3.1 (see Definition 3.6, Lemma 3.1, Remark 3.5, Corollary 3.1, and Proposition 3.1) in the particular (delay-free) case to the slow subsystem (4.28), we have the following proposition. Proposition 4.2 Let the condition (4.10) be fulfilled in the interval [0, tc ]. Then, the slow subsystem (4.28), associated with the auxiliary system (4.1), (4.2), (4.17), and (4.18), is completely controllable at the time instant tc , if and only if the matrix Ws (tc ) is nonsingular, i.e., det Ws (tc ) = 0. Proceed to the fast subsystem (4.29)–(4.30) and (4.31) of the auxiliary system (4.1), (4.2), (4.17), and (4.18). First, based on the equivalent form (4.40)–(4.42) of the auxiliary system, we can rewrite equivalently its fast subsystem as follows: dωf (ξ )  = A4j (t, 0)ωf (ξ − hj ) + dξ N

j =0



0 −h

G4 (t, η, 0)ωf (ξ + η)dη +B2 vf (ξ ),

  ζf (ξ ) = Ωf ωf (ξ ), ξ ≥ 0, Ωf = Im , Om×r , where t ≥ 0 is a parameter and   ωf (ξ ) = col yf (ξ ), uf (ξ ) ,

ξ ≥ −h.

ξ ≥ 0,

(4.52) (4.53)

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231

Definition 4.8 For a given t ≥ 0, the system (4.52)–(4.53) is said to be Euclidean space output controllable if for any ω0 ∈ E m+r , ϕωf (·) ∈ L2 [−h, 0; E m+r ], and ζc ∈ E m , there exist a number ξc > 0, independent of ω0 , ϕωf (·), and ζc , and a control function vf (·) ∈ L2 [0, ξc ; E r ], for which the solution ωf (ξ ), ξ ∈ [0, ξc ], of the system (4.52) with the initial conditions ωf (η) = ϕωf (η), η ∈ [−h, 0), and ωf (0) = ω0 satisfies the terminal condition Ωf ωf (ξc ) = ζc . Let, for any given t ≥ 0, the (m + r) × (m + r)-matrix-valued function Ψf (ξ, t) be the unique solution of the following initial-value problem: T dΨf (ξ )   = A4j (t, 0) Ψf (ξ − hj ) dξ N

 +

j =0

0 −h

 T G4 (t, η, 0) Ψf (ξ + η)dη, ξ > 0,

Ψf (ξ ) = 0, ξ < 0,

Ψf (0) = Im+r .

(4.54)

Consider the m × m-matrix-valued function  Wf (ξ, t) = Ωf 0

ξ

ΨfT (ρ, t)B2 B2T Ψf (ρ, t)dρΩfT ,

ξ ≥ 0,

t ≥ 0.

(4.55)

Proposition 4.3 For a given t ≥ 0, the fast subsystem (4.29)–(4.30) and (4.31) of the auxiliary system (4.1), (4.2), (4.17), and (4.18) is Euclidean space output controllable if and only if there exists a number ξc > 0 such that the matrix Wf (ξc , t) is nonsingular, i.e., det Wf (ξc , t) = 0. Proof By virtue of the results of Sect. 3.3.1 (see Definition 3.5, Lemma 3.1, Remark 3.5, and Corollary 3.1) and Sect. 2.2.4 (see Remark 2.4), the system (4.52)– (4.53) is Euclidean space output controllable if and only if det Wf (ξc , t) = 0 for some ξc > 0. Since this system is equivalent to the fast subsystem (4.29)–(4.30) and (4.31), then, by Definitions 4.6 and 4.8, the latter subsystem is also Euclidean space output controllable if and only if det Wf (ξc , t) = 0. Thus, the proposition is proven.  

4.3.3 Linear Control Transformation in the Original System with Small State Delays Let us transform the control u(t) in the system (4.1)–(4.2) as follows: u(t) = K(t)y(t) + w(t),

(4.56)

232

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

where w(t) ∈ E r is a new control and K(t) is some r × m-matrix-valued function 1,2 given for t ≥ −ε0 h and K(·) ∈ Wloc [−ε0 h, +∞; E r×m ]. Due to this transformation, the system (4.1)–(4.2) becomes as $ dx(t)  # = A1j (t, ε)x(t − εhj ) + AK 2j (t, , ε)y(t − εhj ) dt N

 + +

N 

j =0

0 −h



G1 (t, η, ε)x(t + εη) + GK 2 (t, η, ε)y(t + εη) dη 

B1j (t, ε)w(t − εhj ) +

j =0

0

−h

H1 (t, η, ε)w(t + εη)dη, t ≥ 0,

(4.57)

$ dy(t)  # A3j (t, ε)x(t − εhj ) + AK (t, ε)y(t − εh ) = j 4j dt N

ε

 + +

N 

j =0

0 −h



G3 (t, η, ε)x(t + εη) + GK 4 (t, η, ε)y(t + εη) dη 

B2j (t, ε)w(t − εhj ) +

j =0

0

−h

H2 (t, η, ε)w(t + εη)dη, t ≥ 0,

(4.58)

where AK 2j (t, ε) = A2j (t, ε) + B1j (t, ε)K(t − εhj ), j = 0, 1, . . . , N, AK 4j (t, ε) = A4j (t, ε) + B2j (t, ε)K(t − εhj ), j = 0, 1, . . . , N, GK 2 (t, η, ε) = G2 (t, η, ε) + H1 (t, η, ε)K(t + εη), GK 4 (t, η, ε) = G4 (t, η, ε) + H2 (t, η, ε)K(t + εη).

(4.59)

Lemma 4.3 For a given ε ∈ (0, ε0 ], the system (4.1)–(4.2) is completely Euclidean space controllable at the time instant tc , if and only if the system (4.57)–(4.58) is completely Euclidean space controllable at this time instant. Proof (Necessity) Let us assume that, for some ε ∈ (0, ε0 ], the system (4.1)–(4.2) is completely Euclidean space controllable at tc . Let x0 ∈ E n , y0 ∈ E m , ϕx (·) ∈ L2 [−εh, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ], xc ∈ E n , and yc ∈ E m be arbitrary given. Let also w0 ∈ E r and ϕw (·) ∈ L2 [−εh, 0; E r ] be arbitrary given. Let u0 = K(0)y0 + w0 ,

ϕu (η) = K(η)ϕy (η) + ϕw (η), η ∈ [−εh, 0).

(4.60)

4.3 Preliminary Results

233

Since K(·) ∈ W 1,2 [−εh, 0; E r×m ], ϕy (·) ∈ L2 [−εh, 0; E m ], and ϕw (·) ∈ L2 [−εh, 0; E r ], then ϕu (·) ∈ L2 [−εh, 0; E r ]. By virtue of Definition 4.1, there exists a control function u(·) ∈ W 1,2 [0, tc ; E r ], satisfying u(0) = u0 , for which the system (4.1)–(4.2) subject to the initial and   terminal conditions (4.3)–(4.5) has a solution col x(t), y(t) , t ∈ [0, tc ]. For the abovementioned control function u(t), t ∈ [0, tc ], the system (4.1)–(4.2) can be rewritten in the equivalent form $ dx(t)  # = A1j (t, ε)x(t − εhj ) + AK 2j (t, , ε)y(t − εhj ) dt N

j =0

 +

0 −h

+

N 

B1j (t, ε)[u(t − εhj ) − K(t − εhj )y(t − εhj )]

j =0

 +

G1 (t, η, ε)x(t + εη) + GK 2 (t, η, ε)y(t + εη) dη

0 −h

H1 (t, η, ε)[u(t + εη) − K(t + εη)y(t + εη)]dη, t ≥ 0,

(4.61)

$ dy(t)  # A3j (t, ε)x(t − εhj ) + AK = 4j (t, ε)y(t − εhj ) dt N

ε

 +

j =0

0 −h

+  +



G3 (t, η, ε)x(t + εη) + GK 4 (t, η, ε)y(t + εη) dη

N 

B2j (t, ε)[u(t − εhj ) − K(t − εhj )y(t − εhj )]

j =0 0 −h

H2 (t, η, ε)[u(t + εη) − K(t + εη)y(t + εη)]dη, t ≥ 0.

(4.62)

Denote w(t) = u(t) − K(t)y(t), t ∈ [−εh, tc ].

(4.63)

Since u(·) ∈ W 1,2 [0, tc ; E r ] and y(t) is the component of the solution to the initialvalue problem (4.1)–(4.2) and (4.3)–(4.4), then y(t) ∈ W 1,2 [0, tc ; E m ]. Due to this observation and the inclusion K(·) ∈ W 1,2 [0, tc ; E r×m ], we obtain that w(·) ∈ W 1,2 [0, tc ; E r ].

(4.64)

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4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Moreover, due to (4.60), w(0) = w0 ,

w(η) = ϕw (η), η ∈ [−εh, 0).

(4.65)

Now, using the equivalent form (4.61)–(4.62) of the system (4.1)–(4.2), we directly have that the control function w(t), given by (4.63) in the interval (0, tc ], and satisfying (4.64)–(4.65), provides the existence of solution to the system (4.57)– (4.58) subject to the first two conditions in (4.3) and the conditions (4.4) and (4.5). The latter, along with Definition 4.1, implies the complete Euclidean space controllability of the system (4.57)–(4.58) at the time instant tc . Thus, the necessity is proven. Sufficiency The sufficiency is proven similarly to the necessity.   Now, let us decompose asymptotically the singularly perturbed system (4.57)– (4.58) into the slow and fast subsystems. This decomposition is carried out similarly to that for the system (4.1)–(4.2). Thus, the slow subsystem, associated with (4.57)– (4.58), is the differential-algebraic system dxs (t) = A1s (t)xs (t) + AK 2s (t)ys (t) + B1s (t)ws (t), t ≥ 0, dt

(4.66)

0 = A3s (t)xs (t) + AK 4s (t)ys (t) + B2s (t)ws (t), t ≥ 0,

(4.67)

where xs (t) ∈ E n and ys (t) ∈ E m are state variables, ws (t) ∈ E r is a control, the matrices A1s (t), A3s (t), B1s (t), and B2s (t) are given in (4.8)–(4.9), and AK ls (t)

=

N  j =0

 AK lj (t, 0) +

0 −h

GK l (t, η, 0)dη,

l = 2, 4.

(4.68)

If det AK 4s (t) = 0,

t ≥ 0,

(4.69)

the differential-algebraic system (4.66)–(4.67) can be converted to an equivalent system consisting of the explicit expression for ys (t)  −1   A3s (t)xs (t) + B2s (t)ws (t) , ys (t) = − AK 4s (t)

t ≥ 0,

and the differential equation with respect to xs (t) dxs (t) ¯ = A¯ K s (t)xs (t) + B dt

K s (t)ws (t),

t ≥ 0,

(4.70)

4.3 Preliminary Results

235

where  K −1 K A3s (t), A¯ K s (t) = A1s (t) − A2s (t) A4s (t) B¯

K s (t)

 K −1 = B1s (t) − AK B2s (t). 2s (t) A4s (t)

The fast subsystem, associated with (4.57)–(4.58), is the differential equation with state and control delays dyf (ξ )  K = A4j (t, 0)yf (ξ − hj ) + dξ N

j =0

+

N  j =0

 B2j (t, 0)wf (ξ − hj ) +

0

−h



0 −h

GK 4 (t, η, 0)yf (ξ + η)dη

H2 (t, η, 0)wf (ξ + η)dη, ξ ≥ 0.

(4.71)

Note that in (4.71), t ≥ 0 is a parameter, while ξ is an independent variable.  Moreover, in this equation, yf (ξ ) ∈ E m ; yf (ξ ), yf (ξ + η) , η ∈ [−h, 0) is a state variable; and wf (ξ ) ∈ E r (wf (ξ ) is a control). Lemma 4.4 The system (4.6)–(4.7) is impulse-free controllable with respect to xs (t) at the time instant tc if and only if the system (4.66)–(4.67) is impulse-free controllable with respect to xs (t) at this time instant. Lemma 4.5 Let the condition (4.69) be satisfied. Then, the system (4.66)–(4.67) is impulse-free controllable with respect to xs (t) at the time instant tc , if and only if the system (4.70) is completely controllable at this time instant. Lemma 4.6 Let the conditions (4.10) and (4.69) be valid. Then, the system (4.11) is completely controllable at the time instant tc if and only if the system (4.70) is completely controllable at this time instant. Lemmas 4.4, 4.5, and 4.6 are proven similarly to Lemmas 3.3, 3.4, and 3.5, respectively (see Sect. 3.3.2.2 for the details). Lemma 4.7 For a given t ≥ 0, the system (4.13) is completely Euclidean space controllable if and only if the system (4.71) is completely Euclidean space controllable. Proof First of all, let us note that the following transformation of the control in Eq. (4.13): uf (ξ ) = K(t)yf (ξ ) + wf (ξ ), where wf (ξ ) is a new control, converts this equation to Eq. (4.71). Based on this observation, the lemma is proven similarly to Lemma 4.3.  

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4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

4.3.4 Stabilizability of a Parameter-Dependent System with State and Control Delays by a Memory-Less Feedback Control In this section, we derive sufficient conditions, subject to which the system (4.13) can be stabilized by the memory-less state-feedback control uf (ξ ) = K(t)yf (ξ ),

ξ ≥ 0,

(4.72)

where K(t), t ∈ [0, tc ], is a parameter-dependent gain matrix of the dimension r × m. Also we show that, the gain K(t) can be chosen as a continuously differentiable function in the interval [0, tc ]. Substitution of the control (4.72) into the system (4.13) yields the closed-loop system dyf (ξ )  K = A4j (t, 0)yf (ξ − hj ) + dξ N

j =0



0

−h

GK 4 (t, η, 0)yf (ξ + η)dη, ξ ≥ 0, (4.73)

where, due to (4.59), AK 4j (t, 0) = A4j (t, 0) + B2j (t, 0)K(t), j = 0, 1, . . . , N, GK 4 (t, η, 0) = G4 (t, η, 0) + H2 (t, η, 0)K(t). Definition 4.9 For a given t ∈ [0, tc ], the system (4.13) is called stabilizable by the memory-less state-feedback control (4.72) if for any y0 ∈ E m , ϕyf (·) ∈ L2 [−h, 0; E m ], there exist positive values af (t) and βf (t) such that the solution yf (ξ ), ξ ≥ 0 of the closed-loop system (4.73) subject to the initial conditions yf (η) = ϕyf (η), η ∈ [−h, 0),

yf (0) = y0 ,

(4.74)

satisfies the inequality 2   # y0 yf (ξ ) ≤ af (t) exp − βf (t)ξ + ϕyf (·)

2 $1/2 L2

, ξ ≥ 0. (4.75)

If this inequality is valid for all t ∈ [0, tc ] and the values af (t) and βf (t) are independent of t ∈ [0, tc ], then the system (4.13) is called stabilizable by the memory-less state-feedback control (4.72) uniformly with respect to t ∈ [0, tc ].

4.3 Preliminary Results

237

In what follows, we assume that (AI)

the matrix-valued function G4 (t, η, 0) satisfies the inequality  T G4 (t, η, 0) G4 (t, η, 0) < CG (t),

t ∈ [0, tc ],

η ∈ [−h, 0],

where the m × m-matrix CG (t) is symmetric and positive definite for any t ∈ [0, tc ], and this matrix is a continuous function of t ∈ [0, tc ]; (AII)

the matrix-valued function H2 (t, η, 0) satisfies the inequality  T H2 (t, η, 0) H2 (t, η, 0) < CH (t),

t ∈ [0, tc ],

η ∈ [−h, 0],

where the r × r-matrix CH (t) is symmetric and positive semi-definite for any t ∈ [0, tc ], and this matrix is a continuous function of t ∈ [0, tc ]. Let P and Qj (j = 1, . . . , N ) be any symmetric matrices of dimension m × m. Let V be any matrix of dimension r × m. Based on these matrices, we construct the following matrices for t ∈ [0, tc ]: T (t, 0) Γ (P , Q1 , . . . , QN , V , t) = A40 (t, 0)P + P AT40 (t, 0) + B20 (t, 0)V + V T B20

+

N    T B2j (t, 0)B2j (t, 0) + Qj + 2hIm , j =1

(4.76) ! " Δ(P , V , t) = A41 (t, 0)P , . . . , A4N (t, 0)P , V T , P ,

(4.77)

!  −1  −1 " , Λ(Q1 , . . . , QN , t) = diag Q1 , . . . , QN , NIr + CH (t) , hCG (t) (4.78)  Θ(P , Q1 , . . . , QN , V , t) =

 Δ(P , V , t) Γ (P , Q1 , . . . , QN , t) , − Λ(Q1 , . . . , QN , t) ΔT (P , V , t) (4.79)

" ! Υ (P , Q1 , . . . , QN , V , t)=diag Θ(P , Q1 , . . . , QN , V , t), −P , −Q1 , . . . , −QN . (4.80) m× Remark 4.3 Note that the matrix Γ (P , Q1 , . . . , QN , t) is of the dimension  m, the matrix Δ(P , V , t) is of the dimension m × (N + 1)m + r , the matrix

238

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

    Λ(Q1 , . . . , QN , t) is of the dimension (N + 1)m  + r × (N + 1)m  + r , the matrix  Θ(P , Q1 , . . . , QN , V , t) is of the dimension (N + 2)m + r  × (N + 2)m +r , and  the matrix Υ(P , Q1 , . . . , QN , V , t) is of the dimension (2N + 3)m + r × (2N + 3)m + r . Lemma 4.8 Let the assumptions (AI)–(AII) be valid. Let, for any t ∈ [0, tc ], the following Linear Matrix Inequality (LMI) with respect to the symmetric m × mmatrices P and Q1 , . . . , QN and the r × m-matrix V : Υ (P , Q1 , . . . , QN , V , t) < 0

(4.81)

have a solution. Then, there exists an r × m-matrix-valued function K(t), continuously differentiable in the interval [0, tc ], such that the system (4.13) is stabilizable by the memory-less state-feedback control (4.72) uniformly with respect to t ∈ [0, tc ]. Proof of the lemma is presented in Sect. 4.3.5 Corollary 4.1 Let the assumptions (AI)–(AII) be valid. Let, for any t ∈ [0, tc ], the Linear Matrix Inequality (4.81) with respect to the symmetric m × m-matrices P and Q1 , . . . , QN and the r × m-matrix V have a solution. Then, there exists an r × m-matrix-valued function K(t), continuously differentiable for t ∈ [−ε0 h, tc ], such that all roots λ(t) of the equation ⎤ ⎡  0 N  ⎦ AK GK det ⎣λIm − 4j (t, 0) exp(−λhj ) − 4 (t, η, 0) exp(λη)dη = 0 −h

j =0

(4.82) satisfy the inequality Reλ(t) < −2βK

t ∈ [0, tc ],

(4.83)

K where AK 4j (t, ε) (j = 0, 1, . . . , N) and G4 (t, η, 0) are given in (4.59) and βK > 0 is some constant.

Proof First of all, let us note that, for any t ∈ [0, tc ], Eq. (4.82) is the characteristic equation of the system (4.73) (for the details on the characteristic equations, see Remark 2.2 in Sect. 2.2.4). Due to Lemma 4.8 and Definition 4.9, there exists an r × m-matrix-valued function K(t), continuously differentiable for t ∈ [0, tc ], such that the solution of the initial-value problem (4.73) and (4.74) satisfies the inequality (4.78) for all t ∈ [0, tc ]. In this inequality, af > 0 and βf > 0 are some constants. This observation, along with the results of [4] (Theorem 5.3), directly yields that all the roots λ(t) of the characteristic equation (4.82) of the system (4.73) satisfy the inequality (4.83). Now, a smooth extension of the matrix-valued function K(t) from the interval [0, tc ] to the interval [−ε0 h, tc ] completes the proof of the corollary.  

4.3 Preliminary Results

239

4.3.5 Proof of Lemma 4.8 The proof consists of three stages. Stage I At this stage, we show the existence of a positive number ν such that the following LMI with respect to the matrices P , Qj (j = 1, . . . , N), and V : Υ (P , Q1 , . . . , QN , V , t) ≤ −4νIl , l = (2N + 3)m + r

(4.84)

has a solution for all t ∈ [0, tc ]. Indeed, let us assume that this statement is wrong. Then, there exist sequences * ++∞ ++∞ * of symmetric m × m-matrices Pk k=1 and Qj,k k=1 (j = 1, . . . , N), a sequence * ++∞ +∞ of r × m-matrices Vk k=1 , and two sequences of numbers {tk }+∞ k=1 and {νk }k=1 , satisfying the following conditions: (i) (ii) (iii) (iv)

tk ∈ [0, tc ], (k = 1, 2, . . .); νk > 0, (k = 1, 2, . . .) and limk→+∞ νk = 0; Υ (Pk , Q1,k , . . . , QN,k , Vk , tk ) < 0, (k = 1, 2, . . .); and Υ (Pk , Q1,k , . . . , QN,k , Vk , tk ) > −4νk Il , (k = 1, 2, . . .).

Due to the condition (i), the sequence {tk }+∞ k=1 is bounded. Hence, there exists a convergent subsequence of this sequence. For the sake of simplicity (but without loss of generality), we assume that the sequence {tk }+∞ k=1 itself is convergent. Denote

t¯ = limk→+∞ tk . It is clear that t¯ ∈ [0, tc ]. * ++∞ * ++∞ * ++∞ For the sequences Pk k=1 , Qj,k k=1 (j = 1, . . . , N ), and Vk k=1 , two cases can be distinguished: (a) all these sequences are bounded and (b) at least one of these sequences is unbounded. Let us start with the case (a). In this case there exist convergent subsequences of all these sequences of the matrices. Moreover, we can choose the same set of the indexes in each of these subsequences. Therefore, for the sake of simplicity (but without loss of generality), we assume that all these sequences themselves are convergent. Denote P¯ = lim Pk , k→+∞

¯j = Q lim Qj,k , j = 1, . . . , N ; k→+∞

V¯ = lim Vk . k→+∞

Calculating limits for k → +∞ in the inequalities of the items (iii) and (iv), and taking into account the item (ii), we obtain ¯ N , V¯ , t¯) = 0. ¯ 1, . . . , Q Υ (P¯ , Q

(4.85)

This equality, along with Eqs. (4.77), (4.79), and (4.80), yields P¯ = 0,

¯ j = 0, j = 1, . . . , N ; Q

V¯ = 0.

(4.86)

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4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Substitution of (4.86) into (4.85) and use of (4.76)–(4.80) directly yield ¯ 1, . . . , Q ¯ N , V¯ , t¯)= Θ(P¯ , Q

-

N T ¯ ¯ j =1 B2j (t , 0)B2j (t , 0)+2hIm

O((N +1)m+r)×m

Om×((N +1)m+r) Λ¯

 ,

where !  −1  −1 " . Λ¯ = diag Om×m , . . . , Om×m , NIr + CH (t¯) , hCG (t¯) 1 23 4 N

The latter, along with the positive definiteness of the matrix CG (t¯) and the positive semi-definiteness of the matrix CH (t¯) (see the assumptions (AI) and (AII)), means ¯ 1, . . . , Q ¯ N , V¯ , t¯) = 0, which contradicts Eqs. (4.80) and (4.85). that Θ(P¯ , Q Proceed to the case (b). In this case at least one of the sequences of numbers ++∞ * ++∞ ++∞ * * Pk k=1 , Qj,k k=1 (j = 1, . . . , N ), and Vk k=1 is unbounded. Let us denote αk = Pk + N j =1 Qj,k + Vk > 0, (k = 1, 2, . . .). Thus, there exists a subsequence of the sequence {αk }+∞ k=1 , which tends to +∞ for k → +∞. For the sake of simplicity (but without loss of generality), we assume that limk→+∞ αk = * ++∞ * ++∞ +∞. It is clear that the sequences αk−1 Pk k=1 , αk−1 Qj,k k=1 (j = 1, . . . , N), * ++∞ and αk−1 Vk k=1 are bounded. Similarly to the case (a), we can assume that all these sequences are convergent. Denote j = = = lim αk−1 Pk , Q lim αk−1 Qj,k , j = 1, . . . , N; V lim αk−1 Vk . P k→+∞

k→+∞

k→+∞

* ++∞ * ++∞ Since at least one of the sequences Pk k=1 , Qj,k k=1 (j = 1, . . . , N), and ++∞ , Q j (j = 1, . . . , N), and V is Vk k=1 is unbounded, at least one of the matrices P non-zero. Now, multiplying the inequalities in the conditions (iii) and (iv) by αk−1 , and calculating the limit for k → +∞ in the resulting inequalities, we obtain

*

  , Q 1 , . . . , Q N , V , t¯ = 0, P Υ where " !     , Q 1 , . . . , Q N , V , t¯ =diag Θ , Q 1 , . . . , Q N , V , t¯ , −P , −Q 1 , . . . , −Q N , P P Υ

  , Q 1 , . . . , Q N , V , t¯ = P Θ

   , Q 1 , . . . , Q N , V , t¯ Γ P   , V , t ΔT P

   , V , t¯ Δ P   , 1 , . . . , Q N Q −Λ

4.3 Preliminary Results

241

  , Q 1 , . . . , Q N , V , t¯ = A40 (t¯, 0)P + P AT40 (t¯, 0) Γ P T ¯ +V T B20 (t , 0) + +B20 (t¯, 0)V

N 

j , Q

j =1

" !   1 , . . . , Q 1 , . . . , Q N = diag Q N , Or×r , Om×m . Q Λ = 0, Q j = 0 (j = 1, . . . , N ), From these five equations, we directly obtain that P and V = 0. However, due to the above made observation, at least one of these matrices is non-zero. This contradiction, along with the contradiction obtained in the case (a), proves the solvability of the LMI (4.84) for all t ∈ [0, tc ] with some constant ν > 0. Stage II At this stage, we replace the LMI (4.84) with a more general LMI depending on a parameter δ ≥ 0. Then, based on the solvability of (4.84), we show the solvability of the new LMI. For any given t ∈ [0, tc ] and δ ≥ 0, let us construct the following matrices: T Γδ (P , Q1 , . . . , QN , V , t) = A40 (t, 0)P + P AT40 (t, 0) + B20 (t, 0)V + V T B20 (t, 0)

+

N  

 T exp(2δhj )B2j (t, 0)B2j (t, 0) + Qj + 2h exp(2δh)Im + 2δP ,

j =1

(4.87) ! " Δδ (P , V , t) = exp(δh1 )A41 (t, 0)P , . . . , exp(δhN )A4N (t, 0)P , V T , P , (4.88)  Θδ (P , Q1 , . . . , QN , V , t) =

 Δδ (P , V , t) Γδ (P , Q1 , . . . , QN , t) , − Λ(Q1 , . . . , QN , t) ΔTδ (P , V , t) (4.89)

" ! Υδ (P , Q1 , . . . , QN , V , t)=diag Θδ (P , Q1 , . . . , QN , V , t), −P , −Q1 , . . . , −QN . (4.90) The matrix Υδ (P , Q1 , . . . , QN , V , t) is continuous with respect to δ ≥ 0, and Υδ (P , Q1 , . . . , QN , V , t)δ=0 = Υ (P , Q1 , . . . , QN , V , t). Also, remember that the matrix-valued functions A4j (t, 0), B2j (t, 0) (j = 0, 1, . . . , N), CG (t), and CH (t) are continuous in the interval t ∈ [0, tc ]. These observations, along with the solvability of the LMI (4.84) for any t ∈ [0, tc ], imply the existence of a positive

242

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

value δ, such that the following LMI with respect to the symmetric m × m-matrices P and Qj (j = 1, . . . , N ) and the r × m-matrix V : Υδ (P , Q1 , . . . , QN , V , t) ≤ −2νIl has a solution for any t ∈ [0, tc ]. Now, by virtue of the results of [2] (Theorem 1 and its proof), we obtain immediately the existence of a polynomial solution {P (t), Q1 (t), . . . , QN (t), V (t)}, t ∈ [0, tc ], to the LMI Υδ (P , Q1 , . . . , QN , V , t) ≤ −νIl .

(4.91)

Stage III At this stage, we show that the system (4.13) is stabilizable by the statefeedback control (4.72) with the gain K(t) = V (t)P −1 (t),

t ∈ [0, tc ],

(4.92)

and this stabilizability is uniform with respect to t ∈ [0, tc ]. First of all, let us note the following. Since V (t) and P (t) are matrix-valued polynomials of t ∈ [0, tc ], and P (t) is a positive definite matrix for all t ∈ [0, tc ], the matrix-valued function K(t), given by (4.92), is continuously differentiable. For any t ∈ [0, tc ] and ξ ≥ 0, consider the Lyapunov–Krasovskii-like functional 5



F yf,ξ , t = Fi yf,ξ , t , i=1

 + * where yf,ξ = yf (ξ ), yf (θ ) , θ ∈ [ξ − h, ξ ) ; for any ξ ≥ −h, yf (ξ ) ∈ E m ; yf (ξ ) is locally absolutely continuous for ξ ≥ 0; yf (η) ∈ L2 [−h, 0; E m ]; and

F1 yf,ξ , t = yfT (ξ )P −1 (t)yf (ξ ), N

 F2 yf,ξ , t =



ξ

j =1 ξ −hj

(4.93)

  exp 2δ(κ − ξ ) yfT (κ)P −1 (t)Qj (t)P −1 (t)yf (κ)dκ, (4.94)



F3 yf,ξ , t =



0



  exp 2δ(κ − ξ ) yfT (κ)CG (t)yf (κ)dκdη,

(4.95)

  exp 2δ(κ − ξ ) yfT (κ)K T (t)K(t)yf (κ)dκ,

(4.96)

ξ

−h ξ +η

N

 F4 yf,ξ , t =



ξ

j =1 ξ −hj

4.3 Preliminary Results



F5 yf,ξ , t =



0

243



ξ

−h ξ +η

  exp 2δ(κ − ξ ) yfT (κ)K T (t)CH (t)K(t)yf (κ)dκdη. (4.97)

Let, for a given t ∈ [0, tc ], the vector-valued function yf K (ξ ), ξ ≥ −h, be the solution of the initial-value problem (4.13) and (4.74), generated by the

state-feedback control (4.72) and (4.92). Let uf K (ξ ) = K(t)yf K (ξ ), ξ ≥ −h. By FKi (ξ,

t) (i = 1, 2, . . . , 5), let us denote the realizations of the functionals Fi yf,ξ , t (i = 1, 2, . . . , 5) along the solution yf K (ξ ), i.e., FKi (ξ, t) =

Fi yf K,ξ , t (i = 1, 2, . . . , 5). Thus, the realization of the functional F yf,ξ , t along yf K (ξ ) is FK (ξ, t) = F (yf K,ξ , t) =

5 

FKi (ξ, t).

(4.98)

i=1

Let us observe that the matrix-valued coefficients in the functionals (4.93)–(4.95) are positive definite for all t ∈ [0, tc ], and such coefficients in the functionals (4.96)– (4.97) are at least positive semi-definite. Moreover, all these coefficients are continuous functions of t ∈ [0, tc ]. Therefore, there exist two constants a1 > 0 and a2 > 0 (a1 < a2 ) such that the function (4.98) satisfies the inequality #  2 $ 2  2  a1 yf K (ξ ) ≤ FK (ξ, t ≤ a2 yf K (ξ ) + yf K (θ ) L2 , θ ∈ [ξ − h, ξ ),

ξ ≥ 0.

(4.99)

Equation (4.98) yields  5  dFK (ξ, t dFKi (ξ, t) = , dξ dξ

ξ ≥ 0.

(4.100)

i=1

Let us calculate and estimate each of the derivatives dFKi (ξ, t)/dξ (i = 1, 2, . . . , 5). Using Eq. (4.93) and the definition of yf K (ξ ), we obtain the expression of dFK1 (ξ, t)/dξ for any t ∈ [0, tc ]   dyf K (ξ ) dyf K (ξ ) T −1 dFK1 (ξ, t) P (t)yf K (ξ ) + yfT K (ξ )P −1 (t) = dξ dξ dξ

T (t, 0) P −1 (t)y = yfT K (ξ )P −1 (t) A40 (t, 0)P + P AT40 (t, 0) + B20 (t, 0)V + V T B20 f K (ξ ) +2yfT K (ξ )P −1 (t)

N  j =1

A4j (t, 0)yf K (ξ − hj )

244

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

+2yfT K (ξ )P −1 (t) +2yfT K (ξ )P −1 (t) +2yfT K (ξ )P −1 (t)

 0 −h

N 

B2j (t, 0)uf K (ξ − hj )

j =1

 0 −h

G4 (t, η, 0)yf K (ξ + η)dη

H2 (t, η, 0)uf K (ξ + η)dη,

ξ ≥ 0. (4.101)

Now, we are going to estimate some of the addends in the right-hand side of (4.101). For this purpose, we use the following two inequalities. Namely, let L ∈ E p×p be any symmetric positive definite matrix. Let γ1 ∈ E p and γ2 ∈ E p be any vectors. Let γ (ξ ) ∈ L2 [ξ1 , ξ2 ; E p ] be any vector-valued function, where ξ2 ≥ ξ1 . Then, 2γ1T γ2 ≤ γ1T L γ1 + γ2T L −1 γ2 , 

T

ξ2

γ (ξ )dξ

 L

ξ1



ξ2

γ (ξ )dξ



ξ2

≤ (ξ2 −ξ1 )

ξ1

(4.102)

γ T (ξ )L γ (ξ )dξ.

(4.103)

ξ1

The first of these inequalities is a straightforward generalization of the well-known inequality 2a1 a2 ≤ a12 + a22 for any real numbers a1 and a2 . Proof of the second inequality can be found in [12]. Let us start the abovementioned estimate with the expression 2yfT K (ξ )P −1 (t)A4j (t, 0)yf K (ξ − hj ).



In this expression, we set AT4j (t, 0)P −1 (t)yf K (ξ ) = γ1 , yf K (ξ − hj ) = γ2 . Using

these notations and the inequality (4.102) with L = exp(2δhj )P (t)Q−1 j (t)P (t), we obtain for any j ∈ {1, . . . , N} 2yfT K (ξ )P −1 (t)A4j (t, 0)yf K (ξ − hj ) ≤ T −1 (t)yf K (ξ ) exp(2δhj )yfT K (ξ )P −1 (t)A4j (t, 0)P (t)Q−1 j (t)P (t)A4j (t, 0)P

+ exp(−2δhj )yfT K (ξ − hj )P −1 (t)Qj (t)P −1 (t)yf K (ξ − hj ). This inequality yields 2yfT K (ξ )P −1 (t)

N  j =1

A4j (t, 0)yf K (ξ − hj ) ≤

⎡ ⎤ N  yfT K (ξ )P −1 (t) ⎣ exp(2δhj )A4j (t, 0)P (t)Q−1 P (t)AT4j (t, 0)⎦ P −1 (t)yf K (ξ ) j

j =1

4.3 Preliminary Results

+

N 

245

exp(−2δhj )yfT K (ξ − hj )P −1 (t)Qj (t)P −1 (t)yf K (ξ − hj ).

j =1

(4.104) Proceed with the expression 2yfT K (ξ )P −1 (t)B2j (t, 0)uf K (ξ − hj ).



T (t, 0)P −1 (t)y Here, we set B2j f K (ξ ) = γ1 , uf K (ξ − hj ) = γ2 . Using these notations and the inequality (4.102) with L = exp(2δhj )Ir , we obtain for any j ∈ {1, . . . , N }

2yfT K (ξ )P −1 (t)B2j (t, 0)uf K (ξ − hj ) ≤ T exp(2δhj )yfT K (ξ )P −1 (t)B2j (t, 0)B2j (t, 0)P −1 (t)yf K (ξ )

+ exp(−2δhj )uTf K (ξ − hj )uf K (ξ − hj ), which yields 2yfT K (ξ )P −1 (t)

N 

B2j (t, 0)uf K (ξ − hj ) ≤

j =1

⎡ ⎤ N  T yfT K (ξ )P −1 (t) ⎣ exp(2δhj )B2j (t, 0)B2j (t, 0)⎦ P −1 (t)yf K (ξ ) j =1

+

N 

exp(−2δhj )uTf K (ξ − hj )uf K (ξ − hj ).

(4.105)

j =1

The next expression is 2yfT K (ξ )P −1 (t)



0

−h

G4 (t, η, 0)yf K (ξ + η)dη.

To estimate this expression, first, we apply the inequality (4.102) with γ1 = .0 P −1 (t)yf K (ξ ), γ2 = −h G4 (t, η, 0)yf K (t + η)dη, and L = h exp(2δh)Im . Then, we apply the inequality (4.103) and the assumption (AI). Thus, we have 2yfT K (ξ )P −1 (t)

 0 −h

G4 (t, η, 0)yf K (ξ + η)dη ≤ h exp(2δh)yfT K P −2 (t)yf K (ξ )

246

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

1 + exp(−2δh) h

 0 −h

T  G4 (t, η, 0)yf K (ξ + η)dη

0 −h

 G4 (t, η, 0)yf K (ξ + η)dη

≤ h exp(2δh)yfT K P −2 (t)yf K (ξ ) + exp(−2δh)

 0 −h

yfT K (ξ + η)GT4 (t, η, 0)G4 (t, η, 0)yf K (ξ + η)dη

≤ h exp(2δh)yfT K P −2 (t)yf K (ξ ) + exp(−2δh)

 0 −h

yfT K (ξ + η)CG (t)yf K (ξ + η)dη. (4.106)

Similarly to (4.106), we treat the expression  0 H2 (t, η, 0)uf K (ξ + η)dη, 2yfT K (ξ )P −1 (t) −h

which yields 2yfT K (ξ )P −1 (t)



0

−h

H2 (t, η, 0)uf K (ξ + η)dη ≤ h exp(2δh)yfT K P −2 (t)yf K (ξ )  + exp(−2δh)

0 −h

uTf K (ξ + η)CH (t)uf K (ξ + η)dη. (4.107)

Using Eq. (4.101) and the inequalities (4.104), (4.105), (4.106), and (4.107), we obtain the estimate of dFK1 (ξ, t)/dξ 

dFK1 (ξ, t ≤ yfT K (ξ )P −1 (t) A40 (t, 0)P (t) + P (t)AT40 (t, 0) + B20 (t, 0)V (t) dξ T +V T (t)B20 (t, 0) +

N 

T exp(2δhj )A4j (t, 0)P (t)Q−1 j P (t)A4j (t, 0)

j =1

+

N 

 T exp(2δhj )B2j (t, 0)B2j (t, 0) + 2h exp(2δh)Im P −1 (t)yf K (ξ )

j =1

+

N 

exp(−2δhj )yfT K (ξ − hj )P −1 (t)Qj (t)P −1 (t)yf K (ξ − hj )

j =1

+

N 

exp(−2δhj )uTf K (ξ − hj )uf K (ξ − hj )

j =1

+ exp(−2δh)



0 −h

yfT K (ξ + η)CG (t)yf K (ξ + η)dη

4.3 Preliminary Results

247

 + exp(−2δh)

0 −h

uTf K (ξ + η)CH (t)uf K (ξ + η)dη,

ξ ≥ 0, t ∈ [0, tc ]. (4.108)

Proceed to the derivatives dFKi (ξ, t)/dξ (i = 2, . . . , 5). Using Eqs. (4.92) and (4.94)–(4.97), we obtain for all ξ ≥ 0 and t ∈ [0, tc ] dFK2 (ξ, t)  T = yf K (ξ )P −1 (t)Qj (t)P −1 (t)yf K (ξ ) dξ N

j =1

−

N 

exp(−2δhj )yfT K (ξ − hj )P −1 (t)Qj (t)P −1 (t)yf K (ξ − hj ) − 2δFK2 (ξ, t),

j =1

(4.109) dFK3 (ξ, t) = hyfT K (ξ )CG (t)yf K (ξ ) dξ

 −

0 −h

exp(2δη)yfT K (ξ + η)CG (t)yf K (ξ + η)dη − 2δFK3 (ξ, t)

 − exp(−2δh)

≤ hyfT K (ξ )P −1 (t)P (t)CG (t)P (t)P −1 (t)yf K (ξ ) 0 −h

yfT K (ξ + η)CG (t)yf K (ξ + η)dη − 2δFK3 (ξ, t),

(4.110)

dFK4 (ξ, t) = NyfT K (ξ )K T (t)K(t)yf K (ξ ) dξ −

N 

exp(−2δhj )yfT K (ξ − hj )K T (t)K(t)yf K (ξ − hj ) − 2δFK4 (ξ, t),

j =1

(4.111)

 −

dFK5 (ξ, t) = hyf K (ξ )K T CH (t)K(t)yf K (ξ ) dξ 0 −h

exp(2δη)yf K (ξ + η)K T CH (t)K(t)yf K (ξ + η)dη − 2δFK5 (ξ, t)

 − exp(−2δh)

≤ hyf K (ξ )P −1 (t)V T CH (t)V (t)P −1 (t)yf K (ξ ) 0 −h

uf K (ξ + η)CH (t)uf K (ξ + η)dη − 2δFK5 (ξ, t). (4.112)

248

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Using Eqs. (4.92), (4.93), (4.98), (4.100), and (4.108)–(4.112), we obtain the inequality for all ξ ≥ 0 and t ∈ [0, tc ]

dFK (ξ, t) + 2δFK (ξ, t) ≤ yfT K (ξ )P −1 (t) A40 (t, 0)P (t) + P (t)AT40 (t, 0) dξ T +B20 (t, 0)V (t) + V T (t)B20 (t, 0) +

N  

T exp(2δhj )B2j (t, 0)B2j (t, 0) + Qj (t)



j =1

+2h exp(2δh)Im + 2δP (t) +

N 

T exp(2δhj )A4j (t, 0)P (t)Q−1 j P (t)A4j (t, 0)

j =1

   +V T (t) NIr + hCH (t) V (t) + hP (t)CG (t)P (t) P −1 (t)yf K (ξ ). Using the notations (4.78) and (4.87)–(4.89), we can rewrite this inequality in the following form for all ξ ≥ 0 and t ∈ [0, tc ]:

  dFK (ξ, t) T −1 + 2δFK (ξ, t) ≤ yf K (ξ )P (t) Γδ P (t), Q1 (t), . . . , QN (t), V (t), t dξ    −1   −1  T +Δδ P (t), V (t), t Λ Q1 (t), . . . , QN (t), t Δδ P (t), V (t), t P (t)yf K (ξ ). (4.113) + * Remember that P (t), Q1 (t), . . . , QN (t), V (t) is the solution of the LMI (4.91). Therefore, due to Eq. (4.90), we obtain that   Θδ P (t), Q1 (t), . . . , QN (t), V (t), t < 0, t ∈ [0, tc ]. This inequality, along with Eq. (4.89) and the Schur Complement Theorem [3], yields the inequality   Γδ P (t), Q1 (t), . . . , QN (t), V (t), t       +Δδ P (t), V (t), t Λ−1 Q1 (t), . . . , QN (t), t ΔTδ P (t), V (t), t < 0. Using this inequality and the inequality (4.113), we obtain that dFK (ξ, t) + 2δFK (ξ, t) ≤ 0, dξ

ξ ≥ 0, t ∈ [0, tc ],

implying FK (ξ, t) ≤ FK (0, t) exp(−2δξ ),

ξ ≥ 0, t ∈ [0, tc ].

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

249

The latter, along with Eq. (4.74) and the inequality (4.99), yields 5 yf K (ξ ) ≤

# 2  a2 exp(−δξ ) y0 + ϕyf (·) a1

2 $1/2 L2

,

ξ ≥ 0, t ∈ [0, tc ].

√ This inequality coincides with the inequality (4.75) for af (t) ≡ a2 /a1 and βf (t) ≡ δ, t ∈ [0, tc ], meaning the stabilizability of the system (4.13) by the memory-less state-feedback control (4.72) and (4.92) uniformly with respect to t ∈ [0, tc ]. This conclusion, along with the abovementioned existence of the continuous derivative of the gain (4.92) in the interval [0, tc ], proves the lemma.

4.4 Parameter-Free Controllability Conditions for Systems with Small State Delays In this section, using the preliminary results, various ε-independent conditions, providing the complete Euclidean space controllability of the original system (4.1)– (4.2) for all sufficiently small ε > 0, are derived.

4.4.1 Case of the Standard System (4.1)–(4.2) In this subsection, we assume that the condition (4.10) holds for all t ∈ [0, tc ]. In the literature, singularly perturbed systems with such a feature are called standard (see, e.g., [5, 16]). In what follows, we also assume that (AIII)

(AIV)

(AV)

the matrix-valued functions Aij (t, ε) and Bkj (t, ε) (i = 1, . . . , 4; j = 0, 1, . . . , N ; k = 1, 2) are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ]; the matrix-valued functions Gi (t, η, ε) and Hk (t, η, ε) (i = 1, . . . , 4; k = 1, 2) are piecewise continuous with respect to η ∈ [−h, 0] for each (t, ε) ∈ [0, tc ] × [0, ε0 ], and these functions are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in η ∈ [−h, 0]; all roots λ(t) of the equation ⎡ det ⎣λIm −

N  j =0

 A4j (t, 0) exp(−λhj ) −

⎤ 0 −h

G4 (t, η, 0) exp(λη)dη⎦ = 0 (4.114)

satisfy the inequality Reλ(t) < −2β for all t ∈ [0, tc ], where β > 0 is some constant.

250

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

It should be noted that, similarly to Remark 3.8 (see Sect. 3.4.1), the fulfillment of the assumption (AV) directly yields the fulfillment of the condition (4.10) for all t ∈ [0, tc ]. Lemma 4.9 (Main Lemma) Let the assumptions (AIII)–(AV) be valid. Let the system (4.28) be completely controllable at the time instant tc . Let, for t = tc , the system (4.29)–(4.30) and (4.31) be Euclidean space output controllable. Then, there exists a positive number ε1 (ε1 ≤ ε0 ), such that for all ε ∈ (0, ε1 ], the singularly perturbed system (4.1), (4.2), (4.17), and (4.18) is Euclidean space output controllable at the time instant tc . Proof of the lemma is presented in Sect. 4.4.3. Remark 4.4 Note that the Euclidean space output controllability for singularly perturbed systems with small state delays was studied in Chap. 3. In this chapter, the case of the standard original system was treated in Sect. 3.4.1, where different ε-free sufficient conditions for the Euclidean space output controllability of the original system were formulated in Theorems 3.1–3.6. These conditions depend considerably on relations between the dimensions of the slow Euclidean state variable, the fast Euclidean state variable, and the output of the system. However, due to the specific form (4.19) of the matrix of the coefficients Z in the output equation of the system (4.1), (4.2), (4.17), and (4.18), only Theorem 3.1, and only in the very specific case n ≤ r, is applicable to this system. Therefore, in Sect. 4.4.3, we present the proof of Lemma 4.9 which is not based on Theorems 3.1–3.6. In particular, this proof is uniformly valid for all relations between the dimensions of the abovementioned variables of the system (4.1), (4.2), (4.17), and (4.18). Theorem 4.1 Let the assumptions (AIII)–(AV) be valid. Let the system (4.11) be completely controllable at the time instant tc . Let, for t = tc , the system (4.13) be completely Euclidean space controllable. Then, for all ε ∈ (0, ε1 ], the singularly perturbed system (4.1)–(4.2) is completely Euclidean space controllable at the time instant tc . Proof Based on Lemma 4.1, Remark 4.1, and Lemma 4.2, the theorem directly follows from Lemma 4.9.   Now, we consider the case where the assumption (AV) is violated. The theorem, presented below, is based on less restrictive assumptions than the assumption (AV). However, the proof of this theorem is essentially based on Theorem 4.1. Theorem 4.2 Let the assumptions (AI)–(AIV) be valid. Let, for any t ∈ [0, tc ], the Linear Matrix Inequality (4.81) with respect to the symmetric m × m-matrices P and Q1 , . . . , QN and the r × m-matrix V have a solution. Let the system (4.11) be completely controllable at the time instant tc . Let, for t = tc , the system (4.13) be completely Euclidean space controllable. Then, there exists a positive number ε2 (ε2 ≤ ε0 ), such that for all ε ∈ (0, ε2 ], the singularly perturbed system (4.1)–(4.2) is completely Euclidean space controllable at the time instant tc .

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

251

Proof First of all, let us note the following. Since the assumptions (AI)–(AII) are valid and the LMI (4.81) has the solution for any t ∈ [0, tc ], then, by virtue of Corollary 4.1, there exists the continuously differentiable r × m-matrix-valued function K(t), t ∈ [−ε0 h, 0], such that all the roots λ(t) of Eq. (4.82) satisfy the inequality (4.83). Now, in the system (4.1)–(4.2), let us make the control transformation (4.56) with the abovementioned matrix K(t). Due to this transformation, we obtain the system (4.57)–(4.58). Since the matrix-valued function K(t) is continuously differentiable, the coefficients (4.59) of this system have the properties, similar to the assumptions (AIII) and (AIV) on the coefficients of the system (4.1)–(4.2). The slow subsystem, associated with (4.57)–(4.58), has the form (4.66)–(4.67), where the AK 4s (t) is given in (4.68). Since all the roots λ(t) of Eq. (4.82) satisfy the inequality (4.83), the matrix AK 4s (t) is invertible (the inequality (4.69) is valid). Thus, the slow subsystem (4.66)–(4.67) is reduced to the differential equation (4.70). Moreover, due to Lemma 4.6, the system (4.70) is completely controllable at the time instant tc . The fast subsystem, associated with (4.57)–(4.58), has the form (4.71). The latter, by virtue of Corollary 4.1, satisfies the property similar to the assumption (AV) on the coefficients of the system (4.13) in Theorem 4.1. Moreover, due to Lemma 4.7, the system (4.71) for t = tc is completely Euclidean space controllable. Thus, the transformed system (4.57)–(4.58) satisfies all the conditions of Theorem 4.1, yielding the existence of a positive number ε2 (ε2 ≤ ε0 ), such that for all ε ∈ (0, ε2 ], this system is completely Euclidean space controllable at the time instant tc . The latter, along with Lemma 4.3, means the complete Euclidean space controllability at the time instant tc of the system (4.1)–(4.2) for all ε ∈ (0, ε2 ]. This completes the proof of the theorem.  

4.4.2 Case of the Nonstandard System (4.1)–(4.2) In this subsection, in contrast with the previous one, we consider the case where the condition (4.10) does not hold at least for one value of t ∈ [0, tc ]. In the literature, singularly perturbed systems with such a feature are called nonstandard (see, e.g., [5, 16]). Since the condition (4.10) is not satisfied for some t¯ ∈ [0, tc ], then det A4s (t¯) = 0. The latter, along with Eq. (4.8), means that one of the roots λ(t¯) of Eq. (4.114) equals zero. Thus, in the case of the nonstandard system (4.1)–(4.2) the assumption (AV) is not valid. Therefore, in this subsection, we replace this assumption with the assumptions (AI)–(AII). Theorem 4.3 Let the assumptions (AI)–(AIV) be valid. Let, for any t ∈ [0, tc ], the Linear Matrix Inequality (4.81) with respect to the symmetric m×m-matrices P and Q1 , . . . , QN and the r × m-matrix V have a solution. Let the system (4.6)–(4.7) be impulse-free controllable with respect to xs (t) at the time instant tc . Let, for t = tc ,

252

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

the system (4.13) be completely Euclidean space controllable. Then, there exists a positive number ε3 (ε3 ≤ ε0 ), such that for all ε ∈ (0, ε3 ], the singularly perturbed system (4.1)–(4.2) is completely Euclidean space controllable at the time instant tc . Proof For a given ε ∈ (0, ε0 ] in the system (4.1)–(4.2), let us make the control transformation (4.56), where K(t) is the matrix-valued function mentioned in Corollary 4.1. As a result of this transformation, we obtain the system (4.57)– (4.58). The slow and fast subsystems, associated with the latter, are (4.66)–(4.67) and (4.71), respectively. Since, for t = tc , the system (4.13) is completely Euclidean space controllable, then, due to Lemma 4.7, the system (4.71) for t = tc is completely Euclidean space controllable. Furthermore, since the system (4.6)–(4.7) is impulse-free controllable with respect to xs (t) at the time instant tc , then, due to Lemma 4.4, the system (4.66)–(4.67) is impulse-free controllable with respect to xs (t) at the time instant tc . By virtue of Corollary 4.1, the inequality (4.69) is valid. Thus, the slow subsystem (4.66)–(4.67) is reduced to the differential equation (4.70). Therefore, due to Lemma 4.5, the abovementioned impulse-free controllability of the system (4.66)–(4.67) yields the complete controllability of the system (4.70) at the time instant tc . Now, by application of Theorem 4.1 to the system (4.57)– (4.58), we directly obtain the existence of a positive number ε3 (ε3 ≤ ε0 ), such that for all ε ∈ (0, ε3 ], this system is completely Euclidean space controllable at the time instant tc . Finally, using Lemma 4.3 yields the complete Euclidean space controllability of the system (4.1)–(4.2) at the time instant tc for all ε ∈ (0, ε3 ], which completes the proof of the theorem.  

