Stability, Control and Differential Games: Proceedings of the International Conference Stability, Control, Differential Games Scdg2019 [1 ed.] 3030428303, 9783030428303

Table of contents :
Organization
International Program Committee (IPC)
National Organizing Committee (NOC)
Preface
Contents
1 On Unbounded Limit Sets of Dynamical Systems
1.1 Introduction
1.2 Unboundedness of the Connectivity Components of a Limit Set
1.3 On the Number of Connectivity Components of Limit Sets of Planar Systems
1.4 On the Number of Connectivity Components of the ω-Limit Set in Analytic and Polynomial Systems
1.5 Conclusion
References
2 About One Problem of Optimal Control of String Oscillations with Non-separated Multipoint Conditions at Intermediate Moments of Time
2.1 Introduction
2.2 The Formulation of the Problem
2.3 The Problem Solution
2.4 Example
2.5 Conclusion
References
3 Application of Discrete-Time Optimal Control to Forest Management Problems
3.1 Introduction
3.2 Problem Statement
3.2.1 Discrete Dynamics of Forest Areas
3.2.2 Modeling Economic Decision-Making
3.2.3 Optimal Control Problem
3.3 Solution to the Bilinear Optimal Control Problem
3.4 Illustrative Example
3.5 Conclusions
References
4 Program and Positional Control Strategies for the Lotka–Volterra Competition Model
4.1 Introduction
4.2 Model
4.3 Optimal Controls
4.3.1 Killing Cancer Cells
4.3.2 Inhibiting Cancer Cells Proliferation
4.3.3 Cytotoxic and Cytostatic Actions Are Simultaneously Applied
4.3.4 Numerical Results
4.4 Positional Control Strategies
4.4.1 Case 1: Unrestricted Control u and Constant Control w
4.4.2 Case 2: Bounded Control u in[0,umax] and Constant Control w
References
5 Construction of Dynamically Stable Solutions in Differential Network Games
5.1 Introduction
5.2 Game Formulation and Worth of Coalition
5.3 Imputation of Cooperative Network Gains
5.3.1 Time-Consistent Dynamic Shapley Value Imputation
5.3.2 Instantaneous Characteristic Function and IDP
5.4 Conclusions
References
6 On a Differential Game in a System Described by a Functional Differential Equation
6.1 Introduction
6.2 Statement of the Game Problem
6.3 Solvability Conditions of the Game Problem
6.4 Application to Partial Functional Differential Equations
References
7 The Program Constructions in Abstract Retention Problem
7.1 Introduction
7.2 General Notions and Designations
7.3 The Program Operators and Quasistrategies
7.4 The Direct Iterated Procedure
7.5 Main Problem
7.6 Conclusion
References
8 UAV Path Planning in Search and Rescue Operations
8.1 Introduction
8.2 Problem Statement
8.2.1 UAV