4.4.3 Proof of Main Lemma (Lemma 4.9) First of all, let us make one important observation. Namely, due to the results of Sect. 4.3.2.1, the system (4.1), (4.2), (4.17), and (4.18) is equivalent to the system (4.40)–(4.41) and (4.42). Moreover, due to the results of Sect. 4.3.2.3, the fast subsystem (4.29)–(4.30) and (4.31) of the system (4.1), (4.2), (4.17), and (4.18) is equivalent to the fast subsystem (4.52) and (4.53) of the system (4.40)–(4.41) and (4.42). Based on this observation, we can conclude the following. In order to prove the lemma, it is necessary and sufficient to show that the complete controllability at tc of the system (4.28) and the Euclidean space output controllability of the system (4.52) and (4.53) for t = tc yield the Euclidean space output controllability at tc of the system (4.40)–(4.41) and (4.42) for all ε ∈ (0, ε1 ], where ε1 ≤ ε0 is some positive number. In the proof of Main Lemma, the following two auxiliary propositions are used.

4.4.3.1

Auxiliary Propositions

For any given t ∈ [0, tc ] and any complex number μ, let us consider the matrix

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

W (t, μ) =

N 

 A4j (t, 0) exp(−μhj ) +

j =0

0

−h

G4 (t, η, 0) exp(μη)dη,

253

(4.115)

where A4j (t, ε) (j = 0, 1, . . . , N) and G4 (t, η, ε) are given in (4.35)–(4.36) and (4.38), respectively. Proposition 4.4 Let the assumption (AV) be valid. Then, all roots μ(t) of the equation

det μIm+r − W (t, μ) = 0

(4.116)

satisfy the inequality Reμ(t) < −2γ for all t ∈ [0, tc ], where γ = min{β, 1/4}. Proof Using (4.35)–(4.36), (4.38), and (4.115), we obtain for all t ∈ [0, tc ] ⎡ det ⎣μIm −

N 

 A41 (t, 0) exp(−λhj ) −

j =0

det μIm+r − W (t, μ) = ⎤ 0 −h

G4 (t, η, 0) exp(μη)dη⎦ (μ + 1)r ,

meaning that for any t ∈ [0, tc ], the set of all roots μ(t) of Eq. (4.116) consists of all roots of Eq. (4.114) and the root μ(t) ≡ −1 of the multiplicity r. This observation, along with the assumption (AV), directly yields the statement of the proposition.   Let us partition the matrix-valued function Ψ (σ, ε), given by the terminal-value problem (4.47)–(4.48), into blocks as  Ψ (σ, ε) =

 Ψ1 (σ, ε) Ψ2 (σ, ε) , Ψ3 (σ, ε) Ψ4 (σ, ε)

(4.117)

where the blocks Ψ1 (σ, ε), Ψ2 (σ, ε), Ψ3 (σ, ε), and Ψ4 (σ, ε) are of the dimensions n × n, n × (m + r), (m + r) × n, and (m + r) × (m + r), respectively. Proposition 4.5 Let the assumptions (AIII)–(AV) be valid. Then, there exists a positive number ε0 (ε0 ≤ ε0 ), such that for all ε ∈ (0, ε0 ], the matrixvalued functions Ψ1 (σ, ε), Ψ2 (σ, ε), Ψ3 (σ, ε), and Ψ4 (σ, ε) satisfy the following inequalities for all ε ∈ (0, ε0 ] : Ψ1 (σ, ε) − Ψ1s (σ ) ≤ aε,

Ψ2 (σ, ε) ≤ a, σ ∈ [0, tc ],



Ψ3 (σ, ε) − εΨ3s (σ ) ≤ aε ε + exp(−γ (tc − σ )/ε) ,   Ψ4 (σ, ε) − Ψ4f (tc − σ )/ε ≤ aε,

σ ∈ [0, tc ],

σ ∈ [0, tc ],

(4.118)

(4.119)

(4.120)

254

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

where ⎞ ⎛  −1 − AT4s (σ ) AT2s (σ ) ⎟ ⎜ Ψ1s (σ ) = Ψs (σ ), Ψ3s (σ ) = ⎝ ⎠ Ψs (σ ),  T ¯ Bs (σ )

σ ∈ [0, tc ],

Ψ4f (ξ ) = Ψf (ξ, tc ),

ξ ≥ 0;

the matrix-valued functions A2s (t) and A4s (t) are given in (4.8); the matrix-valued function B¯ s (t) is given in (4.12); the matrix-valued functions Ψs (σ ) and Ψf (ξ, t) are given by the terminal-value problem (4.51) and the initial-value problem (4.54), respectively; and a > 0 is some constant independent of ε. Proof Based on Proposition 4.4, the validity of the inequalities (4.118)–(4.120) directly follows from Theorem 2.1 (Sect. 2.2.4).   Remark 4.5 By virtue of Proposition 4.4 and the results of [14], we have the inequality Ψ4f (ξ ) ≤ a exp(−2γ ξ ),

ξ ≥ 0,

(4.121)

where a > 0 is some constant.

4.4.3.2

Main Part of the Proof

Due to Proposition 4.1 and Remark 4.2, in order to prove Main Lemma, it is necessary and sufficient to show the existence of a positive number ε1 such that det WZ (tc , ε) = 0 ∀ε ∈ (0, ε1 ],

(4.122)

where the (n + m) × (n + m)-matrix WZ (tc , ε) is defined by Eqs. (4.49)–(4.50). Let, for a given ε ∈ (0, ε0 ], the matrix W1 (tc , ε) of the dimension n×n, the matrix W2 (tc , ε) of the dimension n × (m + r), and the matrix W3 (tc , ε) of the dimension (m + r) × (m + r) be the upper left-hand, upper right-hand, and lower right-hand blocks, respectively, of the symmetric matrix W (tc , ε), given by Eq. (4.49). Thus,  W (tc , ε) =

 W1 (tc , ε) W2 (tc , ε) . W2T (tc , ε) W3 (tc , ε)

(4.123)

Using (4.49), the block representations of the matrices B(ε) and Ψ (σ, ε) (see Eqs. (4.44) and (4.117)), and the block representation of the matrix W (tc , ε), we obtain similarly to Eqs. (3.120)–(3.122) (see Sect. 3.4.3)  W1 (tc , ε) = 0

tc

#

Ψ1T (σ, ε)P1 Ψ1 (σ, ε) + (1/ε)Ψ3T (σ, ε)P2T Ψ1 (σ, ε)

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

$ +(1/ε)Ψ1T (σ, ε)P2 Ψ3 (σ, ε) + (1/ε2 )Ψ3T (σ, ε)P3 Ψ3 (σ, ε) dσ, 

tc

W2 (tc , ε) =

#

0



tc

0

#

(4.124)

Ψ1T (σ, ε)P1 Ψ2 (σ, ε) + (1/ε)Ψ3T (σ, ε)P2T Ψ2 (σ, ε)

$ +(1/ε)Ψ1T (σ, ε)P2 Ψ4 (σ, ε) + (1/ε2 )Ψ3T (σ, ε)P3 Ψ4 (σ, ε) dσ,

W3 (tc , ε) =

255

(4.125)

Ψ2T (σ, ε)P1 Ψ2 (σ, ε) + (1/ε)Ψ4T (σ, ε)P2T Ψ2 (σ, ε)

$ +(1/ε)Ψ2T (σ, ε)P2 Ψ4 (σ, ε) + (1/ε2 )Ψ4T (σ, ε)P3 Ψ4 (σ, ε) dσ,

(4.126)

where, due to (4.44), P1 = B1 B1T = On×n ,

P2 = B1 B2T = On×(m+r) , 

P3 = B2 B2T =

Om×m Om×r Or×m Ir

 .

(4.127)

The latter, along with (4.124)–(4.126), yields 

tc

W1 (tc , ε) = (1/ε2 ) 0



tc

W2 (tc , ε) = (1/ε2 ) 0



tc

W3 (tc , ε) = (1/ε ) 2

0

Ψ3T (σ, ε)P3 Ψ3 (σ, ε)dσ,

Ψ3T (σ, ε)P3 Ψ4 (σ, ε)dσ,

Ψ4T (σ, ε)P3 Ψ4 (σ, ε)dσ.

Let us estimate the matrices W1 (tc , ε), W2 (tc , ε), and W3 (tc , ε). Using Proposition 4.5 and the inequality (4.121), similarly to the proof of Theorem 3.1 (see Sect. 3.4.3.1), we obtain the existence of a positive number ε¯ 0 (¯ε0 ≤ ε0 ), such that for all ε ∈ (0, ε¯ 0 ], the following inequalities are satisfied:

W1 (tc , ε) − W1s ≤ aε, W2 (tc , ε) ≤ a, εW3 (tc , ε) − W3f ≤ aε, (4.128)

256

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

where  W1s = 0

tc

T Ψ1s (σ )B¯ s (σ )B¯ s T (σ )Ψ1s (σ )dσ,



W3f = 0

+∞

T Ψ4f (ξ )P3 Ψ4f (ξ )dξ,

(4.129)

and B¯ s (t) is given in (4.12). By virtue of the inequality (4.121), the integral in the expression for W3f converges. Due to the complete controllability of the system (4.28) at the time instant tc , we obtain (by Proposition 4.2) that det W1s = 0.

(4.130)

Now, let us proceed to the analysis of the matrix WZ (tc , ε). Using the expression for WZ (tc , ε) in Eq. (4.50) and the block form of the matrices Z(t, ε) in (4.19) and W (tc , ε) in (4.123), we obtain the following block representation of the matrix WZ (tc , ε):   W1 (tc , ε) W21 (tc , ε) (4.131) WZ (tc , ε) = T (t , ε) W (t , ε) , W21 c 31 c where W21 (tc , ε) is the left-hand block of the dimension n × m of the matrix W2 (tc , ε), while W31 (tc , ε) is the upper left-hand block of the dimension m × m of the matrix W3 (tc , ε). By virtue of the second and the third inequalities in (4.128), we immediately have that

W21 (tc , ε) ≤ a,

εW31 (tc , ε) − W3f,1 ≤ aε, ε ∈ (0, ε¯ 0 ],

(4.132)

where W3f,1 is the upper left-hand block of the dimension m×m of the matrix W3f . Let us show that det W3f,1 ≥ b,

(4.133)

where b > 0 is some number. Note that W3f,1 can be represented as W3f,1 = Ωf W3f ΩfT ,

(4.134)

where Ωf is given in (4.53). Comparison of the expressions for Wf (ξ, t) and W3f,1 (see Eqs. (4.55) and (4.134)) and use of the expression for W3f (see Eqs. (4.129) and (4.127)) and the equality Ψ4f (ξ ) = Ψf (ξ, tc ) yield that

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

W3f,1 = lim Wf (ξ, tc ).

257

(4.135)

ξ →+∞

Let us observe that, for any ξ > 0 and t ∈ [0, tc ], the matrix Wf (ξ, t) is positive semi-definite. Moreover, since the system (4.29)–(4.30) and (4.31) is Euclidean space output controllable for t = tc , then, by virtue of Proposition 4.3, det Wf (ξc , tc ) = 0 with some ξc > 0. Therefore, det Wf (ξc , tc ) > 0, and Wf (ξc , tc ) is a positive definite matrix. For any ξ > ξc , we have  Wf (ξ, tc ) = Wf (ξc , tc ) + Ωf

ξ ξc

ΨfT (ρ, tc )B2 B2T Ψf (ρ, tc )dρΩfT ,

and the second addend in the right-hand side of this equation is a positive semidefinite matrix. Hence, by use of the results of [1], we obtain that det Wf (ξ, tc ) ≥ det Wf (ξc , tc ) > 0,

ξ > ξc .

The latter, along with the equality (4.135), directly yields the inequality (4.133), where b = det Wf (ξc , tc ). Now, we proceed to the proof of the inequality (4.122). Let us introduce into the consideration the matrix   On×m In √ . L(ε) = Om×n εIm For any ε > 0, det L(ε) > 0. Using Eq. (4.131), we obtain ⎛ L(ε)WZ (tc , ε)L(ε) = ⎝

W1 (tc , ε)

√

εW21 (tc , ε)

√ T εW21 (tc , ε) εW31 (tc , ε)

⎞ ⎠.

Calculating the limit of the determinant of this matrix as ε → +0 and using the first inequality in (4.128) and the inequalities (4.130), (4.132), and (4.133), we obtain  ! " W1s lim det L(ε)WZ (tc , ε)L(ε) = det ε→+0 0

0 W3f,1



= det W1s det W3f,1 = 0. This inequality, along with the inequality det L(ε) > 0, ε > 0, implies the existence of a positive number ε1 such that the inequality (4.122) is valid. This completes the proof of Main Lemma.

258

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

4.4.4 Alternative Approach to Controllability Analysis of the Nonstandard System (4.1)–(4.2) In this subsection, we present another approach to analysis of the complete Euclidean space controllability of the nonstandard system (4.1)–(4.2). This approach is not based on the assumptions (AI)–(AII) and the assumption of the solvability of the LMI (4.81).

4.4.4.1

Linear Control Transformation in the Auxiliary System (4.40)–(4.42)

Let us transform the control v(t) in the auxiliary system (4.40)–(4.41) and (4.42) as follows:  0 v(t) = K1 (t)ω(t) + K2 (t, η)ω(t + εη)dη + w (t), (4.136) −h

where w (t) ∈ E r is a new control; K1 (t) and K2 (t, η) are any specified matrixvalued functions of the dimension r × (m + r) given for t ≥ 0, η ∈ [−h, 0]; K1 (t) is continuous for t ≥ 0; and K2 (t, η) is continuous with respect to t ≥ 0 uniformly in η ∈ [−h, 0], and this function is piecewise continuous in η ∈ [−h, 0] for any t ≥ 0. Due to this transformation and Eq. (4.39), the dynamic part (4.40)–(4.41) of the system (4.40)–(4.41) and (4.42) becomes as $ dx(t)  # = A1j (t, ε)x(t − εhj ) + A2j (t, ε)ω(t − εhj ) dt N

 +

j =0

0 −h

# $ G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)ω(t + εη) dη,

(4.137)

$ dω(t)  # = A3j (t, ε)x(t − εhj ) + A4jK (t, ε)ω(t − εhj ) dt N

ε  +

j =0

0 −h

# $ G3 (t, η, ε)x(t + εη) + G4K (t, η, ε)ω(t + εη) dη + B2 w (t),

(4.138)

where t ≥ 0, and K (t, ε) = A40 (t, ε) + B2 K1 (t), A40

A4jK (t, ε) = A4j (t, ε), j = 1, . . . , N, G4K (t, η, ε) = G4 (t, η, ε) + B2 K2 (t, η). (4.139)

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

259

Proposition 4.6 For a given ε ∈ (0, ε0 ], the system (4.1)–(4.2) is completely Euclidean space controllable at the time instant tc , if and only if the system (4.137)– (4.138) and (4.42) is Euclidean space output controllable at this time instant. Proof First of all, let us note that, due to Lemma 3.2 (see Sect. 3.3.2 of Chap. 3), the Euclidean space output controllability of the auxiliary system (4.40)–(4.41) and (4.42) at tc is equivalent to such a controllability of the system (4.137)–(4.138) and (4.42). Remember that the system (4.40)–(4.41) and (4.42) is equivalent to the system (4.1), (4.2), (4.17), and (4.18). These two observations, along with Lemma 4.1, directly yield the statement of the proposition.   Now, let us decompose asymptotically the singularly perturbed system (4.137)– (4.138) and (4.42) into the slow and fast subsystems. This decomposition is carried out similarly to that for the system (4.1),(4.2), (4.17), and (4.18). Thus, the slow subsystem, associated with (4.137)–(4.138) and (4.42), consists of the differentialalgebraic system dxs (t) = A1s (t)xs (t) + A2s (t)ωs (t), t ≥ 0, dt

(4.140)

0 = A3s (t)xs (t) + A4sK (t)ωs (t) + B2 w s (t), t ≥ 0,

(4.141)

and the output equation (4.25). In (4.140)–(4.141) and (4.25), xs (t) ∈ E n and ωs (t) ∈ E m+r are state variables, w s (t) ∈ E r is a control, ζs (t) ∈ E n is an output, and Ais (t) =

N 

 Aij (t, 0) +

j =0

A4sK (t) =

N  j =0

0

−h

Gi (t, η, 0)dη, 

A4jK (t, 0) +

0

−h

i = 1, 2, 3,

G4K (t, η, 0)dη.

(4.142)

(4.143)

If det A4sK (t) = 0,

t ≥ 0,

(4.144)

the differential-algebraic system (4.140)–(4.141) can be reduced to the differential equation with respect to xs (t) dxs (t) ¯ K ws (t), = A¯ K s (t)xs (t) + B s (t) dt

t ≥ 0,

where  K −1 A3s (t), A¯ K s (t) = A1s (t) − A2s (t) A4s (t)  K −1 B2 . B¯ K s (t) = −A2s (t) A4s (t)

(4.145)

260

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

The fast subsystem, associated with (4.137)–(4.138) and (4.42), consists of the differential equation with state delays dωf (ξ )  K = A4j (t, 0)ωf (ξ − hj ) + dξ N

j =0



0 −h

G4K (t, η, 0)ωf (ξ + η)dη +B2 w f (ξ ), ξ ≥ 0,

(4.146)

and the output equation (4.53). Note that in (4.146) and (4.53), t ≥ 0 is a parameter, while ξ is an independent variable. Moreover, in this system, ωf (ξ ) ∈ E m+r ;   ωf (ξ ), ωf (ξ + η) , η ∈ [−h, 0) is a state variable; w f (ξ ) ∈ E r , ( wf (ξ ) is a m control); and ζf (ξ ) ∈ E (ζf (ξ ) is an output). Remark 4.6 By the same arguments as in Remark 4.1, we conclude that the output controllability of the slow subsystem in the form (4.140)–(4.141) and (4.25) is the impulse-free controllability of the system (4.140)–(4.141) with respect to xs (t). Similarly, the output controllability of the slow subsystem in the form (4.145) and (4.25) is the complete controllability of the system (4.145). Lemma 4.10 The system (4.66)–(4.67) is impulse-free controllable with respect to xs (t) at the time instant tc if and only if the system (4.140)–(4.141) is impulse-free controllable with respect to xs (t) at this time instant. Proof of the lemma is presented in Sect. 4.4.4.2. Based on Lemmas 4.4 and 4.10, we directly obtain the following corollary. Corollary 4.2 The system (4.6)–(4.7) is impulse-free controllable with respect to xs (t) at the time instant tc if and only if the system (4.140)–(4.141) is impulse-free controllable with respect to xs (t) at this time instant. Quite similarly to Lemmas 4.5 and 4.7, we have the following propositions. Proposition 4.7 Let the condition (4.144) be satisfied. Then, the system (4.140)– (4.141) is impulse-free controllable with respect to xs (t) at the time instant tc , if and only if the system (4.145) is completely controllable at this time instant. Proposition 4.8 For a given t ≥ 0, the system (4.52)–(4.53) is Euclidean space output controllable if and only if the system (4.146) and (4.53) is Euclidean space output controllable. Based on Lemma 4.2 and Proposition 4.8, as well as on the equivalence of the systems (4.29)–(4.30), (4.31) and (4.52)–(4.53), we directly obtain the following corollary. Corollary 4.3 For a given t ≥ 0, the system (4.13) is completely Euclidean space controllable if and only if the system (4.146) and (4.53) is Euclidean space output controllable.

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

4.4.4.2

261

Proof of Lemma 4.10

First of all, we are going to transform equivalently the differential-algebraic systems mentioned in the lemma. Let us start with the system (4.66)–(4.67). Substitution of (4.59) into Eqs. (4.68) and use of (4.8)–(4.9) yield after a routine algebra AK 2s (t) = A2s (t) + B1s (t)K(t),

AK 4s (t) = A4s (t) + B2s (t)K(t), t ≥ 0.

K Due to these expressions for AK 2s (t) and A4s (t), the system (4.66)–(4.67) can be rewritten as

  dxs (t) = A1s (t)xs (t) + A2s (t) + B1s (t)K(t) ys (t) + B1s (t)ws (t), t ≥ 0, dt   0 = A3s (t)xs (t) + A4s (t) + B2s (t)K(t) ys (t) + B2s (t)ws (t), t ≥ 0. (4.147) Proceed to the system (4.140)–(4.141). Substitution of (4.139) into (4.143) directly yields A4sK (t) = A40 (t, 0) + B2 K1 (t) +  +

N 

A4j (t, 0)

j =1 0 −h



G4 (t, η, 0) + B2 K2 (t, η) dη, t ≥ 0.

Let us partition the matrices K1 (t) and K2 (t, η) into blocks as ! " K1 (t) = K11 (t) , K12 (t) ,

! " K2 (t, η) = K21 (t, η) , K22 (t, η) ,

where the blocks K11 (t) and K21 (t, η) are of the dimension r × m, while the blocks K12 (t) and K22 (t, η) are of the dimension r × r. Using these block representations, as well as the block representations of the matrices A4j (t, ε) (j = 0, 1, . . . , N), G4 (t, η, ε), and B2 (see Eqs. (4.35), (4.36), (4.38), and (4.39)), we can rewrite the above obtained expression for A4sK (t) in the following block form: ⎛⎜ A4sK (t) = ⎝

N j =0 A4j (t, 0)

-N

j =0 B2j (t, 0)

⎞ ⎟ ⎠

K11 (t) K12 (t) − Ir ⎞ .0 −h G4 (t, η, 0)dη −h H2 (t, η, 0)dη ⎜ ⎟ +⎝ ⎠ , t ≥ 0. .0 .0 −h K21 (t, η)dη −h K22 (t, η)dη ⎛.0

262

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Finally, by virtue of (4.8)–(4.9), we can convert this block form expression to the following one:  A4sK (t) =

A4s (t) B2s (t) K1 (t) K2 (t) − Ir

 ,

t ≥ 0,

(4.148)

where  K1 (t) = K11 (t) +

0

−h

 K21 (t, η)dη, K2 (t) = K12 (t) +

0 −h

K22 (t, η)dη.

Furthermore, using (4.8)–(4.9) and (4.33)–(4.34), (4.37)–(4.38), we can rewrite the matrices Ais (t) (i = 1, 2, 3), given by Eq. (4.142), as   ! " A3s (t) A1s (t) = A1s (t), A2s (t) = A2s (t), B1s (t) , A3s (t) = . Or×n (4.149)   Now, let us partition the vector ωs (t) into blocks as ωs (t) = col ys (t) , us (t) , where ys (t) ∈ E m and us (t) ∈ E r . Using this block representation, as well as Eqs. (4.39), (4.148), and (4.149), we can rewrite the system (4.140)–(4.141) in the following form: dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) + B1s (t)us (t), t ≥ 0, dt 0 = A3s (t)xs (t) + A4s (t)ys (t) + B2s (t)us (t), t ≥ 0, 0 = K1 (t)ys (t) + K2 (t)us (t) − us (t) + w s (t), t ≥ 0. Finally, we can rewrite this system as   dxs (t) = A1s (t)xs (t) + A2s (t) + B1s (t)K1 (t) ys (t) dt ws (t), t ≥ 0, +B1s (t)K2 (t)us (t) + B1s (t)   0 = A3s (t)xs (t) + A4s (t) + B2s (t)K1 (t) ys (t) +B2s (t)K2 (t)us (t) + B2s (t) ws (t), t ≥ 0, us (t) = K1 (t)ys (t) + K2 (t)us (t) + w s (t), t ≥ 0.

(4.150)

Now, based on the equivalent forms (4.147) and (4.150) of the systems (4.66)– (4.67) and (4.140)–(4.141), respectively, we are going to show the validity of the lemma’s statement. Let us start with the sufficiency. Namely, we assume that the system (4.150) is impulse-free controllable with respect to xs (t) at the time instant tc . This means that for any x0 ∈ E n and xc ∈ E n , there exists a control

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

263

  function w s (·) ∈ L2 [0, tc ; E r ], for which (4.150) has a solution col xs , ys , us =   col x˜s (t), y˜s (t), u˜ s (t) ∈ L2 [0, tc ; E n+m+r ] satisfying the initial and terminal conditions x˜s (0) = x0 and x˜s (tc ) = xc . In the system (4.147), let us choose the control ws (t) ∈ L2 [0, tc ; E r ] as   ws (t) = K1 (t) − K(t) y˜s (t) + K2 (t)u˜ s (t) + w s (t), t ∈ [0, tc ]. Substituting this control into (4.147), we obtain   dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) + B1s (t)K(t) ys (t) − y˜s (t) dt

+B1s (t) K1 (t)y˜s (t) + K2 (t)u˜ s (t) + w s (t) , t ∈ [0, tc ],   0 = A3s (t)xs (t) + A4s (t)ys (t) + B2s (t)K(t) ys (t) − y˜s (t)

+B2s (t) K1 (t)y˜s (t) + K2 (t)u˜ s (t) + w s (t) , t ∈ [0, tc ].

(4.151)

Comparing this system with the first two equations of the system (4.150), we conclude immediately that the vector-valued function col x˜s (t), y˜s (t) , t ∈ [0, tc ] is the solution of (4.151). Therefore, the above chosen control ws (t) provides the existence of the solution to the system (4.147) subject to the initial and terminal conditions xs (0) = x0 and xs (tc ) = xc , where x0 ∈ E n and xc ∈ E n are any vectors. Moreover, this solution belongs to L2 [0, tc ; E n+m ]. These observations mean the impulse-free controllability with respect to xs (t) at the time instant tc of the system (4.147). Thus, the sufficiency is proven. The necessity, claimed in the lemma, is proven similarly to the sufficiency, which completes the proof of the lemma.

4.4.4.3

Hybrid Set of Riccati-Type Matrix Equations

Let us denote

S22 = B2 B2T . Consider the following set, consisting of one algebraic and two differential equations (ordinary and partial) for matrices P, Q, and R: T (t, 0)P(t)−P(t)S22 P(t)+Q(t, 0)+Q T (t, 0)+Im+r = 0, P(t)A40 (t, 0)+A40 (4.152)

" dQ(t, η) ! T = A40 (t, 0) − P(t)S22 Q(t, η) + P(t)G4 (t, η, 0) dη

264

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

+

N −1 

P(t)A4j (t, 0)δ(η + hj ) + R(t, 0, η),

(4.153)

j =1



 ∂ ∂ R(t, η, χ ) = G4T (t, η, 0)Q(t, χ ) + ∂η ∂χ

+Q T (t, η)G4 (t, χ , 0) +

N −1 

A4jT (t, 0)Q(t, χ )δ(η + hj )

j =1

+

N −1 

Q T (t, η)A4j (t, 0)δ(χ + hj ) − Q T (t, η)S22 Q(t, χ ),

(4.154)

j =1

where t ∈ [0, tc ] is a parameter, η ∈ [−h, 0] and χ ∈ [−h, 0] are independent variables, and δ(·) is the Dirac delta-function. The set of Eqs. (4.152)–(4.154) is subject to the boundary conditions Q(t, −h) = P(t)A4N (t, 0), T R(t, −h, η) = A4N (t, 0)Q(t, η),

R(t, η, −h) = Q T (t, η)A4N (t, 0). (4.155)

Remember that A4j (t, ε) (j = 0, 1, . . . , N ) are given by Eqs. (4.35)–(4.36), G4 (t, η, ε) is given in (4.38), and B2 is given in (4.39). For the sake of the further analysis of the set (4.152)–(4.155), similarly to Sect. 3.3.3, we introduce the following definition. For a given t ∈ [0, tc ], consider the following state-feedback control in the system (4.52):   1f (t)ωf (ξ ) + v˜f ωf,ξ = K



0

−h

2f (t, η)ωf (ξ + η)dη, K

(4.156)

* + 1f (t) and K 2f (t, η) are an r × (m + r)where ωf,ξ = ωf (ξ + η), η ∈ [−h, 0] , K 2f (t, η) matrix and an r × (m + r)-matrix-valued function of η, respectively, and K is piecewise continuous in the interval [−h, 0]. Remember that the system (4.52) is the dynamic part of the fast subsystem (4.52)–(4.53), associated with the auxiliary system (4.40)–(4.42). Definition 4.10 For a given t ∈ [0, tc ], the system (4.52) is called L2 -stabilizable if there exists the state-feedback control (4.156) such that for any given ω0 ∈ 2 m+r ], the solution ωf (ξ ) of (4.52) with vf (ξ ) = E m+r  , ϕωf (·) ∈ L [−h, 0; E ωf (ξ ) ∈ v˜f ωf,ξ and subject to the initial conditions (4.32) satisfies the inclusion L2 [0, +∞; E m+r ].

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

265

The following proposition is a direct consequence of the results of [4]. Proposition 4.9 Let the matrix-valued function G4 (t, η, 0) be piecewise continuous with respect to η ∈ [−h, 0] for any t ∈ [0, tc ]. Let, for any t ∈ [0, tc ], the system (4.52) be L2 -stabilizable. Then, for any t ∈ [0, tc ], the set of Eqs. (4.152)–(4.154) subject to the boundary conditions (4.155)+ has the unique * solution P(t), Q(t, η), R(t, η, χ ), (η, χ ) ∈ [−h, 0] × [−h, 0] such that (a) P T (t) = P(t); (b) the matrix-valued function Q(t, η) is piecewise absolutely continuous in η ∈ [−h, 0] with the bounded jumps at η = −hj (j = 1, . . . , N − 1); (c) the matrix-valued function R(t, η, χ ) is piecewise absolutely continuous in η ∈ [−h, 0] and χ ∈ [−h, 0] with the bounded jumps at η = −hj1 and χ = −hj2 (j1 = 1, . . . , N − 1; j2 = 1, . . . , N − 1), moreover, R T (t, η, χ ) = R(t, χ , η); (d) all roots λ(t) of the equation  N ! "  det λIm − A40 (t, 0) − S22 P(t) − A4j (t, 0) exp(−λhj ) j =1

 −

0 −h

!

 " G4 (t, η, 0) − S22 Q(t, η) exp(λη)dη = 0

(4.157)

satisfy the inequality Reλ(t) < −2γ (t),

t ∈ [0, tc ],

where γ (t) > 0 is some function of t. By virtue of the results of Sect. 3.3.3 (Lemmas 3.7, 3.8, and 3.9), we directly have the following two propositions. Proposition 4.10 Let the assumption (AIII) with respect to A4j (t, ε) and B2j (t, ε) (j = 0, 1, . . . , N) be valid. Let the assumption (AIV) with respect to G4 (t, η, ε) and H2 (t, η, ε) be valid. Let, for any t ∈ [0, tc ], the system (4.52) be L2 -stabilizable. Then, the matrices P(t), Q(t, η), and R(t, η, χ ) are continuous functions of t ∈ [0, tc ] uniformly in (η, χ ) ∈ [−h, 0]×[−h, 0]. Moreover, the derivatives dP(t)/dt, ∂Q(t, η)/∂t, and ∂R(t, η, χ )/∂t exist and are continuous functions of t ∈ [0, tc ] uniformly in (η, χ ) ∈ [−h, 0] × [−h, 0]. Proposition 4.11 Let the assumption (AIII) with respect to A4j (t, ε) and B2j (t, ε) (j = 0, 1, . . . , N) be valid. Let the assumption (AIV) with respect to G4 (t, η, ε) and H2 (t, η, ε) be valid. Let, for any t ∈ [0, tc ], the system (4.52) be L2 -stabilizable. Then, there exists a positive number γ¯ such that all roots λ(t) of Eq. (4.157) satisfy the inequality Reλ(t) < −2γ¯ , t ∈ [0, tc ].

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4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

4.4.4.4

Parameter-Free Controllability Conditions of the Nonstandard System (4.1)–(4.2)

As it was noted in Sect. 4.4.2, in the case of the nonstandard system (4.1)–(4.2), the assumption (AV) is violated. In Sect. 4.4.2, this assumption was replaced with the assumptions (AI)–(AII). Here, we replace the assumption (AV) with another assumption. Namely, we assume that (AVI) for all t ∈ [0, tc ] and any complex number λ with Reλ ≥ 0, the following equality is valid: $ # rank FA (t, λ) − λIm , FB (t, λ) = m, where FA (t, λ) =

N 

 A4j (t, 0) exp(−λhj ) +

j =0

FB (t, λ) =

N 

0

−h

 B2j (t, 0) exp(−λhj ) +

j =0

G4 (t, η, 0) exp(λη)dη,

0

−h

H2 (t, η, 0) exp(λη)dη.

Lemma 4.11 Let the assumption (AVI) be valid. Then, for all t ∈ [0, tc ] and any complex number λ with Reλ ≥ 0, the following equality is valid: rank  +

 N

A4j (t, 0) exp(−λhj )

j =0

0 −h

 G4 (t, η, 0) exp(λη)dη − λIm+r , B2 = m + r.

(4.158)

Proof Using the block form of the matrices A4j (t, ε) (j = 0, 1, . . . , N ), G4 (t, η, ε), and B2 (see Eqs. (4.35), (4.36), (4.38), and (4.39)), we can rewrite the block matrix in the left-hand side of (4.158) as follows:  N j =0

 A4j (t, 0) exp(−λhj ) +

0

−h

 G4 (t, η, 0) exp(λη)dη − λIm+r , B2



FA (t, λ) − λIm FB (t, λ) Om×r Or×m − (λ + 1)Ir Ir

=  .

This equation, along with the assumption (AVI), directly yields Eq. (4.158), which completes the proof of the lemma.  

4.4 Parameter-Free Controllability Conditions for Systems with Small State. . .

267

Corollary 4.4 Let the assumption (AVI) be valid. Then, for any t ∈ [0, tc ], the system (4.52) is L2 -stabilizable. Proof The corollary is a direct consequence of Lemma 4.11 and the results of [17].   Theorem 4.4 Let the assumptions (AIII)–(AIV) and (AVI) be valid. Let the system (4.6)–(4.7) be impulse-free controllable with respect to xs (t) at the time instant tc . Let, for t = tc , the system (4.13) be completely Euclidean space controllable. Then, there exists a positive number ε∗ (ε∗ ≤ ε0 ), such that for all ε ∈ (0, ε∗ ], the singularly perturbed system (4.1)–(4.2) is completely Euclidean space controllable at the time instant tc . Proof Let us start with the auxiliary system (4.40)–(4.41) and (4.42). Due to Eqs. (4.33)–(4.39), the matrix-valued coefficients of this system satisfy the smoothness conditions similar to the assumptions (AIII) and (AIV) on the matrix-valued functions Aij (t, ε) and Gi (t, η, ε) (i = 1, . . . , 4; j = 0, 1, . . . , N). For a given ε ∈ (0, ε0 ] in the auxiliary system (4.40)–(4.41) and (4.42), let us make the control transformation (4.136), where K1 (t) = −B2T P(t), K2 (t, η) = −B2T Q(t, η), t ∈ [0, tc ], η ∈ [−h, 0], (4.159) and P(t) and Q(t, η) are the components of the solution to the problem (4.152)– (4.154) and (4.155) mentioned in Proposition 4.9. As a result of this transformation, we obtain the system (4.137)–(4.138) and (4.42). By virtue of Corollary 4.4 and Propositions 4.9 and 4.10, the matrix-valued coefficients of this system satisfy the smoothness conditions similar to the assumptions (AIII) and (AIV) on the matrixvalued functions Aij (t, ε) and Gi (t, η, ε) (i = 1, . . . , 4; j = 0, 1, . . . , N). The slow and fast subsystems, associated with (4.137)–(4.138) and (4.42), are (4.140)–(4.141) and (4.146), (4.53), respectively. Since the system (4.6)– (4.7) is impulse-free controllable with respect to xs (t) at the time instant tc , then, due to Corollary 4.2, the system (4.140)–(4.141) is impulse-free controllable with respect to xs (t) at the time instant tc . Furthermore, since, for t = tc , the system (4.13) is completely Euclidean space controllable, then, due to Corollary 4.3, the system (4.146) and (4.53) for t = tc is Euclidean space output controllable. By virtue of Corollary 4.4 and Propositions 4.9 and 4.11, we can conclude that, for any t ∈ [0, tc ], the value λ = 0 is not a root of Eq. (4.157). Hence, the matrix A4sK (t), given by (4.143) and (4.159), is invertible for all t ∈ [0, tc ]. Thus, the slow subsystem (4.140)–(4.141) is reduced to the differential equation (4.145). Therefore, due to Proposition 4.7, the abovementioned impulse-free controllability of the system (4.140)–(4.141) yields the complete controllability of the system (4.145) at the time instant tc . Now, by application of Lemma 4.9 to the system (4.137)–(4.138) and (4.42), we directly obtain the existence of a positive number ε∗ (ε∗ ≤ ε0 ), such that for all ε ∈ (0, ε∗ ], this system is Euclidean space output controllable at the time instant tc . Finally, using Proposition 4.6 yields the complete Euclidean space controllability of the system (4.1)–(4.2) at the time instant tc for all ε ∈ (0, ε∗ ], which completes the proof of the theorem.  

268

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

4.5 Examples: Systems with Small State and Control Delays In this section, we present several examples, illustrating the theoretic results obtained in the previous sections for the systems with the small state delays.

4.5.1 Example 1 Consider the following system, a particular case of (4.1)–(4.2):

 + ε

dx(t) = x(t) − 4y(t) + 5y(t − ε) dt 0 −2

ηx(t + εη)dη + (t − 5)u(t) − tu(t − ε),

t ≥ 0,

dy(t) = 3x(t) + (t − 5)y(t) − x(t − ε) − x(t − 2ε) + y(t − ε) dt +(t − 2)u(t) + tu(t − ε),

t ≥ 0,

(4.160)

where x(t), y(t), and u(t) are scalars, i.e., n = m = r = 1; h1 = 1 and h2 = h = 2. We study the complete Euclidean space controllability of the system (4.160) at the time instant tc = 2 for all sufficiently small ε > 0. For this purpose, let us write down the slow and fast subsystems associated with (4.160). Begin with the slow subsystem. For the system (4.160), the matrix A4s (t), given in (4.8), becomes a scalar and has the form A4s (t) = t − 4. Thus, the condition (4.10) is satisfied for all t ∈ [0, 2], meaning that the slow subsystem associated with (4.160) can be reduced to the differential equation (4.11), i.e., dxs (t) t −3 7t − 22 = xs (t) + us (t), dt 4−t 4−t

t ∈ [0, 2].

(4.161)

Due to (4.13), the fast subsystem associated with the system (4.160) is dyf (ξ ) = (t −5)yf (ξ )+yf (ξ −1)+(t −2)uf (ξ )+tuf (ξ −1), ξ ≥ 0, dξ

(4.162)

where t ∈ [0, 2] is a parameter. It should be noted the following. Although the delay in the original system (4.160) is 2ε, the delay in the fast subsystem is 1 (but not 2), meaning that in this subsystem the coefficients for the terms with the delay 2 equal zero. Therefore, in what follows, it is sufficient to analyze the fast subsystem with the delay 1. It is seen directly that the assumptions (AIII)–(AIV) are satisfied for the system (4.160). Let us show the fulfillment of the assumption (AV) for this system.

4.5 Examples: Systems with Small State and Control Delays

269

Indeed, Eq. (4.114) becomes as λ − t + 5 − exp(−λ) = 0.

(4.163)

For Reλ ≥ −0.5, one obtains the following:   Re λ − t + 5 − exp(−λ) ≥ 2.85 − t > 0 ∀t ∈ [0, 2], meaning that all roots λ(t) of Eq. (4.163) satisfy the inequality Reλ(t) < −0.5, t ∈ [0, 2]. Thus, for the system (4.160) and tc = 2, the assumption (AV) is satisfied with β = 0.25. Since the assumptions (AIII)–(AIV) are also satisfied for the system (4.160) and tc = 2, one can try to use Theorem 4.1 in order to find out whether the system (4.160) is completely Euclidean space controllable at tc = 2 for all sufficiently small values of ε > 0. For this purpose, proper kinds of controllability of the systems (4.161) and (4.162) should be analyzed. Let us start with the system (4.161). Since the coefficient for us (t) in (4.161) differs from zero for t ∈ [0, 2], this system is completely controllable at the time instant tc = 2. Proceed to the system (4.162). Due to Lemma 4.2, for the given t = tc = 2, this system is completely Euclidean space controllable if for this value of t, the auxiliary system, consisting of the differential equations (4.162) and (4.30) with the scalar control vf (ξ ) and the output equation (4.31), is Euclidean space output controllable. For t = 2, the system (4.162), (4.30), and (4.31) becomes d dξ



yf (ξ ) uf (ξ )



=A



yf (ξ ) uf (ξ )



+H



yf (ξ − 1) uf (ξ − 1)



f (ξ ), + Bv

  yf (ξ ) , ζf (ξ ) = Z uf (ξ )

(4.164)

where       = − 3 0 , H = 1 2 , B = 0 , Z = (1 , 0). A 0 1 0 0 1 Note that the Euclidean dimension of the state variable in (4.164) is nf =2, while such dimensions of the control and the output are rf = 1 and qf = 1, respectively. To verify the Euclidean space output controllability of the system (4.164), we apply the algebraic criterion for such a controllability of a time-invariant differentialdifference system (see [18]). Using this criterion, we are going to show that the system (4.164) is Euclidean space output controllable at any given instant ξc ∈ (1, 2] of the stretched time ξ . For this purpose, we construct the following matrices:

270

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .







= − 3 0 , A 0 = A 1 = A 0 1 0 = I2 , E

  1 = O2×2 , I2 , E 0 = I2 , C

0 O2×2 A A 0 H 0 = Z, Z

0 = B, B



⎛

⎞ −3 0 0 0 ⎜ 0 −1 0 0⎟ ⎟, =⎜ ⎝ 1 2 −3 0⎠ 0 0 0 −1 1 = Z E 1 = (0, 0, 1, 0), Z

1 = C 1 B, B

where 1 = C



I2   0 C 0 exp A



⎛

⎞ 1 0 ⎜0 ⎟ 1 ⎟. =⎜ ⎝ exp(−3) 0 ⎠ 0 exp(−1)

Hence, ⎛

⎞ 0 ⎜ ⎟ ⎟. 1 = ⎜ 1 B ⎝0 ⎠ exp(−1) Due to the results of [18], the system (4.164) is Euclidean space output controllable at a given value ξc ∈ (1, 2] of the independent variable ξ , if and only if the rank of the following matrix equals qf :   = Z 0 B 0 A 1 B 1 A 0 , . . . , Z nf −1 B 0 , Z 1 , . . . , Z 2nf −1 B 1 . D 0 1 is scalar and qf = 1, it is sufficient to show that at Since each block of the matrix D least one block in this matrix differs from zero. Remember that nf = 2. Therefore, 1 A Calculating this block, we obtain Z 1 A 1 B 1 is a block of D. 1 B 1 = 2 = 0, Z = qf = 1. Thus, the system (4.164) is Euclidean space output meaning that rankD controllable with any given value ξc ∈ (1, 2] mentioned in Definition 4.6. Hence, the system (4.162) is completely Euclidean space controllable. Therefore, by virtue of Theorem 4.1, the system (4.160) is completely Euclidean space controllable at tc = 2 robustly with respect to ε > 0 for all its sufficiently small values.

4.5.2 Example 2 Consider the following system, a particular case of (4.1)–(4.2): dx(t) = 2(t − 1)x(t) + 4y(t) − 2tx(t − ε) − y(t − ε) dt

4.5 Examples: Systems with Small State and Control Delays

 +tu(t) − u(t − ε) + ε

271

0 −1

2tηu(t + εη)dη, t ≥ 0,

dy(t) = 4x(t) + y(t) − 2x(t − ε) − y(t − ε) + 2u(t) − u(t − ε), t ≥ 0, dt (4.165)

where x(t), y(t), and u(t) are scalars, i.e., n = m = r = 1; h = 1. In this example, like in the previous one, we study the complete Euclidean space controllability of the considered system. We study this controllability at the time instant tc = 2 for all sufficiently small ε > 0. The asymptotic decomposition of the system (4.165) yields the slow and fast subsystems, respectively, dxs (t) = −2xs (t) + 3ys (t) − us (t), dt

t ≥ 0,

0 = 2xs (t) + us (t),

t ≥ 0,

(4.166)

and dyf (ξ ) = yf (ξ ) − yf (ξ − 1) + 2uf (ξ ) − uf (ξ − 1), dξ

ξ ≥ 0.

(4.167)

It is seen that the assumptions (AIII)–(AIV) are satisfied for the system (4.165). The condition (4.10) is not satisfied for this system, meaning that (4.165) is a nonstandard system, and it does not satisfy the assumption (AV). Let us show the fulfillment of the assumptions of Corollary 4.1 for the system (4.165). Indeed, since in this system G4 (t, η, ε) ≡ 0 and H2 (t, η, ε) ≡ 0, the assumptions (AI) and (AII) are fulfilled with any positive number CG and any nonnegative number CH . Let us choose CG = 0.1 and CH = 0. In such a case, the LMI (4.81) can be rewritten as the following set of LMIs with respect to scalar unknowns P , Q1 , and V : ⎛

2P + Q1 + 4V + 3 ⎜ −P ⎜ ⎝ V P

P V 0 − Q1 0 −1 0 0

⎞ P 0⎟ ⎟ < O4×4 , 0 ⎠ − 10 P > 0,

Q1 > 0.

(4.168)

It is verified directly that the values P = 0.1, Q1 = 0.1, and V = −2 satisfy the LMIs (4.168). Moreover, using Eq. (4.92), we obtain the gain K = −20 mentioned in Lemma 4.8 and Corollary 4.1. Now, let us find out whether the systems (4.166) and (4.167) are controllable in the sense mentioned in Theorem 4.3. We start with (4.166). Let x0 and xc be

272

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

any given numbers. Let ϑ = (xc − x0 )/6. One can verify immediately that for the numbers x0 and xc , there exists a control us (t) ∈ L2 [0, 2; E 1 ], namely, us (t) = −2x0 − 3ϑt 2 , such that the system (4.166), subject to the initial xs (0) = x0 and the terminal xs (2) = xc conditions, has a continuous solution, namely, xs (t) = x0 + 1.5ϑt 2 ,

ys (t) = ϑt.

Thus, the system (4.166) is impulse-free controllable with respect to xs (t) at the time instant tc = 2. Proceed to (4.167). The complete Euclidean space controllability of this system is shown similarly to such a kind of controllability of the system (4.162) in the previous example. Finally, using Theorem 4.3, we obtain the complete Euclidean space controllability of the system (4.165) at the time instant tc = 2 robustly with respect to ε > 0 for all its sufficiently small values. Now, let us prove the complete Euclidean space controllability of the system (4.165) by application of Theorem 4.4. For this purpose, we should show the fulfillment of the assumption (AVI) for the system (4.165). The matrix, mentioned in this assumption, becomes as

F1 (λ) = 1 − exp(−λ) − λ , 2 − exp(−λ) . For λ = 0, the rank of this matrix equals the Euclidean dimension of the fast subsystem m = 1. Since λ = 0 is a single root with the nonnegative real part of the equation λ − 1 + exp(−λ) = 0, the rank of the matrix F1 (λ) equals m = 1 for all complex λ with Reλ ≥ 0. Thus, the assumption (AVI) is fulfilled for the system (4.165). The latter means that, by Theorem 4.4, the system (4.165) is completely Euclidean space controllable at the time instant tc = 2 robustly with respect to ε > 0 for all its sufficiently small values.

4.5.3 Example 3 Consider the following particular case of (4.1)–(4.2): dx(t) =2(t − 1)x(t)+4y(t)−2tx(t−ε)−y(t−ε)+u(t)−2u(t−ε), t ≥ 0, dt dy(t) ε =4x(t)+y(t)−2x(t−ε)−y(t−ε)+2u(t)− exp(t−2)u(t−ε), t ≥ 0, dt (4.169)

4.5 Examples: Systems with Small State and Control Delays

273

where x(t), y(t), and u(t) are scalars, i.e., n = m = r = 1; h = 1. Let us analyze the complete Euclidean space controllability of the system (4.169) at the time instant tc = 2. Decomposing asymptotically this system, we obtain the slow subsystem dxs (t) = −2xs (t) + 3ys (t) − us (t), t ≥ 0, dt ! " 0 = 2xs (t) + 2 − exp(t − 2) us (t), t ≥ 0,

(4.170)

and the fast subsystem dyf (ξ ) = yf (ξ ) − yf (ξ − 1) + 2uf (ξ ) − exp(t − 2)uf (ξ − 1), ξ ≥ 0, dξ (4.171) where t ≥ 0 is a parameter. Similarly to Sect. 4.5.2, the assumptions (AIII)–(AIV) are fulfilled for the system (4.169), while the condition (4.10) and, therefore, the assumption (AV) are not satisfied for this system. Thus, Theorems 4.1 and 4.2 are not applicable to the analysis of the controllability of the system (4.169). Let us show that Theorem 4.3 is applicable to this analysis. For this purpose, first we establish (like in Sect. 4.5.2) the fulfillment of the assumptions of Corollary 4.1. Since in the system (4.169), the coefficients G4 (t, η, ε) and H2 (t, η, ε) are identically zero, the assumptions (AI) and (AII) are fulfilled with any positive number CG and any nonnegative number CH . In this example, we choose CG = CH = 0.1. Thus, the LMI (4.81) can be rewritten as the following set of LMIs with respect to scalar unknowns P , Q1 , and V : ⎛

⎞ −P V P 2P + Q1 + 4V + exp(2t − 4) + 2 ⎜ −P ⎟ 0 − Q1 0 ⎜ ⎟ < O4×4 , ⎝ V ⎠ 0 − 0.909 0 P 0 0 − 10 P > 0,

Q1 > 0. (4.172)

Based on the inequality 0 < exp(2t − 4) ≤ 1, t ∈ [0, 2], one can verify directly that the values P = 0.1, Q = 0.1, and V = −1.5 satisfy the LMIs (4.172) for all t ∈ [0, 2]. Hence, using Eq. (4.92), we obtain that the gain K(t) mentioned in Lemma 4.8 and Corollary 4.1 is K(t) = −15, t ∈ [0, 2]. Now, let us establish the impulse-free controllability of the system (4.170) with respect to xs (t) at the time instant tc = 2. Let x0 and xc be any given numbers. Let us denote

274

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

  4 1 11 − exp(−4) 2 − exp(−2) + exp(−4) = 0, φ1 = 12 3 4 ! " φ2 φ2 = xc − x0 exp(−2) 2 − exp(−2) , φ3 = , φ1

exp(2 − t) = 0 ∀t ∈ [0, 2], 2 − exp(t − 2) " ! " 4!     exp 3(t − 2) − exp(−6) f2 (t) = 2 exp 2(t − 2) − exp(−4) − 3 "   1! + exp 4(t − 2) − exp(−8) . 4

f1 (t) =

It is verified directly by routine calculations that the continuous control function us (t) = −

2[(xc − φ2 )f1 (t) + φ3 f1 (t)f2 (t)] , t ∈ [0, 2], 2 − exp(t − 2)

transfers the system (4.170) from the initial position xs (0) = x0 to the terminal position xs (2) = xc along the continuous trajectory xs (t) = (xc − φ2 )f1 (t) + φ3 f1 (t)f2 (t), ys (t) =

φ3 , t ∈ [0, 2], 3f1 (t)

meaning that the system (4.170) is impulse-free controllable with respect to xs (t) at the time instant tc = 2. Proceed to the system (4.171). For t = tc = 2, this system coincides with the system (4.167). Then, due to Sect. 4.5.2, the system (4.171) for t = 2 is completely Euclidean space controllable. Thus, all the conditions of Theorem 4.3 are fulfilled, implying the complete Euclidean space controllability of the system (4.169) at the time instant tc = 2 robustly with respect to ε > 0 for all its sufficiently small values. Now, let us show the complete Euclidean space controllability of the system (4.169) by application of Theorem 4.4. This means that we should verify the fulfillment of the assumption (AVI) for (4.169). The matrix, mentioned in this assumption, becomes as

F2 (t, λ) = 1 − exp(−λ) − λ , 2 − exp(t − 2 − λ) , and we should show that rankF2 (t, λ) = m = 1 for all t ∈ [0, tc ] = [0, 2] and all complex λ with nonnegative real parts. By the same arguments as in Example 2, it is sufficient to show that the rank of F2 (t, 0) equals 1 for all t ∈ [0, 2]. The latter is verified immediately, because exp(t − 2) ≤ 1 for all t ∈ [0, 2]. Hence, the assumption (AVI) is fulfilled for the system (4.169). Therefore, by Theorem 4.4, the system (4.169) is completely Euclidean space controllable at the time instant tc = 2 robustly with respect to ε > 0 for all its sufficiently small values.

4.6 Systems with State Delays of Two Scales: Main Notions and Definitions

275

4.6 Systems with State Delays of Two Scales: Main Notions and Definitions 4.6.1 Original System In this section we consider another type of singularly perturbed systems with state and control delays. Namely,  dx(t)  = A1i (t, ε)x(t − gi ) + A2j (t, ε)y(t − εhj ) dt  + +

N 

0 −g

M

N

i=0

j =0



G1 (t, ρ, ε)x(t + ρ)dρ + 

B1j (t, ε)u(t − εhj ) +

j =0

ε

−h

−h

G2 (t, η, ε)y(t + εη)dη

H1 (t, η, ε)u(t + εη)dη, t ≥ 0,

(4.173)

 dy(t)  = A3i (t, ε)x(t − gi ) + A4j (t, ε)y(t − εhj ) dt  +

+

0

0

N  j =0

0 −g

M

N

i=0

j =0



G3 (t, ρ, ε)x(t + ρ)dρ + 

B2j (t, ε)u(t − εhj ) +

0

−h

0

−h

G4 (t, η, ε)y(t + εη)dη

H2 (t, η, ε)u(t + εη)dη, t ≥ 0,

(4.174)

where x(t) ∈ E n , y(t) ∈ E m , and u(t) ∈ E r (u(t) is a control); ε > 0 is a small parameter; N ≥ 1 and M ≥ 1 are integers; 0 = g0 < g1 < . . . < gM = g and 0 = h0 < h1 < . . . < hN = h are some given ε-independent constants; Aki (t, ε), Alj (t, ε), Gk (t, ρ, ε), Gl (t, η, ε), Bpj (t, ε), and Hp (t, η, ε) (i = 0, 1, . . . , M; j = 0, 1, . . . , N; k = 1, 3; l = 2, 4; p = 1, 2) are matrixvalued functions of corresponding dimensions, given for t ≥ 0, ρ ∈ [−g, 0], η ∈ [−h, 0], and ε ∈ [0, ε0 ]; the functions Aki (t, ε), Alj (t, ε), and Bpj (t, ε) (i = 0, 1, . . . , M; j = 0, 1, . . . , N; k = 1, 3; l = 2, 4; p = 1, 2) are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ]; the functions Gk (t, ρ, ε) (k = 1, 3) are piecewise continuous in ρ ∈ [−g, 0] for any (t, ε) ∈ [0, +∞) × [0, ε0 ], and they are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in ρ ∈ [−g, 0]; and the functions Gl (t, η, ε) and Hp (t, η, ε) (l = 2, 4; p = 1, 2) are piecewise continuous in η ∈ [−h, 0] for any (t, ε) ∈ [0, +∞) × [0, ε0 ], and they are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0].

276

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Like in Sect. 3.7, in the present section we assume that the positive number ε0 satisfies the inequality ε0 ≤ g1 / h. Similarly to the system (4.1)–(4.2), for a given u(·) ∈ L2loc [−εh, +∞; E r ], the system (4.173)–(4.174) is a linear time-dependent nonhomogeneous functionaldifferential (infinite-dimensional) system, and this system is singularly perturbed. However, in contrast with (4.1)–(4.2), the state variables of (4.173)–(4.174) are     x(t), x(t + ρ) , ρ ∈ [−g, 0), and y(t), y(t + εη) , η ∈ [−h, 0). Let tc ≥ g be a given time instant independent of ε. Definition 4.11 For a given ε ∈ (0, ε0 ], the system (4.173)–(4.174) is said to be completely Euclidean space controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , u0 ∈ E r , ϕx (·) ∈ L2 [−g, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ], ϕu (·) ∈ L2 [−εh, 0; E r ], xc ∈ E n , and yc ∈ E m , there exists a control function u(·) ∈ W 1,2 [0, tc ; E r ] satisfying u(0) = u0 , for which the system (4.173)–(4.174) subject to the initial and terminal conditions x(ρ) = ϕx (ρ), ρ ∈ [−g, 0); y(τ ) = ϕy (τ ), u(τ ) = ϕu (τ ), τ ∈ [−εh, 0); x(0) = x0 ,

y(0) = y0 ;

x(tc ) = xc ,

y(tc ) = yc

has a solution.

4.6.2 Asymptotic Decomposition of the Original System Now, we will decompose the original system (4.173)–(4.174) into the slow and fast subsystems. This decomposition is carried out quite similarly to such a decomposition of the system (4.1)–(4.2) in Sect. 4.2.2. Thus, the slow subsystem, associated with the system (4.173)–(4.174), is dxs (t)  = A1i (t, 0)xs (t − gi ) + dt M



0

−g

i=0

G1 (t, ρ, 0)xs (t + ρ)dρ

+A2s (t)ys (t) + B1s (t)us (t), t ≥ 0,

0=

M  i=0

 A3i (t, 0)xs (t − gi ) +

0 −g

(4.175)

G3 (t, ρ, 0)xs (t + ρ)dρ

+A4s (t)ys (t) + B2s (t)us (t), t ≥ 0,

(4.176)

4.6 Systems with State Delays of Two Scales: Main Notions and Definitions

277

  where xs (t) ∈ E n and ys (t) ∈ E m ; xs (t), xs (t + ρ) , ρ ∈ [−g, 0), and ys (t) are state variables; us (t) ∈ E r is a control; and Als (t) =

N 

 Alj (t, 0) +

−h

j =0

Bks (t) =

N 

 Bkj (t, 0) +

j =0

0

Gl (t, η, 0)dη,

l = 2, 4,

Hk (t, η, 0)dη,

k = 1, 2.

0 −h

The slow subsystem (4.175)–(4.176) is a descriptor (differential-algebraic) system with state delays, and it is ε-free. Note that the slow subsystem (4.6)–(4.7), associated with the small state delay system (4.1)–(4.2), does not have delays in the state and control variables. If det A4s (t) = 0, t ≥ 0, the time delay differential-algebraic system (4.175)– (4.176) can be converted to an equivalent system. This new system consists of the explicit expression for ys (t) ys (t) = −A−1 4s (t)

 M

 A3i (t, 0)xs (t − gi ) +

i=0

0 −g

G3 (t, ρ, 0)xs (t + ρ)dρ

 +B2s (t)us (t) ,

t ≥ 0,

and the time delay differential equation with respect to xs (t) dxs (t)  ¯ = Ai,s (t)xs (t − gi ) + dt M



i=0

0 −g

¯ s (t, ρ)xs (t + ρ)dρ G

+B¯ s (t)us (t),

t ≥ 0,

(4.177)

where A¯ i,s (t) = A1i (t, 0) − A2s (t)A−1 4s (t)A3i (t, 0), ¯ s (t, ρ) = G1 (t, ρ, 0) − A2s (t)A−1 (t)G3 (t, ρ, 0), G 4s B¯ s (t) = B1s (t) − A2s (t)A−1 4s (t)B2s (t). The differential equation (4.177) is also called the slow subsystem, associated with the original system (4.173)–(4.174). The fast subsystem, associated with the system (4.173)–(4.174), has the form dyf (ξ )  = A4j (t, 0)yf (ξ − hj ) + dξ N

j =0



0 −h

G4 (t, η, 0)yf (ξ + η)dη

278

+

4 Complete Euclidean Space Controllability of Linear Systems with State and. . . N  j =0

 B2j (t, 0)uf (ξ − hj ) +

0

−h

H2 (t, η, 0)uf (ξ + η)dη, ξ ≥ 0,

(4.178)

  where t ≥ 0 is a parameter; yf (ξ ) ∈ E m ; yf (ξ ), yf (ξ + η) , η ∈ [−h, 0) is a state variable; and uf (ξ ) ∈ E r (uf (ξ ) is a control). The system (4.178) coincides with the fast subsystem, associated with the system (4.1)–(4.2). Definition 4.12 Subject to the inequality det A4s (t) = 0, t ∈ [0, tc ], the system (4.177) is said to be completely Euclidean space controllable at the time instant tc if for any x0 ∈ E n , ϕx (·) ∈ L2 [−g, 0; E n ], and xc ∈ E n , there exists a control function us (·) ∈ L2 [0, tc ; E r ], for which (4.177) has a solution xs (t), t ∈ [0, tc ], satisfying the initial and terminal conditions x(ρ) = ϕx (ρ), ρ ∈ [−g, 0), xs (0) = x0 ;

xs (tc ) = xc .

(4.179)

Definition 4.13 The system (4.175)–(4.176) is said to be impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc if for any x0 ∈ E n , ϕx (·) ∈ L2 [−g, 0; E n ], and xc ∈ E n , there exists a control function us (·) ∈ L2 [0, tc ; E r ], for which the differential-algebraic system (4.175)–(4.176) has a solution col(xs (t), ys (t)) ∈ L2 [0, tc ; E n+m ] satisfying the initial and terminal conditions (4.179). Now, let us proceed to a definition of controllability of the fast subsystem (4.178). Since this subsystem coincides with the fast subsystem, associated with the system (4.1)–(4.2), the definition of its complete Euclidean space controllability is quite similar to Definition 4.4 for such a kind of the controllability of the system (4.13). For the sake of the further controllability analysis of the system (4.175)–(4.176), in the next section an auxiliary system with only state delays is introduced, and some properties of this system are studied.

4.7 Auxiliary System with State Delays of Two Scales and Delay-Free Control 4.7.1 Description of the Auxiliary System and Some of Its Properties Similarly to Sects. 4.3.1 and 4.3.2.1, we consider the following system:  dx(t)  = A1i (t, ε)x(t − gi ) + A2j (t, ε)ω(t − εhj ) dt M

N

i=0

j =0

4.7 Auxiliary System with State Delays of Two Scales and Delay-Free Control

 +

0 −g

 G1 (t, ρ, ε)x(t + ρ)dρ +

ε  +

0 −g

0

−h

279

G2 (t, η, ε)ω(t + εη)dη + B1 v(t), t ≥ 0,

 dω(t)  = A3i (t, ε)x(t − gi ) + A4j (t, ε)ω(t − εhj ) dt M

N

i=0

j =0

 G3 (t, ρ, ε)x(t + ρ)dρ +

0

−h

G4 (t, η, ε)ω(t + εη)dη + B2 v(t), t ≥ 0, (4.180)

  ζ (t) = Zcol x(t), ω(t) ,

t ≥ 0,

(4.181)

    where ω(t) = col y(t), u(t) , y(t) ∈ E m , u(t) ∈ E r , t ≥ −εh; x(t), x(t + ρ) , ρ ∈ [−g, 0), and ω(t), ω(t + τ ) , τ ∈ [−εh, 0), are state variables; v(t) ∈ E r is a control; ζ (t) ∈ E n+m is an output; the matrices Aki (t, ε) and Gk (t, ρ, ε) (k = 1, 3; i = 1, . . . , M) have the form  A1i (t, ε) = A1i (t, ε),

A3j (t, ε) =

 A3j (t, ε) , i = 0, 1, . . . , M, t ≥ 0, Or×n 

G1 (t, ρ, ε) = G1 (t, ρ, ε),

G3 (t, ρ, ε) =

 G3 (t, ρ, ε) , t ≥ 0, ρ ∈ [−g, 0]; Or×n

the matrices Alj (t, ε), Gl (t, η, ε) (l = 2, 4; j = 1, . . . , N ), B1 , and B2 have the form given in (4.33) and (4.35)–(4.39); and the matrix Z has the form given in (4.19). Note that the system (4.180)–(4.181) is a direct extension of the system (4.40)– (4.42) to the case of state delays of two scales in the original system. Adapting Definition 3.8 (see Sect. 3.7.1) to the system (4.180)–(4.181), we have the following definition. Definition 4.14 For a given ε ∈ (0, ε0 ], the system (4.180)–(4.181) is said to be Euclidean space output controllable at the time instant tc if for any x0 ∈ E n , ω0 ∈ E m+r , ϕx (·) ∈ L2 [−g, 0; E n ], ϕω (·) ∈ L2 [−εh, 0; E m+r ], and ζc ∈ E n+m , there  exists a control function v(·) ∈ L2 [0, tc ; E r ], for which the solution col x(t), ω(t) , t ∈ [0, tc ], of the system (4.180) with the initial conditions x(ρ)=ϕx (ρ), ρ ∈ [−g, 0); ω(τ )=ϕω (τ ), τ ∈ [−εh, 0); x(0)=x0 , ω(0)=ω0 satisfies the terminal condition   Zcol x(tc ), ω(tc ) = ζc .

280

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Quite similarly to Lemma 4.1, we have the following lemma. Lemma 4.12 For a given ε ∈ (0, ε0 ], the system (4.173)–(4.174) is completely Euclidean space controllable at the time instant tc , if and only if the system (4.180)– (4.181) is Euclidean space output controllable at this time instant. Now, let us derive a criterion of the Euclidean space output controllability for the system For this purpose, we introduce the block vector  (4.180)–(4.181).  z(t) = col x(t), ω(t) , t ≥ −g. Also, for a given ε ∈ (0, ε0 ], we consider the block matrices   A1i (t, ε) 0 Ax,i (t, ε) = 1 , i = 0, 1, . . . , M, t ≥ 0, ε A3i (t, ε) 0   0 A2j (t, ε) , j = 0, 1, . . . , N, t ≥ 0, Ay,j (t, ε) = 0 1ε A4j (t, ε)   G (t, ρ, ε) 0 , t ≥ 0, ρ ∈ [−g, 0], Gx (t, ρ, ε) = 1 1 ε G3 (t, ρ, ε) 0   0 G2 (t, η, ε) , t ≥ 0, η ∈ [−h, 0], Gy (t, η, ε) = 0 1ε G4 (t, η, ε)

ρ ∈ [−g, −εh), Gx (t, ρ, ε), t ≥ 0, Gx,y (t, ρ, ε) = 1 Gx (t, ρ, ε) + ε Gy (t, ρ/ε, ε), ρ ∈ [−εh, 0],   B B(ε) = 1 1 . ε B2 Using the above introduced block vector and block matrices, we can rewrite the system (4.180)–(4.181) in the following equivalent form for all ε ∈ (0, ε0 ] and t ≥ 0:  dz(t)  = Ax,i (t, ε)z(t − gi ) + Ay,j (t, ε)z(t − εhj ) dt M

N

j =0

i=0

 +

0 −g

Gx,y (t, ρ, ε)z(t + ρ)dρ + B(ε)v(t), ζ (t) = Zz(t).

(4.182)

The Euclidean space output controllability of the system (4.182) is defined quite similarly to such a controllability of the system (4.180)–(4.181). Moreover, the Euclidean space output controllability of the latter is equivalent to such a controllability of the former. For any given ε ∈ (0, ε0 ], we consider the following terminal-value problem with respect to (n + m + r) × (n + m + r)-matrix-valued function K (σ ):

4.7 Auxiliary System with State Delays of Two Scales and Delay-Free Control

281

M  T  dK (σ ) =− Ax,i (σ + gi , ε) K (σ + gi ) dσ i=0

−  −

N 

 T Ay,j (σ + εhj , ε) K (σ + εhj )

j =0 0 −g

 T Gx,y (σ − ρ, ρ, ε) K (σ − ρ)dρ, K (tc ) = In+m ,

σ ∈ [0, tc ),

K (σ ) = 0, σ > tc .

In this problem, it is assumed that the blocks of the matrices Ax,i (t, ε) (i = 0, 1, . . . , M), Ay,j (t, ε) (j = 0, 1, . . . , N ), and Gx,y (t, ρ, ε) satisfy the following equalities: Aki (t, ε) = Aki (tc , ε),

Alj (t, ε) = Alj (tc , ε), t > tc , ε ∈ [0, ε0 ],

Gk (t, ρ, ε) = Gk (tc , ρ, ε), t > tc , ρ ∈ [−g, 0], ε ∈ [0, ε0 ], Gl (t, η, ε) = Gl (tc , η, ε), t > tc , η ∈ [−h, 0], ε ∈ [0, ε0 ], k = 1, 3, l = 2, 4, i = 0, 1, . . . , M, j = 0, 1, . . . , N. By virtue of the results of [13] (Sect. 4.3), this terminal-value problem has the unique solution K (σ ) = K (σ, ε). Proposition 4.12 For a given ε ∈ (0, ε0 ], the system (4.180)–(4.181) is Euclidean space output controllable at the time instant tc if and only if   det Z

tc

 K (σ, ε)B(ε)B (ε)K (σ, ε)dσ Z T

T

T

= 0.

0

Proof The proposition directly follows from the results of Sect. 3.3.1 (see Corollary 3.1) and the equivalence of the Euclidean space output controllability of the systems (4.182) and (4.180)–(4.181).  

4.7.2 Asymptotic Decomposition of the Auxiliary System (4.180)–(4.181) For the sake of further analysis, let us decompose asymptotically the system (4.180)–(4.181) into the slow and fast subsystems. These subsystems are constructed similarly to the results of Sect. 3.7.2. We start with the slow subsystem. The dynamic part of this subsystem is obtained from (4.180) by setting formally ε = 0 in these equations. Thus, using the fact that B1 = 0, we have

282

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

dxs (t)  = A1i (t, 0)xs (t − gi ) + A2s (t)ωs (t) dt M

i=0



+ 0=  +

0 −g

M 

G1 (t, ρ, 0)xs (t + ρ)dρ,

t ≥ 0,

A3i (t, 0)xs (t − gi ) + A4s (t)ωs (t)

i=0 0 −g

G3 (t, ρ, 0)xs (t + ρ)dρ + B2 vs (t),

t ≥ 0,

  n m r where xs (t) ∈ E  , ωs (t) = col ys (t), us (t) , ys (t) ∈ E , and us (t) ∈r E ; xs (t), xs (t + ρ) , ρ ∈ [−g, 0), and ωs (t) are state variables; vs (t) ∈ E is a control; and Als (t) =

N 

 Alj (t, 0) +

j =0

0

−h

Gl (t, η, 0)dη,

l = 2, 4.

This system is a time delay differential-algebraic system, and it is ε-free. Using the expressions for Alj (t, ε) and Gl (t, η, ε), (l = 2, 4; j = 0, 1, . . . , N) (see Eqs. (4.33) and (4.35)–(4.38)), as well as the expressions for Als (t), (l = 2, 4) and Bks (t), (k = 1, 2) (see Sect. 4.6.2), we can rewrite the matrices Als (t), (l = 2, 4) as   ! " A4s (t) B2s (t) . A2s (t) = A2s (t) , B1s (t) , A4s (t) = − Ir Or×m The output part of the slow subsystem is obtained from (4.181) by  removing formally the term with the Euclidean part ω(t) of the fast state variable ω(t), ω(t +  εη) , η ∈ [−h, 0) (see Sect. 2.2 for the details on the slow and fast state variables). Thus, we have ζs (t) = xs (t),

t ≥ 0.

(4.183)

where ζs (t) ∈ E n is an output. Eliminating the block us (t) of the state variable ωs (t) from the above obtained differential-algebraic system and using the block form of its matrix-valued coefficient, we can reduce this system to the following one: dxs (t)  = A1i (t, 0)xs (t − gi ) + dt M

i=0



0

−g

G1 (t, ρ, 0)xs (t + ρ)dρ

+A2s (t)ys (t) + B1s (t)vs (t), t ≥ 0,

4.7 Auxiliary System with State Delays of Two Scales and Delay-Free Control

0=

M 

 A3i (t, 0)xs (t − gi ) +

0

−g

i=0

283

G3 (t, ρ, 0)xs (t + ρ)dρ

+A4s (t)ys (t) + B2s (t)vs (t), t ≥ 0.

(4.184)

  In this time delay differential-algebraic system, xs (t), xs (t + ρ) , ρ ∈ [−g, 0), and ys (t) are state variables, and vs (t) ∈ E r is a control. The system (4.184) is a simplified dynamic part of the slow subsystem, associated with the auxiliary system (4.180)–(4.181). If det A4s (t) = 0, then the system (4.184) can be reduced to the following time delay differential system with respect to xs (t): dxs (t)  ¯ = Ai,s (t)xs (t − gi ) + dt M



i=0

0 −g

¯ s (t, ρ)xs (t + ρ)dρ G

+B¯ s (t)vs (t),

t ≥ 0.

(4.185)

Note that the expressions for the matrix-valued coefficients of this equation can be found in Sect. 4.6.2. The system (4.185) and (4.183) is also called the slow subsystem of the auxiliary system (4.180)–(4.181). Remark 4.7 Similarly to Remark 4.1, we can conclude that the output controllability of the slow subsystem in the form (4.184) and (4.183) is the impulse-free controllability of its dynamic part (4.184) with respect to xs (t). The Euclidean space output controllability of the slow subsystem in the form (4.185) and (4.183) is the complete Euclidean space controllability of its dynamic part (4.185). Moreover, since the system (4.184) differs from the system (4.175)–(4.176) only by the notation of the control, the impulse-free controllability of (4.184) with respect to xs (t) coincides with such a kind of the controllability of (4.175)–(4.176). Similarly, the complete Euclidean space controllability of the system (4.185) coincides with such a kind of the controllability of the system (4.177). Proceed to the fast subsystem, associated with the auxiliary system (4.180)– (4.181). This subsystem has the form similar to the system (3.193)–(3.194) (see Sect. 3.7.2), namely, dωf (ξ )  = A4j (t, 0)ωf (ξ − hj ) dξ N

 +

j =0

0 −h

G4 (t, η, 0)ωf (ξ + η)dη + B2 vf (ξ ), ξ ≥ 0,

ζf (ξ ) = Ωf ωf (ξ ), ξ ≥ 0,

  Ωf = Im , Om×r ,

(4.186) (4.187)

284

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

  where t ≥ 0 is a parameter; ωf (ξ ) ∈ E m+r ; ωf (ξ ), ωf (ξ + η) , η ∈ [−h, 0) is a state variable; vf (ξ ) ∈ E r is a control; and ζf (ξ ) ∈ E m is an output. The system (4.186)–(4.187) coincides with the fast subsystem (4.52)–(4.53), associated with the system (4.40)–(4.42). Remember that the Euclidean space output controllability of the system (4.52)–(4.53) is defined in Sect. 4.3.2.3 (see Definition 4.8). Proposition 4.13 For a given t ≥ 0, the system (4.178) is completely Euclidean space controllable if and only if the system (4.186)–(4.187) is Euclidean space output controllable. Proof The proposition is proven similarly to Lemma 4.2 (Sect. 4.3.1).

 

In the completion of this subsection, we are going to derive criterions of the complete Euclidean space controllability for the system (4.185) and the Euclidean space output controllability for the system (4.186)–(4.187). We start with the system (4.185). Let the n × n-matrix-valued function Ks (σ ), σ ∈ [0, tc ], be the unique solution of the terminal-value problem  T dKs (σ ) =− A¯ i,s (σ + gi ) Ks (σ + gi ) dσ M

 −

i=0

0 −g

  ¯ s (σ − ρ, ρ) T Ks (σ − ρ)dρ, G Ks (tc ) = In ;

σ ∈ [0, tc ),

Ks (σ ) = 0, σ > tc .

In this problem, it is assumed that the matrices A¯ i,s (t) (i = 0, 1, . . . , M) and ¯ s (t, ρ) satisfy the following equalities: G A¯ i,s (t)=A¯ i,s (tc ), i = 0, 1, . . . , M,

¯ s (t, ρ)=G ¯ s (tc , ρ), G

t > tc , ρ ∈ [−g, 0].

Applying the results of Sect. 3.3.1 (see Proposition 3.1) to the system (4.185), we directly obtain the following assertion. Proposition 4.14 The system (4.185) is completely Euclidean space controllable at the time instant tc , if and only if  det 0

tc

KsT (σ )B¯ s (σ )B¯ Ts (σ )Ks (σ )dσ

 = 0.

Proceed to the system (4.186)–(4.187). Let, for any given t ≥ 0, the (m + r) × (m + r)-matrix-valued function Kf (ξ, t) be the unique solution of the following initial-value problem: T dKf (ξ )   = A4j (t, 0) Kf (ξ − hj ) dξ N

j =0

4.7 Auxiliary System with State Delays of Two Scales and Delay-Free Control

 +

0 −h

 T G4 (t, η, 0) Kf (ξ + η)dη,

Kf (ξ ) = 0, ξ < 0;

285

ξ > 0,

Kf (0) = Im+r .

Application of the results of Sect. 3.3.1 (see Corollary 3.1) to the system (4.186)– (4.187) yields immediately the following proposition. Proposition 4.15 For a given t ≥ 0, the system (4.186)–(4.187) is Euclidean space output controllable at a prescribed instant ξ = ξc > 0 if and only if   det Ωf

ξc 0

 KfT (ρ, t)B2 B2T Kf (ρ, t)dρΩfT

= 0.

4.7.3 Linear Control Transformation in the Auxiliary System (4.180)–(4.181) Let us make the control transformation (4.136) in the auxiliary system (4.180)– (4.181). Due to this transformation and Eq. (4.39), the dynamic part (4.180) of this system becomes as  dx(t)  = A1i (t, ε)x(t − gi ) + A2j (t, ε)ω(t − εhj ) dt M

N

j =0

i=0

 +

0 −g

 G1 (t, ρ, ε)x(t + ρ)dρ +

0 −h

G2 (t, η, ε)ω(t + εη)dη,

 dω(t)  = A3j (t, ε)x(t − gi ) + A4jK (t, ε)ω(t − εhj ) ε dt M

N

j =0

i=0

 +

0 −g

 G3 (t, ρ, ε)x(t + ρ)dρ +

0 −h

G4K (t, η, ε)ω(t + εη)dη + B2 w (t), (4.188)

where t ≥ 0 and the matrix-valued functions A4jK (t, ε) (j = 0, 1, . . . , N) and G4K (t, η, ε) have the form of (4.139). Quite similarly to Proposition 4.6, we obtain the following assertion. Proposition 4.16 For a given ε ∈ (0, ε0 ], the system (4.173)–(4.174) is completely Euclidean space controllable at the time instant tc , if and only if the system (4.188) and (4.181) is Euclidean space output controllable at this time instant.

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4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

Now, we are going to decompose asymptotically the singularly perturbed system (4.188) and (4.181) into the slow and fast subsystems. This decomposition is carried out similarly to that for the system (4.180)–(4.181). Thus, the slow subsystem, associated with (4.188) and (4.181), consists of the differential-algebraic system dxs (t)  = A1i (t, 0)xs (t − gi ) + dt M



0 −g

i=0

G1 (t, ρ, 0)xs (t + ρ)dρ

+A2s (t)ωs (t), 0=

M 

 A3i (t, 0)xs (t − gi ) +

i=0

0 −g

t ≥ 0,

G3 (t, ρ, 0)xs (t + ρ)dρ

s (t), +A4sK (t)ωs (t) + B2 w

t ≥ 0,

(4.189)

and (4.183). In (4.189) and (4.183), xs (t) ∈ E n , ωs (t) ∈ E m+r ;  the output equation  xs (t), xs (t + ρ) , ρ ∈ [−g, 0), and ωs (t) are state variables; w s (t) ∈ E r is a control; ζs (t) ∈ E n is an output; and the matrix-valued functions A2s (t) and A4sK (t) have the forms presented in (4.142) and (4.143), respectively. If det A4sK (t) = 0, t ≥ 0, the differential-algebraic system (4.189) can be reduced to the differential equation with respect to xs (t) dxs (t)  ¯ K A i,s (t)xs (t−gi )+ = dt M

i=0



0 −g

¯ K ws (t), t ≥ 0, G¯ K s (t, ρ)xs (t+ρ)dρ+B s (t) (4.190)

where  K −1 A3i (t, 0), i = 0, 1, . . . , M, A¯ K i,s (t) = A1i (t, 0) − A2s (t) A4s (t)  K −1 G3 (t, ρ, 0), G¯ K s (t, ρ) = G1 (t, ρ, 0) − A2s (t) A4s (t)  −1 K B¯ K B2 . s (t) = −A2s (t) A4s (t) Remark 4.8 Similarly to Remark 4.6, we can conclude that the Euclidean space output controllability of the slow subsystem in the form (4.189) and (4.183) is the impulse-free Euclidean space controllability of the system (4.189) with respect to xs (t). Similarly, the Euclidean space output controllability of the slow subsystem in the form (4.190) and (4.183) is the complete Euclidean space controllability of the system (4.190). Proceed to the fast subsystem, associated with (4.188) and (4.181). This subsystem consists of the differential equation with state delays

4.8 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

dωf (ξ )  K = A4j (t, 0)ωf (ξ − hj ) + dξ N

j =0



0 −h

287

G4K (t, η, 0)ωf (ξ + η)dη +B2 w f (ξ ), ξ ≥ 0,

(4.191)

and the output equation (4.187). Note that the system (4.191) and (4.187) coincides with the system (4.146) and (4.53). Similarly to Corollary 4.2, Proposition 4.7, and Corollary 4.3 (see Sect. 4.4.4.1), we have the following assertions. Proposition 4.17 The system (4.175)–(4.176) is impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc if and only if the system (4.189) is impulse-free Euclidean space controllable with respect to xs (t) at this time instant. Proposition 4.18 Let det A4sK (t) = 0 for all t ∈ [0, tc ]. Then, the system (4.189) is impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc , if and only if the system (4.190) is completely Euclidean space controllable at this time instant. Proposition 4.19 For a given t ≥ 0, the system (4.178) is completely Euclidean space controllable if and only if the system (4.191) and (4.187) is Euclidean space output controllable.

4.8 Parameter-Free Controllability Conditions for Systems with Delays of Two Scales In this section, based on the study of the auxiliary system (4.180)–(4.181), various ε-independent conditions, providing the complete Euclidean space controllability of the original system (4.173)–(4.174) for all sufficiently small ε > 0, are derived. In what follows of this section, we assume that (AVII)

(AVIII)

(AIX)

the matrix-valued functions Aki (t, ε) and Alj (t, ε) (i = 0, 1, . . . , M; j = 0, 1, . . . , N ; k = 1, 3; l = 2, 4) are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ]; the matrix-valued functions Gk (t, ρ, ε) (k = 1, 3) are piecewise continuous with respect to ρ ∈ [−g, 0] for each (t, ε) ∈ [0, tc ] × [0, ε0 ], and these functions are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in ρ ∈ [−g, 0]; the matrix-valued functions Gl (t, η, ε) and Hp (t, η, ε) (l = 2, 4; p = 1, 2) are piecewise continuous with respect to η ∈ [−h, 0] for each (t, ε) ∈ [0, tc ] × [0, ε0 ], and these functions are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in η ∈ [−h, 0].

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4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

In what follows, we consider two case: (i) the assumption (AV) is valid for the original system (4.173)–(4.174) and (ii) the assumption (AV) is violated, while the assumption (AVI) is valid for the original system (4.173)–(4.174). Remember that the assumptions (AV) and (AVI) are introduced in Sects. 4.4.1 and 4.4.4.4.

4.8.1 Case of the Validity of the Assumption (AV) Lemma 4.13 Let the assumptions (AVII)–(AIX) and (AV) be valid. Let the system (4.185) be completely Euclidean space controllable at the time instant tc . Let, for t = tc , the system (4.186)–(4.187) be Euclidean space output controllable. Then, there exists a positive number ε0 (ε0 ≤ ε0 ), such that for all ε ∈ (0, ε0 ], the singularly perturbed system (4.180)–(4.181) is Euclidean space output controllable at the time instant tc . Proof Using Propositions 4.12, 4.14, and 4.15 and the results of Sect. 2.3.4 (see Theorems 2.2 and 2.3), the lemma is proven similarly to Lemma 4.9 (see Sects. 4.4.1 and 4.4.3).   Theorem 4.5 Let the assumptions (AVII)–(AIX) and (AV) be valid. Let the system (4.177) be completely Euclidean space controllable at the time instant tc . Let, for t = tc , the system (4.178) be completely Euclidean space controllable. Then, for all ε ∈ (0, ε0 ], the singularly perturbed system (4.173)–(4.174) is completely Euclidean space controllable at the time instant tc . Proof Based on Lemma 4.12, Remark 4.7, and Proposition 4.13, the theorem directly follows from Lemma 4.13.  

4.8.2 Case of the Validity of the Assumption (AVI) Theorem 4.6 Let the assumptions (AVII)–(AIX) and (AVI) be valid. Let the system (4.175)–(4.176) be impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc . Let, for t = tc , the system (4.178) be completely Euclidean space controllable. Then, there exists a positive number ε∗ (ε∗ ≤ ε0 ), such that for all ε ∈ (0, ε∗ ], the singularly perturbed system (4.173)–(4.174) is completely Euclidean space controllable at the time instant tc . Proof Based on the results of Sect. 4.7.3 (see Proposition 4.16, Remark 4.8, and Propositions 4.17–4.19), the theorem is proven similarly to Theorem 4.4 (see Sect. 4.4.4.4).  

4.8 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

289

4.8.3 Example 4 Consider the following particular case of (4.173)–(4.174): dx(t) = x(t) − 4y(t) + 2x(t − 0.5) − x(t − 1.5) + ty(t − ε) dt  0 ηu(t + εη)dη, t ≥ 0, +(t − 4)u(t) − 6 ε

dy(t) = 3x(t) + (t − 5)y(t) − dt



−1

0 −1.5

tρx(t + ρ)dρ + y(t − ε)

+(t − 2)u(t) + tu(t − ε),

t ≥ 0,

(4.192)

where x(t), y(t), and u(t) are scalars, i.e., n = m = r = 1; g1 = 0.5, g2 = g = 1.5, and h1 = h = 1. Moreover, M = 2 and N = 1; ε ∈ (0, 0.5]. We study the complete Euclidean space controllability of the system (4.192) at t = tc = 2 for all sufficiently small ε > 0. Asymptotic decomposition of (4.192) yields the slow subsystem in the pure differential form and the fast subsystem, respectively: dxs (t) = −2xs (t) + 2xs (t − 0.5) − xs (t − 1.5) dt  0 + tρxs (t + ρ)dρ + (1 − t)us (t), t ≥ 0, −1.5

and dyf (ξ ) = (t − 5)yf (ξ ) + yf (ξ − 1) + (t − 2)uf (ξ ) + tuf (ξ − 1), dξ

ξ ≥ 0.

It is seen directly that the assumptions (AVII)–(AIX) are satisfied for the system (4.192). Moreover, the above obtained fast subsystem coincides with the fast subsystem (4.162) in Example 1. Therefore, due to the results of Sect. 4.5.1, the assumption (AV) is also valid for the system (4.192), and the fast subsystem of this system is completely Euclidean space controllable. Proceed to the slow subsystem. Since the coefficient for us (t) in the slow subsystem equals zero only at t = 1, then, by virtue of Remark 4.7 and Proposition 4.14, this subsystem is completely Euclidean space controllable at t = tc = 2. Thus, due to Theorem 4.5, the system (4.192) is completely Euclidean space controllable at tc = 2 robustly with respect to ε > 0 for all its sufficiently small values.

290

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

4.8.4 Example 5 Consider the following particular case of the system (4.173)–(4.174): dx(t) = 2tx(t) − y(t) + (2 − t)x(t − 1) − x(t − 2) + 2y(t − ε) dt  0 (t − 2ρ)x(t + ρ)dρ + tu(t) − u(t − ε), t ≥ 0, − −2

ε

dy(t) = −x(t) − 2y(t) + tx(t − 1) + x(t − 2) + 2y(t − ε) dt +(t − 4)u(t) + (6 − t)u(t − ε),

t ≥ 0,

(4.193)

where x(t), y(t), and u(t) are scalars, i.e., n = 1, m = 1, and r = 1. Moreover, M = 2, N = 1, g1 = 1, g2 = g = 2, h1 = h = 1, and ε ∈ (0, 1]. We study the complete Euclidean space controllability of the system (4.193) at the time instant tc = 4 robust with respect to ε for all its sufficiently small values. Decomposing asymptotically the system (4.193), we obtain its slow subsystem in the differential-algebraic form and the fast subsystem, respectively: dxs (t) = 2txs (t) + ys (t) + (2 − t)xs (t − 1) − xs (t − 2) dt  0 − (t − 2ρ)xs (t + ρ)dρ + (t − 1)us (t), t ≥ 0, −2

0 = −xs (t) + txs (t − 1) + xs (t − 2) + 2us (t),

t ≥ 0,

and dyf (ξ ) = −2yf (ξ ) + 2yf (ξ − 1) + (t − 4)uf (ξ ) + (6 − t)uf (ξ − 1), dξ

ξ ≥ 0.

It is seen directly that the assumptions (AVII)–(AIX) are satisfied for the system (4.193). However, the assumption (AV) is not satisfied for this system. Indeed, in this example, Eq. (4.114) becomes as λ + 2 − 2 exp(−λ) = 0. This equation has the root with nonnegative real part λ = 0, meaning that the assumption (AV) is not valid for the system (4.193). Let us verify the validity of the assumption (AVI). The matrix, mentioned in this assumption, becomes as

F3 (t, λ) = − 2 + 2 exp(−λ) − λ , t − 4 + (6 − t) exp(−λ) .

4.8 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

291



For λ = 0, this matrix becomes F3 (t, 0) = 0 , 2 , and rankF3 (t, 0) = m = 1 for all t ∈ [0, tc ] = [0, 4]. Since λ = 0 is the unique  root with nonnegative real part of the quasi-polynomial − 2 + 2 exp(−λ) − λ , then rankF3 (t, λ) = m = 1 for all t ∈ [0, 4] and all λ with nonnegative real parts. This means the validity of the assumption (AVI). Now, let us verify whether the above derived slow subsystem is impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc = 4. For this purpose, we make in this system the invertible control transformation us (t) = ys (t) + vs (t), where vs (t) is a new control. Due to this transformation, the slow subsystem becomes as dxs (t) = 2txs (t) + tys (t) + (2 − t)xs (t − 1) − xs (t − 2) dt  0 − (t − 2ρ)xs (t + ρ)dρ + (t − 1)vs (t), t ≥ 0, −2

0 = −xs (t) + 2ys (t) + txs (t − 1) + xs (t − 2) + 2vs (t),

t ≥ 0.

Eliminating ys (t) from this system, we obtain the time delay differential equation with respect to xs (t) dxs (t) = 2.5txs (t) + (2 − t − 0.5t 2 )xs (t − 1) − (1 + 0.5t)xs (t − 2) dt  0 − (t − 2ρ)xs (t + ρ)dρ − vs (t), t ≥ 0. −2

Applying Proposition 4.14 to this equation, we directly obtain its complete Euclidean space controllability at tc = 4. Then, due to Proposition 4.18, the slow subsystem of the system (4.193) is impulse-free Euclidean space controllable with respect to xs (t) at the time instant tc = 4. Proceed to the fast subsystem of the system (4.193) calculated for t = tc = 4. To show its complete Euclidean space controllability, we are going to prove the Euclidean space output controllability of the system (4.186)–(4.187) calculated for t = tc = 4. The matrix-valued coefficients of the latter in this example become as  A40 (tc , 0) =

 −2 0 , 0 −1

G4 (tc , η, 0) ≡ 0,

B2 =

 A41 (tc , 0) =   0 , 1

 2 2 , 0 0

Ωf = (1 , 0).

Quite similarly to Example 1 in Sect. 4.5.1, one can prove the Euclidean space output controllability of the abovementioned system (4.186)–(4.187). Therefore, due to Proposition 4.13, the fast subsystem of the system (4.193) calculated for

292

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

t = tc = 4 is completely Euclidean space controllable. Hence, by virtue of Theorem 4.6, the singularly perturbed system (4.193) is completely Euclidean space controllable at tc = 4 robustly with respect to ε > 0 for all its sufficiently small values.

4.8.5 Example 6: Analysis of Car-Following Model with State and Control Delays In this example we are going to analyze the system (1.27), which is a singularly perturbed controlled system with state delays of two scales and a small control delay. This system models a car-following process of three vehicles, which follow each other in one lane in the shape of a simple open curve (for details, see Sect. 1.1.5). For the sake of the book’s reading convenience, we write this system here once again dx(θ ) = −x(θ − g) + y(θ − εh1 ), dθ dy(θ ) = −y(θ − εh1 ) + u(θ − εh2 ), ε dθ

(4.194)

where θ ∈ [0, θc ] is an independent variable, ε > 0 is a small parameter, the value θc > 0 is independent of ε, and g > 0, h1 > 0, and h2 > 0 are constants independent of ε. In what follows of this example, we assume that θc ≥ g. The system (4.194) is a particular case of the system (4.173)–(4.174). We study the complete Euclidean space controllability of the system (4.194) at the nondimensional time instant θc for all sufficiently small ε > 0. Asymptotic decomposition of the system (4.194) yields its slow subsystem in the differential-algebraic form dxs (θ ) = −xs (θ − g) + ys (θ ), dθ

θ ∈ [0, θs ],

0 = −ys (θ ) + us (θ ),

θ ∈ [0, θs ],

(4.195)

θ ∈ [0, θs ].

(4.196)

and in the pure differential form dxs (θ ) = −xs (θ − g) + us (θ ), dθ

The fast subsystem, associated with the system (4.194), is dyf (ξ ) = −yf (ξ − h1 ) + uf (ξ − h2 ), dξ

ξ ≥ 0.

(4.197)

4.8 Parameter-Free Controllability Conditions for Systems with Delays of Two. . .

293

The assumptions (AVII)–(AIX) are satisfied for the system (4.194). Let us check up whether the assumption (AV) is satisfied for this system. Equation (4.114) becomes in the present example as follows: λ + exp(−λh1 ) = 0.

(4.198)

First, we assume that h1
0 such that all roots λ of Eq. (4.198) satisfy the inequality Reλ < −2β. The latter means that the assumption (AV) is also satisfied for the system (4.194). Now, let us analyze the complete Euclidean space controllability of the slow (4.196) and the fast (4.197) subsystems. We start with the slow one. Since the coefficient for the control in this subsystem is nonzero, it is completely Euclidean space controllable at θ = θc . Proceed to the fast subsystem. Due to Proposition 4.13 and the system (4.186)–(4.187), the complete Euclidean space controllability of the fast subsystem is equivalent to the Euclidean space output controllability of the following system: dωf (ξ )  = A4j ω(ξ − hj ) + B2 vf (ξ ), dξ

ξ ≥ 0,

ζf (ξ ) = Ωf ωf (ξ ),

ξ ≥ 0,

2

j =0

(4.199)

where ω(ξ ) ∈ E 2 , vf (ξ ) is scalar, Ωf = (1, 0), and  A40 =

 0 0 , 0 −1

 A41 =

 −1 0 , 0 0

 A42 =

 0 1 , 0 0

B2 =

  0 . 1

Let us show the Euclidean space output controllability of the system (4.199). We do this by contradiction. Namely, we assume that this system is not Euclidean space output controllable. Due to this assumption and Proposition 4.15, the following equality is valid:  Ωf 0

ξc

KfT (ρ)B2 B2T Kf (ρ)dρΩfT = 0 ∀ξc > 0,

(4.200)

where the 2 × 2-matrix-valued function Kf (ξ ) is the unique solution of the initialvalue problem

294

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

dKf (ξ )  T = A4j Kf (ξ − hj ), dξ 2

ξ > 0;

Kf (ξ ) = 0, ξ < 0;

Kf (0) = I2 .

j =0

Representing the matrix Kf (ξ ) in the form  Kf (ξ )=

 Kf 1 (ξ ) Kf 2 (ξ ) , Kf 3 (ξ ) Kf 4 (ξ )

where Kf i (ξ ) (i = 1, . . . , 4) are scalar functions, we can rewrite the equality (4.200) as 

ξc

 2 Kf 3 (ρ) dρ = 0 ∀ξc > 0.

(4.201)

0

From the initial-value problem for Kf (ξ ) and the block-form of this matrix, we obtain the initial-value problems for Kf 1 (ξ ) and Kf 3 (ξ ) dKf 1 (ξ ) = −Kf 1 (ξ − h1 ), dξ

ξ > 0;

Kf 1 (ξ ) = 0,

dKf 3 (ξ ) = −Kf 3 (ξ ) + Kf 1 (ξ − h2 ), dξ

ξ > 0;

ξ < 0;

Kf 1 (0) = 1,

Kf 3 (ξ ) = 0,

ξ ≤ 0. (4.202)

This initial-value problem means that Kf 3 (ξ ) is a continuous function in the interval [0, +∞). This observation, along with the equality (4.201) and the initial condition for Kf 3 (ξ ) in the second equation of (4.202), yields Kf 3 (ξ ) = 0 for all ξ ∈ (−∞, +∞). Using the latter and the second differential equation in (4.202), we obtain Kf 1 (ξ −h2 ) = 0 for all ξ ∈ (−∞, +∞), which contradicts the first equation of this system. This contradiction means that the equality (4.200) is wrong. Hence, there exists a number ξc > 0 such that the value in the left-hand side of (4.200) is nonzero meaning the Euclidean space output controllability of the system (4.199) and, therefore, the complete Euclidean space controllability of the fast subsystem associated with the system (4.194). Thus, all the conditions of Theorem 4.5 are fulfilled for (4.194) implying its complete Euclidean space controllability at θc for all sufficiently small ε > 0. Remark 4.9 Due to Proposition 4.12 and its proof, as well as due to the proof of Lemma 3.1 (see Sect. 3.3.4), the control v(t), transferring the system (4.186)– (4.187) from a given initial state-space position to a given terminal Euclidean output position, can be chosen as a continuous function in the interval [0, tc ]. In this case, by virtue of Lemma 4.12, the control u(t), transferring the system (4.173)– (4.174) from given initial state-space and control-space positions to a given terminal Euclidean state position, is a smooth function in the interval [0, tc ]. Moreover, if the initial control function ϕu (·) is continuous in the interval [−εh, 0] and the vector

4.9 Concluding Remarks and Literature Review

295

u0 equals ϕu (0) (for detail on ϕu (·) and u0 , see Definition 4.11), then the control u(t) is a continuous function in the interval [−εh, tc ]. Taking into account this observation, we can conclude the following. If the initial control function u(θ ) in the system (4.194) is a continuous function for θ ∈ [−εh2 , 0], then the control u(θ ), transferring this system from a given state-space position and the abovementioned initial control position to a given terminal Euclidean state position, can be chosen as a continuous function in the interval [0, θc ]. Therefore, the control u(θ ) is a continuous function in the entire interval [−εh2 , θc ], i.e., this control is feasible because it is the speed of the leading vehicle (for details, see Sect. 1.1.5). Proceed with the case h1 ≥ π/2. In this case, Eq. (4.198) can have roots with nonnegative real parts. For instance, if h1 = π/2, then λ = i (i is the imaginary unit) is a root of this equation. Therefore, if h1 ≥ π/2, the assumption (AV) can be violated. Let us verify whether the assumption (AVI) is valid in the present example. The matrix, appearing in this assumption, becomes as

F4 (λ) = − exp(−λh1 ) − λ , exp(−λh2 ) . Hence, for all complex λ : Reλ ≥ 0, we have that rankF4 (λ) = 1. Since the Euclidean dimension of the fast mode in (4.194) equals 1, the assumption (AVI) is valid for this system. Remember that the complete Euclidean space controllability of the fast subsystem (4.197) has already been shown above. Let us show the impulse-free Euclidean space controllability with respect to xs (t) of the slow subsystem (4.195). Indeed, since the slow subsystem in the pure differential form (4.196) is completely Euclidean space controllable at θ = θc , then, by Proposition 4.18, the slow subsystem in the differential-algebraic form (4.195) is impulse-free Euclidean space controllable with respect to xs (t) at θ = θc . Thus, all the conditions of Theorem 4.6 are fulfilled for the system (4.194). Therefore, this singularly perturbed system is completely Euclidean space controllable at θ = θc robustly with respect to ε > 0 for all its sufficiently small values. By the same arguments as in Remark 4.9, the corresponding transferring control can be chosen as a continuous function in the interval [0, θc ], i.e., to be feasible.

4.9 Concluding Remarks and Literature Review In this chapter, singularly perturbed linear time-dependent controlled differential systems with time delays (multiple point-wise and distributed) in the state and control variables were considered. Two cases of the state delays were treated. In the first case, the state delays are proportional to a small positive multiplier ε for a part of the derivatives in the differential equations, i.e., the state delays are small of the order of ε. In the second case, the state delays are of two scales. The delays in the slow state variable are nonsmall (of order of 1), while the delays in the fast state variable are small (proportional to ε). In both cases, the control delays are

296

4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

proportional to ε. The complete Euclidean space controllability of the considered systems, robust with respect to the small parameter ε, was studied. This study uses the asymptotic decomposition of the original system into two ε-free subsystems, the slow and fast ones. The slow subsystem is a differential-algebraic system. In the first case of the state delays, this system is delay-free, while in the second case this system is a time delay one. The fast subsystem, in both cases, is a differential time delay system. For the slow subsystem, the notions of the impulse-free controllability (in the first case) and the impulse-free Euclidean space controllability (in the second case) at a given time instant were presented. For the fast subsystem, the notion of the complete Euclidean space controllability was introduced. The method proposed in this chapter is based on a transformation of the complete Euclidean space controllability of a singularly perturbed system with state and control delays to the Euclidean space output controllability of a larger dimension auxiliary singularly perturbed system with only state delays. To obtain the main result (ε-independent conditions for the controllability of the original system), several preliminary results were derived. Namely, Lemma 4.1 establishes the equivalence of the complete Euclidean space controllability of the original system with state and control delays and the Euclidean space output controllability of the auxiliary system with only state delays. This lemma was formulated for the first time in [9]. However, in this paper, the lemma was not proven. The proof of Lemma 4.1, presented in this chapter, is a generalization of the result of [8] where a much simpler original system with single pointwise delays in the state and the control was analyzed. Using the asymptotic decomposition of the original and auxiliary systems into the corresponding slow and fast subsystems, Lemma 4.2 establishes the equivalence of the complete Euclidean space controllability of the fast subsystem, associated with the original system, and the Euclidean space output controllability of the fast subsystem, associated with the auxiliary system. This result was presented for the first time in [9], and it is a generalization of the corresponding result of [8]. Lemmas 4.3–4.7 establish the invariance of the controllability properties for the original system, as well as its slow and fast subsystems, with respect to a linear control transformation. Particular cases of these lemmas were obtained (formulated and proven) in [6, 7], where a singularly perturbed system with state delays only was considered, and in [8], where a singularly perturbed system with single point-wise delays in the state and the control was studied. Lemma 4.8 and Corollary 4.1 establish conditions for the stabilizability of a parameter-dependent linear system with multiple point-wise and distributed state and control delays by a memory-less linear state-feedback control. The gain in this control is a smooth function with respect to the parameter appearing in the coefficients of the system. Lemma 4.8 was formulated and proven for the first time in [11]. A particular case of Lemma 4.8 was formulated and proven in [8] for a system with single point-wise delays in the state and the control. The latter result was obtained, using the LMI-based stabilizability conditions derived in [15] for a linear constant coefficient system with a single point-wise delay, as well as a distributed delay, in state and control variables. Lemma 4.8 presents the novel

4.9 Concluding Remarks and Literature Review

297

LMI-based conditions for the memory-less stabilizability of a linear parameterdependent system with multiple point-wise and distributed state and control delays. The smoothness of the gain in the stabilizing state-feedback control for this system was established using the results of [2] on properties of a solution to a parameterdependent LMI. Based on the asymptotic decomposition of the auxiliary system, ε-free sufficient conditions for its Euclidean space output controllability were established in Lemma 4.9. This lemma generalizes the corresponding result of the paper [8]. Moreover, the proof of Lemma 4.9 (see Sect. 4.4.3) is simpler than the proof of the corresponding lemma in [8]. Using the above mentioned asymptotic decomposition of the original singularly perturbed system and the preliminary results, as well as Lemma 4.9, different ε-free sufficient conditions for the complete Euclidean space controllability of this system were established in Theorems 4.1, 4.2, and 4.3. The first two theorems consider the standard original system, while the third theorem deals with the nonstandard original system. These theorems generalize the corresponding theorems of the paper [8], which were formulated and proven for the original singularly perturbed system with single point-wise delays in the state and control variables. The alternative approach to the controllability analysis of the nonstandard original system with the small state and control delays was considered in Sect. 4.4.4. For the first time, this approach was proposed in [9]. In the present chapter, the alternative approach is presented in a more detailed form. In Sects. 4.6–4.8, the complete Euclidean space controllability of the singularly perturbed system with the state delays of two scales and the small control delays was studied. The results of these sections are a straightforward extension of the abovementioned results on singularly perturbed systems with the small delays in the state and control variables. Some of the results of Sects. 4.6–4.8 for the standard system were presented for the first time in [10] in a brief form and without detailed proofs of the assertions formulated in this paper. In the present chapter, as well as in the previous chapter, we studied the controllability for the systems having one important property. Namely, these systems can be decomposed asymptotically into the slow and fast subsystems, and each of these subsystems contains the control variable. This circumstance allows us to make the assumptions on the proper kinds of the controllability for the slow and fast subsystems. Based on these assumptions, we have derived the parameter-free sufficient conditions providing the Euclidean space controllability (either complete or output) of the original system for all sufficiently small values of the parameter of singular perturbation. In the next chapter, we consider another type of controlled systems. The systems of this type can also be decomposed asymptotically into the slow and fast subsystems. However, only the slow subsystem contains the control variable, while the fast subsystem does not. Due to this feature, we cannot apply the methods of Chaps. 3 and 4 to the controllability analysis of the system considered in the next chapter, which requires to develop another approach to such an analysis.

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4 Complete Euclidean Space Controllability of Linear Systems with State and. . .

References 1. Bellman, R.: Introduction to Matrix Analysis. SIAM, Philadelphia (1997) 2. Bliman, P.-A.: An existence result for polynomial solutions of parameter-dependent LMIs. Syst. Control Lett. 51, 165–169 (2004) 3. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in Systems and Control Theory. SIAM, Philadelphia (1994) 4. Delfour, M.C., McCalla, C., Mitter, S.K.: Stability and the infinite-time quadratic cost problem for linear hereditary differential systems. SIAM J. Control 13, 48–88 (1975) 5. Gajic, Z., Lim, M.T.: Optimal Control of Singularly Perturbed Linear Systems and Applications. High Accuracy Techniques. Marcel Dekker, New York (2001) 6. Glizer, V.Y.: Controllability of singularly perturbed linear time-dependent systems with small state delay. Dynam. Control 11, 261–281 (2001) 7. Glizer, V.Y.: Controllability of nonstandard singularly perturbed systems with small state delay. IEEE Trans. Automat. Control 48, 1280–1285 (2003) 8. Glizer, V.Y.: Controllability conditions of linear singularly perturbed systems with small state and input delays. Math. Control Signals Syst. 28(1), 1–29 (2016) 9. Glizer, V.Y.: Euclidean space controllability conditions for singularly perturbed linear systems with multiple state and control delays. Axioms. 8, Paper No. 36 (2019) 10. Glizer, V.Y.: Euclidean space controllability conditions of singularly perturbed systems with multiple state and control delays. In: Proceedings of the 15th IEEE Conference on Control and Automation, pp. 1144–1149. Edinburgh, Scotland (2019) 11. Glizer, V.Y.: Uniform stabilizability of parameter-dependent systems with state and control delays by smooth-gain controls. J. Optim. Theory Appl. 183, 50–65 (2019) 12. Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE Conference on Decision and Control, pp. 2805–2810. Sydney, Australia (2000) 13. Halanay, A.: Differential Equations : Stability, Oscillations, Time Lags. Academic Press, New York (1966) 14. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) 15. Hien, L.V., Thi, H.V.: Exponential stabilization of linear systems with mixed delays in state and control. Differ. Equ. Control Process Electron. J. (2), 11 (2009) 16. Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic, London (1986) 17. Pritchard, A.J., Salamon, D.: The linear-quadratic control problem for retarded systems with delays in control and observation. IMA J. Math. Control Inform. 2, 335–362 (1985) 18. Zmood, R.B.: On Euclidean space and function space controllability of control systems with delay. Technical Report. The University of Michigan, Ann Arbor, MI, p. 99 (1971)

Chapter 5

First-Order Euclidean Space Controllability Conditions for Linear Systems with Small State Delays

5.1 Introduction In this chapter, a singularly perturbed linear time-dependent controlled system with multiple point-wise state delays and a distributed state delay is considered. The delays are small of the order of a small parameter ε > 0 multiplying a part of the derivatives in the system. The presence of this parameter in the system represents its singular perturbation. The feature of this system is that its slow mode is controlled directly, while the fast mode is controlled through the slow one. Due to this feature, the asymptotic decomposition of the original system yields a controlled slow subsystem and an uncontrolled fast subsystem. The latter means that the results of Chap. 3 are not applicable to the analysis of the complete Euclidean space controllability of the original system, considered in the present chapter. It should be noted that the derivation of the ε-free condition of the Euclidean space output/complete controllability for the original system, considered in Chap. 3, is based on the zero-order asymptotic analysis of the necessary and sufficient conditions for such a controllability. In this analysis, the assumptions on proper types of controllability of both, slow and fast, subsystems are crucial. Therefore, in the present chapter we propose another approach to obtain ε-free conditions of the complete Euclidean space controllability of the original system. This approach is based on a more deep asymptotic analysis (than the zero-order one) of the necessary and sufficient conditions for this controllability. The following main notations are applied in this chapter: 1. E n is the n-dimensional real Euclidean space. 2. The Euclidean norm of either a vector or a matrix is denoted by · . 3. The upper index T denotes the transposition either of a vector x (x T ) or of a matrix A (AT ).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Y. Glizer, Controllability of Singularly Perturbed Linear Time Delay Systems, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-65951-6_5

299

300

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

4. In denotes the identity matrix of dimension n. 5. The notation On1 ×n2 is used for the zero matrix of the dimension n1 × n2 , excepting the cases where the dimension of a zero matrix is obvious. In such cases, we use the notation 0 for the zero matrix. 6. L2 [t1 , t2 ; E n ] denotes the linear space of all vector-valued functions x(·) : [t1 , t2 ] → E n square integrable in the interval [t1 , t2 ]. 7. L2loc [t¯, +∞; E n ] denotes the linear space of all vector-valued functions x(·) : [t¯, +∞) → E n square integrable in any subinterval [t1 , t2 ] ⊂ [t¯, +∞). 8. AC 2 [t1 , t2 ; E n ] denotes the linear space of all absolutely continuous vectorvalued functions x(·) : [t1 , t2 ] → E n with the first-order derivatives belonging to L2 [t1 , t2 ; E n ]. 9. col(x, y), where x ∈ E n , y ∈ E m , denotes the column block vector of the dimension n + m with the upper block x and the lower block y, i.e., col(x, y) = (x T , y T )T . 10. Reλ denotes the real part of a complex number λ.

5.2 Singularly Perturbed System: Main Notions and Definitions 5.2.1 Original System Consider the system

dx(t)  = A1j (t, ε)x(t − εhj ) + A2j (t, ε)y(t − εhj ) dt N

 +

j =0

0 −h

G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)y(t + εη) dη + B1 (t, ε)u(t), t ≥ 0, (5.1)

dy(t)  = A3j (t, ε)x(t − εhj ) + A4j (t, ε)y(t − εhj ) dt N

ε  +

j =0

0 −h



G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη,

t ≥ 0,

(5.2)

where x(t) ∈ E n , y(t) ∈ E m , and u(t) ∈ E r (u(t) is a control); ε > 0 is a small parameter; N ≥ 1 is an integer; 0 = h0 < h1 < h2 < . . . < hN = h are some given constants independent of ε; Aij (t, ε), Gi (t, η, ε), and B1 (t, ε) (i = 1, . . . , 4; j = 0, 1, . . . , N ) are matrix-valued functions of corresponding dimensions, given for t ≥ 0, η ∈ [−h, 0] and ε ∈ [0, ε0 ], (ε0 > 0); the

5.2 Singularly Perturbed System: Main Notions and Definitions

301

functions Aij (t, ε) (i = 1, . . . , 4; j = 0, . . . , N) and B1 (t, ε) are continuous in (t, ε) ∈ [0, +∞) × [0, ε0 ]; and the functions Gi (t, η, ε) (i = 1, . . . , 4) are piecewise continuous in η ∈ [−h, 0] for any (t, ε) ∈ [0, +∞) × [0, ε0 ], and these functions are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0]. By virtue of the results of Sect. 2.2, for any given ε ∈ (0, ε0 ] and u(·) ∈ 2 Lloc [0, +∞; E r ], the system (5.1)–(5.2) is a linear time-dependent nonhomogeneous functional-differential system.   This system is infinite-dimensional with the state variables x(t), x(t + εη) and y(t), y(t + εη) , η ∈ [−h, 0). Moreover, the system (5.1)–(5.2) is a singularly perturbed system. Equation (5.1) is the slow mode of this system, while Eq. (5.2) is its fast mode. Remark 5.1 In contrast with the original differential systems (3.1)–(3.2) and (4.1)– (4.2) of Chaps. 3 and 4, respectively, the fast mode of the system (5.1)–(5.2) does not contain a control variable, meaning that this mode is not controlled directly but only through the slow mode. Let tc > 0 be a given time instant independent of ε. Definition 5.1 For a given ε ∈ (0, ε0 ], the system (5.1)–(5.2) is said to be completely Euclidean space controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , ϕx (·) ∈ L2 [−εh, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ], xc ∈ E n , and yc ∈ E m , there exists a control function u(·) ∈ L2 [0, tc ; E r ], for which the system (5.1)–(5.2) with the initial and terminal conditions x(τ ) = ϕx (τ ), y(τ ) = ϕy (τ ),

τ ∈ [−εh, 0),

x(0) = x0 ,

y(0) = y0 ,

x(tc ) = xc ,

y(tc ) = yc

has a solution.

5.2.2 Asymptotic Decomposition of the Original System Now, we are going to decompose asymptotically the original singularly perturbed system (5.1)–(5.2) into two much simpler ε-free subsystems, the slow and fast ones. Let us start with the slow subsystem. This subsystem is obtained from (5.1)–(5.2) by setting formally ε = 0 in these functional-differential equations, which yields dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) + B1 (t, 0)us (t), dt 0 = A3s (t)xs (t) + A4s (t)ys (t),

t ≥ 0,

t ≥ 0,

(5.3) (5.4)

302

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

where xs (t) ∈ E n and ys (t) ∈ E m are state variables and us (t) ∈ E r is a control; Ais (t) =

N 

 Aij (t, 0) +

j =0

0 −h

Gi (t, η, 0)dη,

i = 1, . . . , 4.

(5.5)

The slow subsystem (5.3)–(5.4) is a descriptor (differential-algebraic) system, and it is delay-free and ε-free. One more important feature of the slow subsystem is that its algebraic mode (5.4) does not contain the control variable us (t). If det A4s (t) = 0,

t ≥ 0,

(5.6)

the slow subsystem (5.3)–(5.4) can be converted to an equivalent system, consisting of the explicit expression for ys (t) ys (t) = −A−1 4s (t)A3s (t)xs (t),

t ≥0

and the differential equation with respect to xs (t) dxs (t) = A¯ s (t)xs (t) + B1 (t, 0)us (t), t ≥ 0, dt

(5.7)

A¯ s (t) = A1s (t) − A2s (t)A−1 4s (t)A3s (t).

(5.8)

where

Note that the differential equation (5.7) is also called a slow subsystem, associated with the original system (5.1)–(5.2). Proceed to the construction of the fast subsystem, associated with the original system (5.1)–(5.2). This subsystem is derived from the fast  mode (5.2) in the following way: (a) the terms containing the state variable x(t), x(t + εη) , η ∈ [−h, 0), are removed from (5.2) and (b) the transformations of the variables t =

t1 + εξ and y(t1 + εξ ) = yf (ξ ) are made in the resulting system, where t1 ≥ 0 is any fixed time instant. Thus, we obtain the system dyf (ξ )  = A4j (t1 + εξ, ε)yf (ξ − hj ) + dξ N

j =0



0

−h

G4 (t1 + εξ, η, ε)yf (ξ + η)dη.

Finally, setting formally ε = 0 in this system and replacing t1 with t, we obtain the fast subsystem

5.3 Auxiliary Results

303

dyf (ξ )  = A4j (t, 0)yf (ξ − hj ) + dξ N

j =0



0

−h

G4 (t, η, 0)yf (ξ + η)dη, ξ ≥ 0, (5.9)

  where t ≥ 0 is a parameter; yf (ξ ) ∈ E m ; and yf (ξ ), yf (ξ + η) , η ∈ [−h, 0), is a state variable. Like in Chap. 2, the new independent variable ξ is called the stretched time, and it is expressed by the original time t in the form ξ = (t − t1 )/ε. Thus, for any t > t1 , ξ → +∞ as ε → +0. Remark 5.2 The fast subsystem (5.9) is a differential equation with state delays. It is of a lower Euclidean dimension than the original system (5.1)–(5.2), and it is εfree. In contrast with Chaps. 3 and 4, the fast subsystem (5.9) does not contain any control, i.e., it is uncontrolled. Hence, it is not controllable in any sense. Due to this feature of (5.9), the results of Chaps. 3 and 4 are not applicable to the system (5.1)– (5.2). Therefore, in the present chapter, we propose another approach to the analysis of this system, which does not require a controllability of its fast subsystem (5.9). Definition 5.2 Subject to (5.6), the slow subsystem (5.7) is said to be completely controllable at the time instant tc if for any x0 ∈ E n and xc ∈ E n , there exists a control function us (·) ∈ L2 [0, tc ; E r ], for which (5.7) has a solution xs (t), t ∈ [0, tc ], satisfying the initial and terminal conditions xs (0) = x0 ,

xs (tc ) = xc .

5.3 Auxiliary Results In this section, the estimates of solutions to some initial-value and terminal-value problems for singularly perturbed linear time delay matrix differential equations are derived. Necessary and sufficient conditions for the controllability of the original systems and its slow subsystem are presented. Based on these results, in the next section, parameter-free conditions for the complete Euclidean space controllability of the original singularly perturbed system (5.1)–(5.2) are established.

5.3.1 Estimates of Solutions to Some Singularly Perturbed Linear Time Delay Matrix Differential Equations In what follows, we assume that (AI)

the matrix-valued functions Aij (t, ε) (i = 1, . . . , 4; j = 0, 1, . . . , N ) are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ];

304

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

(AII)

(AIII)

the matrix-valued functions Gi (t, η, ε) (i = 1, . . . , 4) are piecewise continuous with respect to η ∈ [−h, 0] for each (t, ε) ∈ [0, tc ]×[0, ε0 ], and they are continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in η ∈ [−h, 0]; and all roots λ(t) of the equation ⎡ det ⎣λIm −

N 

 A4j (t, 0) exp(−λhj ) −

j =0

⎤ 0 −h

G4 (t, η, 0) exp(λη)dη⎦ = 0 (5.10)

satisfy the inequality Reλ(t) < −2β for all t ∈ [0, tc ], where β > 0 is some constant. Remark 5.3 If the assumption (AIII) is valid, then for each t ∈ [0, tc ], λ(t) = 0 is not a root of Eq. (5.10). Therefore, for all t ∈ [0, tc ], the following inequality is satisfied: ⎤ ⎡   0 N   A4j (t, 0) exp(−λhj ) − G4 (t, η, 0) exp(λη)dη⎦  = 0, det ⎣λIm − −h

j =0

λ=0

meaning the fulfillment of the inequality (5.6) for all t ∈ [0, tc ]. Consider the block matrices:   A1j (t, ε) A2j (t, ε) Aj (t, ε) = 1 , j = 0, 1, . . . , N, 1 ε A3j (t, ε) ε A4j (t, ε)  G(t, η, ε) =

 G1 (t, η, ε) G2 (t, η, ε) . 1 1 ε G3 (t, η, ε) ε G4 (t, η, ε)

(5.11)

Using these matrices, for any given ε ∈ (0, ε0 ] and σ ∈ [0, tc ], let us consider the initial-value problem for the (n + m) × (n + m)-matrix-valued function Φ(t) dΦ(t)  = Aj (t, ε)Φ(t − εhj ) + dt N

j =0



0

−h

G(t, η, ε)Φ(t + εη)dη, t > σ,

Φ(τσ ) = 0, σ − εh ≤ τσ < σ ;

Φ(σ ) = In+m . (5.12)

Based on Proposition 2.1 and Lemma 2.2 (for the details, see Sects. 2.2.3 and 2.2.4), we have the following assertion. Proposition 5.1 Let the assumptions (AI)–(AIII) be valid. Then, for any given σ ∈ [0, tc ] and ε ∈ (0, ε0 ], the initial-value problem (5.12) has the unique solution

5.3 Auxiliary Results

305

Φ(t, σ, ε), t ∈ [0, tc ]. Moreover, there exists a positive number ε1 (ε1 ≤ ε0 ) such that for all ε ∈ (0, ε1 ], this solution satisfies the following inequalities: Φl (t, σ, ε) ≤ a, l = 1, 3, Φ2 (t, σ, ε) ≤ aε,  

Φ4 (t, σ, ε) ≤ a ε + exp − β(t − σ )/ε , 0 ≤ σ ≤ t ≤ tc , where the matrices Φ1 (t, σ, ε), Φ2 (t, σ, ε), Φ3 (t, σ, ε) and Φ4 (t, σ, ε) are the upper left-hand, upper right-hand, lower left-hand, and lower right-hand blocks of the matrix Φ(t, σ, ε) of the dimensions n × n, n × m, m × n, and m × m, respectively, and a > 0 is some constant independent of ε. Let, for any given ε ∈ (0, ε1 ], the (n + m) × (n + m)-matrix-valued function Ψ (σ, ε), σ ∈ [0, tc ] be a solution of the terminal-value problem  T dΨ (σ, ε) =− Aj (σ + εhj , ε) Ψ (σ + εhj , ε) dσ N

 −

j =0

0 −h

 T G(σ − εη, η, ε) Ψ (σ − εη, ε)dη, Ψ (tc , ε) = In+m ,

σ ∈ [0, tc ),

Ψ (σ, ε) = 0, σ > tc .

(5.13)

In (5.13) we assume that, for all σ > tc , η ∈ [−h, 0] and ε ∈ [0, ε1 ], Aj (σ, ε) = Aj (tc , ε), j = 0, 1, . . . , N, G(σ, η, ε) = G(tc , η, ε).

(5.14)

Due to Remark 2.3 (see Sect. 2.2.4), the solution Ψ (σ, ε), σ ∈ [0, tc ], of (5.13) exists and is unique. Note that the solution Φ(t, σ, ε), 0 ≤ σ ≤ t ≤ tc , of the initial-value problem (5.12) is the fundamental matrix of the original system (5.1)–(5.2), while the solution Ψ (σ, ε), σ ∈ [0, tc ], of the terminal-value problem (5.13) is the adjoint matrix of the fundamental matrix Φ(t, σ, ε) at the time instant tc (for the details, see Sects. 2.2.3 and 2.2.4). Let the matrix-valued functions Ψ1s (σ ) and Ψ4f (ξ ) be the solutions of the following problems:  T dΨ1s (σ ) = − A¯ s (σ ) Ψ1s (σ ), σ ∈ [0, tc ), Ψ1s (tc ) = In , dσ T dΨ4f (ξ )   = A4j (tc , 0) Ψ4f (ξ − hi ) dξ N

j =0

(5.15)

306

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

 +

0 −h

 T G4 (tc , η, 0) Ψ4f (ξ + η)dη, ξ > 0, Ψ4f (ξ ) = 0, ξ < 0,

Ψ4f (0) = Im .

(5.16)

Remember that A¯ s (t) is defined by (5.8). From the assumption (AIII) and the results of [3], we directly obtain

Ψ4f (ξ ) ≤ a exp(−2βξ ), ξ ≥ 0,

(5.17)

where a > 0 is some constant. Consider the following matrix-valued functions: N +∞   

 Ψ2f (ξ ) = ξ

 +

0 −h



G3 (tc , η, 0)

A3j (tc , 0)

T

Ψ4f (τ − hj )

j =0

T

 Ψ4f (τ + η)dη dτ,

ξ ≥ 0,

Ψ2s (σ ) = Ψ1s (σ )Ψ2f (0), σ ∈ [0, tc ]; Ψ2s (σ ) = Ψ2s (tc ), σ > tc ,  T −1 Ψ3s (σ ) = − A4s (σ ) AT2s (σ )Ψ1s (σ ), σ ∈ [0, tc ]; Ψ3s (σ ) = Ψ3s (tc ), σ > tc ,  −1 Ψ4s (σ ) = − AT4s (σ ) AT2s (σ )Ψ2s (σ ), σ ∈ [0, tc ]; Ψ4s (σ ) = Ψ4s (tc ), σ > tc . (5.18) Remark 5.4 Due to the inequality (5.17), the integral in the expression of Ψ2f (ξ ) converges for any ξ ≥ 0. Moreover, the matrix-valued function Ψ2f (ξ ) satisfies the inequality

Ψ2f (ξ ) ≤ a exp(−2βξ ), ξ ≥ 0,

(5.19)

where a > 0 is some constant. Lemma 5.1 Let the assumptions (AI)–(AIII) be valid. Let the matrices Ψ1 (σ, ε), Ψ2 (σ, ε), Ψ3 (σ, ε), and Ψ4 (σ, ε) be the upper left-hand, upper right-hand, lower left-hand, and lower right-hand blocks of the matrix Ψ (σ, ε) of the dimensions n×n, n × m, m × n, and m × m, respectively. Then, there exists a positive number ε2 (ε2 ≤ ε1 ) such that for all σ ∈ [0, tc ] and ε ∈ (0, ε2 ], these matrices satisfy the following inequalities:   Ψ1 (σ, ε) − Ψ1s (σ ) ≤ aε, Ψ2 (σ, ε) − Ψ2s (σ ) + Ψ2f (tc − σ )/ε ≤ aε,  

Ψ3 (σ, ε) − εΨ3s (σ ) ≤ aε ε + exp − β(tc − σ )/ε ,    

Ψ4 (σ, ε) − Ψ4f (tc − σ )/ε − εΨ4s (σ ) ≤ aε ε + exp − β(tc − σ )/ε , (5.20)

5.3 Auxiliary Results

307

where a > 0 is some constant independent of ε. The lemma is proven in the next subsection.

5.3.2 Proof of Lemma 5.1 Let us set Ψ2f (ξ ) = Ψ2f (0)

∀ξ < 0.

The further proof of the lemma uses one technical proposition.

5.3.2.1

Technical Proposition

Consider the following (n + m) × (n + m)-matrix-valued functions:  Θ(σ, ε) =

Γ (σ, ε) =  +

   Ψ1s (σ ) Ψ2s (σ ) − Ψ2f (tc − σ )/ε   , εΨ3s (σ ) Ψ4f (tc − σ )/ε + εΨ4s (σ )

σ ≥ 0,

(5.21)

N T dΘ(σ, ε)   + Aj (σ + εhj , ε) Θ(σ + εhj , ε) dσ j =0

0 −h



T G(σ − εη, η, ε) Θ(σ − εη, ε)dη,

σ ∈ [0, tc ].

Let us partition the matrix Γ (σ, ε) into blocks as   Γ1 (σ, ε) Γ2 (σ, ε) , Γ (σ, ε) = Γ3 (σ, ε) Γ4 (σ, ε)

(5.22)

(5.23)

where the matrices Γ1 (σ, ε), Γ2 (σ, ε), Γ3 (σ, ε), and Γ4 (σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Proposition 5.2 Let the assumptions (AI)–(AIII) be valid. Then, for all σ ∈ [0, tc ] and ε ∈ (0, ε1 ], the following inequalities are satisfied: Γl (σ, ε) ≤ aε, l = 1, 3,  

Γp (σ, ε) ≤ a ε + exp − β(tc − σ )/ε , p = 2, 4,

(5.24)

where ε1 is introduced in Proposition 5.1 and a > 0 is some constant independent of ε.

308

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

Proof We prove here the inequalities for Γ2 (σ, ε) and Γ3 (σ, ε). The other inequalities are proven similarly. Let us start with the inequality for Γ2 (σ, ε). Substitution of (5.11), (5.21), and (5.23) into (5.22) yields after a routine algebra the expression for Γ2 (σ, ε) Γ2 (σ, ε) = Γ21 (σ, ε) + Γ22 (σ, ε),

(5.25)

where N  T dΨ2s (σ )   A1j (σ + εhj , ε) Ψ2s (σ + εhj ) Γ21 (σ, ε) = + dσ j =0

T  + A3j (σ + εhj , ε) Ψ4s (σ + εhj )  +

0 −h





 T G1 (σ − εη, η, ε) Ψ2s (σ − εη)

  T + G3 (σ − εη, η, ε) Ψ4s (σ − εη) dη,

Γ22 (σ, ε) = −

1 dΨ2f (ξ )   ξ =(tc −σ )/ε ε dξ

N     T  ε A1j (σ + εhj , ε) Ψ2f (tc − σ − εhj )/ε j =0

 T   − A3j (σ + εhj , ε) Ψ4f (tc − σ − εhj )/ε  −

(5.26)

0 −h



   T  ε G1 (σ − εη, η, ε) Ψ2f (tc − σ + εη)/ε

   T   − G3 (σ − εη, η, ε) Ψ4f (tc − σ + εη)/ε dη .

(5.27)

Now, we are going to estimate each of the addends in the right-hand side of (5.25). Let us start with Γ21 (σ, ε). To estimate this matrix-valued function, we consider the following two cases: (i) 0 < σ + εh ≤ tc and (ii) σ + εh > tc . In the case (i), using the assumptions (AI)–(AII), the problem (5.15) for Ψ1s (σ ) and the expressions for Ψ2s (σ ) and Ψ4s (σ ) (see Eq. (5.18)), we directly obtain Alj (σ + εhj , ε) = Alj (σ, 0) + ΔAlj (σ, ε), l = 1, 3, j = 0, 1, .., N, Gl (σ − εη, η, ε) = Gl (σ, η, 0) + ΔGl (σ, η, ε), l = 1, 3,

5.3 Auxiliary Results

309

Ψps (σ + εhj ) = Ψps (σ ) + Δ1 Ψps (σ, ε), p = 2, 4, Ψps (σ − εη) = Ψps (σ ) + Δ2 Ψps (σ, η, ε), p = 2, 4, (5.28) where ΔAlj (σ, ε), ΔGl (σ, η, ε), Δ1 Ψps (σ, ε), and Δ2 Ψps (σ, η, ε) are matrixvalued functions of corresponding dimensions, satisfying the following inequalities for all ε ∈ (0, ε1 ], σ ∈ [0, tc − εh], η ∈ [−h, 0]:

ΔAlj (σ, ε) ≤ aε, ΔGl (σ, η, ε) ≤ aε, l = 1, 3, j = 0, 1, .., N,

Δ1 Ψps (σ, ε) ≤ aε, Δ2 Ψps (σ, η, ε) ≤ aε, p = 2, 4,

(5.29)

where a > 0 is some constant independent of ε. Using Eqs. (5.28) and (5.5) and the inequalities in (5.29), we can rewrite the expression (5.26) in the following form for all ε ∈ (0, ε1 ], σ ∈ [0, tc − εh]: Γ21 (σ, ε) =  +

 N  T  T dΨ2s (σ )   A1j (σ, 0) Ψ2s (σ ) + A3j (σ, 0) Ψ4s (σ ) + dσ j =0

0 −h

   T  T G1 (σ, η, 0) Ψ2s (σ ) + G3 (σ, η, 0) Ψ4s (σ ) dη + ΔΓ21 (σ, ε)

⎡ ⎤  0 N  T T dΨ2s (σ ) ⎣  + A1j (σ, 0) + G1 (σ, η, 0) dη⎦ Ψ2s (σ ) = dσ −h j =0

⎡

 N   T ⎣ A3j (σ, 0) + + j =0

=

0 −h

⎤  T G3 (σ, η, 0) dη⎦ Ψ4s (σ ) + ΔΓ21 (σ, ε)

dΨ2s (σ ) + AT1s (σ )Ψ2s (σ ) + AT3s (σ )Ψ4s (σ ) + ΔΓ21 (σ, ε), dσ (5.30)

where the n × m-matrix-valued function ΔΓ21 (σ, ε) satisfies the inequality

ΔΓ21 (σ, ε) ≤ aε, ε ∈ [0, ε1 ], σ ∈ [0, tc − εh],

(5.31)

where a > 0 is some constant independent of ε. Substitution of the expressions for Ψ2s (σ ) and Ψ4s (σ ) (see Eq. (5.18)) into the right-hand side of (5.30) and use of Eq. (5.8) and the problem (5.15) for Ψ1s (σ ) yield Γ21 (σ, ε) = ΔΓ21 (σ, ε), ε ∈ [0, ε1 ], σ ∈ [0, tc − εh],

310

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

which, due to the inequality (5.31), implies

Γ21 (σ, ε) ≤ aε, ε ∈ [0, ε1 ], σ ∈ [0, tc − εh],

(5.32)

where a > 0 is the same constant as in (5.31). To complete obtaining the estimate of Γ21 (σ, ε), we should treat the case (ii) where σ +εh > tc . In this case, using the assumptions (AI)–(AII) and the definitions of the matrix-valued functions Aj (σ, ε) (j = 0, 1, . . . , N), G(σ, η, ε), Ψ2s (σ ), and Ψ4s (σ ) in the interval σ > tc (see Eqs. (5.14) and (5.18)), we have the following inequalities for all ε ∈ (0, ε1 ], σ ∈ (tc − εh, tc ], η ∈ [−h, 0]:

Alj (σ + εhj , ε) − Alj (tc , 0) ≤ aε,

l = 1, 3, j = 0, 1, . . . , N,

Gl (σ − εη, η, ε) − Gl (tc , η, 0) ≤ aε,

Ψps (σ + εhj ) − Ψps (tc ) ≤ aε,

l = 1, 3,

(5.33)

p = 2, 4, j = 0, 1, . . . , N,

Ψps (σ − εη) − Ψps (tc ) ≤ aε,

p = 2, 4.

(5.34)

Moreover, due to the expression for Ψ2s (σ ) (see Eq. (5.18)), Remark 5.4 and the problem (5.15), we obtain for all ε ∈ (0, ε1 ], σ ∈ (tc − εh, tc ], T dΨ2s (σ )  ¯ + As (tc ) Ψ2s (tc ) ≤ aε. dσ

(5.35)

In (5.33)–(5.35), a > 0 is some constant independent of ε. Finally, using the inequalities (5.33)–(5.35), we obtain (similarly to the case (i)) that the matrix-valued function Γ21 (σ, ε) satisfies the inequality

Γ21 (σ, ε) ≤ aε,

ε ∈ [0, ε1 ], σ ∈ (tc − εh, tc ],

where a > 0 is some constant independent of ε. The latter, along with (5.32), yields the fulfillment of the inequality

Γ21 (σ, ε) ≤ aε,

ε ∈ [0, ε1 ], σ ∈ [0, tc ],

(5.36)

with some positive constant a independent of ε. Proceed to the estimate of the second addend in the right-hand side of (5.25), given by the expression (5.27). In this expression, we transform the independent variable σ as σ = tc − εξ, where ξ ≥ 0 is a new independent variable.

(5.37)

5.3 Auxiliary Results

311

Due to this transformation, we obtain Γ22 (tc − εξ, ε) = −

1 dΨ2f (ξ ) ε dξ

N     T ε A1j tc − ε(ξ − hj ), ε Ψ2f (ξ − hj ) j =0

 T  − A3j tc − ε(ξ − hj ), ε Ψ4f (ξ − hj )  −

0 −h



   T ε G1 tc − ε(ξ − η), η, ε Ψ2f (ξ + η)

   T  − G3 tc − ε(ξ − η), η, ε Ψ4f (ξ + η) dη . We can rewrite this equation as

1 dΨ2f (ξ ) Γ22 (tc − εξ, ε) = ε dξ  N 

T A3j (tc , 0) Ψ4f (ξ − hj ) + + j =0

0 −h



T G3 (tc , η, 0)) Ψ4f (ξ + η)dη



 N    T A1j tc − ε(ξ − hj ), ε Ψ2f (ξ − hj ) −  + +  +

j =0 0 −h

  T G1 tc − ε(ξ − η), η, ε Ψ2f (ξ + η)dη

N 

T  1  A3j tc − ε(ξ − hj ), ε) − A3j (tc , 0) Ψ4f (ξ − hj ) ε j =0

0 −h





  

T G3 tc − ε(ξ + η), η, ε) − G3 (tc , η, 0)) Ψ4f (ξ + η)dη ,

yielding, by virtue of the expression for Ψ2f (ξ ) (see Eq. (5.18)),

 N    T A1j tc − ε(ξ − hj ), ε Ψ2f (ξ − hj ) Γ22 (tc − εξ, ε) = −  +

j =0 0 −h



  T G1 tc − ε(ξ − η), η, ε Ψ2f (ξ + η)dη



312

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

N 

T  1  + A3j tc − ε(ξ − hj ), ε) − A3j (tc , 0) Ψ4f (ξ − hj ) ε  +

j =0

0 −h

 

T  G3 tc − ε(ξ + η), η, ε) − G3 (tc , η, 0)) Ψ4f (ξ + η)dη .

(5.38)

Using the assumptions (AI)–(AII), Eq. (5.11), and the estimate (5.17), we obtain the following inequalities for all ε ∈ (0, ε1 ], ξ ∈ [0, tc /ε], η ∈ [−h, 0]:  

T A3j tc − ε(ξ − hj ), ε) − A3j (tc , 0) Ψ4f (ξ − hj ) ≤ aε exp(−βξ ),  

T G3 tc − ε(ξ + η), η, ε) − G3 (tc , η, 0)) Ψ4f (ξ + η) ≤ aε exp(−βξ ), (5.39) where a > 0 is some constant independent of ε. The expression (5.38) for the matrix-valued function Γ22 (tc − εξ, ε), along with the inequalities (5.19) and (5.39), directly yields the estimate

Γ22 (tc − εξ, ε) ≤ a exp(−βξ ) for all ε ∈ (0, ε1 ], ξ ∈ [0, tc /ε] with some positive constant a independent of ε. This estimate and Eq. (5.37) imply immediately the estimate  

Γ22 (σ, ε) ≤ a exp − β(tc − σ )/ε ,

σ ∈ [0, tc ], ε ∈ (0, ε1 ].

The latter, along with Eq. (5.25) and the inequality (5.36), yields the inequality for the matrix-valued function Γ2 (σ, ε) stated in the proposition (see Eq. (5.24)). Let us prove the inequality in (5.24) for the matrix-valued function Γ3 (σ, ε). Substitution of (5.11), (5.21), and (5.23) into (5.22) yields after a routine algebra the expression for Γ3 (σ, ε) N  T dΨ3s (σ )   A2j (σ + εhj , ε) Ψ1s (σ + εhj ) + Γ3 (σ, ε) = ε dσ j =0

T  + A4j (σ + εhj , ε) Ψ3s (σ + εhj )  +

0 −h



  T G2 (σ − εη, η, ε) Ψ1s (σ − εη)

  T + G4 (σ − εη, η, ε) Ψ3s (σ − εη) dη.

(5.40)

5.3 Auxiliary Results

313

Based on the problem (5.15) for the matrix-valued function Ψ1s (σ ) and the expression for the matrix-valued function Ψ3s (σ ) (see Eq. (5.18)), and using Eq. (5.40), the abovementioned inequality for Γ3 (σ, ε) is proven similarly to the   inequality (5.36) for Γ21 (σ, ε).

5.3.2.2

Main Part of the Proof

Now, we transform the variables in the problem (5.13) as follows: Ψ (σ, ε) = Δ(σ, ε) + Θ(σ, ε),

(5.41)

where Δ(σ, ε) is a new unknown matrix-valued function; the matrix-valued function Θ(σ, ε) is given in (5.21). By (5.41) and (5.21), the problem (5.13) becomes N  T  dΔ(σ, ε) =− Aj (σ + εhj , ε) Δ(σ + εhj , ε) dσ

 −

j =0

0 −h

 T G(σ − εη, η, ε) Δ(σ − εη, ε)dη − Γ (σ, ε), σ ∈ [0, tc ),  Δ(σ, ε) = −

 0 0 , σ ≥ tc . εΨ3s (tc ) εΨ4s (tc )

(5.42)

Using the results of [2] (Section 4.3), the problem (5.42) can be rewritten in the equivalent integral form  tc +εh Δ(σ, ε) = Φ T (tc , σ, ε)Δ(tc , ε) + Λ(σ, ρ, ε)Δ(ρ, ε)dρ tc



σ

−

Φ T (t, σ, ε)Γ (t, ε)dt, σ ∈ [0, tc ],

(5.43)

tc

where Φ(t, σ, ε), 0 ≤ σ ≤ t ≤ tc , is given by the problem (5.12); Λ(σ, ρ, ε) =

N 

Φ T (ρ − εhj , σ, ε)ATj (ρ, ε)

j =1

 +

(tc −ρ)/ε −h

Φ T (ρ + εη, σ, ε)GT (ρ, η, ε)dη,

and Φ(ρ, σ, ε) = 0 for ρ > tc .

(5.44)

314

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

Let Λ1 (σ, ρ, ε), Λ2 (σ, ρ, ε), Λ3 (σ, ρ, ε), and Λ4 (σ, ρ, ε) be the upper lefthand, upper right-hand, lower left-hand, and lower right-hand blocks of the matrix Λ(σ, ρ, ε) of the dimensions n × n, n × m, m × n, and m × m, respectively, i.e.,   Λ1 (σ, ρ, ε) Λ2 (σ, ρ, ε) Λ(σ, ρ, ε) = . Λ3 (σ, ρ, ε) Λ4 (σ, ρ, ε) Using Proposition 5.1 and Eq. (5.44), we obtain the following inequalities for all ε ∈ (0, ε1 ], ρ ∈ [tc , tc + εh], σ ∈ [0, tc ]:    

Λ1 (σ, ρ, ε) ≤ a 1 + (ρ − tc )/ε , Λ2 (σ, ρ, ε) ≤ (a/ε) 1 + (ρ − tc )/ε ,  

Λ3 (s, ρ, ε) ≤ a ε + (ρ − tc ) + exp − β(ρ − σ )/ε ,  

Λ4 (σ, ρ, ε) ≤ (a/ε) ε + (ρ − tc ) + exp − β(ρ − σ )/ε , (5.45) where a > 0 is some constant independent of ε. Now, we partition the matrix Δ(σ, ε) into blocks as   Δ1 (σ, ε) Δ2 (σ, ε) , Δ(σ, ε) = Δ3 (σ, ε) Δ4 (σ, ε)

(5.46)

where the matrices Δ1 (σ, ε), Δ2 (σ, ε), Δ3 (σ, ε), and Δ4 (σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Estimating the blocks of the matrix Δ(σ, ε) by using Eq. (5.43) for this matrix, the terminal condition in (5.42), as well as Propositions 5.1 and 5.2, and the inequalities in (5.45), we obtain the following inequalities for all σ ∈ [0, tc ] and ε ∈ (0, ε1 ]:

Δl (σ, ε) ≤ aε, l = 1, 2, 

Δp (σ, ε) ≤ aε ε + exp − β(tc − σ )/ε , p = 3, 4,



(5.47)

where a > 0 is some constant independent of ε. Finally, Eqs. (5.21) and (5.41) and the inequalities in (5.47) directly yield the inequalities in (5.20) claimed in the lemma.

5.3.3 Complete Controllability of the Original System and Its Slow Subsystem: Necessary and Sufficient Conditions Denote

B(t, ε) =



 B1 (t, ε) , Om×r

t ≥ 0, ε ∈ [0, ε0 ],

(5.48)

5.4 Parameter-Free Controllability Conditions

315

where B1 (t, ε) is the matrix-valued coefficient for the control in Eq. (5.1). Based on the results of Sect. 3.3.1 (for the details, see Proposition 3.1), we have the following two assertions. Proposition 5.3 For any given ε ∈ (0, ε0 ], the original system (5.1)–(5.2) is completely Euclidean space controllable at the time instant tc if and only if the following matrix is nonsingular:



tc

W (tc , ε) =

Ψ T (σ, ε)B(σ, ε)B T (σ, ε)Ψ (σ, ε)dσ,

(5.49)

0

i.e., det W (tc , ε) = 0, where the matrix-valued function Ψ (σ, ε) is the solution of the problem (5.13). Proposition 5.4 The slow subsystem (5.7), associated with the original system (5.1)–(5.2), is completely controllable at the time instant tc if and only if the following matrix is nonsingular:



Ws (tc ) =

tc

0

T Ψ1s (σ )B1 (σ, 0)B1T (σ, 0)Ψ1s (σ )dσ,

(5.50)

i.e., det Ws (tc ) = 0, where the matrix-valued function Ψ1s (σ ) is the solution of the problem (5.15).

5.4 Parameter-Free Controllability Conditions In this section, using the results of the previous sections, different ε-independent conditions, providing the complete Euclidean space controllability of the original system (5.1)–(5.2) for all sufficiently small ε > 0, are derived.

5.4.1 Formulation of Main Assertions Denote



+∞

M(tc ) = 0

T Ψ2f (ξ )B1 (tc , 0)B1T (tc , 0)Ψ2f (ξ )dξ,

(5.51)

where the matrix-valued function Ψ2f (ξ ) is given in (5.18). Note that, due to the inequality (5.19), the integral in the right-hand side of (5.51) converges. In what follows, we assume that (AIV)

the matrix-valued function B1 (t, ε) is continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ].

316

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

Theorem 5.1 Let the assumptions (AI)–(AIV) be valid. Let the slow subsystem (5.7), associated with the system (5.1)–(5.2), be completely controllable at the time instant tc . Let the matrix M(tc ) be nonsingular, i.e., det M(tc ) = 0. Then, there exists a positive number ε1∗ (ε1∗ ≤ ε0 ) such that for all ε ∈ (0, ε1∗ ], the original singularly perturbed system (5.1)–(5.2) is completely Euclidean space controllable at the time instant tc . Proof of the theorem is presented in Sect. 5.4.2. Remark 5.5 The conditions of Theorem 5.1 are obtained by using the first-order asymptotic expansion with respect to ε of the Euclidean space controllability matrix W (tc , ε) (see Eq. (5.49)) for the system (5.1)–(5.2). Therefore, these conditions can be called the ε-free first-order conditions of the complete Euclidean space controllability for (5.1)–(5.2). The following lemma shows how the condition det M(tc ) = 0 of Theorem 5.1 can be interpreted from the control theory viewpoint in one particular case of the original system (5.1)–(5.2). Lemma 5.2 Let the following conditions be satisfied: A3j (tc , 0) − A4j (tc , 0)A−1 4s (tc )A3s (tc ) = 0, G3 (tc , η, 0) − G4 (tc , η, 0)A−1 4s (tc )A3s (tc ) = 0,

j = 1, . . . , N,

(5.52)

η ∈ [−h, 0].

(5.53)

Then, the matrix M(tc ) is nonsingular if and only if the following system is completely Euclidean space controllable at some ξ = ξc > 0: d y(ξ ˜ )  = A4j (tc , 0)y(ξ ˜ − hj ) + dξ N

j =0



0

−h

G4 (tc , η, 0)y(ξ ˜ + η)dη

˜ ), +A−1 4s (tc )A3s (tc )B1 (tc , 0)u(ξ

ξ ≥ 0,

(5.54)

  ˜ ) ∈ E m ; y(ξ ˜ ), y(ξ ˜ + η) , η ∈ where A3s (t) and A4s (t) are given in (5.5); y(ξ [−h, 0) is a state variable; and u(ξ ˜ ) ∈ E r is a control. Proof of the lemma is presented in Sect. 5.4.3. Remark 5.6 The complete Euclidean space controllability of the system (5.54) at a given instant ξ = ξc > 0 is defined similarly to such a controllability of the original system (5.1)–(5.2) (for the details, see Definition 5.1 in Sect. 5.2.1). Remark 5.7 Subject to the conditions (5.52)–(5.53), the system (5.54) is called the first-order fast subsystem, associated with the original system (5.1)–(5.2), in contrast with the system (5.9), which can be called the zero-order fast subsystem associated with (5.1)–(5.2).

5.4 Parameter-Free Controllability Conditions

317

The following corollary is a direct consequence of Theorem 5.1 and Lemma 5.2. Corollary 5.1 Let the assumptions (AI)–(AIV) be satisfied. Let the conditions (5.52)–(5.53) be valid. Let the slow subsystem (5.7), associated with the system (5.1)–(5.2), be completely controllable at the time instant tc . Let the firstorder fast subsystem (5.54), associated with the system (5.1)–(5.2), be completely Euclidean space controllable at some ξ = ξc > 0. Then, for all ε ∈ (0, ε1∗ ], the original singularly perturbed system (5.1)–(5.2) is completely Euclidean space controllable at the time instant tc . Now, we consider the case where the assumption (AIII) is violated. In this case, another ε-free assumption is proposed below. This new assumption, along with the assumptions (AI), (AII), and (AIV), provides the complete Euclidean space controllability of the original system (5.1)–(5.2) for all sufficiently small ε > 0. Denote   On×m On×n 6 Aj (t) = , j = 0, 1, . . . , N, A3j (t, 0) A4j (t, 0)   On×n On×m 6 η) = G(t, . (5.55) G3 (t, η, 0) G4 (t, η, 0) Using these matrices and the matrix B(t, ε), given by (5.48), we consider the following ε-free time delay system: d zˆ (ξ )  6 = Aj (t)ˆz(ξ − hj ) dξ N

j =0

 +

0 −h

6 η)ˆz(ξ + η)dη + B(t, 0)u(ξ G(t, ˆ ), ξ ≥ 0,

(5.56)

  where t ∈ [0, tc ] is a parameter; zˆ (ξ ) ∈ E n+m ; zˆ (ξ ), zˆ (ξ + η) , η ∈ [−h, 0) is a state variable; and u(ξ ˆ ) ∈ E r is a control. In what follows, we use the new assumption instead of the assumption (AIII): (AV)

For all t ∈ [0, tc ] and any complex number λ with Reλ ≥ 0, the following equality is valid: rank  +

 N

6j (t) exp(−λhj ) A

j =0

0 −h

 6 G(t, η) exp(λη)dη − λIn+m , B(t, 0) = n + m.

(5.57)

Theorem 5.2 Let the assumptions (AI), (AII), (AIV), and (AV) be valid. Let, for t = tc , the system (5.56) be completely Euclidean space controllable at some ξ = ξc >

318

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

0. Then, there exists a positive number ε2∗ , (ε2∗ ≤ ε0 ), such that for all ε ∈ (0, ε2∗ ], the original singularly perturbed system (5.1)–(5.2) is completely Euclidean space controllable at the time instant tc . Proof of Theorem 5.2 is presented in Sect. 5.4.4.

5.4.2 Proof of Theorem 5.1 Due to Proposition 5.3, to prove the theorem we should show the existence of a number 0 < ε1∗ ≤ ε0 such that for all ε ∈ (0, ε1∗ ], the symmetric matrix W (tc , ε), given by Eq. (5.49), is positive definite. Let us partition the matrix W (tc , ε) into blocks as ⎛ W (tc , ε) = ⎝

W1 (tc , ε) W2 (tc , ε) W2T (tc , ε)

⎞ ⎠,

(5.58)

W3 (tc , ε)

where the matrices W1 (tc , ε), W2 (tc , ε), and W3 (tc , ε) are of the dimensions n × n, n × m, and m × m, respectively. Using the expression for the matrix W (tc , ε), as well as the block representations of the matrices Ψ (σ, ε) and B(t, ε) (see Lemma 5.1 and Eq. (5.48)), we obtain by a routine algebra the expressions for the matrices Wk (tc , ε), (k = 1, 2, 3) 

tc

W1 (tc , ε) = 

0 tc

W2 (tc , ε) = 

0 tc

W3 (tc , ε) = 0

Ψ1T (σ, ε)S1 (σ, ε)Ψ1 (σ, ε)dσ, Ψ1T (σ, ε)S1 (σ, ε)Ψ2 (σ, ε)dσ, Ψ2T (σ, ε)S1 (σ, ε)Ψ2 (σ, ε)dσ,

(5.59)

where S1 (σ, ε) = B1 (σ, ε)B1T (σ, ε).

(5.60)

Denote



Ll (ε) =

tc

0

T Ψ1s (σ )S1 (σ, ε)Δl (σ, ε)dσ,



M1 (ε) = 0

tc

l = 1, 2,

  T Ψ1s (σ )S1 (σ, ε)Ψ2f (tc − σ )/ε dσ,

5.4 Parameter-Free Controllability Conditions



tc

M2 (ε) =



Ql (ε) = 0

0 tc

319

    T (tc − σ )/ε S1 (σ, ε)Ψ2f (tc − σ )/ε dσ, Ψ2f

  T (tc − σ )/ε S1 (σ, ε)Δl (σ, ε)dσ, Ψ2f



tc

R1 (ε) =



0 tc

R2 (ε) =



0 tc

R3 (ε) = 0

l = 1, 2,

ΔT1 (σ, ε)Δ1 (σ, ε)dσ, ΔT1 (σ, ε)Δ2 (σ, ε)dσ, ΔT2 (σ, ε)Δ2 (σ, ε)dσ,

(5.61)

where Ψ1s (σ ) is the solution of the problem (5.15), Ψ2f (ξ ) is given in (5.18), Δ1 (σ ) and Δ2 (σ ) are the upper left-hand block and the upper right-hand block, respectively, of the matrix Δ(σ ) (see Eq. (5.46)), and the matrix Δ(σ ) is given by Eq. (5.41). Now, we are going to express the matrices, given in (5.59), by the matrices, given in (5.61). Remember that, due to Lemma 5.1 and Eqs. (5.21), (5.41), and (5.46), the matrix-valued functions Ψ1 (σ, ε) and Ψ2 (σ, ε), appearing in (5.59), can be represented as Ψ1 (σ, ε) = Δ1 (σ, ε) + Ψ1s (σ ), σ ∈ [0, tc ], ε ∈ (0, ε2 ],   Ψ2 (σ, ε) = Δ2 (σ, ε) + Ψ2s (σ ) − Ψ2f (tc − σ )/ε , σ ∈ [0, tc ], ε ∈ (0, ε2 ], (5.62) where Ψ2s (σ ) = Ψ1s (σ )Ψ2f (0) (see Eq. (5.18)). Thus, the substitution of (5.62) into (5.59) and the use of (5.61) yield the new expressions for the matrices Wk (tc , ε) (k = 1, 2, 3), ε ∈ (0, ε2 ]: W1 (tc , ε) = Ws (tc ) + L1 (ε) + L1T (ε) + R1 (ε), W2 (tc , ε) = Ws (tc )Ψ2f (0) + L1T (ε)Ψ2f (0) + L2 (ε) − M1 (ε) − Q1T (ε) + R2 (ε), T T T (0)Ws (tc )Ψ2f (0) + Ψ2f (0)L2 (ε) + L2T (ε)Ψ2f (0) − Ψ2f (0)M1 (ε) W3 (ε) = Ψ2f

−M1T (ε)Ψ2f (0) + M2 (ε) − Q2 (ε) − Q2T (ε) + R3 (ε), where the matrix Ws (tc ) is given by Eq. (5.50). Based on these expressions and the block form of the matrix W (tc , ε) (see Eq. (5.58)), we rewrite this matrix as

320

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

W (tc , ε) = B T Ws (tc )B + L (ε)B + B T L T (ε) + M3 (ε)B +B T M3T (ε) + M4 (ε) − Q(ε) + R(ε),

ε ∈ (0, ε2 ],

(5.63)

where   T  T  B = In , Ψ2f (0) , L (ε) = L1 (ε), L2 (ε) , M3 (ε) = On×n , M1 (ε) ,     On×n On×m On×n Q1T (ε) , Q(ε) = , M4 (ε) = Om×n M2 (ε) Q1 (ε) Q2 (ε) + Q2T (ε)   R1 (ε) R2 (ε) . R(ε) = R2T (ε) R3 (ε) (5.64) Let χ ∈ E n+m be any vector such that

χ = 1.

(5.65)

Let us partition this vector into blocks as χ = col(χ1 , χ2 ),

χ 1 ∈ E n , χ2 ∈ E m .

(5.66)

Denote

φ = Bχ = χ1 + Ψ2f (0)χ2 .

(5.67)

It is clear that φ ∈ E n . Using (5.63)–(5.64), (5.66) and (5.67), we obtain χ T W (tc , ε)χ = φ T Ws (tc )φ + 2χ1T L1 (ε)φ + 2χ2T L2 (ε)φ + 2χ2T M1 (ε)φ   +χ2T M2 (ε)χ2 − 2χ1T Q1T (ε)χ2 − χ2T Q2 (ε) + Q2T (ε) χ2 +χ1T R1 (ε)χ1 + 2χ1T R2 (ε)χ2 + χ2T R3 (ε)χ2 , ε ∈ (0, ε2 ]. (5.68) Let us estimate the quadratic form of (5.68). For this purpose, we estimate, first, the matrices given in (5.61). Using the estimates for Ψ1 (σ, ε) and Ψ2 (σ, ε) in (5.20), the assumption (AIV), as well as the inequality (5.19), and Eqs. (5.51) and (5.60), we obtain the existence of a positive number ε3 (ε3 ≤ ε2 ) such that for all ε ∈ (0, ε3 ], the following inequalities are satisfied:

5.4 Parameter-Free Controllability Conditions

321

Ll ≤ aε,

l = 1, 2,

M1 (ε) − εM1 ≤ aε2 , M2 (ε) − εM(tc ) ≤ aε2 ,

Ql (ε) ≤ aε2 ,

Rk (ε) ≤ aε2 ,

l = 1, 2

k = 1, 2, 3,

(5.69)

where a > 0 is some constant independent of ε; 

+∞

M1 = S1 (tc , 0)

Ψ2f (ξ )dξ.

(5.70)

0

Now, using Eqs. (5.65)–(5.67), (5.68), and (5.70) and the second, third, and fourth inequalities in (5.69) yields

χ T W (tc , ε)χ − F1 (χ , ε) ≤ aε2 ,

ε ∈ (0, ε3 ],

(5.71)

where a > 0 is some constant independent of ε; F1 (χ , ε) = φ T Ws (tc )φ + 2χ1T L1 (ε)φ + 2χ2T L2 (ε)φ +2εχ2T M1 φ + εχ2T M(tc )χ2 .

(5.72)

Let us treat the function F1 (χ , ε). Since the slow subsystem (5.7) is completely controllable at the time instant tc , then by virtue of Proposition 5.4, the matrix Ws (tc ) is positive definite. The latter means the existence of a positive number α1 such that the following inequality is satisfied: φ T Ws (tc )φ ≥ α1 φ 2

∀φ ∈ E n .

(5.73)

Similarly, based on Eq. (5.51) and the assumption of the theorem that det M(tc ) = 0, we obtain the existence of a positive number α2 such that the following inequality is satisfied: χ2T M(tc )χ2 ≥ α2 χ2 2

∀χ2 ∈ E m .

(5.74)

Moreover, due to Eqs. (5.65)–(5.67) and (5.70) and the first inequality in (5.69), we have the inequality 2χ1T L1 (ε)φ + 2χ2T L2 (ε)φ + 2εχ2T M1 φ ≤ 2 χ1

L1 (ε)

φ

+2 χ2

L2 (ε)

φ + 2ε χ2

M1

φ ≤ aε φ

where a > 0 is some constant independent of ε.

∀φ ∈ E n , ε ∈ (0, ε3 ], (5.75)

322

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

Applying the inequalities (5.73)–(5.75) to Eq. (5.72), we obtain the estimate of the function F1 (χ , ε) from below: F1 (χ , ε) ≥ F2 (χ , ε)

∀χ ∈ E n+m : χ = 1, ε ∈ (0, ε3 ],

(5.76)

where F2 (χ , ε) = α1 φ 2 − aε φ + εα2 χ2 2 .

(5.77)

Remember that χ2 ∈ E m is the lower block of the vector χ , and the vector φ has the form (5.67). Let us show that, for any number ν > 1, there exists a number ε(ν) > 0 (ε(ν) ≤ ε3 ) such that F2 (χ , ε) ≥ εν

∀χ ∈ E n+m : χ = 1, ε ∈ (0, ε(ν)].

(5.78)

We prove (5.78) by the contradiction method; namely, let us assume that (5.78) is wrong. This means that there exist a number ν > 1, a sequence of numbers {εq }+∞ q=1 , and a sequence of vectors {χ q }+∞ , satisfying the conditions: q=1 q = 1, 2, . . . ;

εq > 0,

χ q ∈ E n+m ,

lim εq = 0,

(5.79)

χ q = 1, q = 1, 2, . . . ,

(5.80)

q→+∞

F2 (χ q , εq ) < εqν , q = 1, 2, . . .

(5.81)

Since {χ q } is bounded, there exists a convergent subsequence of this sequence. For the sake of simplicity (but without a loss of generality), we assume that the sequence {χ q }+∞ q=1 itself is convergent. Let q

q

χ q = (χ1 , χ2 ),

q

q

χ1 ∈ E n , χ2 ∈ E m , q = 1, 2, . . .

(5.82)

Denote

q

χ¯ l = lim χl , q→+∞

l = 1, 2.

(5.83)

Due to Eqs. (5.80), (5.82) and (5.83),

χ¯ 1 2 + χ¯ 2 2 = 1.

(5.84)

5.4 Parameter-Free Controllability Conditions

323

Equation (5.77) and the inequality (5.81) yield F2 (χ q , εq )

φ q 2 q = α1 − a φ q + α2 χ2 2 < εqν−1 , q = 1, 2, . . . , εq εq

(5.85)

where q

q

φ q = χ1 + Ψ2f (0)χ2 ,

q = 1, 2, . . .

(5.86)

By virtue of (5.83) and (5.86), there exists a finite limq→+∞ φ q = μφ ≥ 0. Let μφ > 0. Then due to (5.79), for the left-hand side of the inequality in (5.85) we obtain limq→+∞ F2 (χ q , εq )/εq = +∞. However, for the right-hand side of this inequality we have limq→+∞ εqν−1 = 0, which yields the contradiction +∞ ≤ 0. Now, let μφ = 0. Hence, by virtue of (5.83) and (5.86), χ¯ 1 = −Ψ2f (0)χ¯ 2 yielding by (5.84) that Ψ2f (0)χ¯ 2 2 + χ¯ 2 2 = 1. The latter directly leads to the inequality 0 < χ¯ 2 ≤ 1. Using the inequality (5.85), we obtain q

−a φ q + α2 χ2 2 < εqν−1 , q = 1, 2, . . . , yielding by the limit passage for q → +∞ that α2 χ¯ 2 2 ≤ 0. Thus, χ¯ 2 = 0. The latter contradicts the above obtained inequality χ¯ 2 > 0. This contradiction and the contradiction obtained in the case μφ > 0 prove the inequality (5.78). Let us choose 1 < ν < 2. This choice, along with the inequalities (5.71), (5.76) and (5.78), implies the existence of a positive number ε1∗ ≤ ε3 such that the following inequality is satisfied for all ε ∈ (0, ε1∗ ]: χ T W (tc , ε)χ ≥ εν (1 − aε2−ν ) > 0,

(5.87)

where χ ∈ E n+m is any unit vector and a is the constant appearing in the inequality (5.71). The inequality (5.87) directly implies the positive definiteness of the matrix W (tc , ε) for all ε ∈ (0, ε1∗ ], which completes the proof of the theorem.

5.4.3 Proof of Lemma 5.2 Due to the results of Sect. 3.3.1 (see Definition 3.6 and Proposition 3.1 for the details), the system (5.54) is completely Euclidean space controllable at ξ = ξc if and only if the following matrix is nonsingular:  Wf (ξc )= 0

ξc

 −1 T T T T Ψ4f (ξ )A−1 4s (tc )A3s (tc )B1 (tc , 0)B1 (tc , 0)A3s (tc ) A4s (tc ) Ψ4f (ξ )dξ.

324

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

Let us note that the integrand in the integral of this equation is at least a positive semi-definite matrix for all ξ ≥ 0. Also, by virtue of the inequality (5.17), the following integral is convergent: f = W



+∞ 0

 −1 T T T T Ψ4f (ξ )A−1 4s (tc )A3s (tc )B1 (tc , 0)B1 (tc , 0)A3s (tc ) A4s (tc ) Ψ4f (ξ )dξ. (5.88)

The above made observations imply immediately that the system (5.54) is f completely Euclidean space controllable at ξ = ξc if and only if the matrix W is nonsingular (and, therefore, invertible). Let us show that the matrix M(tc ), given by (5.51), coincides with the matrix f . For this purpose, we are going to transform equivalently the matrix M(tc ). W Remember that M(tc ) depends on the matrix Ψ2f (ξ ) and, consequently, the matrix Ψ4f (ξ ). Begin with some treatment of the latter. Integrating the differential equation in (5.16) from any ξ ≥ 0 to +∞, and using Eq. (5.5) and the inequality (5.17), we obtain after a routine algebra  Ψ4f (ξ ) = −AT4s (tc )

+∞

Ψ4f (θ )dθ −

ξ 0 −h

  T G4 (tc , η, 0)

Resolving this equation with respect to Remark 5.3, we obtain  ξ

+∞

 AT4j (tc , 0)

ξ ξ −hj

j =1

 −

N 



ξ ξ +η

. +∞ ξ

Ψ4f (θ2 )dθ2 dη,

+

j =1

0 −h

ξ ≥ 0.

Ψ4f (θ )dθ and taking into account

  N   −1 Ψ4f (ξ ) + Ψ4f (θ )dθ = − AT4s (tc ) AT4j (tc , 0) 

Ψ4f (θ1 )dθ1

  T G4 (tc , η, 0)

ξ ξ −hj

Ψ4f (θ1 )dθ1 

ξ ξ +η



Ψ4f (θ2 )dθ2 dη . (5.89)

Now, we proceed to the matrix Ψ2f (ξ ). Using Eq. (5.5), the expression for this matrix in (5.18) can be rewritten as follows:  Ψ2f (ξ ) =

AT3s (tc )

+∞ ξ

 +

Ψ4f (θ )dθ +

N 

 AT3j (tc , 0)

j =1

0 −h

  GT3 (tc , η, 0)

ξ ξ +η

ξ ξ −hj

Ψ4f (θ1 )dθ1

 Ψ4f (θ2 )dθ2 dη,

ξ ≥ 0.

5.4 Parameter-Free Controllability Conditions

325

Substitution of (5.89) into this equation yields after a routine rearrangement  −1 Ψ2f (ξ ) = −AT3s (tc ) AT4s (tc ) Ψ4f (ξ )  N   T −1 A3j (tc , 0) − A4j (tc , 0)A4s (tc )A3s (tc ) +  +

0



−h

j =1

 T G3 (tc , η, 0) − G4 (tc , η, 0)A−1 4s (tc )A3s (tc )



ξ ξ −hj

Ψ4f (θ1 )dθ1

 Ψ4f (θ2 )dθ2 dη, ξ ≥ 0.

ξ

ξ +η

The latter, subject to the conditions (5.52)–(5.53), becomes  −1 Ψ2f (ξ ) = −AT3s (tc ) AT4s (tc ) Ψ4f (ξ ),

ξ ≥ 0.

Now, substitution of this expression for Ψ2f (ξ ) into (5.51) and use of (5.88) directly f , meaning that the system (5.54) is completely yield the equality M(tc ) = W Euclidean space controllable at ξ = ξc if and only if the matrix M(tc ) is invertible. This completes the proof of the lemma.

5.4.4 Proof of Theorem 5.2 The proof of the theorem is based on one technical proposition presented in the next subsection.

5.4.4.1

Euclidean Space Controllability of a Pure Fast System

Consider the following time delay system:  d y(t) ˜ j (t, ε)y(t = ˜ − εhj ) + A dt N

ε

j =0



0 −h

η, ε)y(t G(t, ˜ + εη)dη

ε)u(t), +B(t, ˜

t ≥ 0,

(5.90)

˜ ∈ E r (u(t) ˜ is a control); ε > 0 is a small parameter; where y(t) ˜ ∈ E m ; u(t) N ≥ 1 is an integer; 0 = h0 < h1 < h2 < . . . < hN = h are some given j (t, ε) (j = 0, 1, . . . , N), G(t, η, ε), and B(t, ε) are constants independent of ε; A matrix-valued functions of corresponding dimensions, given for t ≥ 0, η ∈ [−h, 0] j (t, ε) (j = 0, . . . , N) and B(t, ε) are continuous in and ε ∈ [0, ε0 ] (ε0 > 0); A 0 (t, ε) ∈ [0, +∞) × [0, ε ]; and the function G(t, η, ε) is piecewise continuous in η ∈ [−h, 0] for any (t, ε) ∈ [0, +∞) × [0, ε0 ], and this function is continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0].

326

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

The system (5.90) is a singularly perturbed system with small delays. However, in contrast with the system (5.1)–(5.2), the system (5.90) consists only of the fast mode. Moreover, this mode is directly controlled by u(t). ˜ In spite of these differences, the complete Euclidean space controllability of the system (5.90) at the time instant tc is defined quite similarly to such a definition for the system (5.1)– (5.2) (for the details, see Definition 5.1 in Sect. 5.2.1). The fast subsystem, associated with the system (5.90), is obtained similarly to such a subsystem for the system (5.1)–(5.2). Thus, the fast subsystem for (5.90) has the form d y˜f (ξ )  j (t, 0)y˜f (ξ − hj ) + = A dξ N



j =0

0 −h

η, 0)y˜f (ξ + η)dη G(t,

0)u˜ f (ξ ), +B(t,

ξ ≥ 0,

(5.91)

where t ≥ 0 is a parameter, ξ ≥ 0 is an independent variable, yf (ξ ) ∈ E m , and u˜ f (ξ ) ∈ E r (u˜ f (ξ ) is a control). Remark 5.8 The fast subsystem (5.91) is an ε-free time delay system. Note that (5.91) is of the same Euclidean dimension m as (5.90). However, for the sake of the uniformity in the terminology, we keep for (5.91) the term “fast subsystem” (not “fast system”). We assume that j (t, ε) (j = 0, 1, . . . , N ) and B(t, ε) are I. the matrix-valued functions A continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ]; η, ε) is piecewise continuous with respect II. the matrix-valued function G(t, to η ∈ [−h, 0] for each (t, ε) ∈ [0, tc ] × [0, ε0 ], and it is continuously differentiable with respect to (t, ε) ∈ [0, tc ] × [0, ε0 ] uniformly in η ∈ [−h, 0]; III. for all t ∈ [0, tc ] and any complex number λ with Reλ ≥ 0, the following equality is valid: rank  +

 N

j (t, 0) exp(−λhj ) A

j =0

0 −h

 G(t, η, 0) exp(λη)dη − λIm , B(t, 0) = m.

As a particular case of Theorem 3.11 (for the details, see Sect. 3.5.1), we have the following proposition. Proposition 5.5 Let the assumptions I–III be valid. Let, for t = tc , the fast subsystem (5.91), associated with the system (5.90), be completely Euclidean space controllable at some instant ξ = ξc > 0. Then, there exists a positive number ε˜ (˜ε ≤ ε0 ) such that for all ε ∈ (0, ε˜ ], the singularly perturbed system (5.90) is completely Euclidean space controllable at the time instant tc .

5.4 Parameter-Free Controllability Conditions

5.4.4.2

327

Main Part of the Proof

Let us transform the control in the original system (5.1)–(5.2) as u(t) = ε−1 v(t),

(5.92)

where v(t) is a new control. Due to this transformation, the system (5.1)–(5.2) can be rewritten in the form dz(t)  = εAj (t, ε)z(t − εhj ) + dt N

ε

j =0



0 −h

εG(t, η, ε)z(t + εη)dη +B(t, ε)v(t), t ≥ 0,

(5.93)

  where z(t) = col x(t), y(t) for t ≥ −εh and the matrices Aj (t, ε) (j = 0, 1, . . . , N), G(t, η, ε), and B(t, ε) are given in (5.11) and (5.48). Since the transformation (5.92) is invertible for any ε ∈ (0, ε0 ], then for all such ε, the original system (5.1)–(5.2) is completely Euclidean space controllable at the time instant tc if and only if the system (5.93) is completely Euclidean space controllable at this time instant. Let us derive ε-free conditions of this controllability for the system (5.93). Using Eq. (5.11) yields  εAj (t, ε) =

εA1j (t, ε) A3j (t, ε)

 εA2j (t, ε) , j = 0, 1, . . . , N, A4j (t, ε) 

εG(t, η, ε) =

εG1 (t, η, ε) G3 (t, η, ε)

 εG2 (t, η, ε) . G4 (t, η, ε)

Based on these matrices and Eq. (5.48), we can directly conclude that the system (5.93) consists only of the fast mode. Therefore, only the fast subsystem can be associated with (5.93). By virtue of the results of the previous subsection (Sect. 5.4.4.1), the system (5.56) represents the fast subsystem associated with (5.93). Let us apply Proposition 5.5 to the system (5.93). It is seen directly that, due to this application, the assumptions I–II of the proposition become the assumptions (AI), (AII), and (AIV) of Theorem 5.2. Moreover, the assumption III of the proposition becomes the assumption (AV) of the theorem. Therefore, by virtue of Proposition 5.5 applied to the system (5.93), there exists a positive number ε2∗ , (ε2∗ ≤ ε0 ), such that this system is completely Euclidean space controllable at the time instant tc for all ε ∈ (0, ε2∗ ]. Thus, the theorem is proven.

328

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

5.5 Examples In this section some examples, illustrating the results of the previous sections, are presented.

5.5.1 Example 1 Consider the following system, a particular case of (5.1)–(5.2): dx(t) = x(t) + 2x(t − εh) + y(t) − 2y(t − εh) + u(t), dt dy(t) ε = x(t − εh) − 2y(t) + y(t − εh), dt

(5.94)

where x(t), y(t), and u(t) are scalars, i.e., n = m = r = 1; t ∈ [0, tc ]. By virtue of the results of Sect. 5.2.2, the slow and fast subsystems associated with (5.94) are, respectively, dxs (t) = 2xs (t) + us (t) dt

(5.95)

dyf (ξ ) = −2yf (ξ ) + yf (ξ − h). dξ

(5.96)

and

It is seen that (5.94) satisfies the assumptions (AI), (AII), and (AIV). Let us verify the fulfillment of the assumption (AIII). In this example, Eq. (5.10) becomes λ + 2 − exp(−λh) = 0.

(5.97)

Denote

β = min{0.25, 0.1/ h} > 0. Thus, for Reλ ≥ −2β, we have the inequality   Re λ + 2 − exp(−λh) ≥ −2β + 2 − exp(2βh) ≥ 0.27 > 0. This inequality means that all roots λ of Eq. (5.97) satisfy the inequality Reλ < −2β, i.e., the assumption (AIII) is fulfilled.

5.5 Examples

329

Furthermore, since the coefficient for the control us (t) in the slow subsystem (5.95) differs from zero, this subsystem is completely controllable at the time instant tc . Let us verify the fulfillment of the condition of Theorem 5.1 on the nonsingularity of the matrix M(tc ). First note that, in this example, the matrixvalued function Ψ4f (ξ ) becomes a scalar function. This function is the unique solution of Eq. (5.96) satisfying the initial conditions Ψ4f (ξ ) = 0, ξ < 0;

Ψ4f (0) = 1.

(5.98)

The matrix-valued function Ψ2f (ξ ) and the matrix M(tc ), respectively, are a scalar function and a scalar value, having the form  +∞ Ψ2f (ξ ) = Ψ4f (θ − h)dθ, ξ

 M(tc ) =

+∞ 

2 Ψ2f (ξ ) dξ = M.

0

These two equations, along with the initial conditions (5.98), yield that M > 0. Otherwise, Ψ2f (ξ ) ≡ 0, ξ ≥ 0 implying Ψ4f (ξ − h) = 0 for almost all ξ ≥ 0. The latter is wrong, because   Ψ4f (ξ − h) = exp(−2ξ ) 1 + exp(2h)(ξ − h) > 0

∀ ξ ∈ [h, 2h).

Thus, all the conditions of Theorem 5.1 are valid, meaning that there exists a positive number ε1∗ such that the system (5.94) is completely Euclidean space controllable at the time instant tc for all ε ∈ (0, ε1∗ ]. Note that, in this example, the condition (5.52) is not valid and, consequently, Corollary 5.1 is not applicable. The next example is an extension of Example 1.

5.5.2 Example 2 Consider the following particular case of the system (5.1)–(5.2) with scalars x(t), y(t), and u(t): dx(t) =A10 (t)x(t)+A11 (t)x(t−εh)+A20 (t)y(t)+A21 (t)y(t − εh)+B1 (t)u(t), dt dy(t) =A30 (t)x(t)+A31 (t)x(t−εh)+A40 (t)y(t) + A41 (t)y(t − εh). ε dt (5.99) Let the functions Aij (t) (i = 1, . . . , 4; j = 0, 1) and B1 (t) be continuously differentiable for t ∈ [0, tc ]. This assumption means the fulfillment of the assumptions (AI) and (AIV) for the system (5.99). Moreover, since there are no

330

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

distributed delays in (5.99), then the assumption (AII) is also fulfilled for this system. Now, let us assume that A40 (t) < 0, |A41 (t)| < |A40 (t)|,

t ∈ [0, tc ],

(5.100)

B1 (tc ) = 0,

(5.101)

|A30 (tc )| + |A31 (tc )| > 0.

(5.102)

The inequalities in (5.100) directly yield that, for the system (5.99), the assumption (AIII) is valid for any h > 0. This claim is proven similarly to such a claim in Example 1. Since the assumption (AIII) is valid, then by virtue of Remark 5.3 and Eq. (5.7), the slow subsystem associated with the system (5.99) can be represented in the pure differential form dxs (t) = A¯ s (t)xs (t) + B1 (t)us (t), t ∈ [0, tc ], dt

(5.103)

where   A¯ s (t) = A10 (t) + A11 (t)   −1   − A20 (t) + A21 (t) A40 (t) + A41 (t) (t) A30 (t) + A31 (t) . Furthermore, due to Proposition 5.4 and the smoothness of the function B1 (t) in the interval [0, tc ], the inequality (5.101) means the complete controllability of the slow subsystem (5.103) at the time instant tc . Moreover, due to the inequalities (5.101) and (5.102), it can be shown similarly to Example 1 that the condition of Theorem 5.1 on the nonsingularity of the matrix M(tc ) is fulfilled for the system (5.99). Thus, under the assumption of the smoothness of the coefficients in (5.99) and the inequalities (5.100)–(5.102), the system (5.99) is completely Euclidean space controllable at the time instant tc for all ε ∈ (0, ε1∗ ] with some sufficiently small ε1∗ > 0.

5.5.3 Example 3 In this example, we consider the following system: dx(t) =x(t)+x(t−εh)−2y(t)−y(t−εh)+ dt ε



0

−h

2ηy(t + εη)dη+tu(t),

dy(t) =(t + ε)x(t)−(t−1−ε)x(t−εh)−(t+2+ε)y(t) + (t − ε)y(t−εh), dt (5.104)

where x(t), y(t), and u(t) are scalars; t ∈ [0, 2], i.e., tc = 2.

5.5 Examples

331

It is clear that the assumptions (AI), (AII), and (AIV) are satisfied for this system. The fulfillment of the assumption (AIII) in this example is justified by the same arguments as in Example 1. Let us check up the fulfillment of the conditions (5.52) and (5.53). For the system (5.104), we have A30 (tc , 0) = 2, A31 (tc , 0) = −1, A40 (tc , 0) = −4, A41 (tc , 0) = 2, A3s (tc ) = 1, A4s (tc ) = −2, G3 (tc , η, 0) = G4 (tc , η, 0) = 0. (5.105) Remember that the expressions for Ais (t) (i = 1, . . . , 4) are given in (5.5). Now, using (5.105), we immediately obtain that the conditions (5.52) and (5.53) are fulfilled. Let us write down the slow subsystem, associated with (5.104). For this purpose, we calculate the values A1s = 2,

A2s = −3 − h2 .

Thus, by virtue of (5.7)–(5.8), we obtain the slow subsystem as dxs (t) = (0.5 − h2 )xs (t) + tus (t), dt

t ∈ [0, 2].

By the same argument as in Example 2 (the inequality (5.101) is valid for (5.104)), this slow subsystem is completely controllable at the time instant tc = 2. Now, let us write down the first-order fast subsystem, associated with (5.104). Due to Eq. (5.54), this subsystem has the form d y(ξ ˜ ) = −4y(ξ ˜ ) + 2y(ξ ˜ − h) − u(ξ ˜ ), dξ

ξ ≥ 0.

(5.106)

Note that, for ξ ∈ [0, h), this system can be considered as an undelayed system with a nonhomogeneous term 2y(ξ ˜ − h). Hence, the complete Euclidean space controllability of (5.106) at ξ = ξc ∈ [0, h) is reduced to the complete controllability of the corresponding undelayed system at the instance ξc . The latter kind of the controllability is obviously valid (due to Proposition 5.4), because the coefficient for the control u(ξ ˜ ) in this system is nonzero. Thus, all the conditions of Corollary 5.1 are valid for the system (5.104) yielding its complete Euclidean space controllability at the time instant tc for all ε ∈ (0, ε1∗ ] with some positive sufficiently small ε1∗ .

332

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

5.5.4 Example 4 Consider the system dx(t) = 4x(t) + 2x(t − εh) − y(t) + 3y(t − εh) + (t + 2)u(t), dt dy(t) ε = −x(t − εh) + y(t) − y(t − εh), dt

(5.107)

where x(t), y(t), and u(t) are scalars, i.e., n = m = r = 1; t ∈ [0, tc ]. Equation (5.10) for this system becomes λ − 1 + exp(−λh) = 0, yielding the solution λ = 0. The latter indicates that the system (5.107) does not satisfy the assumption (AIII). Thus, Theorem 5.1 is not applicable to this system. Let us try to use Theorem 5.2 to verify the complete Euclidean space controllability of (5.107). Assumptions (AI), (AII), and (AIV) are obviously satisfied for (5.107). Using Eqs. (5.48) and (5.55), we obtain the matrices of the coefficients in the system (5.56), associated with (5.107),    0 0 0 0 6 6 , A1 (t) = A1 = , 0 1 −1 − 1   t +2 6 6 . Aj (t) ≡ O2×2 , j = 2, . . . , N, G(t, η) ≡ O2×2 , B(t, 0) = B(t) = 0 (5.108) 60 = 60 (t) = A A



Based on these matrices, we construct the matrix in the left-hand side of Eq. (5.57) 

 −λ 0 t +2 . − exp(−λh) 1 − exp(−λh) − λ 0

For any t ∈ [0, tc ] and any complex numbers λ : Reλ ≥ 0, the first and the third columns of this matrix are linearly independent. Therefore, for all such t and λ, the rank of this matrix equals 2 = n + m, meaning the validity of the assumption (AV) in this example. Now, we are going to verify the complete Euclidean space controllability of the system (5.56) for t = tc and the matrix-valued coefficients given in (5.108). Note that this system is a time-invariant differential-difference system. Hence, its complete Euclidean space controllability can be verified by the algebraic criterion proposed in [6]. Using this criterion, we will show that the system (5.56) and

5.5 Examples

333

(5.108), t = tc , is completely Euclidean space controllable at any given instant ξc ∈ (1, 2]. For this purpose, we construct the following matrices:

60 = 60 = A H

60 = I2 , E





0 0 61 = , H 0 1

  61 = O2×2 , I2 , E



60 O2×2 H 61 H 60 A

60 = I2 , C



⎛

⎞ 0 0 0 0 ⎜ 0 1 0 0 ⎟ ⎟, =⎜ ⎝ 0 0 0 0 ⎠ −1 − 1 0 1

60 = B(tc ), B

61 = C 61 B 60 , B

where

61 = C



I2   60 60 C exp H



⎛

⎞ 1 0 ⎜0 1 ⎟ ⎟. =⎜ ⎝1 0 ⎠ 0 exp(1)

Hence, ⎞ tc + 2 ⎟ ⎜ ⎟ 61 = ⎜ 0 B ⎝ tc + 2 ⎠ . 0 ⎛

Due to the results of [6], the system (5.56) and (5.108), t = tc , is completely Euclidean space controllable at a given value ξc ∈ (1, 2] of the independent variable ξ , if and only if the rank of the following matrix is equal to the Euclidean dimension of this system n: ˆ   ˆ ˆ 6= B 60 , . . . , H 6n−1 61 B 61 H 60 , E 61 , . . . , E 62n−1 61 . D B B 0 1 6 is a two-dimensional vector and nˆ = 2, it is Since each block of the matrix D sufficient to show that some set of twoblocks in this matrix is linearly independent.  + t 6 is B 60 = c 2 , while the fourth block of this matrix is The first block of D 0 61 H 61 B 61 = E



 0 . −(tc + 2)

6 = nˆ = 2. Thus, the These blocks are linearly independent, meaning that rankD system (5.56) and (5.108), t = tc , is completely Euclidean space controllable at any given value ξc ∈ (1, 2]. Therefore, by virtue of Theorem 5.2, the system (5.107) is

334

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

completely Euclidean space controllable at the time instant tc robustly with respect to ε > 0 for all its sufficiently small values.

5.5.5 Example 5 Consider the following linear system with a state delay: dX(t) = A0 (t)X(t) + A1 (t)X(t − μ) + D(t)u(t), t ≥ 0, dt

(5.109)

where X(t) and u(t) are scalars, u is a control; A0 (t), A1 (t), and D(t) are given functions of t ≥ 0; and μ > 0 is a given constant time delay. Definition 5.3 For any given tc > μ, the system (5.109) is called completely functional space controllable at the time instant tc if for any functions f0 (·) ∈ L2 [−μ, 0; E 1 ], fc (·) ∈ AC 2 [tc − μ, tc ; E 1 ] and any constant X0 , there exists a control u(·) ∈ L2 [0, tc ; E 1 ] such that the system (5.109) with the initial conditions X(τ ) = f0 (τ ), τ ∈ [−μ, 0);

X(0) = X0

(5.110)

has a solution X = X(t), t ∈ [0, tc ], satisfying the terminal condition X(t) = fc (t), t ∈ [tc − μ, tc ].

(5.111)

Let K > 0 be an integer. Applying the results of [5], one can approximate the system (5.109) by the following set of undelayed differential equations: dx(t) = A10 (t)x(t) + A20 (t)y(t) + B1 (t)u(t), dt dy(t) = A30 x(t) + A40 y(t), ε dt

t ≥ 0, t ≥ 0,

(5.112)

where x(t) is a scalar (an approximation of X(t)); ε = μ/K; y(t) ∈ E K ; A10 (t) = A0 (t); B1 (t) = D(t); and the matrices A20 (t), A30 , and A40 , having the dimensions 1 × K, K × 1, and K × K, respectively, are of the form

A30

⎛ ⎞ 1 ⎜0⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ . ⎟, ⎜ ⎟ ⎝.⎠ 0

A40

A20 (t) = (0, 0, . . . , A1 (t)), ⎛ ⎞ −1 0 0 ... 0 0 ⎜ 1 − 1 0 ... 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ =⎜ . . . . . . . . . ⎟. ⎜ ⎟ ⎝ . . . . . . . . . ⎠ 0 0 0 ... 1 − 1

(5.113)

5.5 Examples

335

Due to [5], the conditions (5.110)–(5.111) are approximated as follows: x(0) = X0 , y(0) = fy0 , x(tc ) = fc (tc ), y(tc ) = fyc , where ! " fy0 = col f0 (−ε), f0 (−2ε), . . . , f0 (−μ) , ! " fyc = col fc (tc − ε), fc (tc − 2ε), . . . , fc (tc − μ) . Thus, the problem of the complete functional space controllability of the system (5.109) at the time instant tc is formally approximated by the problem of the complete controllability of the system (5.112) at tc . In the sequel of this example, we do not study the issue of accuracy of this approximation. We concentrate on the issue of the controllability of the system (5.112). For a sufficiently large K, the value of ε is sufficiently small. Consequently, the system (5.112) becomes a particular undelayed case of (5.1)–(5.2). We assume that Ak (t) (k = 0, 1) and D(t) are continuously differentiable for t ∈ [0, tc ]. This means that the assumptions (AI), (AII), and (AIV) are valid for (5.112). The K × K-matrix A4 , given in (5.113), has the single eigenvalue λ = −1 of the multiplicity K. This observation means the fulfillment of the assumption (AIII) for the system (5.112). Moreover, since this system is delay-free, the conditions (5.52)– (5.53) are fulfilled. Now, let us write down the slow subsystem and the first-order fast subsystem, associated with (5.109). The slow subsystem has the form  dxs (t)  = A0 (t) + A1 (t) xs (t) + D(t)us (t). dt

(5.114)

It is seen that the slow subsystem can be obtained from the original system (5.109) by setting in the latter formally μ = 0. The first-order fast subsystem for (5.109) has the form d y(ξ ˜ ) = A40 y(ξ ˜ ) + F u(ξ ˜ ), dξ

(5.115)

˜ ) is scalar; where y(ξ ˜ ) ∈ E K and u(ξ F = D(tc )A−1 40 A30 .

(5.116)

In the sequel, we assume that D(tc ) = 0. Similarly to Example 2, this assumption implies the complete controllability of (5.114) at the time instant tc . Let us show that this assumption also provides the complete controllability of (5.115) at some instant ξ = ξc > 0. For this purpose, we use the well-known

336

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

Kalman algebraic criterion of the complete controllability for time-invariant systems [4]. Due to this criterion, the system (5.115) is completely controllable at some ξ = ξc > 0, if and only if the rank of the following matrix equals K: " ! K−1 F . W = F , A40 F , . . . , A40 Using the expressions for the matrices A30 , A40 , and F (see Eqs. (5.113) and (5.116)), and taking into account that D(tc ) is a scalar, we obtain W = D(tc )A−1 40 W1 , where " ! K−1 W1 = A30 , A40 A30 , . . . , A40 A30 . Thus, rankW = K if and only if rankW1 = K. By a routine matrix algebra, it is shown that the matrix W1 is an upper triangular matrix with the unit on the main diagonal. Hence, rankW1 = K, meaning the complete controllability of (5.115) at any instant ξ = ξc > 0. Therefore, due to Corollary 5.1, the system (5.112) is completely controllable at the time instant tc robustly with respect to ε for all its sufficiently small positive values.

5.5.6 Example 6 The previous examples have illustrated the application of Theorems 5.1 and 5.2 and Corollary 5.1 to check up the robust Euclidean space controllability of linear singularly perturbed time delay systems. However, the conditions of these theorems and corollary are in general sufficient (but not necessary) for such a controllability. The present example illustrates this statement with respect to Corollary 5.1, a particular case of Theorem 5.1. For the sake of a technical simplicity, we consider in this example a delay-free system with constant coefficients. Namely, dx1 (t) = a11 x1 (t) + a12 x2 (t) + a13 y(t) + b1 u(t), dt dx2 (t) = a21 x1 (t) + a22 x2 (t) + a23 y(t) + b2 u(t), dt dy(t) = a31 x1 (t) + a32 x2 (t) + a33 y(t), ε dt where x1 (t), x2 (t), y(t), and u(t) are scalars and t ∈ [0, tc ].

(5.117)

5.5 Examples

337

The system (5.117) is a particular undelayed case of (5.1)–(5.2) with  A10 =

a11 a12 a21 a22



 , A20 =

a13 a23



 , B1 =

A30 = (a31 , a32 ),

b1 b2

 ,

A40 = a33 ,

and Aij (·) and Gi (·) (i = 1, . . . , 4; j = 1, . . . , N) to be zero matrices of the corresponding dimensions. In the sequel of this example, we assume that a33 < 0, meaning the existence of 1/a33 . The latter, along with the fact that the system (5.117) is delay-free, means the fulfillment of the conditions (5.52)–(5.53) for this system. The slow subsystem, associated with (5.117), has the form   dxs (t) 1 = A10 − A20 A30 xs (t) + B1 us (t), dt a33

(5.118)

where xs (t) ∈ E 2 , while us (t) is scalar. The first-order fast subsystem, associated with (5.117), has the form 1 d y(ξ ˜ ) = a33 y(ξ ˜ )+ A30 B1 u(ξ ˜ ), dξ a33

(5.119)

where y(ξ ˜ ) and u(ξ ˜ ) are scalars. Assume that A30 = 0,

B1 = 0,

A30 A10 B1 = 0, A30 B1 = 0.

(5.120)

Let us show that, subject to the assumptions in (5.120), the original system (5.117) is completely controllable at the time instant tc for any ε > 0 if and only if the slow subsystem (5.118) is completely controllable at this time instant. For this purpose, we use the algebraic criterion of the complete controllability for time-invariant delay-free systems (see [4]). Consider the following 3 × 3-matrix A(ε), ε > 0, and the vector B ∈ E 3 :  A(ε) =

A20 A10 1 1 ε A30 ε A40



 ,

B=

B1 0

 .

(5.121)

By virtue of the results of [4], for a given ε > 0 the original system (5.117) is completely controllable at the time instant tc if and only if " ! rank B, A(ε)B, A2 (ε)B = 3,

(5.122)

while the slow subsystem (5.118) is completely controllable at the time instant tc if and only if

338

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

!   " rank B1 , A10 − (1/a33 )A20 A30 B1 = 2.

(5.123)

Based on Eqs. (5.120) and (5.121), let us transform the matrices in Eqs. (5.122) and (5.123). We have !

" B A B 1 10 1 B, A(ε)B, A2 (ε)B = 0 0

A210 B1 1 ε A30 A10 B1



" !  " !  B1 , A10 − (1/a33 )A20 A30 B1 = B1 , A10 B1 .

,

(5.124)

(5.125)

The comparison of the matrices in the right-hand side of (5.124) and (5.125) and the use of the third condition in (5.120) yield that the equality (5.122) is valid for any ε > 0 if and only if the equality (5.123) is valid. Hence, subject to the assumptions in (5.120), the complete controllability of the slow subsystem (5.118) at the time instant tc is necessary and sufficient for the complete controllability of the original system (5.117) at tc for all ε > 0. However, due to the fourth condition in (5.120), the first-order fast subsystem (5.119) is uncontrolled and, therefore, uncontrollable at any instant ξc > 0 of the independent variable ξ . This means that the condition on the controllability of the first-order fast subsystem in Corollary 5.1 is in general sufficient but not always necessary for the robust with respect to ε Euclidean space controllability of the original singularly perturbed system (5.1)–(5.2).

5.5.7 Example 7: Analysis of Controlled Car-Following Model in a Simple Open Lane In this example we analyze the system (1.26), which is a controlled singularly perturbed system with small state delays. This system modeled a car-following process of three vehicles, which follow each other in one lane in the shape of a straight line (for the details, see Sect. 1.1.5). For the sake of the book’s reading convenience, we write this system here once again dx(θ ) = −x(θ − εh1 ) + u(θ ), dθ dy(θ ) = x(θ − εh1 ) − y(θ − εh2 ), ε dθ

(5.126)

where θ ∈ [0, θc ], ε > 0 is a small parameter, the value θc > 0 is independent of ε, h1 > 0 and h2 > 0 are constants independent of ε, and u(θ ) is a control function.

5.5 Examples

339

The system (5.126) is a particular case of the system (5.1)–(5.2), and we study the complete Euclidean space controllability of (5.126) at θ = θc for all sufficiently small ε > 0. The asymptotic decomposition of the system (5.126) yields the slow subsystem in the pure differential form dxs (θ ) = −xs (θ ) + us (θ ), dθ

θ ∈ [0, θc ]

(5.127)

and the fast subsystem dyf (ξ ) = −yf (ξ − h2 ), dξ

ξ ≥ 0.

(5.128)

It is seen that the assumptions (AI), (AII), and (AIV) are satisfied for the system (5.126). Let us verify the fulfillment of the assumption (AIII). Equation (5.10) becomes in the present example as follows: λ + exp(λh2 ) = 0.

(5.129)

In what follows of this example, we assume that π . 2

h2
0 such that all roots λ of Eq. (5.129) satisfy the inequality Reλ < −2β, Thus, the assumption (AIII) is also satisfied for the system (5.126). Now, we are going to apply Theorem 5.1 to the controllability analysis of the system (5.126). For this purpose, we should show the complete controllability of the slow subsystem (5.127) at θc , and to show the nonsingularity of the matrix M(θc ). This matrix is obtained from the matrix M(tc ) (see Eq. (5.51)) by replacing tc with θc . Let us start with the controllability of the slow subsystem. Since the coefficient for the control us (θ ) in (5.127) is nonzero, this subsystem is completely controllable at θc . Now, let us verify the nonsingularity of the matrix M(θc ). In the present example, this matrix becomes the scalar  M(θc ) = M = 0

+∞

2 Ψ2f (ξ )dξ,

where, due to (5.18),  Ψ2f (ξ ) = ξ

+∞

Ψ4f (τ − h1 )dτ,

340

5 First-Order Euclidean Space Controllability Conditions for Linear Systems with. . .

while Ψ4f (ξ ), ξ ≥ 0, is the unique solution of Eq. (5.128) satisfying the initial conditions Ψ4f (ξ ) = 0 for ξ < 0 and Ψ4f (0) = 1 (see Eq. (5.16)). Similarly to Example 1 (see Sect. 5.5.1), it is shown that M > 0. Hence, all the conditions of Theorem 5.1 are fulfilled for the system (5.126), implying its complete Euclidean space controllability at θc for all sufficiently small ε > 0. Similarly to Remark 3.17, it can be shown that the transferring control can be chosen as a continuous function in the interval [0, θc ], i.e., to be feasible. Remark 5.9 Note that if h2 ≥ π/2, Eq. (5.129) can have roots with nonnegative real parts. Indeed, if h2 = π/2, then λ = i (i is the imaginary unit) satisfies this equation. Therefore, if h ≥ π/2, the assumption (AIII) can be violated. However, it can be shown that the assumption (AV) is valid for the system (5.126). Its complete Euclidean space controllability at θc for all sufficiently small ε > 0 can be analyzed (similarly to Example 4 in Sect. 5.5.4) by using Theorem 5.2 and the results of the work [6]. We leave this analysis to a reader as an exercise.

5.6 Concluding Remarks and Literature Review In this chapter, a class of singularly perturbed linear time-dependent controlled systems with delays (multiple point-wise and distributed) in the state variables was considered. The delays are proportional to a small positive parameter ε multiplying a part of the derivatives in the systems. The essential feature of a system of this class is that the control function appears only in its slow mode. This means that the slow mode is controlled directly, while the fast mode is controlled through the slow one. For these systems, the complete Euclidean space controllability at a prescribed time instant, robust with respect to ε, was studied. It is important to note that the separation of time-scales concept and consequently the results of the previous chapters on the controllability of a singularly perturbed system, based on this concept, are not applicable to this study. Two types of ε-free conditions, guaranteeing the robust Euclidean space controllability of the considered systems, were derived. The conditions of the first type are based on the assumption of asymptotic stability of the fast subsystem associated with the original singularly perturbed system, while the conditions of the second type are not. Instead, the conditions of the second type are based on the assumption of the complete Euclidean space controllability of some controlled ε-free time delay autonomous system. To obtain the abovementioned conditions for the controllability of the original system, an important auxiliary result was derived. Namely, in Lemma 5.1, there was derived the block-wise first-order asymptotic estimate for the adjoint matrix of the fundamental matrix of the original singularly perturbed system. For the first time, this lemma was formulated and proven in a brief form in [1]. In the present chapter we prove Lemma 5.1 in a much more detailed form.

References

341

Theorems 5.1 and 5.2 presented two different types of ε-free conditions, subject to which the original singularly perturbed time delay system is completely Euclidean space controllable at a given time instant for all sufficiently small positive values of the parameter ε. These results were also obtained for the first time in the paper [1]. In the present chapter, we give another, simpler and at the same time more detailed, proofs of these theorems. In the next chapter, we will consider systems with various types of features, which were not considered in the present chapter and Chaps. 3–5. These features are nonsmall delays in both, slow and fast, state variables, a nonsmall control delay, and nonlinear systems. Each of these features does not allow applying the methods of the previous chapters to the controllability analysis of the corresponding system, thus requiring to develop a new approach to such an analysis. However, the results of the previous chapters are used (as auxiliary results) in the derivation of main results of the next chapter.

References 1. Glizer, V.Y.: Novel controllability conditions for a class of singularly perturbed systems with small state delays. J. Optim. Theory Appl. 137, 135–156 (2008) 2. Halanay, A.: Differential Equations : Stability, Oscillations, Time Lags. Academic Press, New York (1966) 3. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) 4. Kalman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Mat. Mex. 5, 102–119 (1960) 5. Repin, I.M.: On the approximate replacement of systems with lag by ordinary dynamical systems. J. Appl. Math. Mech. 29, 254–264 (1965) 6. Zmood, R.B.: On Euclidean space and function space controllability of control systems with delay. Technical report. The University of Michigan, Ann Arbor, MI, p. 99 (1971)

Chapter 6

Miscellanies

6.1 Introduction In this chapter, several classes of singularly perturbed controlled time delay systems are considered. Thus, in the next section, we consider a class of singularly perturbed linear systems with one point-wise delay and a distributed delay in the state variables. The delays are nonsmall (of order of 1) in both, slow and fast, state variables. The systems of this class are obtained by a proper equivalent transformation of high gain control systems. Based on the asymptotic slow–fast decomposition of the considered system, conditions of its complete Euclidean space controllability, robust with respect to the parameter of singular perturbation, are derived. Section 6.3 is devoted to controllability analysis of singularly perturbed linear time-dependent systems with a point-wise delay only in the control variable. The delay is assumed to be nonsmall (of order of 1). Due to this assumption, the slow– fast decomposition approach is not applicable for the analysis of the considered systems. Therefore, in this section, we propose another approach for deriving parameter-free conditions of the complete Euclidean space controllability of the original singularly perturbed system. This approach is based on an asymptotic analysis of the controllability matrix for the original system. In Sect. 6.4, two classes of singularly perturbed nonlinear systems with multiple point-wise and distributed state delays are considered. The delays are small of order of a small positive parameter multiplying a part of the derivatives in the systems. For these systems, Euclidean space and functional space null controllability conditions, robust with respect to the small parameter, are derived. In Sect. 6.5, some open problems are presented.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Y. Glizer, Controllability of Singularly Perturbed Linear Time Delay Systems, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-65951-6_6

343

344

6 Miscellanies

The following main notations are applied in this chapter: 1. E n is the n-dimensional real Euclidean space. 2. The Euclidean norm of either a vector or a matrix is denoted by · . 3. The upper index T denotes the transposition either of a vector x (x T ) or of a matrix A (AT ). 4. In denotes the identity matrix of dimension n. 5. The notation On1 ×n2 is used for the zero matrix of the dimension n1 × n2 , excepting the cases where the dimension of a zero matrix is obvious. In such cases, we use the notation 0 for the zero matrix. 6. L2 [t1 , t2 ; E n ] denotes the linear space of all vector-valued functions x(·) : [t1 , t2 ] → E n square integrable in the interval [t1 , t2 ]. 7. L2loc [t¯, +∞; E n ] denotes the linear space of all vector-valued functions x(·) : [t¯, +∞) → E n square integrable in any subinterval [t1 , t2 ] ⊂ [t¯, +∞). 8. C[t1 , t2 ; E n ] denotes the linear space of all vector-valued functions x(·) : [t1 , t2 ] → E n continuous in the interval [t1 , t2 ]. 9. col(x, y), where x ∈ E n , y ∈ E m , denotes the column block vector of the dimension n + m with the upper block x and the lower block y, i.e., col(x, y) = (x T , y T )T . 10. Reλ denotes the real part of a complex number λ. 11. W 1,2 [t1 , t2 ; E n ] denotes the corresponding Sobolev space, i.e., the linear space of all vector-valued functions x(·) : [t1 , t2 ] → E n square integrable in the interval [t1 , t2 ] with the first derivatives (generalized) square integrable in this interval.

6.2 Euclidean Space Controllability of Linear Time Delay Systems with High Gain Control 6.2.1 High Gain Control System: Main Notions and Definitions 6.2.1.1

Initial System

Consider the system dZ(t)  = Ai (t)Z(t − hi ) + dt 1

i=0



0

1 G (t, η)Z(t + η)dη + B(t)u(t), ε −h

(6.1)

where t ≥ 0; Z(t) ∈ E n , u(t) ∈ E r (u(t) is a control); r ≤ n; ε > 0 is a small parameter; 0 = h0 < h1 = h, and the latter is some given constants independent of ε; Ai (t), (i = 0, 1), G (t, η), and B(t) are matrix-valued functions of corresponding dimensions, given for t ≥ 0 and η ∈ [−h, 0]; the functions Ai (t), (i = 0, 1) and

6.2 Linear High Gain Control Time Delay Systems

345

B(t) are continuous for t ∈ [0, +∞); the function G (t, η) is piecewise continuous in η ∈ [−h, 0] for any t ∈ [0, +∞), and this function is continuous with respect to t ∈ [0, +∞) uniformly in η ∈ [−h, 0]. Due to the results of Sect. 2.4, for any given ε > 0 and u(·) ∈ L2loc [0, +∞; E r ], the system (6.1) is a linear time-dependent nonhomogeneous functional-differential   system. This system is infinite-dimensional with the state variable Z(t), Z(t + η) , η ∈ [−h, 0]. Moreover, due to the smallness of the parameter ε, the system (6.1) is a high gain control system. Remark 6.1 In contrast with the original differential systems (3.1)–(3.2), (4.1)– (4.2), and (5.1)–(5.2) of Chaps. 3, 4, and 5, respectively, the system (6.1) has only nonsmall (of order of 1) state delays. Let tc > h be a given time instant independent of ε. Definition 6.1 For a given positive ε, the system (6.1) is said to be completely Euclidean space controllable at the time instant tc if for any Z0 ∈ E n , ϕZ (·) ∈ L2 [−h, 0; E n ], and Zc ∈ E n there exists a control function u(·) ∈ L2 [0, tc ; E r ], for which the system (6.1) with the initial and terminal conditions Z(η) = ϕZ (η),

η ∈ [−h, 0);

Z(0) = Z0 , Z(tc ) = Zc ,

(6.2)

has a solution.

6.2.1.2

Transformation of the System (6.1)

In what follows, we assume (AI) (AII) (AIII) (AIV)

The matrix B(t) has the full column rank r for all t ∈ [0, tc ]. The matrix-valued functions Ai (t), (i = 0, 1) are continuously differentiable in the interval t ∈ [0, tc ]. The matrix-valued function B(t) is twice continuously differentiable in the interval t ∈ [0, tc ]. The matrix-valued function G (t, η) is piecewise continuous in η ∈ [−h, 0] for any t ∈ [0, tc ], and this function is continuously differentiable with respect to t ∈ [0, tc ] uniformly in η ∈ [−h, 0].

Let, for any t ∈ [0, tc ], the matrix CB (t) be a complement matrix to the matrix B(t). This  means thatthe dimension of CB (t) is n × (n − r), and the block-form matrix CB (t) , B(t) is nonsingular for all t ∈ [0, tc ]. We assume that (AV)

The matrix-valued function CB (t) is twice continuously differentiable in the interval t ∈ [0, tc ].

For all t ∈ [0, tc ], consider the following matrices:

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6 Miscellanies

 −1 H (t) = B T (t)B(t) B T (t)CB (t),

L (t) = CB (t) − B(t)H (t),   = L (t) , B(t) . R(t) (6.3)

Note that, due to the assumptions (AIII) and (AV), the matrix-valued function R(t) is twice continuously differentiable for t ∈ [0, tc ]. Moreover, due to the results of is invertible for all t ∈ [0, tc ]. [10], the matrix R(t) Let R(t), t ∈ [−h, tc ], be an extension of the matrix R(t), t ∈ [0, tc ], which for t ∈ [0, tc ], the matrixkeeps the properties of R(t). Namely, R(t) = R(t) valued function R(t) is twice continuously differentiable for t ∈ [−h, tc ], and det R(t) = 0 for all t ∈ [−h, tc ]. It is clear that R(t), t ∈ [−h, tc ] exists although it is not unique. Using the matrix R(t), we transform the state in the system (6.1) as Z(t) = R(t)z(t),

t ∈ [−h, tc ],

(6.4)

where z(t) ∈ E n , t ∈ [−h, tc ] is a new state. Since the matrix R(t) is invertible for all t ∈ [−h, tc ], the transformation (6.4) is invertible. Lemma 6.1 Let the assumptions (AI)–(AV) be valid. Then, the transformation (6.4) converts the system (6.1) in the interval t ∈ [0, tc ] to the following new system: dz(t)  = Ai (t)z(t − hi ) + dt 1

i=0



0

1 G(t, η)z(t + η)dη + B(t)u(t), t ∈ [0, tc ], ε −h (6.5)

where

A0 (t) = R −1 (t) A0 (t)R(t) − dR(t)/dt ,

A1 (t) = R −1 (t)A1 (t)R(t − h),   O(n−r)×r . G(t, η) = R −1 (t)G (t, η)R(t + η), B(t) = R −1 (t)B(t) ≡ Ir (6.6)

The matrix-valued functions A(t) and B(t) are continuously differentiable in the interval [0, tc ]. The matrix-valued function G(t, η) is piecewise continuous in η ∈ [−h, 0] for any t ∈ [0, tc ], and this matrix-valued function is continuously differentiable with respect to t ∈ [0, tc ] uniformly in η ∈ [−h, 0]. Proof We start with Eq. (6.6). The first three expressions in this equation are derived by the substitution of (6.4) into the system (6.1) and a straightforward calculation of the coefficients for z(t), z(t − h1 ), and z(t + η) in the transformed system (6.5). Let

6.2 Linear High Gain Control Time Delay Systems

347

us prove the expression forB(t). For this purpose, it is necessary and sufficient to  O(n−r)×r for t ∈ [0, tc ], = B(t), t ∈ [0, tc ]. Since R(t) = R(t) show that R(t) Ir then by virtue of (6.3), we obtain 

   O(n−r)×r O(n−r)×r R(t) = R(t) Ir Ir     O(n−r)×r = On×r + B(t) = B(t), = L (t) , B(t) Ir which proves the expression for B(t) in (6.6). The statements of the lemma on the properties of the matrix-valued functions A(t), B(t), and G(t, η) directly follow from their expressions in (6.6), Eq. (6.3), the assumptions (AII)–(AV), and the smoothness property of the matrix-valued function R(t).   Lemma 6.2 Let the assumptions (AI)–(AV) be valid. Then, for a given ε > 0, the system (6.1) is completely Euclidean space controllable at the time instant tc if and only if the system (6.5) is completely Euclidean space controllable at this time instant. Proof (Sufficiency) Let Z0 ∈ E n , Zc ∈ E n , and ϕZ (·) ∈ L2 [−h, 0; E n ] be any given vectors and vector-valued function. Let z0 = R −1 (0)Z0 ,

zc = R −1 (tc )Zc ,

ϕz (η) = R −1 (η)ϕZ (η), η ∈ [−h, 0].

Now, let us assume that the system (6.5) is completely Euclidean space controllable at the time instant tc . Due to this assumption, the system (6.5) has a solution z(t) satisfying the following initial and terminal conditions: z(η) = ϕz (η),

η ∈ [−h, 0);

z(0) = z0 ;

z(tc ) = zc .

Due to Lemma 6.1 and the invertibility of the transformation (6.4), the function Z(t) = R(t)z(t), t ∈ [−h, tc ] satisfies the system (6.1) in the interval [0, tc ], and the initial and terminal conditions (6.2). The latter means the complete Euclidean space controllability of the system (6.1) at the time instant tc which completes the proof of the sufficiency. Necessity. The necessity is proven similarly to the sufficiency.   Due to Lemma 6.2, in what follows of this section, we deal with the system (6.5) as an original system, for which some ε-free sufficient condition of the complete Euclidean space controllability at the time instant tc will be derived.

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6 Miscellanies

6.2.2 High Dimension Controllability Condition for the System (6.5) Let the n × n-matrix-valued function Λ(σ ) be a solution of the terminal-value problem  T dΛ(σ ) =− Ai (σ + hi ) Λ(σ + hi ) dσ 1

 −

i=0

0 −h

 T G(σ − η, η) Λ(σ − η)dη, Λ(tc ) = In ,

σ ∈ [0, tc ),

Λ(σ ) = 0,

σ > tc .

(6.7)

In this problem, it is assumed that A1 (t) = A1 (tc ), G(t, η) = G(tc , η), t > tc , η ∈ [−h, 0].

(6.8)

Due to the results of [13] (Section 4.3), the solution Λ(σ ) of the problem (6.7) exists and is unique. Let us partition the matrix Λ(σ ) into blocks as  Λ(σ ) =

Λ1 (σ ) Λ3 (σ )

 Λ2 (σ ) , Λ4 (σ )

where the matrices Λ1 (σ ), Λ2 (σ ), Λ3 (σ ), and Λ4 (σ ) are of the dimensions (n − r) × (n − r), (n − r) × r, r × (n − r), and r × r, respectively. Lemma 6.3 For any given ε > 0, the system (6.5) is completely Euclidean space controllable at the time instant tc if and only if the n × n-matrix  W (tc ) = 0

tc



 ΛT3 (σ )Λ3 (σ ) ΛT3 (σ )Λ4 (σ ) dσ ΛT4 (σ )Λ3 (σ ) ΛT4 (σ )Λ4 (σ )

(6.9)

is invertible. Proof By virtue of the results of Sect. 3.3.1 (see Proposition 3.1), for a given ε > 0 the system (6.5) is completely Euclidean space controllable at the time instant tc if and only if the matrix (tc , ε) = 1 W ε2



tc

ΛT (σ )B(σ )B T (σ )Λ(σ )dσ

(6.10)

0

is invertible. Substituting the above introduced block representation for the matrix Λ(σ ) and the block representation for the matrix B(σ ) (see Eq. (6.6)) into the integrand of (6.10), we obtain

6.2 Linear High Gain Control Time Delay Systems



ΛT1 (σ ) ΛT2 (σ )

349

ΛT (σ )B(σ )B T (σ )Λ(σ ) =   "  Λ (σ ) Λ (σ )  ΛT3 (σ ) O(n−r)×r ! 1 2 O = , I r r×(n−r) ΛT4 (σ ) Ir Λ3 (σ ) Λ4 (σ )  T  Λ3 (σ )Λ3 (σ ) ΛT3 (σ )Λ4 (σ ) , σ ∈ [0, tc ]. ΛT4 (σ )Λ3 (σ ) ΛT4 (σ )Λ4 (σ )

This equation, along with Eqs. (6.9) and (6.10), yields (tc , ε) = 1 W (tc ). W ε2  

The latter equation directly implies the statement of the lemma.

Remark 6.2 The condition of the complete Euclidean space controllability for the system (6.5), obtained in Lemma 6.3, is independent of ε. However, this condition requires to deal with the high-dimension (n × n-dimension) controllability matrix W (tc ), including obtaining the solution of the terminal-value problem for the time   Λ3 (σ ) . All this means delay differential equation with respect to the matrix Λ4 (σ ) that the use of the controllability condition, obtained in Lemma 6.3, is a rather complicated task. In what follows of this section, we derive another controllability condition for the system (6.5). This condition deals with a lower dimension ((n − r) × (n − r)-dimension) controllability matrix, which requires to solve a terminal-value problem for a time delay differential equation with respect to an (n − r) × (n − r)-dimensional unknown matrix-valued function. Moreover, this lower dimension controllability condition is independent of ε > 0, while provides the complete Euclidean space controllability of the system (6.5) for all sufficiently small values of this parameter. The derivation of the lower dimension controllability condition is based on an asymptotic decomposition of (6.5).

6.2.3 Asymptotic Decomposition of the System (6.5) Let us partition the vector z(t) and the matrices Ai (t), (i = 0, 1) and G(t, η) into blocks as   (6.11) z(t) = col x(t) , y(t) , t ∈ [−h, tc ], x(t) ∈ E n−r , y(t) ∈ E r ,  Ai (t) =

 A1i (t) A2i (t) , i = 0, 1, A3i (t) A4i (t)

 G(t, η) =

 G1 (t, η) G2 (t, η) , G3 (t, η) G4 (t, η)

t ∈ [0, tc ],

η ∈ [−h, 0], (6.12)

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6 Miscellanies

where the blocks A1i (t), (i = 0, 1), and G1 (t, η) are of the dimension (n − r) × (n − r), the blocks A2i (t), (i = 0, 1), and G2 (t, η) are of the dimension (n − r) × r, the blocks A3i (t), (i = 0, 1), and G3 (t, η) are of the dimension r × (n − r), and the blocks A4i (t), (i = 0, 1), and G4 (t, η) are of the dimension r × r. Using these block representations, as well as the block form of the matrix B(t) (see Eq. (6.6)), we can rewrite the system (6.5) in the following equivalent form:

dx(t)  = A1i (t)x(t − hi ) + A2i (t)y(t − hi ) dt 1

i=0

 +

0 −h



G1 (t, η)x(t + η) + G2 (t, η)y(t + η) dη, t ∈ [0, tc ],

(6.13)

 1

dy(t) =ε ε A3i (t)x(t − hi ) + A4i (t)y(t − hi ) dt  +

i=0

0 −h



G3 (t, η)x(t + η) + G4 (t, η)y(t + η) dη + u(t), t ∈ [0, tc ].

(6.14)

Thus, subject to the assumption on the smallness of the parameter ε > 0, the system (6.13)–(6.14) has the small positive multiplier for the part of its derivatives, i.e., this system is singularly perturbed. We are going to decompose asymptotically this system (and, therefore, the system (6.5)) into two much simpler ε-free subsystems, the slow and fast ones. Let us start with the slow subsystem. This subsystem is obtained by setting formally ε = 0 in the system (6.13)–(6.14), which yields

dxs (t)  = A1i (t)xs (t − hi ) + A2i (t)ys (t − hi ) dt 1

 +

i=0

0 −h



G1 (t, η)xs (t + η) + G2 (t, η)ys (t + η) dη, us (t) = 0,

t ∈ [0, tc ].

t ∈ [0, tc ],

(6.15) (6.16)

Thus, in the system (6.15)–(6.16), the control us (t) is theidentically zero vector valued function, while the state variable ys (t), ys (t + η) , η ∈ [−h, 0) does not satisfy any equation. Therefore, we can choose this variable to satisfy a desirable  property of the system (6.15). The latter means that the variable ys (t), ys (t + η) , (6.15) where η ∈ [−h, 0) canbe considered as a control in (6.15). The system   xs (t), xs (t + η) , η ∈ [−h, 0) is a state variable, while ys (t), ys (t + η) , η ∈ [−h, 0) is a control, also is called the slow subsystem associated with the system (6.13)–(6.14) (and, therefore, with the original system (6.5)).

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351

Remark 6.3 It should be noted that, although the original system (6.5) has only state delays, its slow subsystem (6.15) has delays in both state and control variables. Definition 6.2 The slow subsystem (6.15) is said to be completely Euclidean space controllable at the time instant tc if for any x0 ∈ E n−r , ϕx (·) ∈ L2 [−h, 0; E n−r ], ϕy (·) ∈ L2 [−h, 0; E r ], and xc ∈ E n−r , there exists a control ys (t) ∈ L2 [0, tc ; E r ], for which the system (6.15) with the initial and terminal conditions xs (η) = ϕx (η), η ∈ [−h, 0);

xs (0) = x0 ,

ys (η) = ϕy (η), η ∈ [−h, 0), xs (tc ) = xc , has a solution. Let the (n − r) × (n − r)-matrix-valued function Λs (σ ) be the unique solution of the following terminal-value problem:  T dΛs (σ ) =− A1i (σ + hi ) Λs (σ + hi ) dσ 1

 −

i=0

0 −h

 T G1 (t − η, η) Λs (σ − η)dη, Λs (tc ) = In−r ;

σ ∈ [0, tc ),

Λs (σ ) = 0, σ > tc .

(6.17)

In this problem, we assume A11 (t) = A11 (tc ),

G1 (t, η) = G1 (tc , η), t > tc , η ∈ [−h, 0].

Consider the following (n − r) × r-matrix-valued function: Ds (σ ) =  +

1   T Λs (σ + hi ) A2i (σ + hi ) i=0

0 −h

 T Λs (σ − η) G2 (σ − η, η)dη,

σ ∈ [0, tc ].

In this equation, we assume A21 (t) = A21 (tc ),

G2 (t, η) = G2 (tc , η), t > tc , η ∈ [−h, 0].

(6.18)

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6 Miscellanies

Using the matrix-valued function Ds (σ ), we obtain (as a direct consequence of the results of [4]) the following assertion. Proposition 6.1 The slow subsystem (6.15) is completely Euclidean space controllable at the time instant tc if and only if the matrix  Ms (tc ) =

tc

0

Ds (σ )DsT (σ )dσ

(6.19)

is invertible. Proceed to the fast subsystem. This subsystem is derived from Eq. (6.14) in the following formal two-stage way. At the first stage, the expression, multiplied by ε, is removed from the right-hand side of (6.14) which yields the system ε

dy(t) = u(t), dt

t ∈ [0, tc ].

At the second stage, in this system the transformation of the variables t = εξ ,



y(εξ ) = yf (ξ ), u(εξ ) = uf (ξ ) is made, where ξ is a new independent variable. Thus, we obtain the fast subsystem associated with the singularly perturbed system (6.13)–(6.14) (and, therefore, with the original system (6.5)): dyf (ξ ) = uf (ξ ), dξ

ξ ≥ 0,

(6.20)

where yf (ξ ) ∈ E r is a state variable; uf (ξ ) ∈ E r is a control. Definition 6.3 The fast subsystem (6.20) is said to be completely controllable if for any y0 ∈ E r and yc ∈ E r , there exists a number ξc > 0 (independent of y0 , yc ) and a control uf (ξ ) ∈ L2 [0, ξc ; E r ] such that the system (6.20) with the initial and terminal conditions yf (0) = y0 ,

yf (tc ) = yc ,

has a solution. Let us observe that the dimension of the state variable in (6.20) equals to the dimension of the control variable, and the matrix of the coefficients for the latter is invertible. Using this observation and the results of [14], we have immediately the following assertion. Proposition 6.2 The fast subsystem (6.20) is completely controllable.

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353

6.2.4 Auxiliary Results 6.2.4.1

Linear Control Transformation in the System (6.13)–(6.14) and Some of its Properties

Let us transform the control in the system (6.13)–(6.14) as u(t) = −y(t) + v(t), where v(t) is a new control. Due to this transformation, the system (6.13)–(6.14) becomes the one consisting of Eq. (6.13) and the equation    dy(t) =ε A3i (t)x(t − hi ) + εA40 (t) − Ir y(t) + εA41 (t)y(t − h) dt 1

ε

 +ε

i=0

0

−h

G3 (t, η)x(t + η) + G4 (t, η)y(t + η) dη + v(t), t ∈ [0, tc ]. (6.21)

Lemma 6.4 For a given ε > 0, the system (6.13)–(6.14) is completely Euclidean space controllable at the time instant tc if and only if the system (6.13), (6.21) is completely Euclidean space controllable at this time instant. Proof (Necessity) Suppose that for some ε > 0, the system (6.13)–(6.14) is Euclidean space controllable at tc . Let x0 ∈ E n−r , y0 ∈ E r , ϕx (·) ∈ L2 [−h, 0; E n−r ], ϕy (·) ∈ L2 [−h, 0; E r ], xc ∈ E n−r , and yc ∈ E r be arbitrary given. Then, there exists a control function u(·) ∈ L2 [0, tc ; E r ] for which the system (6.13)–(6.14) subject to the initial and terminal conditions x(η) = ϕx (η),

y(η) = ϕy (η),

η ∈ [−h, 0),

x(0) = x0 ,

y(0) = y0 ,

x(tc ) = xc ,

y(tc ) = yc ,

  has a solution x(t), y(t) , t ∈ [0, tc ]. For the abovementioned control u(t), t ∈ [0, tc ], Eq. (6.14) can be rewritten in the equivalent form    dy(t) =ε A3i (t)x(t − hi ) + εA40 (t) − Ir y(t) + εA41 (t)y(t − h) dt 1

ε

 +ε

i=0 0

−h

  G3 (t, η)x(t + η) + G4 (t, η)y(t + η) dη + u(t) + y(t) . (6.22)

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6 Miscellanies

Let us denote

v(t) = u(t) + y(t). Since u(·) ∈ L2 [0, tc ; E r ] and y(t) is an absolutely continuous function in the interval [0, tc ], then v(·) ∈ L2 [0, tc ; E r ]. This observation and the equivalence of Eqs. (6.14) and (6.22), along with the above assumed complete Euclidean space controllability of the system (6.13)–(6.14), imply the complete Euclidean space controllability of the system (6.13), (6.21) at the time instant tc . Sufficiency The sufficiency is proven similarly to the necessity.

 

Consider the following block matrix: 0 (t, ε) = A



A10 (t) A20 (t) A30 (t) A40 (t) − 1ε Ir

 .

Using this matrix, as well as Eqs. (6.11)–(6.12) and the block form of the matrix B(t) (see Eq. (6.6)), we can rewrite the system (6.13), (6.21) in the equivalent form as

 +

dz(t) 0 (t, ε)z(t) + A1 (t)z(t − h) =A dt 0

1 G(t, η)z(t + η)dη + B(t)v(t), ε −h

t ∈ [0, tc ].

Let, for a given ε > 0, the n × n-matrix-valued function Υ (σ, ε) be the unique solution of the following terminal-value problem:     dΥ (σ, ε) 0 (σ, ε) T Υ (σ, ε) − A1 (σ + h) T Υ (σ + h, ε) =− A dσ  0  T − G(σ − η, η) Υ (σ − η, ε)dη, σ ∈ [0, tc ), −h

Υ (tc , ε) = In ;

Υ (σ, ε) = 0,

σ > tc .

(6.23)

In this problem, it is assumed that the matrix-valued functions A1 (t) and G(t, η) satisfy the equalities in (6.8). Let us partition the matrix Υ (σ, ε) into block as  Υ (σ, ε) =

 Υ1 (σ, ε) Υ2 (σ, ε) , Υ3 (σ, ε) Υ4 (σ, ε)

(6.24)

where the blocks Υ1 (σ, ε), Υ2 (σ, ε), Υ3 (σ, ε), and Υ4 (σ, ε) are of the dimensions (n − r) × (n − r), (n − r) × r, r × (n − r), and r × r, respectively.

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355

Similarly to Lemma 6.3, we have the following lemma. Lemma 6.5 For a given ε > 0, the system (6.13), (6.21) is completely Euclidean space controllable at the time instant tc if and only if the n × n-matrix 

tc

M(tc , ε) = 0



 Υ3T (σ, ε)Υ3 (σ, ε) Υ3T (σ, ε)Υ4 (σ, ε) dσ Υ4T (σ, ε)Υ3 (σ, ε) Υ4T (σ, ε)Υ4 (σ, ε)

(6.25)

is invertible.

6.2.4.2

Asymptotic Decomposition of the Transformed System (6.13), (6.21)

Similarly to the asymptotic decomposition of the original system (6.13)–(6.14), we can decompose the transformed system (6.13), (6.21). Thus, the slow subsystem, associated with the latter, consists of Eq. (6.15) and the equation 0 = −ys (t) + vs (t),

t ∈ [0, tc ],

(6.26)

  where ys (t) ∈ E r , vs (t) ∈ E r ; ys (t), ys (t + η) , η ∈ [−h, 0) is a state variable, while vs (t) is a control. (6.26) is differential-algebraic. Eliminating the state variable  The system (6.15),  ys (t), ys (t + η) , η ∈ [−h, 0) from this system, we obtain the purely differential form of the slow subsystem, associated with (6.13), (6.21):

dxs (t)  = A1i (t)xs (t − hi ) + A2i (t)vs (t − hi ) dt 1

 +

i=0

0 −h



G1 (t, η)xs (t + η) + G2 (t, η)vs (t + η) dη, t ≥ 0,

(6.27)

    where xs (t), xs (t + η) , η ∈ [−h, 0) is a state variable, while vs (t), vs (t + η) , η ∈ [−h, 0) is a control variable. Remark 6.4 The slow subsystem (6.27), associated with the system (6.13), (6.21), coincides with the slow subsystem (6.15), associated with the system (6.13)–(6.14).   Remember that in (6.15), x (t), x (t + η) , η ∈ [−h, 0) is a state variable, while s s   ys (t), ys (t + η) , η ∈ [−h, 0) is a control variable. Therefore, Definition 6.2 and Proposition 6.1 also are valid for the system (6.27). The fast subsystem, associated with the system (6.13), (6.21), is the differential equation dyf (ξ ) = −yf (ξ ) + vf (ξ ), dξ

ξ ≥ 0,

(6.28)

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6 Miscellanies

where yf (ξ ) ∈ E r is a state variable; vf (ξ ) ∈ E r is a control. The complete controllability of the fast subsystem (6.28), associated with the system (6.13), (6.21), is defined similarly to such a controllability of the fast subsystem (6.20), associated with the system (6.13)–(6.14) (see Definition 6.3). Moreover, for the fast subsystem (6.28) the assertion, similar to Proposition 6.2, is valid.

6.2.4.3

Block-Wise Estimate of the Solution to the Terminal-Value Problem (6.23)

Let us denote

Υ1s (σ ) = Λs (σ ),

(6.29)

⎧ -  T ⎪ − 1i=0 A2i (σ + hi ) Λs (σ + hi ) ⎪ ⎪ ⎪ ⎪ ⎨ .  T 0 Υ3s (σ ) = − −h G2 (σ − η, η) Λs (σ − η)dη, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0,

σ ∈ [0, tc ],

(6.30)

σ > tc ,

where Λs (σ ) is the solution of the terminal-value problem (6.17). Let the r × r-matrix-valued function Υ4f (ξ ) be the solution of the problem dΥ4f (ξ ) = −Υ4f (ξ ), dξ Υ4f (0) = Ir ;

Υ4f (ξ ) = 0,

ξ > 0, ξ < 0.

Solving this problem, we obtain Υ4f (ξ ) = exp(−ξ )Im ,

ξ ≥ 0,

Υ4f (ξ ) = exp(−ξ ),

ξ ≥ 0.

(6.31)

As a direct consequence of the results of Sect. 2.4.4 (see Theorems 2.4 and 2.5), we have the following proposition. Proposition 6.3 Let the assumptions AI-AV be valid. Then, there exists a positive number ε0 such that, for all ε ∈ (0, ε0 ], the following inequalities are satisfied: Υ1 (σ, ε) − Υ1s (σ ) ≤ aε,

Υ3 (σ, ε)−εΥ3s (σ ) ≤ aε

Υ2 (σ, ε) ≤ aε,

 

⎧ ⎨ ε+ exp − β(tc −σ )/ε ,

σ ∈ [0, tc ], if tc −h < σ ≤ tc ,

 

⎩ ε+ exp − β(tc −h−σ )/ε , if 0 ≤ σ ≤ tc −h,

6.2 Linear High Gain Control Time Delay Systems

 Υ4 (σ, ε) − Υ4f

tc − σ ε

357

 ≤ aε,

σ ∈ [0, tc ],

where Υk (σ, ε), (k = 1, . . . , 4) are the corresponding blocks of the solution to the terminal-value problem (6.23) (see Eq. (6.24)); a > 0 and 0 < β < 0.5 are some constants independent of ε.

6.2.5 Lower Dimension Parameter-Free Controllability Condition for the System (6.5) Theorem 6.1 Let the assumptions AI-AV be valid. Let the slow subsystem (6.15) be completely Euclidean space controllable at the time instant tc . Then, there exists a positive number ε∗ , (ε∗ ≤ ε0 ) such that, for all ε ∈ (0, ε∗ ], the system (6.13)–(6.14) (and, therefore, the system (6.5)) is completely Euclidean space controllable at the time instant tc . Proof First, let us show that, for all sufficiently small ε > 0, the system (6.13), (6.21) is completely Euclidean space controllable at the time instant tc . Denote  tc Υ3T (σ, ε)Υ3 (σ, ε)dσ, (6.32) M1 (tc , ε) = 0





tc

M2 (tc , ε) = 0



M3 (tc , ε) = 0

tc

Υ3T (σ, ε)Υ4 (σ, ε)dσ,

(6.33)

Υ4T (σ, ε)Υ4 (σ, ε)dσ.

(6.34)

Using these notations, the matrix M(tc , ε), given by (6.25), can be rewritten as  M(tc , ε) =

 M1 (tc , ε) M2 (tc , ε) . M2T (tc , ε) M3 (tc , ε)

Applying Proposition 6.3 and Eq. (6.31) to Eqs. (6.32)–(6.34), we obtain the existence of a positive number ε1 ≤ ε0 such that, for all ε ∈ (0, ε1 ], the following inequalities are satisfied: M1 (tc , ε) − ε2 M¯ 1 (tc ) ≤ aε3 ,

(6.35)

M2 (tc , ε) − ε2 M¯ 2 (tc ) ≤ aε2 ,

(6.36)

M3 (tc , ε) − εM¯ 3 ≤ aε2 ,

(6.37)

358

6 Miscellanies

where M¯ 1 (tc ) =



tc 0

Υ3sT (σ )Υ3s (σ )dσ,

M¯ 2 (tc ) = Υ3sT (tc ),

1 M¯ 3 = Ir , 2

(6.38)

a > 0 is some constant independent of ε. Using the assumption of the theorem on the complete Euclidean space controllability of the slow subsystem (6.15) at the time instant tc , as well as Eq. (6.30) and Proposition 6.1, we obtain that the matrix M¯ 1 (tc ) is invertible. Denote   (1/ε)In−r 0 √ . L(ε) = 0 (1/ ε)Ir It is clear that for all ε > 0 this matrix is invertible. Using the matrix L(ε), the inequalities (6.35)–(6.37), and Eq. (6.38), we directly obtain the existence of a positive number ε2 ≤ ε1 such that, for all ε ∈ (0, ε2 ], the following inequality is valid:       det L(ε)M(tc , ε)L(ε) − (1/2r ) det M¯ 1 (tc )  ≤ aε,

(6.39)

where a > 0 is some constant independent of ε. The inequality (6.39), along with the abovementioned invertibility of the matrices M¯ 1 (tc ) and L(ε), (ε > 0), implies the existence of a positive number ε∗ ≤ ε2 such that, for all ε ∈ (0, ε∗ ], the following inequality is valid:   det M(tc , ε) = 0. Thus, by virtue of Lemma 6.5, the system (6.13), (6.21) is completely Euclidean space controllable at the time instant tc for all ε ∈ (0, ε∗ ]. The latter, along with Lemma 6.4, directly yields the complete Euclidean space controllability of the system (6.13)–(6.14) (and, therefore, the equivalent system (6.5)) at the time instant tc for all ε ∈ (0, ε∗ ]. This completes the proof of the theorem.  

6.2.6 Example Consider a particular case of the system (6.1) where n = 3, r = 1, h = 1, and ⎛

1 0 A0 (t) = ⎝ t (t + 2) t − 4 + (t + 2)−1 2 0

⎞ −1 − (t + 2) ⎠ , 4

6.2 Linear High Gain Control Time Delay Systems

359

⎛

⎞ exp(2t − 1) (t + 1)−1 − exp(−2t + 1) A1 (t) = ⎝ −(t + 2) cos t (t + 2) sin t ⎠ , − (t + 2)(t + 1)−1 − exp(−3t + 1.5) 0 − exp(3t − 1.5) ⎛ ⎞ 0 0 0 G (t, η) = ⎝ (t + 2)(t + η) tη(t + 2)(t + η + 2)−1 t 2 (t + 2) exp(η) ⎠ , 0 0 0 ⎛ ⎞ 0 B(t) = ⎝ t + 2 ⎠ . 0 (6.40) Let us choose tc = 1.5. Also, let us choose the complement matrix CB (t) to the matrix B(t) as ⎛

⎞ 1 0 CB (t) = ⎝ 0 0 ⎠ , 0 1

t ∈ [0, 1.5].

Thus, in this example, all the assumptions (AI)–(AV) are satisfied. Using the matrices CB (t) and B(t), as well as Eq. (6.3), we obtain by a routine algebra: ⎛

H (t) = (0 , 0),

L (t) = CB (t),

⎞ 1 0 0 = ⎝ 0 0 t + 2 ⎠ , t ∈ [0, 1.5]. R(t) 0 1 0

We choose the extension of the matrix-valued function R(t), t ∈ [0, 1.5] as ⎛

⎞ 1 0 0 R(t) = ⎝ 0 0 t + 2 ⎠ , t ∈ [−1, 1.5]. 0 1 0 By the state transformation (6.4) in the system (6.1) with the data (6.40) we obtain the system (6.5) with the following data: ⎛

⎞ 1 −1 0 ⎠, A0 (t) = ⎝ 2 4 0 t −1 t −4 ⎛ ⎞ exp(2t − 1) − exp(−2t + 1) 1 A1 (t) = ⎝ − exp(−3t + 1.5) − exp(3t − 1.5) 0⎠, − cos t sin t −1

t ∈ [0, 1.5],

t ∈ [0, 1.5],

360

6 Miscellanies

⎛

⎞ 0 0 0 G(t, η) = ⎝ 0 0 0 ⎠, 2 t + η t exp(η) tη

⎛ ⎞ 0 ⎝ B(t) = 0 ⎠ , 1

t ∈ [0, 1.5].

(6.41)

In what follows of this example, we study the complete Euclidean space controllability of the system (6.5), (6.41) at the time instant tc = 1.5. Representing the unknown vector-valued function z(t), t ∈ [−1, 1.5]   of the system (6.5), (6.41) in the block form z(t) = col x1 (t), x2 (t), y(t) , where x1 (t), x2 (t), and y(t) are scalar functions, we obtain the particular case of the system (6.13)–(6.14):  dx1 (t) = x1 (t) − x2 (t) + exp 2t − 1)x1 (t − 1) dt − exp(−2t + 1)x2 (t − 1) + y(t − 1),

t ∈ [0, 1.5],

(6.42)

 dx2 (t) = 2x1 (t) + 4x2 (t) − exp − 3t + 1.5)x1 (t − 1) dt − exp(3t − 1.5)x2 (t − 1),

ε

 +ε

0 −1

t ∈ [0, 1.5],

(6.43)

! dy(t) = ε tx1 (t) − x2 (t) − (cos t)x1 (t − 1) dt " +(sin t)x2 (t − 1) + (t − 4)y(t) − y(t − 1)



(t + η)x1 (t + η) + t 2 exp(η)x2 (t + η) + tηy(t + η) dη +u(t),

t ∈ [0, 1.5].

(6.44)

Since the assumptions (AI)–(AV) are satisfied in this example then, by virtue of Theorem 6.1 and Proposition 6.1, the invertibility of the matrix Ms (tc ) (see Eq. (6.19)) yields the complete Euclidean space controllability of the system (6.5), (6.41) for all sufficiently small ε > 0. To calculate Ms (tc ), we need to calculate the matrix-valued function Ds (σ ) (see Eq. (6.18)). In the present example, this matrix-valued function becomes a two-dimensional vector-valued function. It is seen from Eq. (6.18), that Ds (σ ) depends on the solution Λs (σ ) of the terminalvalue problem (6.17). From Eqs. (6.42)–(6.43), we directly have the matrix-valued coefficients appearing in this problem:  A10 (t) = A10 =

 1 −1 , 2 4

t ∈ [0, 1.5],

6.2 Linear High Gain Control Time Delay Systems

 A11 (t) =

361

 exp(2t − 1) − exp(−2t + 1) , t ∈ [0, 1.5], − exp(−3t + 1.5) − exp(3t − 1.5)   exp(2) − exp(−2) , t > 1.5, A11 (t) = − exp(−3) − exp(3)   0 0 , t ≥ 0, η ∈ [−1, 0]. G1 (t, η) = 0 0

(6.45)

First, we consider the problem (6.17) in the interval (tc − h, tc ] = (0.5, 1.5]: dΛ1s (σ ) = −AT10 Λ1s (σ ), dσ

σ ∈ (0.5, 1.5];

Λ1s (1.5) = I2 ,

(6.46)

where the matrix A10 is given in (6.45); the 2 × 2-matrix-valued function Λ1s (σ ) has the block form   1 (σ ) Λ1 (σ ) Λ s,1 s,2 Λ1s (σ ) = . (6.47) Λ1s,3 (σ ) Λ1s,4 (σ ) Solving the problem (6.46), we obtain Λ1s,1 (σ ) = − exp(−3σ + 4.5) + 2 exp(−2σ + 3),

σ ∈ (0.5, 1.5],

Λ1s,2 (σ ) = 2 exp(−3σ + 4.5) − 2 exp(−2σ + 3),

σ ∈ (0.5, 1.5],

Λ1s,3 (σ ) = − exp(−3σ + 4.5) + exp(−2σ + 3),

σ ∈ (0.5, 1.5],

Λ1s,4 (σ ) = 2 exp(−3σ + 4.5) − exp(−2σ + 3),

σ ∈ (0.5, 1.5].

(6.48)

Now, let us consider the problem (6.17) in the interval [0, tc − h] = [0, 0.5]: dΛ2s (σ ) = −AT10 Λ2s (σ ) − AT11 (σ + 1)Λ1s (σ + 1), dσ

σ ∈ (0, 0.5];

Λ2s (0.5) = Λ1s (0.5 + 0),

(6.49)

where the 2 × 2-matrix-valued function Λ2s (σ ) has the block form  Λ2s (σ )

=

Λ2s,1 (σ ) Λ2s,2 (σ ) Λ2s,3 (σ ) Λ2s,4 (σ )

 .

Solving the problem (6.49), and taking into account the form of the matrices A10 , A11 (t) (see Eq. (6.45)) and the form of the matrix Λ1s (σ ) (see Eqs. (6.47), (6.48)), we obtain

362

6 Miscellanies

 Λ2s,1 (σ )

=

   11 − exp(3) exp(−3σ + 1.5) + 2.5 + 2 exp(2) exp(−2σ + 1) 6

1 1 exp(−6σ + 3) + exp(−5σ + 2.5) − 2 exp(−4σ + 2) 6 6 1 −1.5 exp(−σ + 0.5) − exp(σ − 0.5), σ ∈ [0, 0.5], 6     2 Λs,2 (σ ) = 2 exp(3) − 4 exp(−3σ + 1.5) − 3 − 2 exp(2) exp(−2σ + 1) −1 +

1 2 1 − − exp(−6σ + 3) − exp(−5σ + 2.5) + 2 exp(−4σ + 2) 3 3 6 1 + exp(σ − 0.5), σ ∈ [0, 0.5], 6     11 Λ2s,3 (σ ) = − exp(3) exp(−3σ + 1.5) + 1.25 + exp(2) exp(−2σ + 1) 6 1 5 exp(−6σ + 3) + exp(−5σ + 2.5) − 3 exp(−4σ + 2) 12 6 1 −0.5 exp(−σ + 0.5) + exp(σ − 0.5), σ ∈ [0, 0.5], 6     2 Λs,4 (σ ) = 2 exp(3) − 4 exp(−3σ + 1.5) + 1.5 − exp(2) exp(−2σ + 1) −0.5 −

11 1 1 exp(−5σ + 2.5) + 4 exp(−4σ + 2) + + exp(−6σ + 3) − 3 6 6 1 − exp(σ − 0.5), σ ∈ [0, 0.5]. 6 (6.50) Also, from Eqs. (6.42)–(6.43), we obtain the matrix-valued coefficients for the matrix-valued function Λs (σ ) appearing in the expression (6.18) for the matrix Ds (σ ):     0 1 A20 (t) = A20 = , t ∈ [0, 1.5], A21 (t) = A21 = , t ≥ 0, 0 0   0 , t ≥ 0, η ∈ [−1, 0]. G2 (t, η) = G2 = 0 (6.51) Now, using Eqs. (6.18), (6.51), as well as (6.48) and (6.50), we obtain the expression for the vector-valued function Ds (σ ):   0 Ds (σ ) = , σ ∈ (0.5, 1.5], 0   − exp(−3σ + 1.5) + 2 exp(−2σ + 1) , σ ∈ [0, 0.5]. Ds (σ ) = 2 exp(−3σ + 1.5) − 2 exp(−2σ + 1)

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

363

The latter, along with Eq. (6.19), directly yields  Ms (1.5) =

 0.6240 0.6681 , 0.6681 1.2208

and det Ms (1.5) = 0.3154 = 0. Thus, the matrix Ms (1.5) is invertible, meaning the complete Euclidean space controllability of the system (6.5), (6.41) for all sufficiently small values of ε > 0, i.e., robustly with respect to such values of this parameter.

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input Delay 6.3.1 Original System Consider the following controlled system: dx(t) = A11 (t, ε)x(t) + A12 (t, ε)y(t) dt   +B01 (t, ε)u(t) + B11 (t, ε)u t − h(ε) , t ≥ 0,

(6.52)

dy(t) = A21 (t, ε)x(t) + A22 (t, ε)y(t) dt   +B02 (t, ε)u(t) + B12 (t, ε)u t − h(ε) , t ≥ 0,

(6.53)

ε

where x(t) ∈ E n , y(t) ∈ E m are state variables; u(t) ∈ E r is an input (a control); ε > 0 is a small parameter; h(ε) ≥ 0 is a time delay given for ε ∈ [0, ε0 ], (ε0 > 0); u(t), u(t + η) , η ∈ [−h(ε), 0) is a control variable; Aij (t, ε) and Bkl (t, ε), (i = 1, 2; j = 1, 2; k = 0, 1; l = 1, 2), are matrix-valued functions given for t ≥ 0 and ε ∈ [0, ε0 ]; Aij (t, ε) and Bkl (t, ε), (i = 1, 2; j = 1, 2; k = 0, 1; l = 1, 2) are continuous in (t, ε) ∈ [0, +∞) × [0, ε0 ]; h(ε) is continuous in ε ∈ [0, ε0 ]. Due to the presence of the small multiplier ε > 0 for the derivative dy(t)/dt, the system (6.52)–(6.53) is singularly perturbed. Equation (6.52) and the state variable x(t) are the slow mode and the slow state variable, while Eq. (6.53) and the state variable y(t) are the fast mode and the fast state variable. An additional feature of (6.52)–(6.53) is the delay in the control (the input). Let tc > 0 be a given time instant independent of ε. Definition 6.4 For a given ε ∈ (0, ε0 ], the system (6.52)–(6.53) is said to be completely Euclidean space controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , ϕu (·) ∈ L2 [−h(ε), 0; E r ], xc ∈ E n and yc ∈ E m there exists a control

364

6 Miscellanies

function u(·) ∈ L2 [0, tc ; E r ], for which the system (6.52)–(6.53) has a solution  col x(t), y(t) , t ∈ [0, tc ] satisfying the initial and terminal conditions: u(η) = ϕu (η),

 η ∈ − h(ε), 0 ,

x(0) = x0 ,

y(0) = y0 ,

x(tc ) = xc ,

y(tc ) = yc .

6.3.2 Discussion on the Slow–Fast Decomposition of the Original System Let us try to decompose asymptotically the system (6.52)–(6.53) into the ε-free slow and fast subsystems. Similarly to the results of Sect. 4.2.2, the slow subsystem is obtained from (6.52)–(6.53) by setting there formally ε = 0 which yields dxs (t) = A11 (t, 0)xs (t) + A12 (t, 0)ys (t) dt   +B01 (t, 0)us (t) + B11 (t, 0)us t − h(0) , t ≥ 0, 0 = A21 (t, 0)xs (t) + A22 (t, 0)ys (t)   +B02 (t, 0)us (t) + B12 (t, 0)us t − h(0) , t ≥ 0, n m r where xs (t) ∈ E and ys (t) ∈ E are state variables; us (t) ∈ E is a control; us (t), us (t + τ ) , τ ∈ [−h(0), 0) is a control variable. In what follows, we assume that

det A22 (t, 0) = 0,

t ≥ 0.

(6.54)

Due to this assumption, we can eliminate the state variable ys (t) from the above obtained slow subsystem and reduce it to the differential equation   dxs (t) = As (t)xs (t) + B0s (t)us (t) + B1s (t)us t − h(0) , dt

t ≥ 0,

where As (t) = A11 (t, 0) − A12 (t, 0)A−1 22 (t, 0)A21 (t, 0), B0s (t) = B01 (t, 0) − A12 (t, 0)A−1 22 (t, 0)B02 (t, 0), B1s (t) = B11 (t, 0) − A12 (t, 0)A−1 22 (t, 0)B12 (t, 0).

(6.55)

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

365

The differential equation (6.55) also is called the slow subsystem associated with the original system (6.52)–(6.53). This equation has a delay in the control, it is of a lower Euclidean dimension than (6.52)–(6.53) and it is ε-free. The fast subsystem, associated with the original system, is obtained from the fast mode equation (6.53) of the latter in the following way. First, we remove the slow state variable x(t) from (6.53). Then, in the resulting equation, we make the



transformation of the variables ξ = (t − t1 )/ε, yf (ξ ) = y(εξ + t1 ), uf (ξ ) = u(εξ + t1 ), where t1 ≥ 0 is any specified number. Thus, we obtain the equation dyf (ξ ) = A22 (εξ + t1 , ε)yf (ξ ) dξ   +B02 (εξ + t1 , ε)uf (ξ ) + B12 (εξ + t1 , ε)uf ξ − h(ε)/ε . Finally, setting formally ε = 0 and replacing t1 with t, we are supposed to obtain the fast subsystem. It will be so, if there exists a finite limit lim h(ε)/ε = h¯ ≥ 0.

(6.56)

ε→+0

In such a case, the fast subsystem has the form dyf (ξ ) = A22 (t, 0)yf (ξ ) dξ ¯ +B02 (t, 0)uf (ξ ) + B12 (t, 0)uf (ξ − h),

ξ ≥ 0.

In this equation, t ≥ 0 is a parameter, while ξ (called the stretched time) is an independent variable. If h¯ > 0, this equation is with a control delay. Moreover, it is of a lower Euclidean dimension than (6.52)–(6.53) and it is ε-free. In Chap. 4, the case h(ε) = εh¯ was studied, where h¯ > 0 is a given number independent of ε. In this case it was shown, subject to some smoothness assumptions on the coefficients of the and the assumption on the asymptotic stability  original system  of the unforced uf (·) ≡ 0 fast subsystem for all t ∈ [0, tc ], that the complete Euclidean space controllability of both, slow and fast, subsystems yields such a kind of controllability of the original system at t = tc for all sufficiently small values of ε (for the details, see Sects. 4.4 and 4.8). However, if the limit value in (6.56) does not exist or this value is infinite, then the fast subsystem of the original system (6.52)–(6.53) does not exist. In this case, the slow–fast decomposition of the original system cannot be used for the controllability analysis of this system. In this section, we derive novel ε-free controllability conditions for (6.52)–(6.53) subject to the assumption that h(ε) ≥ h0

∀ε ∈ [0, ε0 ],

366

6 Miscellanies

where h0 > 0 is some number. The latter means that the limit in (6.56) equals positive infinity. Therefore, the derivation of the controllability conditions for (6.52)–(6.53) cannot be based on the slow–fast decomposition of this system.

6.3.3 Auxiliary Results 6.3.3.1

Necessary and Sufficient Controllability Conditions of the Original System

For a given ε ∈ (0, ε0 ], we consider the block matrices  A12 (t, ε) A11 (t, ε) , 1 1 ε A21 (t, ε) ε A22 (t, ε)     B01 (t, ε) B11 (t, ε) , B1 (t, ε) = 1 . B0 (t, ε) = 1 ε B02 (t, ε) ε B12 (t, ε) 

A(t, ε) =

(6.57)

Let, for a given ε ∈ (0, ε0 ], the (n+m)×(n+m)-matrix-valued function Ψ (σ, ε), σ ∈ [0, +∞) be the solution of the terminal-value problem dΨ (σ, ε) = −AT (σ, ε)Ψ (σ, ε), σ ∈ [0, tc ), dσ Ψ (tc , ε) = In+m ,

Ψ (σ, ε) = 0, σ > tc .

(6.58)

Consider the following matrix-valued functions: B0 (σ, ε) = Ψ T (σ, ε)B0 (σ, ε),       B1 σ + h(ε), ε = Ψ T σ + h(ε), ε B1 σ + h(ε), ε ,   W (σ, ε) = B0 (σ, ε) + B1 σ + h(ε), ε .

(6.59)

Based on W (σ, ε), we construct the matrix 

tc

W (tc , ε) =

W (σ, ε)W T (σ, ε)dσ.

(6.60)

0

The matrix W (tc , ε) is called the controllability matrix of the system (6.52)–(6.53). As a direct consequence of the results of [4], we obtain the following assertion. Proposition 6.4 For a given ε ∈ (0, ε0 ], the original system (6.52)–(6.53) is completely Euclidean space controllable at the time instant tc if and only if the matrix W (tc , ε) is invertible.

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

6.3.3.2

367

Block-Wise Estimate of the Solution to the Terminal-Value Problem (6.58)

Let us partition the matrix Ψ (σ, ε) into blocks as  Ψ (σ, ε) =

 Ψ1 (σ, ε) Ψ2 (σ, ε) , Ψ3 (σ, ε) Ψ4 (σ, ε)

(6.61)

where the blocks Ψ1 (σ, ε), Ψ2 (σ, ε), Ψ3 (σ, ε), and Ψ4 (σ, ε) are of the dimensions n × n, n × m, m × n, and m × m, respectively. Along with the terminal-value problem (6.58), let us consider the following terminal-value problem for the n × n-matrix-valued function Ψs (σ ), σ ∈ [0, +∞): dΨs (σ ) = −ATs (σ )Ψs (σ ), σ ∈ [0, tc ), dσ Ψs (tc ) = In ;

Ψs (σ ) = 0,

σ > tc .

(6.62)

Also, we consider the following m × m-matrix-valued function: % Ψf (ξ ) =

0, ! " ξ < 0, T exp A22 (tc , 0)ξ , ξ ≥ 0.

(6.63)

In what follows, we assume (AI) (AII) (AIII)

The matrix-valued functions Aij (t, ε), Bkl (t, ε), (i = 1, 2; j = 1, 2; k = 0, 1; l = 1, 2) are continuously differentiable for (t, ε) ∈ [0, tc ] × [0, ε0 ]. The scalar function h(ε) is continuously differentiable for ε ∈ [0, ε0 ]. For all t ∈ [0, tc ], all the eigenvalues λq (t), (q = 1, . . . , m) of the matrix A22 (t, 0) satisfy the inequality Reλq (t) < −2β, where β > 0 is some constant.

The assumption (AIII) directly yields the inequality Ψf (ξ ) ≤ a exp(−2βξ ), ξ ≥ 0,

(6.64)

where a > 0 is some constant. Remark 6.5 It should be noted that the assumption (AIII) yields the fulfilment of the inequality (6.54) for all t ∈ [0, tc ]. By virtue of the results of Sect. 2.2.4 (Theorem 2.1), we have the following assertion. Proposition 6.5 Let the assumptions (AI)–(AIII) be valid. Then, there exists a positive number ε0 , (ε0 ≤ ε0 ), such that for all ε ∈ (0, ε0 ] the matrix-valued functions Ψ1 (σ, ε), Ψ2 (σ, ε), Ψ3 (σ, ε), Ψ4 (σ, ε) satisfy the inequalities:

368

6 Miscellanies

Ψ2 (σ, ε) ≤ a, Ψ1 (σ, ε) − Ψs (σ ) ≤ aε,

s (σ ) ≤ aε ε + exp(−β(tc − σ )/ε) , Ψ3 (σ, ε) − εΨ   Ψ4 (σ, ε) − Ψf (tc − σ )/ε ≤ aε.

(6.65)

Here σ ∈ [0, +∞); a > 0 is some constant independent of ε;   s (σ ) = − AT22 (σ, 0) −1 AT12 (σ, 0)Ψs (σ ). Ψ 6.3.3.3

(6.66)

Asymptotic Analysis of the Controllability Matrix W (tc , ε)

In what follows, we assume that the time instant tc , introduced in Sect. 6.3.1, satisfies the inequality tc > max h(ε). ε∈[0,ε0 ]

(6.67)

Let us partition the symmetric matrix W (tc , ε) into blocks as  W (tc , ε) =

 W1 (tc , ε) W2 (tc , ε) , W2T (tc , ε) W3 (tc , ε)

(6.68)

where the blocks W1 (tc , ε), W2 (tc , ε), and W3 (tc , ε) are of the dimensions n × n, n × m, and m × m, respectively. Consider the following matrix-valued functions: B0s (σ ) = ΨsT (σ )B0s (σ ),       B1s σ + h(0) = ΨsT σ + h(0) B1s σ + h(0) ,   Ws (σ ) = B0s (σ ) + B1s σ + h(0) ,

(6.69)

where B0s (t) and B1s (t) are given in (6.56). Using Ws (σ ), we construct the matrix  Ws (tc ) = 0

tc

Ws (σ )WsT (σ )dσ.

(6.70)

Also, we introduce into the consideration the matrix 

+∞

Wf (tc ) = 0

ΨfT (ξ )Sf (tc )Ψf (ξ )dξ,

(6.71)

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

369

where T T (tc , 0) + B12 (tc , 0)B12 (tc , 0). Sf (tc ) = B02 (tc , 0)B02

(6.72)

Due to (6.64), the integral in (6.71) converges. Lemma 6.6 Let the assumptions (AI)–(AIII) be valid. Then, for all ε ∈ (0, ε0 ], the matrices W1 (tc , ε), W2 (tc , ε), W3 (tc , ε) satisfy the inequalities W1 (tc , ε) − Ws (tc ) ≤ aε,

W2 (tc , ε) ≤ a,

εW3 (tc , ε) − Wf (tc ) ≤ aε,

(6.73)

where a > 0 is some constant independent of ε. Proof To prove the lemma, first we need to derive expressions for the blocks of the matrix W (tc , ε). Remember that the expression for this matrix is given by Eq. (6.60), while its block representation is given by Eq. (6.68). We derive the expressions for W1 (tc , ε), W2 (tc , ε), and W3 (tc , ε) by consecutive analytical calculations. Namely, using the block form of the matrices B0 (t, ε), B1 (t, ε), and Ψ (σ, ε) (see Eqs. (6.57) and (6.61)), as well as the expressions for the matrices B0 (σ, ε) and B1 σ +h(ε), ε (see Eq. (6.59)), we derive the block representations for these two matrices as  B01 (σ, ε) B02 (σ, ε) ⎞ ⎛ T Ψ1 (σ, ε)B01 (σ, ε) + 1ε Ψ3T (σ, ε)B02 (σ, ε) ⎠, =⎝ Ψ2T (σ, ε)B01 (σ, ε) + 1ε Ψ4T (σ, ε)B02 (σ, ε)      B11 σ + h(ε), ε   B1 σ + h(ε), ε = B12 σ + h(ε), ε     ⎞ + h(ε), ε + 1ε Ψ3T σ + h(ε), ε B12 σ + h(ε), ε ⎠.  1 T    + h(ε), ε + ε Ψ4 σ + h(ε), ε B12 σ + h(ε), ε (6.74) 

B0 (σ, ε) =

⎛ =⎝

   Ψ1T σ + h(ε), ε B11 σ    Ψ2T σ + h(ε), ε B11 σ

Based on these block matrices, we obtain the following block representation for the matrix W (σ, ε), given in (6.59):  W (σ, ε) =

  B01 (σ, ε) + B11 σ + h(ε), ε   . B02 (σ, ε) + B12 σ + h(ε), ε

Substituting this block-form matrix into the expression (6.60) for the matrix W (tc , ε) and equating the corresponding blocks of the resulting matrix and

370

6 Miscellanies

the block-form matrix in (6.68), we obtain the blocks W1 (tc , ε), W2 (tc , ε), and W3 (tc , ε) as 

tc

W1 (tc , ε) = 0



  T T B01 (σ, ε)B01 σ + h(ε), ε (σ, ε) + B01 (σ, ε)B11

 T   T 

σ + h(ε), ε dσ, (σ, ε) + B11 σ + h(ε), ε B11 +B11 σ + h(ε), ε B01 (6.75) 

tc

W2 (tc , ε) = 0



  T T B01 (σ, ε)B02 σ + h(ε), ε (σ, ε) + B01 (σ, ε)B12

 T   T 

σ + h(ε), ε dσ, (σ, ε) + B11 σ + h(ε), ε B12 +B11 σ + h(ε), ε B02 (6.76) 

tc

W3 (tc , ε) = 0



  T T (σ, ε) + B02 (σ, ε)B12 B02 (σ, ε)B02 σ + h(ε), ε

 T   T 

σ + h(ε), ε dσ. (σ, ε) + B12 σ + h(ε), ε B12 +B12 σ + h(ε), ε B02 (6.77) Now, let us proceed to the proof of the inequalities in (6.73). We start with the first inequality. Substitution of the expressions for B01 (σ, ε) and B11 σ + h(ε), ε (see Eq. (6.74)) into Eq. (6.75) yields after a routine algebra W1 (tc , ε) = Φ1 (tc , ε) + Φ2 (tc , ε) + Φ2T (tc , ε) + Φ3 (tc , ε), where



tc

Φ1 (tc , ε) = 0

# T Ψ1T (σ, ε)B01 (σ, ε)B01 (σ, ε)Ψ1 (σ, ε)

1 T + Ψ3T (σ, ε)B02 (σ, ε)B01 (σ, ε)Ψ1 (σ, ε) ε 1 T (σ, ε)Ψ3 (σ, ε) + Ψ1T (σ, ε)B01 (σ, ε)B02 ε $ 1 T (σ, ε)Ψ3 (σ, ε) dσ, + 2 Ψ3T (σ, ε)B02 (σ, ε)B02 ε  Φ2 (tc , ε) = 0

tc

#

(6.78)

(6.79)

    T Ψ1T (σ, ε)B01 (σ, ε)B11 σ + h(ε), ε Ψ1 σ + h(ε), ε

    1 T + Ψ3T (σ, ε)B02 (σ, ε)B11 σ + h(ε), ε Ψ1 σ + h(ε), ε ε     1 T σ + h(ε), ε Ψ3 σ + h(ε), ε + Ψ1T (σ, ε)B01 (σ, ε)B12 ε    $ 1 T σ + h(ε), ε Ψ3 σ + h(ε), ε dσ, + 2 Ψ3T (σ, ε)B02 (σ, ε)B12 ε

(6.80)

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

371

Φ3 (tc , ε) =



tc 0

#        T Ψ1T σ + h(ε), ε B11 σ + h(ε), ε)B11 σ + h(ε), ε Ψ1 σ + h(ε), ε

    T    1 σ + h(ε), ε Ψ1 σ + h(ε), ε + Ψ3T σ + h(ε), ε B12 σ + h(ε), ε B11 ε    T    1 T σ + h(ε), ε Ψ3 σ + h(ε), ε + Ψ1 σ + h(ε), ε B11 σ + h(ε), ε B12 ε    T   $ 1 T σ + h(ε), ε Ψ3 σ + h(ε), ε dσ. + 2 Ψ3 σ + h(ε), ε B12 σ + h(ε), ε B12 ε (6.81) Let us estimate these matrices. We start with Φ1 (tc , ε). Represent the matrices Ψ1 (σ, ε) and Ψ3 (σ, ε) in the form s (σ ) + ΔΨ,3 (σ, ε), Ψ3 (σ, ε) = εΨ

Ψ1 (σ, ε) = Ψs (σ ) + ΔΨ,1 (σ, ε),

(6.82)

where ΔΨ,1 (σ, ε) = Ψ1 (σ, ε) − Ψs (σ ),

s (σ ). ΔΨ,3 (σ, ε) = Ψ3 (σ, ε) − εΨ

By virtue of Proposition 6.5 (see Eq. (6.65)), we have ΔΨ,1 (σ, ε) ≤ aε,  

ΔΨ,3 (σ, ε) ≤ aε ε + exp − β(tc − σ )/ε , σ ∈ [0, +∞),

ε ∈ (0, ε0 ].

(6.83)

Substituting (6.82) into (6.79), we can rewrite the matrix Φ1 (tc , ε) as Φ1 (tc , ε) = Γ1 (tc , ε) + Δ11 (tc , ε) + Δ13 (tc , ε) + ΔT13 (tc , ε) + Δ33 (tc , ε), (6.84) where 

tc

Γ1 (tc , ε) = 0

# T ΨsT (σ )B01 (σ, ε)B01 (σ, ε)Ψs (σ )

T ,s (σ )B02 (σ, ε)B01 +Ψ (σ, ε)Ψs (σ ) T

T s (σ ) (σ, ε)Ψ +ΨsT (σ )B01 (σ, ε)B02 $ T ,s T (σ )B02 (σ, ε)B02 s (σ ) dσ, +Ψ (σ, ε)Ψ

372

6 Miscellanies



tc

Δ11 (tc , ε) = 0

 0

 Δ33 (tc , ε) = 0

T  T (σ, ε)Ψs (σ ) + ΔΨ,1 (σ, ε) B01 (σ, ε)B01 $ T +ΨsT (σ )B01 (σ, ε)B01 (σ, ε)ΔΨ,1 (σ, ε) dσ,

tc

Δ13 (tc , ε) =

tc

# T T ΔΨ,1 (σ, ε) B01 (σ, ε)B01 (σ, ε)ΔΨ,1 (σ, ε)

#1 T T ΔΨ,3 (σ, ε) B02 (σ, ε)B01 (σ, ε)ΔΨ,1 (σ, ε) ε T 1 T + ΔΨ,3 (σ, ε) B02 (σ, ε)B01 (σ, ε)Ψs (σ ) ε $ T sT (σ )B02 (σ, ε)B01 +Ψ (σ, ε)ΔΨ,1 (σ, ε) dσ, #1 T T (σ, ε)ΔΨ,3 (σ, ε) ΔΨ,3 (σ, ε) B02 (σ, ε)B02 2 ε T 1 T s (σ ) + ΔΨ,3 (σ, ε) B02 (σ, ε)B02 (σ, ε)Ψ ε $ 1 T T s (σ )B02 (σ, ε)B02 (σ, ε)ΔΨ,3 (σ, ε) dσ. + Ψ ε

Since B01 (σ, ε) and B02 (σ, ε) are smooth matrix-valued functions in the domain [0, tc ] × [0, ε0 ] (see the assumption (AI)), then Γ1 (tc , ε) satisfies the inequality Γ1 (tc , ε) − Φ1,s (tc ) ≤ a1 ε,

ε ∈ [0, ε0 ],

(6.85)

where a1 > 0 is some constant independent of ε, and

 = 0

Φ1,s (tc ) = Γ1 (tc , 0) tc

# T T sT (σ )B02 (σ, 0)B01 ΨsT (σ )B01 (σ, 0)B01 (σ, 0)Ψs (σ ) + Ψ (σ, 0)Ψs (σ )

$ T T s (σ ) + Ψ sT (σ )B02 (σ, 0)B02 s (σ ) dσ. +ΨsT (σ )B01 (σ, 0)B02 (σ, 0)Ψ (σ, 0)Ψ (6.86) Furthermore, using the first inequality in (6.83), the boundedness of the matrixvalued function B01 (σ, ε) in the domain [0, tc ] × [0, ε0 ], and the inequality ε0 ≤ ε0 , we can estimate the matrix |Δ11 (tc , ε) as Δ11 (tc , ε) ≤ a2 ε,

ε ∈ (0, ε0 ].

(6.87)

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

373

Similarly, using both inequalities in (6.83), we obtain the estimates for Δ13 (tc , ε) and Δ33 (tc , ε): Δ13 (tc , ε) ≤ a2 ε,

Δ33 (tc , ε) ≤ a2 ε,

ε ∈ (0, ε0 ].

(6.88)

In (6.87) and (6.88), a2 > 0 is some constant independent of ε. Equation (6.84), along with the inequalities (6.85) and (6.87)–(6.88), implies immediately Φ1 (tc , ε) − Φ1,s (tc ) ≤ aε,

ε ∈ (0, ε0 ],

(6.89)

where a = a1 + 4a2 . Taking into account the assumption (AII), we obtain (similarly to (6.89)) the following inequalities: Φp (tc , ε) − Φp,s (tc ) ≤ aε,

p = 2, 3,

ε ∈ (0, ε0 ],

(6.90)

where a > 0 is some constant independent of ε, and  Φ2,s (tc ) = 0

tc

#

    T ΨsT (σ )B01 (σ, 0)B11 σ + h(0), 0 Ψs σ + h(0)

    T sT (σ )B02 (σ, 0)B11 σ + h(0), 0 Ψs σ + h(0) +Ψ     T s σ + h(0) +ΨsT (σ )B01 (σ, 0)B12 σ + h(0), 0 Ψ    $ T s σ + h(0) dσ, sT (σ )B02 (σ, 0)B12 σ + h(0), 0 Ψ +Ψ 

tc

Φ3,s (tc ) = 0

(6.91)

#        T ΨsT σ + h(0) B11 σ + h(0), 0)B11 σ + h(0), 0 Ψs σ + h(0)

    T    sT σ + h(0) B12 σ + h(0), 0 B11 σ + h(0), 0 Ψs σ + h(0) +Ψ     T    s σ + h(0) +ΨsT σ + h(0) B11 σ + h(0), 0 B12 σ + h(0), 0 Ψ     T   $ s σ + h(0) dσ. sT σ + h(0) B12 σ + h(0), 0 B12 σ + h(0), 0 Ψ +Ψ (6.92) Let us transform the expressions for Φp,s (tc ), (p = 1, 2, 3) (see Eqs. (6.86), (6.91), (6.92)). Substitution of Eq. (6.66) into these expressions, and use of the   expressions for B0s (σ ), B1s (σ ), B0s (σ ), B1s σ + h(0) , Ws (σ ), Ws (tc ) (see Eqs. (6.56), (6.69), (6.70)) yield after a routine algebra:

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6 Miscellanies

 Φ1,s (tc ) =  Φ2,s (tc ) = 0



tc

Φ3,s (tc ) = 0

0 tc

tc

T B0s (σ )B0s (σ )dσ,

  T σ + h(0) dσ, B0s (σ )B1s

  T  σ + h(0) dσ, B1s σ + h(0) B1s

T Φ1,s + Φ2,s + Φ2,s + Φ3,s = Ws (tc ).

The latter equality, along with Eq. (6.78) and the inequalities (6.89), (6.90), immediately yields the first inequality in (6.73) for all ε ∈ (0, ε0 ]. Now, let us prove the in (6.73). Substitution of the expressions  third inequality  for B02 (σ, ε) and B12 σ + h(ε), ε (see Eq. (6.74)) into Eq. (6.77) yields after a routine algebra εW3 (tc , ε) = Θ1 (tc , ε) + Θ2 (tc , ε) + Θ2T (tc , ε) + Θ3 (tc , ε), where

 Θ1 (tc , ε) = 0

tc

#

(6.93)

T εΨ2T (σ, ε)B01 (σ, ε)B01 (σ, ε)Ψ2 (σ, ε)

T +Ψ4T (σ, ε)B02 (σ, ε)B01 (σ, ε)Ψ2 (σ, ε) T +Ψ2T (σ, ε)B01 (σ, ε)B02 (σ, ε)Ψ4 (σ, ε) $ 1 T + Ψ4T (σ, ε)B02 (σ, ε)B02 (σ, ε)Ψ4 (σ, ε) dσ, ε



tc

Θ2 (tc , ε) = 0

(6.94)

#     T εΨ2T (σ, ε)B01 (σ, ε)B11 σ + h(ε), ε Ψ2 σ + h(ε), ε

    T σ + h(ε), ε Ψ2 σ + h(ε), ε +Ψ4T (σ, ε)B02 (σ, ε)B11     T +Ψ2T (σ, ε)B01 (σ, ε)B12 σ + h(ε), ε Ψ4 σ + h(ε), ε    $ 1 T + Ψ4T (σ, ε)B02 (σ, ε)B12 σ + h(ε), ε Ψ4 σ + h(ε), ε dσ, ε

(6.95)

Θ3 (tc , ε) = 

tc 0

#     T    εΨ2T σ + h(ε), ε B11 σ + h(ε), ε B11 σ + h(ε), ε Ψ2 σ + h(ε), ε     T    σ + h(ε), ε Ψ2 σ + h(ε), ε +Ψ4T σ + h(ε), ε B12 σ + h(ε), ε B11

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

375

    T    σ + h(ε), ε Ψ4 σ + h(ε), ε +Ψ2T σ + h(ε), ε B11 σ + h(ε), ε B12     T   $ 1 + Ψ4T σ + h(ε), ε B12 σ + h(ε), ε B12 σ + h(ε), ε Ψ4 σ + h(ε), ε dσ. ε (6.96) Let us estimate the matrices Θ1 (tc , ε), Θ2 (tc , ε), and Θ3 (tc , ε). We start with Θ1 (tc , ε). Represent the matrix Ψ4 (σ, ε) as   (6.97) Ψ4 (σ, ε) = Ψf (tc − σ )/ε + ΔΨ,4 (σ, ε), where   ΔΨ,4 (σ, ε) = Ψ4 (σ, ε) − Ψf (tc − σ )/ε . By virtue of Proposition 6.5 (see Eq. (6.65)), we have ΔΨ,4 (σ, ε) ≤ aε,

σ ∈ [0, +∞),

ε ∈ (0, ε0 ].

(6.98)

Also remember that, due to Proposition 6.5, the matrix-valued function Ψ2 (σ, ε) is uniformly bounded in the domain [0, +∞) × (0, ε0 ]. Substituting (6.97) into (6.94), we rewrite the matrix Θ1 (tc , ε) in the form Θ1 (tc , ε) = Ξ1 (tc , ε) + Δ22 (tc , ε) + Δ24 (tc , ε) + ΔT24 (tc , ε) + Δ44 (tc , ε), (6.99) where Ξ1 (tc , ε) =

1 ε



tc 0

    T ΨfT (tc − σ )/ε B02 (σ, ε)B02 (σ, ε)Ψf (tc − σ )/ε dσ, 

tc

Δ22 (tc , ε) = ε 0



tc

Δ24 (tc , ε) = 0



tc

+ 0

Δ44 (tc , ε) =

T Ψ2T (σ, ε)B01 (σ, ε)B01 (σ, ε)Ψ2 (σ, ε)dσ,

T ΔΨ,4 (σ, ε)B02 (σ, ε)B01 (σ, ε)Ψ2 (σ, ε)dσ

  T ΨfT (tc − σ )/ε B02 (σ, ε)B01 (σ, ε)Ψ2 (σ, ε)dσ,

1 ε



tc 0

T T (σ, ε)ΔΨ,4 (σ, ε) ΔΨ,4 (σ, ε)B02 (σ, ε)B02

T +ΔTΨ,4 (σ, ε)B02 (σ, ε)B02 (σ, ε)Ψf (σ, ε)

T +ΨfT (σ, ε)B02 (σ, ε)B02 (σ, ε)ΔΨ,4 (σ, ε) dσ.

376

6 Miscellanies

Let us treat the matrix Ξ1 (tc , ε). In the integral in the expression for this matrix we change the variable as ξ = (tc − σ )/ε. This yields the expression 

tc /ε

Ξ1 (tc , ε) = 0

T ΨfT (ξ )B02 (tc − εξ, ε)B02 (tc − εξ, ε)Ψf (ξ )dξ.

Using this expression, as well as the properties (6.63)–(6.64) of Ψf (ξ ), the smoothness of B02 (σ, ε) (see the assumption (AI)), and the inequality ε0 ≤ ε0 , we obtain the following inequality: Ξ1 (tc , ε) − Θ1,f ≤ a¯ 1 ε,

ε ∈ (0, ε0 ],

(6.100)

where a¯ 1 > 0 is some constant independent of ε, and 

+∞

Θ1,f = 0

T ΨfT (ξ )B02 (tc , 0)B02 (tc , 0)Ψf (ξ )dξ.

Proceed to the matrices Δ22 (tc , ε), Δ24 (tc , ε), and Δ44 (tc , ε). Due to abovementioned boundedness of Ψ2 (σ, ε) and to the boundedness of B01 (σ, ε) (see the assumption (AI)), we directly have Δ22 (tc , ε) ≤ a¯ 2 ε,

ε ∈ (0, ε0 ],

(6.101)

where a¯ 2 > 0 is some constant independent of ε. To estimate the matrix Δ24 (tc , ε), we consider separately each of the two integrals constituting this matrix. Due to the inequality (6.98), and the boundedness of B01 (σ, ε), B02 (σ, ε), and Ψ2 (σ, ε), we obtain 

tc 0

T ΔΨ,4 (σ, ε)B02 (σ, ε)B01 (σ, ε)Ψ2 (σ, ε)dσ ≤ a¯ 31 ε,

ε ∈ (0, ε0 ],

where a¯ 31 > 0 is some constant independent of ε. In the second integral we change the variable as ξ = (tc − σ )/ε, yielding  0



tc /ε

ε 0

tc

  T ΨfT (tc − σ )/ε B02 (σ, ε)B01 (σ, ε)Ψ2 (σ, ε)dσ =

T ΨfT (ξ )B02 (tc − εξ, ε)B01 (tc − εξ, ε)Ψ2 (tc − εξ, ε)dξ.

The latter, along with the properties (6.63)–(6.64) of Ψf (ξ ) and the boundedness of B01 (σ, ε), B02 (σ, ε), Ψ2 (σ, ε), yields  0

tc

  T ΨfT (tc − σ )/ε B02 (σ, ε)B01 (σ, ε)Ψ2 (σ, ε)dσ ≤ a¯ 32 ε,

ε ∈ (0, ε0 ],

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

377

where a¯ 32 > 0 is some constant independent of ε. Therefore, Δ24 (tc , ε) ≤ a¯ 3 ε,

ε ∈ (0, ε0 ],

(6.102)

where a¯ 3 = a¯ 31 + a¯ 32 . The matrix Δ44 (tc , ε) is estimated similarly to the matrices Δ22 (tc , ε) and Δ24 (tc , ε), yielding Δ44 (tc , ε) ≤ a¯ 4 ε,

ε ∈ (0, ε0 ],

(6.103)

where a¯ 4 > 0 is some constant independent of ε. Equation (6.99), along with the inequalities (6.100), (6.101), (6.102), (6.103), implies Θ1 (tc , ε) − Θ1,f ≤ aε, ¯

ε ∈ (0, ε0 ],

(6.104)

where a¯ = a¯ 1 + a¯ 2 + 2a¯ 3 + a¯ 4 . Taking into account the assumption (AII), we obtain (similarly to (6.104)) the inequalities Θ2 (tc , ε) ≤ aε, ¯

Θ3 (tc , ε) − Θ3,f ≤ aε, ¯

ε ∈ (0, ε0 ],

(6.105)

where a¯ > 0 is some constant independent of ε, and  Θ3,f = 0

+∞

T ΨfT (ξ )B12 (tc , 0)B12 (tc , 0)Ψf (ξ )dξ.

Using the expressions of Θ1,f and Θ3,f , as well as Eqs. (6.71)–(6.72), we obtain Θ1,f + Θ3,f = Wf (tc ). This equation, along with Eq. (6.93) and the inequalities (6.104), (6.105), yields the third inequality in (6.73) for all ε ∈ (0, ε0 ]. Similarly to the proofs of the first and third inequalities in (6.73), one can prove the validity of the second inequality in (6.73) for all ε ∈ (0, ε0 ]. This completes the proof of the lemma.  

6.3.4 Parameter-Free Controllability Conditions Theorem 6.2 Let the assumptions (AI)–(AIII) be valid. Let det Ws (tc ) = 0,

det Wf (tc ) = 0.

(6.106)

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6 Miscellanies

Then, there exists a positive number ε∗ , (ε∗ ≤ ε0 ), such that for all ε ∈ (0, ε∗ ] the original system (6.52)–(6.53) is completely Euclidean space controllable at the time instant tc . Proof For a given ε > 0, consider the (n + m) × (n + m)-matrix  L(ε) =

In 0 √ εIm 0

 .

(6.107)

det L(ε) = 0 ∀ε > 0.

(6.108)

It is clear that

Using Eqs. (6.68) and (6.107) yields  L(ε)W (tc , ε)L(ε) =

 √ εW12 (tc , ε) W11 (tc , ε) √ T . εW12 (tc , ε) εW22 (tc , ε)

Calculating the limit of the determinant of this matrix as ε → +0, and using the inequalities (6.73), we obtain   lim det L(ε)W (tc , ε)L(ε) = det Ws (tc ) det Wf (tc ).

ε→+0

This equation, along with the inequalities (6.106) and (6.108), directly implies the existence of a number 0 < ε∗ ≤ ε1 , such that det W (tc , ε) = 0 for all ε ∈ (0, ε∗ ]. The latter, along with Proposition 6.4, yields the statement of the theorem.   Remark 6.6 Note that the first inequality in (6.106) is equivalent to the complete Euclidean space controllability at the time instant tc of the slow subsystem (6.55). The second inequality in (6.106) cannot, in general, be interpreted as a controllability condition of a system with m-dimensional state and r-dimensional control c ) such that variables. However, if there exists an m × r-matrix B(t T (tc ) = Sf (tc ), c )B B(t

(6.109)

then the second inequality in (6.106) is equivalent to the complete controllability of the system d y(ξ ˜ ) c )u(ξ = A22 (tc , 0)y(ξ ˜ ) + B(t ˜ ), ξ ≥ 0 dξ

(6.110)

at some instant ξ = ξc > 0, where y(ξ ˜ ) ∈ E m is a state variable; u(ξ ˜ ) ∈ E r is a control.

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

379

The following assertion is a direct consequence of Theorem 6.2 and Remark 6.6. Corollary 6.1 Let the assumptions (AI)–(AIII) be valid. Let there exist an m × r c ) satisfying the condition (6.109). Let the slow subsystem (6.55) be matrix B(t completely Euclidean space controllable at the time instant tc . Let the system (6.110) be completely controllable at some instant ξ = ξc > 0. Then, there exists a positive number ε∗ , (ε∗ ≤ ε1 ), such that for all ε ∈ (0, ε∗ ] the original system (6.52)–(6.53) is completely Euclidean space controllable at the time instant tc .

6.3.5 Example 1 Consider a particular case of the original system (6.52)–(6.53) with the following data: n = 2, m = 2, r = 1, h(ε) = 1 + sin(ε), and  1−ε t2 + 3 , A11 (t, ε) = ε2 − 2 − (t + 1)2     t +ε −1 −t 1 + ε A12 (t, ε) = , A21 (t, ε) = , 2−ε t +1 2 t +1   εt − 1 0 , A22 (t, ε) = 0 εt − 1     ε − 2t (t + 1)2 , B11 (t, ε) = , B01 (t, ε) = 2 3 exp(ε) t +t −4     2−ε −(t + 2) B02 (t, ε) = , B12 (t, ε) = . −t exp(−ε) 

(6.111)

We study the complete Euclidean space controllability of the system (6.52)– (6.53) with the data (6.111) at the time instant tc = 2 for all sufficiently small ε > 0. It is clear that for any given ε0 ∈ (0, π/2), the inequality (6.67) is valid and the assumptions (AI)–(AIII) are fulfilled. Due to Eqs. (6.56) and (6.111), we obtain  As (t) =

 1 0 , 0 2

  t B0s (t) = , 0

 B1s (t) =

0 −t

 .

Solving the terminal-value problem (6.62) with the above obtained matrix of the coefficients As (t), we have

380

6 Miscellanies

Ψs (σ ) =

⎧  ⎪ exp(2 − σ ) 0 ⎪   ⎪ , ⎪ ⎪ exp 2(2 − σ ) ⎨ 0

σ ∈ [0, 2],

  ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ , ⎩ 0 0

σ > 2.

Using this solution, as well as Eqs. (6.69)–(6.70) and the above obtained matrices B0s (t), B1s (t), we have ⎧  ⎪ σ exp(2 − σ ) ⎪   ⎪ , σ ∈ [0, 1], ⎪ ⎪ ⎨ −(σ + 1) exp 2(1 − σ ) Ws (σ ) =   ⎪ ⎪ ⎪ σ exp(2 − σ ) ⎪ ⎪ , σ ∈ (1, 2], ⎩ 0  Ws (2) =

 10.3995 − 7.1911 . −7.1911 34.5488

Hence, det Ws (2) = 0, i.e., the first inequality in (6.106) is fulfilled. Proceed to the second inequality in (6.106). From Eqs. (6.63), (6.71)–(6.72), and (6.111), we obtain Ψf (ξ ) = exp(−ξ )I2 , ξ ≥ 0,   20 − 8 Sf (2) = , −8 4   10 − 4 . Wf (2) = −4 2 Thus, det Wf (2) = 0, i.e., the second inequality in (6.106) also is fulfilled. Therefore, by virtue of Theorem 6.2, there exists a positive number ε∗ , such that for all ε ∈ (0, ε∗ ] the system (6.52)–(6.53), (6.111) is completely Euclidean space controllable at the time instant tc = 2. Completing this example, let us note the following. Since the above obtained matrix Sf (2) is invertible, then there does not exist a 2 × 1-matrix (a 2-vector) B(2) B T (2) = Sf (2) meaning that the condition (6.109) is not fulfilled in such that B(2) this example. Therefore, in this example, Corollary 6.1 is not valid.

6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input. . .

381

6.3.6 Example 2 Consider a particular case of the original system (6.52)–(6.53) with the following data: n = 1, m = 2, r = 1, h(ε) = 1 + ε, and A11 (t, ε) = −  A21 (t, ε) =

1 , t +1−ε

 t − 2ε , 2t 2 (t + 2) 

B02 (t, ε) =

A12 (t, ε) = (2t 2 − ε , −t), 

A22 (t, ε) =

 0 εt 2 − 1 , 0 − (t + 2)

B01 (t, ε) = t − ε, B11 (t, ε) = 5 exp(ε),    1+ε 4 + ln(1 − ε) , B12 (t, ε) = . t +2 2t (t + 2)

(6.112)

Like in the previous example, we study the complete Euclidean space controllability of the system (6.52)–(6.53) with the data (6.112) at the time instant tc = 2 for all sufficiently small ε > 0. It is clear that for any given ε0 ∈ (0, 1), the inequality (6.67) is valid and the assumptions (AI)–(AIII) are fulfilled. Using Eqs. (6.56) and (6.112) yields As (t) = −

1 , t +1

B0s (t) = 2t 2 ,

B1s (t) = 6t 2 + 5.

Solving the terminal-value problem (6.62) with the above obtained coefficient As (t), we have

(σ + 1)/3, σ ∈ [0, 2], Ψs (σ ) = 0, σ > 2. Using this solution, as well as Eq. (6.69) and the above obtained matrices B0s (t), B1s (t), we have Ws (σ ) =

  ⎧ 2 ⎨ 2σ (σ + 1)/3 + 6(σ + 1)2 + 5 (σ + 2)/3, ⎩

2σ 2 (σ + 1)/3,

σ ∈ [0, 1], σ ∈ (1, 2].

The function Ws (σ ) is not identical zero in both intervals [0, 1] and (1, 2]. Therefore, by virtue of Eq. (6.70), we have that W (tc ) = 0. Proceed to the calculation of the matrix Sf (tc ). Using Eqs. (6.72) and (6.112), we obtain   17 68 Sf (tc ) = . 68 272

382

6 Miscellanies

This matrix can be represented in the form (6.109), where c) = B(t

 √ √17 . 4 17

Let us establish the complete controllability of the system (6.110) in this example. For this purpose, we use the results of [14]. Due to these results, we should show that the rank of the following matrix:   c) = c ) , A22 (tc , 0)B(t c) A(t B(t equals m. Calculating this matrix, we obtain c) = A(t

√ 17 √ 4 17

√  − 17 √ . − 16 17

c ) = 2 = m. Hence, by virtue of Corollary 6.1, the system (6.52)– Thus, rankA(t (6.53) with the data (6.112) is completely Euclidean space controllable at the time instant tc = 2 for all sufficiently small ε > 0.

6.4 Functional Null Controllability of Some Nonlinear Systems with Small State Delays 6.4.1 Problem Formulation Consider the controlled system

dx(t)  = A1j (t, ε)x(t − εhj ) + A2j (t, ε)y(t − εhj ) dt N

 +

j =0

0 −h



 

G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)y(t + εη) dη + f1 z(t), t   +f2 z(t − εh1 ), . . . , z(t − εhN ), t + B1 (t, ε)u(t),

dy(t)  = A3j (t, ε)x(t − εhj ) + A4j (t, ε)y(t − εhj ) dt N

ε

t ≥ 0,

j =0

(6.113)

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

 +

0 −h



383

 

G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη + f3 z(t), t   +f4 z(t − εh1 ), . . . , z(t − εhN ), t + B2 (t, ε)u(t),

t ≥ 0,

(6.114)

  where x(t) ∈ E n , y(t) ∈ E m , z(·) = col x(·), y(·) , u(t) ∈ E r (u(t) is a control); ε > 0 is a small parameter; N ≥ 1 is an integer; 0 = h0 < h1 < . . . < hN = h are some given ε-independent constants; Aij (t, ε) and Bk (t, ε), (i = 1, . . . , 4; j = 0, 1, . . . , N; k = 1, 2) are matrix-valued functions of corresponding dimensions, continuously differentiable for (t, ε) ∈ [0, +∞) × [0, ε0 ], (ε0 > 0); Gi (t, η, ε), (i = 1, . . . , 4) are matrix-valued functions of corresponding dimensions, piecewise continuous in η ∈ [−h, 0] for any (t, ε) ∈ [0, +∞) × [0, ε0 ] and continuously differentiable with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0]; fl (z, t), (l = 1, 3) are given vector-valued functions of corresponding dimensions, continuous for (z, t) ∈ E n+m ×[0, +∞); fp (q1 , . . . , qN , t), (p = 2, 4) are given vector-valued functions of corresponding dimensions, continuous for (q1 , . . . , qN , t) ∈ E n+m × . . . × E n+m × [0, +∞). Moreover, the functions fl (z, t) and fp (q1 , . . . , qN , t), (l = 1, 3; p = 2, 4) satisfy the following limit conditions uniformly in t ∈ [0, tˆ]: lim

α1 →+∞

lim

α2 →+∞

fl (z, t) = 0, α1

fp (q1 , . . . , qN , t) = 0, α2

l = 1, 3,

(6.115)

p = 2, 4,

(6.116)

where tˆ > 0 is any given time instant, and



α1 = z , α2 = col(q1 , . . . , qN ) , z ∈ E n+m , qj ∈ E n+m , j = 1, . . . , N. For any given ε ∈ (0, ε0 ] and piecewise continuous control function u(t), t ∈ [0, +∞), the system (6.113)–(6.114) is a nonlinear time-dependent functionaldifferential system. It is infinite-dimensional with the state variables x(t + εη), y(t + εη), η ∈ [−h, 0]. Moreover, for 0 < ε 0 be a given time instant independent of ε. Definition 6.5 For a given ε ∈ (0, ε0 ], the system (6.113)–(6.114) is said to be functional null controllable at the time instant tc if for any ϕx (·) ∈ C[−εh, 0; E n ], and ϕy (·) ∈ C[−εh, 0; E m ] there exists a piecewise continuous control function

384

6 Miscellanies

  u(t), t ∈ [0, tc + εh], for which the solution col x(t, ε), y(t, ε) , t ∈ [0, tc + εh] of the system (6.113)–(6.114) with the initial conditions x(τ ) = ϕx (τ ),

y(τ ) = ϕy (τ ), τ ∈ [−εh, 0]

(6.117)

satisfies the terminal conditions x(t, ε) = 0,

y(t, ε) = 0 ∀ t ∈ [tc , tc + εh].

(6.118)

In what follows, we study two particular types of the system (6.113)–(6.114). In the system of the first type, the slow mode is uncontrolled and this mode does not contain the delayed state variables. In the system of the second type, the fast mode is uncontrolled and it does not contain the delayed state variables. For each of the abovementioned types of the system (6.113)–(6.114), we derive ε-free conditions of the functional null controllability, which are valid for all sufficiently small values of ε.

6.4.2 Preliminary Results 6.4.2.1

Euclidean Space Controllability of Singularly Perturbed Linear Time Delay System

In this subsection we consider the linear system associated with the original system (6.113)–(6.114). Namely,

dx(t)  = A1j (t, ε)x(t − εhj ) + A2j (t, ε)y(t − εhj ) dt N

j =0

 +

0 −h



G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)y(t + εη) dη +B1 (t, ε)u(t),

t ≥ 0,

(6.119)

dy(t)  = A3j (t, ε)x(t − εhj ) + A4j (t, ε)y(t − εhj ) dt N

ε

 +

j =0

0 −h



G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη +B2 (t, ε)u(t),

t ≥ 0.

(6.120)

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

385

Definition 6.6 For a given ε ∈ (0, ε0 ], the system (6.119)–(6.120) is said to be completely Euclidean space controllable at the time instant tc if for any ϕx (·) ∈ C[−εh, 0; E n ], ϕy (·) ∈ C[−εh, 0; E m ], xc ∈ E n , and yc ∈ E m there exists a piecewise continuous control function u(t), t ∈ [0, tc ], for which the initial-value problem (6.119)–(6.120), (6.117) has the solution satisfying the terminal conditions: x(tc ) = xc ,

6.4.2.2

y(tc ) = yc .

(6.121)

Euclidean Space Controllability of the Original System (6.113)–(6.114)

Definition 6.7 For a given ε ∈ (0, ε0 ], the system (6.113)–(6.114) is said to be completely Euclidean space controllable at the time instant tc if for any ϕx (·) ∈ C[−εh, 0; E n ], ϕy (·) ∈ C[−εh, 0; E m ], xc ∈ E n , and yc ∈ E m there exists a piecewise continuous control function u(t), t ∈ [0, tc ], for which the boundaryvalue problem (6.113)–(6.114), (6.117), (6.121) has a solution. Definition 6.8 For a given ε ∈ (0, ε0 ], the system (6.113)–(6.114) is said to be Euclidean space null controllable at the time instant tc if for any ϕx (·) ∈ C[−εh, 0; E n ], and ϕy (·) ∈ C[−εh, 0; E m ] there exists a piecewise continuous control function u(t), t ∈ [0, tc ], for which the initial-value problem (6.113)– (6.114), (6.117) has the solution satisfying the terminal conditions: x(tc ) = 0,

y(tc ) = 0.

Due to Definitions 6.7 and 6.8, we directly have the following assertion. Proposition 6.6 If for a given ε ∈ (0, ε0 ] the system (6.113)–(6.114) is completely Euclidean space controllable at the time instant tc , then this system is Euclidean space null controllable at the time instant tc . Based on the limit conditions (6.115)–(6.116) we have, as a direct consequence of the results of [1], the following assertion. Proposition 6.7 Let, for a given ε ∈ (0, ε0 ], the linear system (6.119)–(6.120) be completely Euclidean space controllable at the time instant tc . Then, for this ε, the nonlinear system (6.113)–(6.114) is completely Euclidean space controllable at the time instant tc . By virtue of Propositions 6.6 and 6.7, we have the following assertion. Corollary 6.2 Let, for a given ε ∈ (0, ε0 ], the linear system (6.119)–(6.120) be completely Euclidean space controllable at the time instant tc . Then, for this ε, the nonlinear system (6.113)–(6.114) is Euclidean space null controllable at the time instant tc .

386

6 Miscellanies

6.4.3 System of the First Type: Formulation and Some Auxiliary Results In this subsection, we consider the following particular case of the system (6.113)– (6.114):   dx(t) = A10 (t, ε)x(t) + A20 (t, ε)y(t) + f1 z(t), t , dt

t ≥ 0,

(6.122)

dy(t)  = A3j (t, ε)x(t − εhj ) + A4j (t, ε)y(t − εhj ) dt N

ε  +

j =0

0 −h



 

G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη + f3 z(t), t   +f4 z(t − εh1 ), . . . , z(t − εhN ), t + B2 (t, ε)u(t),

t ≥ 0.

(6.123)

The linear system, corresponding to (6.122)–(6.123), is a particular case of the system (6.119)–(6.120), and it has the form dx(t) = A10 (t, ε)x(t) + A20 (t, ε)y(t), dt

t ≥ 0,

(6.124)

dy(t)  = A3j (t, ε)x(t − εhj ) + A4j (t, ε)y(t − εhj ) dt N

ε

 +

j =0

0 −h



G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη +B2 (t, ε)u(t),

6.4.3.1

t ≥ 0.

(6.125)

Asymptotic Decomposition of the System (6.124)–(6.125)

Let us decompose asymptotically the system (6.124)–(6.125) into two much simpler ε-free subsystems, the slow and fast ones. The slow subsystem is obtained from (6.124)–(6.125) by setting there formally ε = 0. Thus, the slow subsystem has the form dxs (t) = A10 (t, 0)xs (t) + A20 (t, 0)ys (t), dt 0 = A3s (t)xs (t) + A4s (t)ys (t) + B2 (t, 0)us (t),

t ≥ 0, t ≥ 0,

(6.126) (6.127)

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

387

where xs (t) ∈ E n , ys (t) ∈ E m , us (t) ∈ E m , (us is a control); Ais (t) =

N 

 Aij (t, 0) +

j =0

0 −h

Gi (t, η, 0)dη,

i = 3, 4.

(6.128)

The slow subsystem (6.126)–(6.127) is a differential-algebraic system. In what follows, we assume that det A4s (t) = 0,

t ≥ 0.

(6.129)

Subject to this assumption, the slow subsystem (6.126)–(6.127) can be reduced to the differential equation with respect to xs (t) dxs (t) = As (t)xs (t) + Bs (t)us (t), dt

t ≥ 0,

(6.130)

where As (t) = A10 (t, 0) − A20 (t, 0)A−1 4s (t)A3s (t), Bs (t) = −A20 (t, 0)A−1 4s (t)B2 (t, 0).

(6.131)

The differential equation (6.130) also is called the slow subsystem, associated with the system (6.124)–(6.125). The fast subsystem is derived from the fast mode (6.125) of the system (6.124)– (6.125) in the following way: (a) the terms containing the state variable x(t + εη), η ∈ [−h, 0] are removed from (6.125); (b) the transformations of the variables



t = t1 + εξ , y(t1 + εξ ) = yf (ξ ), u(t1 + εξ ) = uf (ξ ) are made in the resulting system, where t1 ≥ 0 is any fixed time instant, ξ ≥ 0 is a new independent variable. Thus, we obtain the system dyf (ξ )  = A4j (t1 + εξ, ε)yf (ξ − hj ) dξ N

 +

j =0

0 −h

G4 (t1 + εξ, η, ε)yf (ξ + η)dη + B2 (t1 + εξ, ε)uf (ξ ).

Now, setting formally ε = 0 in this system and replacing t1 with t yield the fast subsystem dyf (ξ )  = A4j (t, 0)yf (ξ − hj ) dξ N

 +

j =0

0 −h

G4 (t, η, 0)yf (ξ + η)dη + B2 (t, 0)uf (ξ ),

ξ ≥ 0,

(6.132)

388

6 Miscellanies

where t ≥ 0 is a parameter; yf (ξ ) ∈ E m ; yf (ξ + η), η ∈ [−h, 0] is a state variable; uf (ξ ) ∈ E r (uf (ξ ) is a control). The new independent variable ξ is called the stretched time, and it is expressed by the original time t in the form ξ = (t − t1 )/ε. Thus, for any t > t1 , ξ → +∞ as ε → +0. The fast subsystem (6.132) is a differential equation with state delays. It is of a lower Euclidean dimension than the system (6.124)–(6.125). Definition 6.9 The slow subsystem (6.130) is said to be completely controllable at the time instant tc if for any x0 ∈ E n and xc ∈ E n there exists a piecewise continuous control function us (t), t ∈ [0, tc ], for which (6.130) has a solution xs (t), t ∈ [0, tc ], satisfying the initial and terminal conditions xs (0) = x0 ,

xs (tc ) = xc .

Definition 6.10 For a given t ≥ 0, the fast subsystem (6.132) is said to be completely Euclidean space controllable if for any ϕyf (·) ∈ C[−h, 0; E m ] and yc ∈ E m there exist a number ξc > 0, independent of ϕyf (·) and yc , and a piecewise continuous control function uf (ξ ), ξ ∈ [0, ξc ], for which the system (6.132) with the initial and terminal conditions yf (τ ) = ϕyf (τ ),

τ ∈ [−h, 0], yf (ξc ) = yc ,

has a solution.

6.4.3.2

Controllability Conditions for the Slow and Fast Subsystems

Let the n × n-matrix-valued function Ψs (σ ), σ ∈ [0, tc ] be the unique solution of the terminal-value problem dΨs (σ ) = −ATs (σ )Ψs (σ ), σ ∈ [0, tc ), dσ

Ψs (tc ) = In .

Let, for any given t ≥ 0, the m × m-matrix-valued function Ψf (ξ, t) be the unique solution of the initial-value problem dΨf (ξ )  T = A4j (t, 0)Ψf (ξ − hj ) + dξ N

j =0



0 −h

GT4 (t, η, 0)Ψf (ξ + η)dη,

Ψf (ξ ) = 0, ξ < 0;

ξ > 0,

Ψf (0) = Im .

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

Consider the matrices  Ws (tc ) =

tc 0



ξ

Wf (ξ, t) = 0

389

ΨsT (σ )Bs (σ )BsT (σ )Ψs (σ )dσ,

ΨfT (ρ, t)B2 (t, 0)B2T (t, 0)Ψf (ρ, t)dρ,

(6.133)

ξ ≥ 0,

t ≥ 0. (6.134)

By virtue of the results of Sect. 3.3.1 (see Proposition 3.1 and Remark 3.7), we have the following two assertion. Proposition 6.8 The slow subsystem (6.130) is completely controllable at the time instant tc , if and only if det Ws (tc ) = 0. Proposition 6.9 For a given t ≥ 0, the fast subsystem (6.132) is completely Euclidean space controllable if and only if there exists an instant of the stretched time ξ = ξc > 0 such that det Wf (ξc , t) = 0.

6.4.3.3

Parameter-Free Conditions for the Complete Euclidean Space Controllability of the Linear System (6.124)–(6.125)

Here, we assume (I1 ) All roots λ(t) of the equation ⎡ det ⎣λIm −

N  j =0

 A4j (t, 0) exp(−λhj ) −

⎤ 0 −h

G4 (t, η) exp(λη)dη⎦ = 0 (6.135)

satisfy the inequality Reλ(t) < −β for all t ∈ [0, tc ], where β > 0 is some constant. By virtue of the results of Sect. 3.5.1 (see Theorem 3.9), we have the following assertion. Proposition 6.10 Let the assumption (I1 ) be valid. Let the slow subsystem (6.130) be completely controllable at the time instant tc . Let, for t = tc , the fast subsystem (6.132) be completely Euclidean space controllable. Then, there exists a positive number ε1 ≤ ε0 such that, for all ε ∈ (0, ε1 ], the singularly perturbed linear system (6.124)–(6.125) is completely Euclidean space controllable at the time instant tc . Using Propositions 6.8–6.9, Proposition 6.10 can be reformulated as follows.

390

6 Miscellanies

Proposition 6.11 Let the assumption (I1 ) be valid. Let det Ws (tc ) = 0. Let there exist a number ξc > 0 such that det Wf (ξc , tc ) = 0. Then, there exists a positive number ε1 ≤ ε0 such that, for all ε ∈ (0, ε1 ], the singularly perturbed linear system (6.124)–(6.125) is completely Euclidean space controllable at the time instant tc .

6.4.3.4

Euclidean Space Null Controllability of the Nonlinear System (6.122)–(6.123): Parameter-Free Conditions

Based on Corollary 6.2 and Proposition 6.11, we directly obtain the following assertion. Proposition 6.12 Let the assumption (I1 ) be valid. Let det Ws (tc ) = 0. Let there exist a number ξc > 0 such that det Wf (ξc , tc ) = 0. Then, there exists a positive number ε1 ≤ ε0 such that, for all ε ∈ (0, ε1 ], the singularly perturbed nonlinear system (6.122)–(6.123) is Euclidean space null controllable at the time instant tc .

6.4.4 System of the First Type: Parameter-Free Conditions for the Functional Null Controllability In this subsection, we assume that r = m, i.e., u(t) ∈ E m and B2 (t, ε) is an m × mmatrix. Also, we assume (II1 ) The matrix B2 (tc , 0) is invertible. (III1 ) f1 (0, t) = 0 for all t ∈ [tc , tc + ε0 h]. (IV1 ) The vector-valued functions f1 (z, t) and f3 (z, t) satisfy the local Lipschitz condition with respect to z ∈ E n+m uniformly in t ∈ [tc , tc + ε0 h]. Lemma 6.7 Let the assumption (I I1 ) be valid. Then, det Wf (ξc , tc ) = 0 for any ξc > 0. Proof The statement of the lemma directly follows from the definition of the matrix Wf (ξ, t) (see Eq. (6.134)).   Theorem 6.3 Let the assumptions (I1 )–(I V1 ) be valid. Let det Ws (tc ) = 0. Then, there exists a positive number ε∗ , (ε∗ ≤ ε0 ), such that for any ε ∈ (0, ε∗ ], the system (6.122)–(6.123) is functional null controllable at the time instant tc . Proof By virtue of Proposition 6.12 and Lemma 6.7, there exists a positive number ε1 ≤ ε0 such that, for all ε ∈ (0, ε1 ], the system (6.122)–(6.123) is Euclidean space null controllable at the time instant tc . Hence, for any ε ∈ (0, ε1 ] and any ϕx (·) ∈ C[−εh, 0; E n ], ϕy (·) ∈ C[−εh, 0; E m ], there exists the piecewise continuous control function u(t) = u(t, ˜ ε), t ∈ [0, tc ], for which the solution    col x(t), y(t) = col x(t, ˜ ε), y(t, ˜ ε) of the initial-value problem (6.122)–(6.123), (6.117) satisfies the zero terminal conditions, i.e.,

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

x(t ˜ c , ε) = 0,

y(t ˜ c , ε) = 0.

391

(6.136)

Note, that any piecewise continuous control u(t), t ∈ [0, tc ], which differs from ˜ ε) only at the single point t = tc , generates the same solution  u(t, col x(t, ˜ ε), y(t, ˜ ε) of the initial-value problem (6.122)–(6.123), (6.117) in the entire interval t ∈ [0, tc ], i.e., (6.136) holds for any such control. Now, for any ε ∈ (0, ε1 ], we consider the initial-value problem (6.122)– (6.123), (6.117) in the interval [0, tc + εh]. For the system (6.122)–(6.123), we choose the control u(t) = u(t, ˜ ε) in the interval [0, tc ), while in the interval [tc , tc + εh] the piecewise continuous control u(t) will be chosen later. Namely, in the interval [tc , tc + εh], the piecewise continuous control u(t) will be chosen in such a way that the solution of (6.122)–(6.123) with the initial conditions x(ρ) = x(ρ, ˜ ε),

y(ρ) = y(ρ, ˜ ε),

ρ ∈ [tc − εh, tc ]

(6.137)

will be zero for all t ∈ (tc , tc + εh]. For this purpose, we introduce in the consideration the following functions:

x(t ˜ − εhj , ε), t ∈ [tc , tc + εhj ), Xj (t, ε) = j = 1, . . . , N, 0, t ∈ [tc + εhj , tc + εh], (6.138)

Yj (t, ε) =

y(t ˜ − εhj , ε), t ∈ [tc , tc + εhj ), 0, t ∈ [tc + εhj , tc + εh],

j = 1, . . . , N, (6.139)

  Zj (t, ε) = col Xj (t, ε), Yj (t, ε) ,

ω(t, ε) =

tc − t , ε

t ∈ [tc , tc + εh],

t ∈ [tc , tc + εh].

j = 1, . . . , N, (6.140)

(6.141)

Furthermore, we represent the integral in the right-hand side of (6.123) as  0

G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη =   +

−h

ω(t,ε) −h

0



G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη

G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη, t ∈ [tc , tc + εh].

ω(t,ε)

(6.142)

392

6 Miscellanies

The first integral in the right-hand side of (6.142) can be rewritten as  

ω(t,ε)

G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη =

−h ω(t,ε) −h

G3 (t, η, ε)x(t ˜ + εη, ε) + G4 (t, η, ε)y(t ˜ + εη, ε) dη, t ∈ [tc , tc + εh]. (6.143)

The second integral in the right-hand side of (6.142) can be rewritten as   t tc



G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη =

0 ω(t,ε)

     χ −t χ −t G3 t, , ε x(χ ) + G4 t, , ε y(χ ) dχ , t ∈ [tc , tc + εh]. ε ε (6.144)

Thus, due to (6.142)–(6.144), we obtain    t tc

0 −h

G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη =

˜ + εη)dη + G4 (t, η, ε)y(t ˜ + εη) dη+ G3 (t, η, ε)x(t

ω(t,ε) −h

     χ −t χ −t , ε x(χ ) + G4 t, , ε y(χ ) dχ , t ∈ [tc , tc + εh]. G3 t, ε ε (6.145)

Also we note that, due to the smoothness of the matrix-valued function B2 (t, ε) and the assumption (II1 ), B2 (t, ε) is invertible in the entire interval t ∈ [tc , tc + εh] for all ε ∈ [0, ε2 ], where 0 < ε2 ≤ ε1 is some sufficiently small number. For any ε ∈ (0, ε2 ], we choose the abovementioned control u(t), t ∈ [tc , tc + εh] as

−B2−1 (t, ε)



u(t) = u(t, ¯ ε) = ω(t,ε)

−h

G3 (t, η, ε)x(t ˜ + εη) + G4 (t, η, ε)y(t ˜ + εη) dη +

N 

A3j (t, ε)Xj (t, ε) + A4j (t, ε)Yj (t, ε)



j =1

   +f3 (0, t) + f4 Z1 (t, ε), . . . , ZN (t, ε), t .

(6.146)

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

393

Substitution of this control into (6.123) and use of Eq. (6.145) yield the following equation in the interval [tc , tc + εh]: dy(t) = A30 (t, ε)x(t) + A40 (t, ε)y(t) dt      t  χ −t χ −t G3 t, , ε x(χ ) + G4 t, , ε y(χ ) dχ + ε ε tc     +f3 z(t), t − f3 (0, t) + Γ z(t − εh1 ), . . . , z(t − εhN ), t, ε , ε

(6.147)

where   Γ z(t − εh1 ), . . . , z(t − εhN ), t, ε = N     A3j (t, ε) , A4j (t, ε) z(t − εhj ) − Zj (t, ε) + j =1

    f4 z(t − εh1 ), . . . , z(t − εhN ), t − f4 Z1 (t, ε), . . . , ZN (t, ε), t .   Remember that z(·) = col x(·), y(·) . Let us solve the system (6.122), (6.147) in the interval [tc , tc + εh] subject to the initial conditions (6.137). To do this, we use the method of steps [2]. Namely, let us consider this initial-value problem in the interval [tc , tc + εh1 ]. By virtue of (6.138)–(6.140), we have   Γ z(t − εh1 ), . . . , z(t − εhN ), t, ε = 0,

t ∈ [tc , tc + εh1 ].

Due to the latter, Eq. (6.147) becomes dy(t) = A30 (t, ε)x(t) + A40 (t, ε)y(t) dt      t  χ −t χ −t G3 t, , ε x(χ ) + G4 t, , ε y(χ ) dχ + ε ε tc   +f3 z(t), t − f3 (0, t), t ∈ [tc , tc + εh1 ]. ε

(6.148)

Moreover, due to (6.136)–(6.137), the point-wise initial conditions for the system (6.122), (6.148) become x(tc ) = 0,

y(tc ) = 0.

(6.149)

Solving the initial-value problem (6.122), (6.148), (6.149) in the interval [tc , tc + εh1 ] and taking into account the assumptions (III1 )–(IV1 ), we directly obtain that its unique solution is

394

6 Miscellanies

x(t) = x(t, ε) = 0,

y(t) = y(t, ε) = 0,

t ∈ [tc , tc + εh1 ].

(6.150)

Now, based on the solution (6.150) of the initial-value problem (6.122), (6.148), (6.149), let us consider Eq. (6.147) in the interval [tc + εh1 , tc + εh¯ 2 ], where h¯ 2 = min{2h1 , h2 }. Similarly to (6.148), we obtain this equation in the following form: dy(t) = A30 (t, ε)x(t) + A40 (t, ε)y(t) dt        t χ −t χ −t + G3 t, , ε x(χ ) + G4 t, , ε y(χ ) dχ ε ε tc +εh1   +f3 z(t), t − f3 (0, t), t ∈ [tc + εh1 , tc + εh¯ 2 ]. ε

(6.151)

The system (6.122), (6.151), along with the initial conditions x(tc + εh1 ) = 0,

y(tc + εh1 ) = 0,

yields the unique solution x(t) = x(t, ε) = 0,

y(t) = y(t, ε) = 0,

t ∈ [tc + εh1 , tc + εh¯ 2 ].

(6.152)

Continuing to solve the system (6.122), (6.147) in the interval [tc , tc + εh] subject to the initial conditions (6.137) by the method of steps, we obtain (similarly to (6.150), (6.152)) the unique solution of this initial-value problem in the entire interval [tc , tc + εh]: x(t) = x(t, ε) = 0,

y(t) = y(t, ε) = 0,

t ∈ [tc , tc + εh].

Thus, for any ε ∈ (0, ε2 ], the control u(t) = u(t, ¯ ε), t ∈ [tc , tc + εh], given by (6.146), generates the unique zeroth solution of the system (6.122)–(6.123) subject to the initial conditions (6.137) in the entire interval [tc , tc + εh]. Therefore, the control

u(t, ˜ ε), t ∈ [0, tc ), u(t) = u(t, ¯ ε), t ∈ [tc , tc + εh] generates the solution x(t) = x(t, ε), y(t) = y(t, ε), t ∈ [0, tc + εh] of the initialvalue problem (6.122)–(6.123), (6.117) satisfying the terminal conditions (6.118).

The latter, along with Definition 6.5 and the notation ε∗ = ε2 , completes the proof of the theorem.  

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

395

6.4.5 System of the Second Type: Formulation and Some Auxiliary Results In this subsection, we consider the following particular case of the system (6.113)– (6.114):

dx(t)  = A1j (t, ε)x(t − εhj ) + A2j (t, ε)y(t − εhj ) dt N

j =0

 +

0 −h

ε



 

G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)y(t + εη) dη + f1 z(t), t   +f2 z(t − εh1 ), . . . , z(t − εhN ), t + B1 (t, ε)u(t),

  dy(t) = A30 (t, ε)x(t) + A40 (t, ε)y(t) + f3 z(t), t , dt

t ≥ 0,

t ≥ 0.

(6.153)

(6.154)

The linear system, corresponding to (6.153)–(6.154), is a particular case of the system (6.119)–(6.120), and it has the form

dx(t)  = A1j (t, ε)x(t − εhj ) + A2j (t, ε)y(t − εhj ) dt N

 +

j =0

0 −h



G1 (t, η, ε)x(t + εη) + G2 (t, η, ε)y(t + εη) dη +B1 (t, ε)u(t),

ε

6.4.5.1

dy(t) = A30 (t, ε)x(t) + A40 (t, ε)y(t), dt

t ≥ 0,

t ≥ 0.

(6.155)

(6.156)

Asymptotic Decomposition of the System (6.155)–(6.156)

Decomposing asymptotically the system (6.155)–(6.156), we obtain its slow subsystem dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) + B1 (t, 0)us (t), dt 0 = A30 (t, 0)xs (t) + A40 (t, 0)ys (t),

t ≥ 0,

t ≥ 0,

(6.157)

(6.158)

396

6 Miscellanies

where xs (t) ∈ E n , ys (t) ∈ E m , us (t) ∈ E n , (us is a control); Ais (t) =

N  j =0

 Aij (t, 0) +

0

−h

Gi (t, η, 0)dη,

i = 1, 2.

Like in Sect. 6.4.3.1, the slow subsystem (6.157)–(6.158) is a differentialalgebraic system. In what follows, we assume that det A40 (t, 0) = 0,

t ≥ 0.

Subject to this assumption, the slow subsystem (6.157)–(6.158) can be reduced to the differential equation with respect to xs (t) dxs (t) = As (t)xs (t) + B1 (t, 0)us (t), dt

t ≥ 0,

(6.159)

where As (t) = A1s (t) − A2s (t)A−1 40 (t, 0)A30 (t, 0). Like in Sect. 6.4.3.1, the differential equation (6.159) also is called the slow subsystem, associated with the system (6.155)–(6.156). The fast subsystem, associated with the system (6.155)–(6.156), is obtained in the way similar to those for obtaining the system (6.132). Thus, we have dyf (ξ ) = A40 (t, 0)yf (ξ ), dξ

ξ ≥ 0,

(6.160)

where t ≥ 0, is a parameter; ξ ≥ 0 is an independent variable; yf (ξ ) ∈ E m is a state variable. The meaning of the new independent variable ξ and its connection with the original independent variable t are the same as in Sect. 6.4.3.1. However, in contrast with the results of Sect. 6.4.3.1 (see Eq. (6.132)), the fast subsystem (6.160) does not contain a control variable, i.e., it is completely uncontrollable. Hence, the method, proposed in Sects. 6.4.3.2–6.4.3.4 for the ε-free analysis of the Euclidean space null controllability of the First Type system (6.122)–(6.123), is not applicable for such an analysis of the Euclidean space null controllability of the Second Type system (6.153)–(6.154). Therefore, in what follows, we apply another method for this analysis, which is based on the results of Chap. 5.

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

6.4.5.2

397

Parameter-Free Conditions for the Complete Euclidean Space Controllability of the Linear System (6.155)–(6.156)

Here, we assume (I2 ) All roots λ(t) of the equation det [λIm − A40 (t, 0)] = 0

(6.161)

satisfy the inequality Reλ(t) < −β for all t ∈ [0, tc ], where β > 0 is some constant. Let the n × n-matrix-valued function Φs (σ ) be the solution of the following terminal-value problems: dΦs (σ ) = −ATs (σ )Φs (σ ), dσ

σ ∈ [0, tc ),

Φs (tc ) = In .

Let

Φf (ξ ) = exp AT40 (tc , 0)ξ ,

ξ ≥ 0.

From the assumption (I2 ), we directly obtain

Φf (ξ ) ≤ a exp(−βξ ),

ξ ≥ 0,

(6.162)

where a > 0 is some constant. Let



Θf (ξ ) = AT30 (tc , 0)

+∞

Φf (κ)dκ,

ξ ≥ 0.

ξ

Due to the inequality (6.162), the integral in the right-hand side of this equation converges and  −1

exp AT40 (tc , 0)ξ , Θf (ξ ) = −AT30 (tc , 0) AT40 (tc , 0) Consider the matrices  tc ΦsT (σ )B1 (σ, 0)B1T (σ, 0)Φs (σ )dσ, Ms (tc ) =

ξ ≥ 0.

(6.163)

0



+∞

Mf (tc ) = 0

ΘfT (ξ )B1 (tc , 0)B1T (tc , 0)Θf (ξ )dξ.

(6.164)

As a direct consequence of the results of Sect. 5.3.3 (see Proposition 5.4) and Sect. 5.4 (see Theorem 5.1), we have the following assertion.

398

6 Miscellanies

Proposition 6.13 Let the assumption (I2 ) be valid. Let det Ms (tc ) = 0 and

det Mf (tc ) = 0.

(6.165)

Then, there exists a positive number ε1 , (ε1 ≤ ε0 ), such that for all ε ∈ (0, ε1 ], the singularly perturbed linear system (6.155)–(6.156) is completely Euclidean space controllable at the time instant tc . Remark 6.7 Due to the results of Sect. 5.4 (see Lemma 5.2), the matrix Mf (tc ) is nonsingular if and only if the following system is completely controllable at some ξ = ξc > 0: d y(ξ ˜ ) = A40 (tc , 0)y(ξ ˜ ) + A−1 ˜ ), 40 (tc , 0)A30 (tc , 0)B1 (tc , 0)u(ξ dξ

ξ ≥ 0, (6.166)

˜ ) ∈ E r is a control variable. where y(ξ ˜ ) ∈ E m is a state variable; u(ξ Note that the complete controllability of the system (6.166) is defined similarly to such a kind of the controllability for the system (6.130) (see Definition 6.9).

6.4.5.3

Euclidean Space Null Controllability of the Nonlinear System (6.153)–(6.154): Parameter-Free Conditions

Based on Corollary 6.2 and Proposition 6.13, we directly obtain the following assertion. Proposition 6.14 Let the assumption (I2 ) be valid. Let the inequalities in (6.165) be satisfied. Then, there exists a positive number ε1 , (ε1 ≤ ε0 ), such that for all ε ∈ (0, ε1 ], the singularly perturbed nonlinear system (6.153)–(6.154) is Euclidean space null controllable at the time instant tc .

6.4.6 System of the Second Type: Parameter-Free Conditions for the Functional Null Controllability In this subsection, we assume that r = n, i.e., u(t) ∈ E n and B1 (t, ε) is an n × nmatrix. Along with this, we assume (II2 ) The matrix B1 (tc , 0) is invertible. (III2 ) f3 (0, t) = 0 for all t ∈ [tc , tc + ε0 h]. (IV2 ) The vector-valued functions f1 (z, t) and f3 (z, t) satisfy the local Lipschitz condition with respect to z ∈ E n+m uniformly in t ∈ [tc , tc + ε0 h].

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

399

Note, that the assumption (IV2 ) is the same as the assumption (IV1 ) in Sect. 6.4.4. However, for the sake of the reader’s convenience, we repeat here this assumption. Lemma 6.8 Let the assumption (I I2 ) be valid. Then, det Ms (tc ) = 0. Proof Due to the smoothness of the matrix-valued function B1 (t, ε) and the assumption (I I2 ), there exists a number t1 ∈ (0, tc ) such that, for all t ∈ [t1 , tc ], the matrix B1 (t, 0) is invertible. Using Eq. (6.163), we can represent the matrix Ms (tc ) as   Ms t c = +



t1

0  tc t1

ΦsT (σ )B1 (σ, 0)B1T (σ, 0)Φs (σ )dσ ΦsT (σ )B1 (σ, 0)B1T (σ, 0)Φs (σ )dσ.

The first integral in the right-hand side of this equation is at least a positive semidefinite matrix, while the second integral is a positive definite matrix. Therefore, the sum of these integrals is a positive definite matrix, which completes the proof of the lemma.   Theorem 6.4 Let the assumptions (I2 )–(I V2 ) be valid. Let det Mf (tc ) = 0. Then, there exists a positive number ε∗ , (ε∗ ≤ ε0 ), such that for any ε ∈ (0, ε∗ ], the system (6.153)–(6.154) is functional null controllable at the time instant tc . Proof Using Proposition 6.14 and Lemma 6.8, the theorem is proven quite similarly to Theorem 6.3.  

6.4.7 Example 1 Consider the following system, a particular case of (6.122)–(6.123): dx1 (t) = −tx1 (t) + (t + 2)x2 (t) − y(t) dt     + sin x1 (t) − x2 (t) + cos ty(t) − 1, t ≥ 0,

(6.167)

dx2 (t) = (t − 1)x1 (t) − (t + 1)x2 (t) + y(t) dt     − ln 1 + x12 (t) + x22 (t) + arctan y(t)/(t + 1) , t ≥ 0,

(6.168)

ε

dy(t) = x1 (t) − x2 (t) − 2y(t) + x1 (t − ε) + tx2 (t − ε) + y(t − ε) dt

400

6 Miscellanies

 +2

0 −1



 1/4 tηx1 (t + εη) − ηx2 (t + εη) dη + t 1 + x12 (t) + x22 (t) + y 2 (t)

  +(t + 2) cos2 x1 (t − ε) + x2 (t − ε) + y(t − ε) + u(t),

t ≥ 0, (6.169)

where x1 (t), x2 (t), y(t), and u(t) are scalars, i.e., n = 2, m = 1, r = 1. In this example, we study the functional null controllability of the system (6.167)–(6.169) at the time instant tc = 1 for all sufficiently small ε > 0. Comparing the systems (6.167)–(6.169) with the system (6.122)–(6.123), we obtain that in this example: N = 1, h1 = h = 1, and  A10 (t, ε) = A10 (t) =

 −t t +2 , t − 1 − (t + 1)

 A20 (t, ε) = A20 =

A30 (t, ε) = A30 = (1 , −1),

 −1 , 1

A40 (t, ε) = A40 = −2,

A31 (t, ε) = A31 (t) = (1 , t),

A41 (t, ε) = A41 = 1,

G3 (t, η, ε) = G3 (t, η) = (2tη , −2η),

G4 (t, η, ε) = G4 = 0,

B2 (t, ε) = B2 = 1,        sin x1 (t) − x2 (t) + cos ty(t) − 1     , f1 z(t), t = − ln 1 + x12 (t) + x22 (t) + arctan y(t)/(t + 1)    1/4 , f3 z(t), t = t 1 + x12 (t) + x22 (t) + y 2 (t)     2 f4 z(t − ε), t = (t + 2) cos x1 (t − ε) + x2 (t − ε) + y(t − ε) , 

  where z(·) = col x1 (·), x2 (·), y(·) . Thus in this example, the limit conditions (6.115)–(6.116) are valid for l = 1, 3, p = 4. The coefficients of x1 (·), x2 (·), y(·), and u(t) in the linear terms are smooth functions for t ≥ 0, η ∈ [−1, 0]. Moreover, the assumptions (II1 )–(IV1 ) also are valid. Let us show the validity of the assumption (I1 ). In this example, Eq. (6.135) becomes λ + 2 − exp(−λ) = 0.

(6.170)

For all complex numbers λ with Reλ ≥ −0.4, we have the inequality   Re λ + 2 − exp(−λ) > 0.108, meaning that such λ are not roots of Eq. (6.170). Therefore, all roots of this equation satisfy the inequality Reλ < −0.4,

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

401

i.e., the assumption (I1 ) is valid for β = 0.4. Using (6.128) and (6.131), we obtain   A3s (t) = 2 − t , t ,  As (t) = As =

 −2 2 , 1 −1

A4s (t) = A4s = −1,  Bs (t) = Bs =

t ∈ [0, 1],

 −1 , 1

t ∈ [0, 1].

Now, let us show that in this example the matrix Ws (tc ), given by (6.133), is invertible. Since the matrices As (t) and Bs (t) are constant, then due to the results of [14], the invertibility of Ws (tc ) is equivalent to the following condition:   rank Bs , As Bs = n = 2.

(6.171)

Calculating the matrix in the left-hand side of (6.171), we obtain 



Bs , As Bs =



 −1 4 , 1 −2

meaning the validity of the condition (6.171) and, therefore, the invertibility of the matrix Ws (tc ). Thus in this example, all the conditions of Theorem 6.3 are fulfilled. This yields the existence of a positive number ε∗ such that, for all ε ∈ (0, ε∗ ], the system (6.167)–(6.169) is functional null controllable at the time instant tc = 1.

6.4.8 Example 2: Analysis of Controlled Sunflower Equation In this example, we analyze the system (1.9). This system is equivalent to the nonlinear controlled sunflower equation (1.8), (for details, see Sect. 1.1.2). For the sake of the book’s reading convenience, we write this system here once again dx(t) = y(t), dt   dy(t) ε = −c1 y(t) − c2 sin x(t − ε) + u(t), dt

(6.172)

where ε > 0 is a small parameter; c1 > 0 and c2 > 0 are constants; u(t) is a control. The system (6.172) is a particular case of the system (6.122)–(6.123). We study the functional null controllability of (6.172) at the time instant t = tc > 0 for all sufficiently small ε > 0.

402

6 Miscellanies

Comparison of the system (6.172) with the system (6.122)–(6.123) yields that in this example         f1 z(t), t ≡ 0, f3 z(t), t ≡ 0, f4 z(t − ε), t = sin x(t − ε) . Hence, the system (6.172) satisfies the limit conditions (6.115)–(6.116) for l = 1, 3, p = 4. Since the constant c1 is positive, then the assumption (I1 ) is valid with β = c1 . Moreover, the assumptions (II1 )–(IV1 ) also are valid. The slow subsystem (6.130) becomes in this example as dxs (t) 1 = us (t), dt c1

t ∈ [0, tc ].

Since in the slow subsystem the coefficient for the control is nonzero, this subsystem is completely controllable at t = tc . Therefore, Ws (tc ), being a scalar in this example, is nonzero. Thus the system (6.172), satisfies all the conditions of Theorem 6.3. This implies the existence of a positive number ε∗ such that, for all ε ∈ (0, ε∗ ], this system is functional null controllable at the time instant tc .

6.4.9 Example 3 Consider the following system, a particular case of (6.153)–(6.154): dx(t) = x(t) + 2y1 (t) + 5t 2 y2 (t) + tx(t − ε) − y1 (t − ε) + y2 (t − ε) dt  0  0 −2t ηx(t + εη)dη − 10t 2 ηy1 (t + εη)dη −1

−1

  + cos t 2 x(t) + y1 (t) − ty2 (t)   − sin2 x(t − ε) + t 2 y1 (t − ε) + y2 (t − ε) + tu(t), t ≥ 0, (6.173) dy1 (t) = 2x(t) − y1 (t) − 5ty2 (t) dt   − ln 1 + tx 2 (t) + y12 (t) + y22 (t) , t ≥ 0,

(6.174)

dy2 (t) = −x(t) + ty1 (t) − y2 (t) dt   + arctan x(t) − ty1 (t) + y2 (t) , t ≥ 0,

(6.175)

ε

ε

where x(t), y1 (t), y2 (t), and u(t) are scalars, i.e., n = 1, m = 2, r = 1.

6.4 Functional Null Controllability of Some Nonlinear Systems with Small. . .

403

In this example, like in the example of Sect. 6.4.7, we study the functional null controllability of the system (6.173)–(6.175) at the time instant tc = 1 for all sufficiently small ε > 0. Comparing the system (6.173)–(6.175) with the system (6.153)–(6.154), we obtain that in the present example: N = 1, h1 = h = 1, and A10 (t, ε) = A10 = 1, A20 (t, ε) = A20 (t) = (2 , 5t 2 ),     2 −1 − 5t A30 (t, ε) = A30 = , A40 (t, ε) = A40 (t) = , −1 t −1 A11 (t, ε) = A11 (t) = t, G1 (t, η, ε) = G1 (t, η) = −2tη,

A21 (t, ε) = A21 = (−1 , 1),

G2 (t, η, ε) = G2 (t, η) = (−10t 2 η , 0),

B1 (t, ε) = B1 (t) = t,   2  f1 z(t), t = cos t x(t) + y1 (t) − ty2 (t) ,     f2 z(t − ε), t = − sin2 x(t − ε) + t 2 y1 (t − ε) + y2 (t − ε) ,      − ln 1 + tx 2 (t) + y12 (t) + y22 (t)   f3 z(t), t = , arctan x(t) − ty1 (t) + y2 (t)   where z(·) = col x(·), y1 (·), y2 (·) . Thus in this example, the limit conditions (6.115)–(6.116) are valid for l = 1, 3, p = 2. The coefficients of x(·), y1 (·), y2 (·), and u(t) in the linear terms are smooth functions for t ≥ 0, η ∈ [−1, 0]. Moreover, the assumptions (II2 )–(IV2 ) also are valid. Let us show the validity of the assumption (I2 ). In this example, Eq. (6.161) becomes (1 + λ)2 + 5t 2 = 0,

(6.176)

yielding the roots λ1 (t) = −1 +

√

5ti,

λ2 (t) = −1 −

√

5ti,

t ≥ 0,

where i is the imaginary unit. Thus, the real parts of the roots of Eq. (6.176) equal −1, meaning the fulfilment of the assumption (I2 ) with any positive β smaller than 1. Now, let us show that det Mf (tc ) = 0. Due to Remark 6.7, this inequality is equivalent to the complete controllability of the system (6.166) at some ξ = ξc > 0, where tc = 1. Since the coefficients of this system are constant, then due to the results of [14], the complete controllability of this system is equivalent to the following condition:

404

6 Miscellanies

  rank A−1 40 (1)A30 B1 (1) , A30 B1 (1) = m = 2.

(6.177)

Calculating the matrix in the left-hand side of (6.177), we obtain   −1 A40 (1)A30 B1 (1) , A30 B1 (1) =



 −7/6 2 , −1/6 − 1

meaning the validity of the condition (6.177) and, therefore, the complete controllability of the system (6.166). Thus in this example, the matrix Mf (tc ) is nonsingular. Hence in this example, all the conditions of Theorem 6.4 are fulfilled. This yields the existence of a positive number ε∗ such that, for all ε ∈ (0, ε∗ ], the system (6.173)– (6.175) is functional null controllable at the time instant tc = 1.

6.5 Some Open Problems 6.5.1 Complete Euclidean Space Controllability of Linear Systems with State Delays and Nonsmall Control Delays First, we consider the system with small state delays:

dx(t)  = A1j (t, ε)x(t − εhj ) + A2j (t, ε)y(t − εhj ) dt N

j =0

 + +

M 

0 −h



G1 (t, η, ε)x(t + εη)dη + G2 (t, η, ε)y(t + εη) dη 

B1l (t, ε)u(t − gl ) +

0 −g

l=0

H1 (t, ρ, ε)u(t + ρ)dρ, t ≥ 0,

(6.178)

dy(t)  = A3j (t, ε)x(t − εhj ) + A4j (t, ε)y(t − εhj ) ε dt N

 + +

M  l=0

j =0

0 −h

G3 (t, η, ε)x(t + εη) + G4 (t, η, ε)y(t + εη) dη 

B2l (t, ε)u(t − gl ) +

0 −g

H2 (t, ρ, ε)u(t + ρ)dρ, t ≥ 0,

(6.179)

6.5 Some Open Problems

405

where x(t) ∈ E n , y(t) ∈ E m , u(t) ∈ E r (u(t) is a control); N ≥ 1 and M ≥ 1 are integers; ε > 0 is a small parameter; 0 = h0 < h1 < h2 < . . . < hN = h and 0 = g0 < g1 < g2 < . . . < gM = g are some given ε-independent constants; Aij (t, ε), Gi (t, η, ε), Bkl (t, ε), Hk (t, ρ, ε), (i = 1, . . . , 4; j = 0, 1, . . . , N; l = 0, 1, . . . , M; k = 1, 2) are matrix-valued functions of corresponding dimensions, given for t ≥ 0, η ∈ [−h, 0], ρ ∈ [−g, 0], ε ∈ [0, ε0 ], (ε0 > 0); Aij (t, ε) and Bkl (t, ε), (i = 1, . . . , 4; j = 0, 1, . . . , N ; l = 0, 1, . . . , M; k = 1, 2) are continuous for (t, ε) ∈ [0, +∞) × [0, ε0 ]; the functions Gi (t, η), (i = 1, . . . , 4) are piecewise continuous in η ∈ [−h, 0] for any (t, ε) ∈ [0, +∞) × [0, ε0 ], and they are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0]; the functions Hk (t, ρ, ε), (k = 1, 2) are piecewise continuous in ρ ∈ [−g, 0] for any (t, ε) ∈ [0, +∞)×[0, ε0 ], and they are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in ρ ∈ [−g, 0]. Due to the results of Sect. 2.2, for a given u(·) ∈ L2loc [−g, +∞; E r ], the system (6.178)–(6.179) is a linear time-dependent nonhomogeneous functionaldifferential with the state variables x(t), x(t +   system. It is infinite-dimensional  εη) and y(t), y(t + εη) , η ∈ [−h, 0). Moreover, the system (6.178)–(6.179) is singularly perturbed. Equation (6.178) isthe slow mode and the Euclidean part  x(t) of the state variable x(t), x(t + εη) is the slow Euclidean state variable.  Equation (6.179) and the entire state variable y(t), y(t +εη) are the fast mode and the fast state variable, respectively, while y(t) is the fast Euclidean state variable. It is important to note that in the system (6.178)–(6.179) the state delays are small (of order of ε), while the control delays are nonsmall (of order of 1). In what follows, we assume that ε0 ∈ (0, g/ h]. Let tc > g be a given time instant independent of ε. Definition 6.11 For a given ε ∈ (0, ε0 ], the system (6.178)–(6.179) is said to be completely Euclidean space controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , u0 ∈ E r , ϕx (·) ∈ L2 [−εh, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ], ϕu (·) ∈ L2 [−g, 0; E r ], xc ∈ E n and yc ∈ E m there exists a control function u(·) ∈ W 1,2 [0, tc ; E r ] satisfying u(0) = u0 , for which the system (6.178)–(6.179) has a solution subject to the following initial and terminal conditions: x(τ ) = ϕx (τ ),

y(τ ) = ϕy (τ ), τ ∈ [−εh, 0),

u(ρ) = ϕu (ρ), ρ ∈ [−g, 0), x(0) = x0 ,

x(tc ) = xc ,

y(0) = y0 ,

y(tc ) = yc .

The attempt to decompose the system (6.178)–(6.179) into slow and fast subsystems meets the result similar to that in Sect. 6.3. Namely, the slow subsystem exists, while the fast subsystem does not exist. In this case, we can propose the

406

6 Miscellanies

following three possible approaches to derivation of ε-free sufficient conditions for the complete Euclidean space controllability of (6.178)–(6.179), robust with respect to ε > 0 for all its sufficiently small values. The first approach is a generalization of the method proposed in Sect. 6.3, which is based on the necessary and sufficient condition for the complete Euclidean space controllability of a system with state and control delays (see [4]). This approach is rather complicated, because in the case of both, state and control, multiple pointwise and distributed delays the abovementioned condition is very complicated. The second approach is similar to the one, proposed in Chap. 4. Remember, that this approach consists in replacing the analysis of the complete Euclidean space controllability of the original system with the analysis of the Euclidean space output controllability of an auxiliary system. Due to the method of Chap. 4, the auxiliary system, associated with the original system (6.178)–(6.179), consists of these differential equations, the additional singularly perturbed differential equation ε

du(t) = −u(t) + v(t), dt

t ≥ 0,

(6.180)

and the algebraic output equation   ζ (t) = Zcol x(t), y(t), u(t) ,

t ≥ 0,

(6.181)

! " where Z is the (n + m) × (n + m + r)-matrix of the block form Z = In+m , 0 . In (6.180), v(t) is a new control, while u(·) becomes a state variable (the fast one) in the system (6.178)–(6.179), (6.180), (6.181). Although the replacement of the analysis of the complete Euclidean space controllability of the system (6.178)–(6.179) with the analysis of the Euclidean space output controllability of the system (6.178)–(6.179), (6.180), (6.181) looks similar to the one of Chap. 4, actually it is not so. Namely, in Chap. 4, the new fast state variable u(·) in the auxiliary dynamic system is with the small delays (of order of ε). However, in the dynamic system (6.178)–(6.179), (6.180), the new fast state variable u(·) is with the nonsmall delays (of order of 1). This circumstance requires to develop a new analysis of the fundamental matrix and its adjoint matrix for the singularly perturbed system (6.178)–(6.179), (6.180). This analysis is very far of being a trivial task. The third approach also is based on the replacing the analysis of the complete Euclidean space controllability of the system (6.178)–(6.179) with the analysis of the Euclidean space output controllability of a new (auxiliary) system. However, in this approach the additional differential equation differs from (6.180). Namely, this equation is du(t) = u(t) + v(t), dt

t ≥ 0.

(6.182)

The output equation for the dynamic system (6.178)–(6.179), (6.182) is Eq. (6.181).

6.5 Some Open Problems

407

Similarly to the system (6.178)–(6.179), (6.180), in the system (6.178)– (6.179), (6.182) the variable u(·) is a new state variable, while v(t) is a new control. However, in contrast with (6.178)–(6.179), (6.180), in (6.178)–(6.179), (6.182) the state variable u(·) is a slow one. Thus, in (6.178)–(6.179), (6.182) only y(·) is a fast state variable and all its delays are small. This circumstance allows us to analyze the fundamental matrix and its adjoint matrix of the system (6.178)–(6.179), (6.182) in the way similar to that in Sect. 2.3. Moreover, the system (6.178)–(6.179), (6.182), (6.181) can be decomposed asymptotically into slow and fast subsystems. The slow subsystems is dxs (t) = A1s (t)xs (t) + A2s (t)ys (t) dt +



M 

B1l (t, 0)us (t − gl ) +

l=0

0

H1 (t, ρ, 0)us (t + ρ)dρ,

−g

t ≥ 0,

0 = A3s (t)xs (t) + A4s (t)ys (t) +

M 

 B2l (t, 0)us (t − gl ) +

0

−g

l=0

H2 (t, ρ, 0)us (t + ρ)dρ, t ≥ 0,

dus (t) = us (t) + vs (t), dt ζs (t) = xs (t),

t ≥ 0, t ≥ 0,

  where xs (t) ∈ E n , ys (t) ∈ E m , us (t) ∈ E r ; xs (t), ys (t) and us (t), us (t + ρ) , ρ ∈ [−g, 0) are state variables; vs (t) ∈ E r is a control; ζs (t) ∈ E n is an output; and Ais (t) =

N 

 Aij (t, 0) +

j =0

0

−h

Gi (t, η, 0)dη,

i = 1, . . . , 4.

The fast subsystem is dyf (ξ )  = A4j (t, 0)yf (ξ − hj ) + dξ N

j =0



0

−h

G4 (t, η, 0)yf (ξ + η)dη,

ξ ≥ 0,

ζf (ξ ) = yf (ξ ),

ξ ≥ 0,

  where t ≥ 0 is a parameter; yf (ξ ) ∈ E m ; yf (ξ ), yf (ξ + η) , η ∈ [−h, 0) is a state variable; ζf (ξ ) ∈ E m is an output. It is seen that the dynamic part of the slow subsystem contains the control vs (t), while the dynamic part of the fast subsystem does not contain a control. This feature of the fast subsystem, associated with the auxiliary system (6.178)–(6.179), (6.182), (6.181), is the same as the feature of the fast subsystem (5.9), associated with the

408

6 Miscellanies

original system (5.1)–(5.2) studied in Chap. 5. Thus, to derive ε-free conditions of the Euclidean space output controllability for the system (6.178)–(6.179), (6.182), (6.181), one can try to extend the method of Chap. 5 to the controllability analysis of this system. Now, let us consider the system with state delays of two scales:  dx(t)  = A1l (t, ε)x(t − gl ) + A2j (t, ε)y(t − εhj ) dt M

N

j =0

l=0

 + +

M 

0 −g

 G1 (t, ρ, ε)x(t + ρ)dρ + 

B1l (t, ε)u(t − gl ) +

l=0

0

−g

0

−h

H1 (t, ρ, ε)u(t + ρ)dρ,

+

M  l=0

(6.183)

N

j =0

l=0

 +

t ≥ 0,

 dy(t)  = A3l (t, ε)x(t − gl ) + A4j (t, ε)y(t − εhj ) dt M

ε

G2 (t, η, ε)y(t + εη)dη

0 −g

 G3 (t, ρ, ε)x(t + ρ)dρ + 

B2l (t, ε)u(t − gl ) +

0

−g

0

−h

G4 (t, η, ε)y(t + εη)dη

H2 (t, ρ, ε)u(t + ρ)dρ,

t ≥ 0,

(6.184)

where x(t) ∈ E n , y(t) ∈ E m , u(t) ∈ E r (u(t) is a control); ε > 0 is a small parameter; 0 = g0 < g1 < g2 < . . . , < gM = g and 0 = h0 < h1 < h2 < . . . , < hN = h are some given ε-independent constants; Ail (t, ε), Akj (t, ε), Gi (t, ρ, ε), Gk (t, η, ε), Bpl (t, ε), Hp (t, η, ε), (i = 1, 3; k = 2, 4; l = 0, 1, . . . , M; j = 0, 1, . . . , N; p = 1, 2) are matrix-valued functions of corresponding dimensions, given for t ≥ 0, ρ ∈ [−g, 0], η ∈ [−h, 0], ε ∈ [0, ε0 ]; Ail (t, ε), Akj (t, ε), and Bpl (t, ε), (i = 1, 3; k = 2, 4; l = 0, 1, . . . , M; j = 0, 1, . . . , N; p = 1, 2) are continuous for (t, ε) ∈ [0, +∞) × [0, ε0 ]; the functions Gi (t, ρ, ε) and Hp (t, ρ, ε), (i = 1, 3; p = 1, 2) are piecewise continuous in ρ ∈ [−g, 0] for any (t, ε) ∈ [0, +∞)×[0, ε0 ], and they are continuous with respect to (t, ε) ∈ [0, +∞)×[0, ε0 ] uniformly in ρ ∈ [−g, 0]; the functions Gk (t, η, ε), (k = 2, 4) are piecewise continuous in η ∈ [−h, 0] for any (t, ε) ∈ [0, +∞) × [0, ε0 ], and they are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in η ∈ [−h, 0]. Similarly to the system (6.178)–(6.179), for a given u(·) ∈ L2loc [−g, +∞; E r ], the system (6.183)–(6.184) is a linear time-dependent nonhomogeneous functional  differential system.It is infinite-dimensional with the state variables x(t), x(t +ρ) ,  ρ ∈ [−g, 0), and y(t), y(t + εη) , η ∈ [−h, 0). Moreover, the system  (6.183)– (6.184) is singularly perturbed. Equation (6.183) and the state variable x(t), x(t +

6.5 Some Open Problems

409

 ρ) are the  slow mode and  the slow state variable. Equation (6.184) and the state variable y(t), y(t +εη) are the fast mode and the fast state variable. It is important to note that the system (6.183)–(6.184) has the delays of two scales, namely, the nonsmall delays (of order of 1) in the slow state variable, the small delays (of order of ε) in the fast state variable, and the nonsmall delays in the control. Definition 6.12 For a given ε ∈ (0, ε0 ], the system (6.183)–(6.184) is said to be completely Euclidean space controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , u0 ∈ E r , ϕx (·) ∈ L2 [−g, 0; E n ], ϕy (·) ∈ L2 [−εh, 0; E m ], ϕu (·) ∈ L2 [−g, 0; E r ], xc ∈ E n , and yc ∈ E m there exists a control function u(·) ∈ W 1,2 [0, tc ; E r ] satisfying u(0) = u0 , for which the system (6.183)–(6.184) has a solution subject to the following initial and terminal conditions: x(ρ) = ϕx (ρ),

u(ρ) = ϕu (ρ), ρ ∈ [−g, 0), y(τ ) = ϕy (τ ),

τ ∈ [−εh, 0),

x(0) = x0 , x(tc ) = xc ,

y(0) = y0 ,

y(tc ) = yc .

To derive ε-free sufficient conditions, providing the complete Euclidean space controllability of (6.183)–(6.184) for all sufficiently small ε > 0, one can try to use each of the abovementioned approaches to the controllability analysis of the system (6.178)–(6.179).

6.5.2 Euclidean Space Controllability of Linear Systems with Nonsmall Delays First, we consider the system with state delays only:

dx(t)  = A1l (t, ε)x(t − gl ) + A2l (t, ε)y(t − gl ) dt M

 +

l=0

0 −g



G1 (t, ρ, ε)x(t + ρ)dρ + G2 (t, ρ, ε)y(t + ρ) dρ +B1 (t, ε)u(t),

t ≥ 0, (6.185)

410

6 Miscellanies

dy(t)  = A3l (t, ε)x(t − gl ) + A4l (t, ε)y(t − gl ) dt M

ε

 +

l=0

0 −g



G3 (t, ρ, ε)x(t + ρ)dρ + G4 (t, ρ, ε)y(t + ρ) dρ +B2 (t, ε)u(t),

t ≥ 0,

(6.186)

where x(t) ∈ E n , y(t) ∈ E m , u(t) ∈ E r (u(t) is a control); ε > 0 is a small parameter; 0 = g0 < g1 < g2 < . . . , < gM = g are some given ε-independent constants; Ail (t, ε), Gi (t, ρ, ε), Bp (t, ε), (i = 1, . . . , 4; l = 0, 1, . . . , M; p = 1, 2) are matrix-valued functions of corresponding dimensions, given for t ≥ 0, ρ ∈ [−g, 0], ε ∈ [0, ε0 ]; Ail (t, ε) and Bp (t, ε), (i = 1, . . . , 4; l = 0, 1, . . . , M; p = 1, 2) are continuous for (t, ε) ∈ [0, +∞) × [0, ε0 ]; the functions Gi (t, ρ, ε), (i = 1, . . . , 4) are piecewise continuous in ρ ∈ [−g, 0] for any (t, ε) ∈ [0, +∞)×[0, ε0 ], and they are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in ρ ∈ [−g, 0]. The system (6.185)–(6.186) is functional-differential and infinite-dimensional.     The state variables of this system have the form x(t), x(t + ρ) , y(t), y(t + ρ) , ρ ∈ [−g, 0). Moreover, this system is singularly perturbed. Equation (6.185) is a  slow mode of this system, and the state variable x(t), x(t + ρ) is slow. The state   variable y(t), y(t + ρ) and Eq. (6.186) are a fast state variable and a fast mode of (6.185)–(6.186), respectively. It is important to note that in the system (6.185)– (6.186) the delays in both, slow and fast, state variables are nonsmall (of order of 1). Remark 6.8 It is important to emphasize that in this subsection we do not make the assumption similar to (2.151) (see Sect. 2.4.1). In the other words, we do not assume that Ail (t, 0), Gi (t, ρ, 0), (i = 3, 4; l = 1, . . . , M) are identically zero. Therefore, for the system (6.185)–(6.186), it does not exist a fast subsystem in an entire time interval [0, tc ], where tc > g be a given number independent of ε. Along with the dynamic system (6.185)–(6.186), we consider the algebraic delay-free output equation ζ (t) = X (t, ε)x(t) + Y (t, ε)y(t),

t ≥ 0,

(6.187)

where ζ (t) ∈ E q , (q ≤ n + m), is an output; X (t, ε) and Y (t, ε) are matrix-valued functions of corresponding dimensions, given for t ≥ 0 and ε ∈ [0, ε0 ]; X (t, ε) and Y (t, ε) are continuous in (t, ε) ∈ [0, +∞) × [0, ε0 ]. Definition 6.13 For a given ε ∈ (0, ε0 ], the system (6.185)–(6.186), (6.187) is said to be Euclidean space output controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , ϕx (·) ∈ L2 [−g, 0; E n ], ϕy (·) ∈ L2 [−g, 0; E m ] and ζc ∈ E q there exists a control function u(·) ∈ L2 [0, tc ; E r ], for which the system (6.185)–(6.186), (6.187) with the initial and terminal conditions

6.5 Some Open Problems

x(ρ) = ϕx (ρ),

411

y(ρ) = ϕy (ρ), ρ ∈ [−g, 0);

x(0) = x0 ,

y(0) = y0 ,

ζ (tc ) = ζc , has a solution. Due to Remark 6.8, the asymptotic decomposition approach is not applicable to the controllability analysis of the system (6.185)–(6.186), (6.187). Instead, we proposed the following possible approach to derivation of ε-free sufficient conditions for the Euclidean space output controllability of this system, robust with respect to ε > 0 for all its sufficiently small values. This approach consists of two stages. At the first stage, estimates of the fundamental matrix and its adjoint matrix, associated with the dynamic system (6.185)–(6.186), should be obtained. At the second stage, using these estimates and Corollary 3.1, the abovementioned ε-free sufficient conditions for the controllability of the system (6.185)–(6.186), (6.187) are derived. Note, that the estimates of the fundamental matrix and its adjoint matrix, associated with (6.185)–(6.186), cannot be derived in the entire interval [0, tc ]. These esti mates should be derived consecutively in the intervals [0, g1 ), g1 , min{2g1 , g2 } , and so on. Now, let us consider the systems with state and control delays

dx(t)  = A1l (t, ε)x(t − gl ) + A2l (t, ε)y(t − gl ) dt M

 + +

M 

l=0

0 −g

G1 (t, ρ, ε)x(t + ρ)dρ + G2 (t, ρ, ε)y(t + ρ) dρ 

B1l (t, ε)u(t − gl ) +

l=0

0

−g

H1 (t, ρ, ε)u(t + ρ)dρ,

t ≥ 0, (6.188)

dy(t)  = A3l (t, ε)x(t − gl ) + A4l (t, ε)y(t − gl ) dt M

ε

 + +

M  l=0

l=0

0 −g



G3 (t, ρ, ε)x(t + ρ)dρ + G4 (t, ρ, ε)y(t + ρ) dρ 

B2l (t, ε)u(t − gl ) +

0 −g

H2 (t, ρ, ε)u(t + ρ)dρ,

t ≥ 0,

(6.189)

412

6 Miscellanies

where x(t) ∈ E n , y(t) ∈ E m , u(t) ∈ E r (u(t) is a control); ε > 0 is a small parameter; 0 = g0 < g1 < g2 < . . . , < gM = g are some given ε-independent constants; Ail (t, ε), Gi (t, ρ, ε), Bpl (t, ε), Hp (t, ρ, ε), (i = 1, . . . , 4; l = 0, 1, . . . , M; p = 1, 2) are matrix-valued functions of corresponding dimensions, given for t ≥ 0, ρ ∈ [−g, 0], ε ∈ [0, ε0 ]; Ail (t, ε) and Bpl (t, ε), (i = 1, . . . , 4; l = 0, 1, . . . , M; p = 1, 2) are continuous for (t, ε) ∈ [0, +∞)×[0, ε0 ]; the functions Gi (t, ρ, ε) and Hp (t, ρ, ε), (i = 1, . . . , 4; p = 1, 2) are piecewise continuous in ρ ∈ [−g, 0] for any (t, ε) ∈ [0, +∞) × [0, ε0 ], and they are continuous with respect to (t, ε) ∈ [0, +∞) × [0, ε0 ] uniformly in ρ ∈ [−g, 0]. Definition 6.14 For a given ε ∈ (0, ε0 ], the system (6.188)–(6.189) is said to be completely Euclidean space controllable at the time instant tc if for any x0 ∈ E n , y0 ∈ E m , u0 ∈ E r , ϕx (·) ∈ L2 [−g, 0; E n ], ϕy (·) ∈ L2 [−g, 0; E m ], ϕu (·) ∈ L2 [−g, 0; E r ], xc ∈ E n and yc ∈ E m there exists a control function u(·) ∈ W 1,2 [0, tc ; E r ] satisfying u(0) = u0 , for which the system (6.188)–(6.189) has a solution subject to the following initial and terminal conditions: x(ρ) = ϕx (ρ),

y(ρ) = ϕy (ρ),

u(ρ) = ϕu (ρ), ρ ∈ [−g, 0), x(0) = x0 ,

x(tc ) = xc ,

y(0) = y0 ,

y(tc ) = yc .

Similarly to Sect. 6.5.1, the analysis of the complete Euclidean space controllability of the system (6.188)–(6.189) can be replaced with the analysis of the Euclidean space output controllability of a new (auxiliary) system. The dynamics of this auxiliary system consists of Eqs. (6.188)–(6.189) and the differential equation (6.180). The output equation has the form (6.181). To derive ε-free sufficient conditions, providing the Euclidean space output controllability of the system (6.188)–(6.189), (6.180), (6.181) for all sufficiently small ε > 0, one can try to use the abovementioned approach to the controllability analysis of the system (6.185)–(6.186), (6.187).

6.5.3 Complete Euclidean Space Controllability of One Class of Nonlinear Systems with Small State Delays Consider the system   dx(t) = f1 x(t), y(t), x(t − εh), y(t − εh), u(t), t, ε , dt ε

t ≥ 0,

(6.190)

    dy(t) = f2 x(t), x(t − εh), t, ε + A1 x(t), x(t − εh), t, ε y(t) dt

6.6 Concluding Remarks and Literature Review

413

    +A2 x(t), x(t − εh), t, ε y(t − εh) + B x(t), x(t − εh), t, ε u(t),

t ≥ 0, (6.191)

where x(t) ∈ E n , y(t) ∈ E m , u(t) ∈ E r (u(t) is a control); ε > 0 is a small parameter; h > 0 is some given ε-independent constant; f1 (x, y, xd , yd , u, t, ε) is a given vector-valued function of corresponding dimension, continuous for (x, y, xd , yd , u, t, ε) ∈ E n ×E m ×E n ×E m ×E r ×[0, +∞)×[0, ε0 ]; f2 (x, xd , t, ε) is a given vector-valued function of corresponding dimension, continuous for (x, xd , t, ε) ∈ E n × E n × [0, +∞) × [0, ε0 ]; A1 (x, xd , t, ε), A2 (x, xd , t, ε) and B(x, xd , t, ε) are given matrix-valued functions of corresponding dimensions, continuous for (x, xd , t, ε) ∈ E n × E n × [0, +∞) × [0, ε0 ]. Let tc > 0 be a given time instant independent of ε. Definition 6.15 For a given ε ∈ (0, ε0 ], the system (6.190)–(6.191) is said to be completely Euclidean space controllable at the time instant tc if for any ϕx (·) ∈ C[−εh, 0; E n ], ϕy (·) ∈ C[−εh, 0; E m ], xc ∈ E n , and yc ∈ E m there exists a piecewise continuous control function u(t), t ∈ [0, tc ], for which the system (6.190)–(6.191) has a solution subject to the following initial and terminal conditions: x(τ ) = ϕx (τ ),

y(τ ) = ϕy (τ ),

x(tc ) = xc ,

τ ∈ [−εh, 0],

y(tc ) = yc .

The problem is to derive ε-free sufficient conditions, providing the complete Euclidean space controllability of the system (6.190)–(6.191) for all sufficiently small values of ε > 0. This problem is a nontrivial extension of the problem solved in [17], where a singularly perturbed nonlinear controlled system without delays was treated. It seems that the approach, proposed in [17], can be helpful in the controllability analysis of the system (6.190)–(6.191).

6.6 Concluding Remarks and Literature Review In this chapter, four different classes of singularly perturbed controlled time delay systems were considered. In Sect. 6.2, a linear time-dependent controlled system with delays (point-wise and distributed) in the state variables was considered. Matrix of the coefficients for the control in this system is multiplied by the value 1/ε, (ε > 0 is a small parameter), meaning that this system is a high gain control one. The complete Euclidean space controllability of this system is under the study. Subject to proper assumptions and by a proper state transformation this system was converted equivalently to a new system consisting of two modes. The first mode does not contain the control variable, while the second mode contains the control. Moreover, the dimension

414

6 Miscellanies

of the second mode coincides with the control dimension, and the matrix of the coefficients for the control is the identity matrix multiplied by 1/ε. Thus, the second mode of the new system is controlled directly, while the first mode is controlled through the second one. It was shown (see Lemma 6.2) the equivalence of the complete Euclidean space controllability of the initially considered system and the new system. Therefore, in the rest part of Sect. 6.2, the new system is considered as an original system. Due to the coefficient 1/ε for the control, the original system is a high gain control system. Multiplication of the second mode of the original system by ε converts this system to the explicit singular perturbation form. The connection between high gain control and singular perturbation, as well as the use of such a connection for suboptimal control design, were studied in the literature for undelayed systems (see e.g. [11, 12, 16] and references therein) and for time delay systems (see e.g. [5, 7] and references therein). After the original system was converted to an explicit form singularly perturbed system, its complete Euclidean space controllability was studied by the slow–fast decomposition approach. The feature of the slow–fast decomposition of the original system is the following. Although the original system has delays only in the state variables, its slow subsystem is a differential equation with delays in both, state and control, variables. The fast subsystem is a differential equation without delays. It was established in the section that the complete Euclidean space controllability of the slow subsystem yields the complete Euclidean space controllability of the original high gain control system for all sufficiently small values of ε, i.e., for all sufficiently large values of the gain. Some of the results of Sect. 6.2 (see Sects. 6.2.2–6.2.5) were initially reported in [6]. In Sect. 6.3, a singularly perturbed linear time-dependent controlled system with a point-wise control delay was considered. The delay depends on the parameter of singular perturbation, and it is nonsmall (of order of 1). The complete Euclidean space controllability of this system, robust with respect to the parameter of singular perturbation, was studied. Sufficient conditions of this controllability were derived. This derivation is not based on the slow–fast decomposition of the considered system, because the latter does not allow such a decomposition. Instead of the slow– fast decomposition, an asymptotic analysis of the controllability matrix was applied to derive the controllability conditions. Although these conditions are independent of the parameter of singular perturbation, they guarantee the complete Euclidean space controllability of the considered system for all sufficiently small values of this parameter. Some of the results of Sect. 6.3 were reported in [9]. However, in the present book the proofs of these results, especially the proof of Lemma 6.6, are given in a more detailed form. In Sect. 6.4, two types of singularly perturbed nonlinear time-dependent controlled systems with time delays (multiple point-wise and distributed) in the state variables were analyzed. The case, where the delays are small, of the order of a small positive multiplier ε for a part of the derivatives in the systems, was treated.

References

415

In the system of the first type, the slow mode equation is uncontrolled, and this equation does not contain the delayed state variables. In the system of the second type, the fast mode equation is uncontrolled, and it does not contain the delayed state variables. The functional null controllability of the considered systems, robust with respect to the small parameter ε, was studied. For each type of the systems, ε-free conditions, guaranteeing the functional null controllability for all sufficiently small values of ε, were derived. These results were based on the ε-free analysis of the Euclidean space null controllability of the considered systems. The idea to establish the functional null controllability of a time delay system, based on its Euclidean space null controllability and using a control variable at the final interval as a proper function of the state variable at the preceding interval, was applied in the literature in a number of works (see [3, 13, 15]). In these works, simplest unperturbed linear differential-difference systems with constant coefficients were considered. The results of Sect. 6.4 are an essential extension of the application of this idea to the functional null controllability analysis of singularly perturbed nonlinear time-dependent systems with multiple point-wise and distributed delays. The results of this section also extend the results of the work [8] where the coefficients for the linear terms (with respect to the state and the control) of the considered systems depend only on the time. In Sect. 6.4, such coefficients depend not only on the time but also on ε. Moreover, in this section, the functional null controllability is defined with the final time interval following the prescribed time instant, but not preceding such a point (as it was done in [8]). This difference in the definition of the functional null controllability allowed to simplify considerably its ε-free analysis for the considered systems. In Sect. 6.5, five open problems are formulated, and some approaches to their solution are proposed.

References 1. Dauer, J.P., Gahl, R.D.: Controllability of nonlinear delay systems. J. Optim. Theory Appl. 21, 59–70 (1977) 2. Fridman, E.: Introduction to Time-Delay Systems. Birkhauser, New York (2014) 3. Gabasov, R., Kirillova, F.M.: The Qualitative Theory of Optimal Processes. Marcel Dekker, New York (1976) 4. Glizer, V.Y.: Cheap quadratic control of linear systems with state and control delays. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19, 277–301 (2012) 5. Glizer, V.Y: Stochastic singular optimal control problem with state delays: regularization, singular perturbation, and minimizing sequence. SIAM J. Control Optim. 50, 2862–2888 (2012) 6. Glizer, V.Y.: Euclidean space controllability conditions and minimum energy problem for time delay system with a high gain control. J. Nonlinear Var. Anal. 2, 63–90 (2018) 7. Glizer, V. Y.: Saddle-point equilibrium sequence in one class of singular infinite horizon zerosum linear-quadratic differential games with state delays. Optimization 68, 349–384 (2019)

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8. Glizer, V.Y.: Conditions of functional null controllability for some types of singularly perturbed nonlinear systems with delays. Axioms 8(3), 80 (2019) 9. Glizer, V.Y.: Novel conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numer. Algebra Control Optim. (2020). https://doi.org/10.3934/naco. 2020027 10. Glizer, V.Y., Fridman, L.M., Turetsky, V.: Cheap suboptimal control of an integral sliding mode for uncertain systems with state delays. IEEE Trans. Automat. Control 52, 1892–1898 (2007) 11. Glizer, V.Y., Kelis, O.: Solution of a zero-sum linear quadratic differential game with singular control cost of minimizer. J. Control Decis. 2, 155–184 (2015) 12. Glizer, V.Y., Kelis, O.: Singular infinite horizon zero-sum linear-quadratic differential game: saddle-point equilibrium sequence. Numer. Algebra Control Optim. 7, 1–20 (2017) 13. Halanay, A.: On the controllability of linear difference-differential systems. In: Kuhn, H.W., Szegö, G.P. (eds.), Mathematical Systems Theory and Economics. Lecture Notes in Operations Research and Mathematical Economics Book Series, 2nd edn., vol. 12, pp. 329–336. Springer, Berlin (1969) 14. Kalman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5, 102–119 (1960) 15. Kirillova, F.M., Churakova, S.V.: The problem of the controllability of linear systems with an after-effect. Differ. Equ. 3, 221–225 (1967) 16. Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic Press, London (1986) 17. Sannuti, P.: On the controllability of some singularly perturbed nonlinear systems. J. Math. Anal. Appl. 64, 579–591 (1978)

Index

A Adjoint matrix, 30, 42, 108, 109, 118, 154, 159, 162, 305, 340, 406, 407, 411 Asymptotic analysis, 299, 343, 368–377, 414 Asymptotic decomposition, 23, 24, 108, 113–116, 121–125, 194–200, 213, 214, 220–222, 271, 276–278, 281–285, 289, 292, 296, 297, 299, 301–303, 339, 349–352, 355–356, 386–388, 395–396, 411 Asymptotic estimate, 340 Asymptotic expansion, 316 Asymptotic stability, 340, 365

B Block form, 82, 100, 104, 113, 223, 256, 261, 262, 266, 282, 294, 319, 345, 350, 354, 360, 361, 369, 370, 406 Block matrix, 150, 162, 201, 266, 354 Block of matrix, 102, 103, 107, 210, 256, 270, 333 Block of vector, 322 Block representation, 52, 57, 63, 81, 84, 96, 99, 102, 107, 155, 159, 163, 254, 256, 261, 262, 318, 348, 350, 369 Block vector, 21, 25, 60, 88, 112, 192, 218, 227, 228, 280, 300, 344 Block-wise estimate, 108, 356–357, 367–368

C Car-following model, 11–17, 76–81, 210–213, 292–295, 338–340

Control delay, 14, 214, 217–297, 341, 365, 404–409, 411, 414 Control feedback, 1, 126, 180, 184, 187, 236–238, 242, 243, 249, 264, 296, 297 Control function, 2, 113, 115–117, 121, 124, 170, 188, 191, 195, 211, 220, 222–224, 226, 229, 231, 233, 234, 274, 276, 278, 279, 294, 295, 301, 303, 338, 340, 345, 353, 383, 385, 388, 390, 405, 409, 410, 412, 413 Control high-gain, 4, 343–363, 413, 414 Control input, 9, 363 Controllability analysis, 190, 212, 258–268, 278, 297, 339, 341, 343, 408, 409, 411, 413, 415 Controllability complete Euclidean space, 2, 3, 116, 164–165, 192, 196, 200, 202–203, 211, 212, 214, 217–297, 299, 303, 315, 316, 326, 332, 339, 340, 343, 349, 358, 360, 363, 379, 404–409, 412–414 Controllability conditions high dimension, 2, 348–349 lower dimension, 349, 357–358 parameter-free, 2–4, 116, 149–164, 200–213, 222, 249–267, 287–295, 297, 303, 315–327, 343, 357–358, 377–379, 389–394, 397–399 Controllability functional null, 3, 382–404, 415 Controllability impulse-free, 116, 178, 225, 252, 260, 263, 267, 273, 283, 296 Controllability impulse-free output, 116, 122, 124, 153, 171, 172, 225

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Y. Glizer, Controllability of Singularly Perturbed Linear Time Delay Systems, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-65951-6

417

418 Controllability matrix, 182, 316, 343, 349, 366, 368–377, 414 Controllability output Euclidean space, 111–214, 230, 250, 252, 259, 269, 280, 281, 286, 291, 293, 294, 296, 297, 406, 408, 411, 412 Controllability special cases, 164–166, 202–204, 214 Controllability with respect to x(t), 116, 165, 178, 196, 203, 212, 225, 263, 295 Controllability with respect to y(t), 166, 203–204, 211 Controlled car-following model, 210–213, 292–295, 338–340 Controlled differential equation, 7 Controlled sunflower equation, 6, 401–402 Controlled system, 2–4, 7, 10, 13, 14, 112, 116, 213, 217, 219, 292, 299, 363, 382, 413, 414 Control memory-less, 236–238, 249, 296, 297 Control reactivity, 9, 189 Control sampled-data, 1 Control shifted, 11, 189 Control state-feedback, 126, 180, 236, 238, 243, 249, 264, 296, 297 Control transformation linear, 119–125, 197–200, 231–235, 258–260, 285–287, 296, 353–355 Control variable, 1–3, 217, 277, 295–297, 301, 302, 343, 351, 352, 355, 363, 364, 396, 398, 414, 415 D Delayed neutrons, 1, 7, 9 Delay-free control, 223–226, 278–287 Delay in feedback power reactivity, 7 Delay in information, 37 Delay in the reaction, 1, 11, 15 Delays of two scales, 3, 57–85, 108, 190–214, 275–295, 297, 408, 409 Differential-difference system, 269, 332, 415 Distributed delay, 2, 3, 87, 108, 111, 217, 296, 330, 343, 406, 415 E Equation algebraic, 2, 114, 188, 213 Equation algebraic output, 3, 113–115, 121, 122, 194, 198, 199, 406 Equation characteristic, 28, 33, 36–38, 40, 43, 47, 71, 72, 74, 77, 238 Equation controlled, 6, 7, 220, 224, 401–402 Equation differential, 2, 5–7, 11, 23, 24, 28, 33, 34, 36, 37, 39, 40, 59, 60, 71, 72, 80, 87,

Index 88, 108, 114, 115, 122, 124, 130, 132, 133, 151, 153, 168, 171, 175, 177, 180, 190, 195, 199, 213, 220–222, 224, 225, 234, 235, 251, 252, 260, 267–269, 277, 286, 291, 294, 295, 301–307, 324, 334, 349, 364, 365, 387, 388, 396, 406, 414 nonhomogeneous, 34 singularly perturbed, 37, 188, 406 time delay, 2, 5, 11, 87, 195, 199, 213, 277, 291, 349 Equation fast mode, 16, 17, 37, 177, 223, 301, 365, 383, 415 Equation homogeneous, 28, 34, 80 Equation integral, 50, 51, 99, 100, 147 Equation linear, 6 Equation nonlinear, 6, 139 Equation quasi-polynomial, 28, 71, 72, 74, 78, 80, 140, 291 Equation slow mode, 17, 22, 58, 86, 301, 363, 383, 405, 409, 410, 415 Estimate of fundamental matrix, 109, 411 Estimate of roots, 36 Estimate of solution, 356–357, 367–368 Euclidean part of state variable, 22, 58, 86, 113, 114, 214, 224, 225, 282, 405 Evader, 179–181, 183–185 F Fast subsystem, 24, 111, 221, 299, 352 Functional-differential equation, 220, 224, 301 Functional-differential system, 22, 112, 219, 301, 345 Functional part of state variable, 22, 23, 58, 86 Fundamental matrix solution, 21, 25–27, 29, 32, 33, 39, 42, 44, 47, 60–65, 70, 71, 74, 78, 88–92, 96, 108, 109, 117, 118, 154, 159, 162, 305, 340, 406, 407, 411 I Infinite-dimensional system, 191, 276 Input delay, 363–382 K Kalman algebraic criterion, 336 L Linearized model, 7, 73, 188 Linear matrix inequality (LMI), 238, 239, 241, 242, 248, 250, 251, 258, 271, 273, 296, 297 Lyapunov-Krasovskii-like functional, 242

Index M Method of mathematical induction, 72, 76, 80 Method of steps, 72, 75, 96, 97, 393, 394 Model of controlled coupled-core nuclear reactor, 9–11 Model of engagement between two flying vehicles, 179, 214 Model of nuclear reactor dynamics, 7–9

N Neurosystem model, 5–6, 41–46 Nonsmall delay, 1–4, 13, 14, 16, 17, 21, 66, 86–109, 341, 406, 409–412 Nonstandard system, 152–153, 251–252, 258–268, 271

P Parameter-free controllability conditions, 149–164, 200–213, 249–267, 287–295, 315–327, 377–379 Parameter of singular perturbation, 1, 3, 4, 22, 27, 66, 109, 111, 217, 297, 343, 414 Point-wise delay, 2–4, 108, 111, 213, 217, 296, 297, 299, 340, 343, 413–415 Propagation delay, 1, 5 Pursuer, 179–181, 183–185, 187 Pursuit-evasion, 179–187

Q Quadratic form, 320

419 R Riccati-type matrix equations, 125–127, 214, 263–265

S Schur complement theorem, 248 Singularly perturbed system, 2, 23, 111, 217, 301, 343 Slow-fast decomposition, 2, 3, 21, 23–24, 58–60, 87–88, 108, 111, 343, 364–366, 414 Slow subsystem, 23, 113, 220, 299, 350 Small delay, 1–3, 6, 9, 13, 14, 16, 17, 21–58, 66, 108, 112–116, 119–125, 145–190, 214, 297, 326, 406, 409 Stabilizability, 236–238, 242, 249, 296, 297 Stabilizable system, 126, 127, 151, 236, 238, 242, 264, 265, 267 Standard system, 59, 149–152, 249–251 State delay, 2–4, 10, 14, 24, 59, 60, 111–214, 217–297, 299–341, 343, 345, 382–409, 412–413 Stretched time, 24, 115, 222, 269, 303, 365, 388, 389 Sunflower equation, 6–7, 46–48, 401–402

T Tracking model with delay, 36–41

V Volterra integral equation, 99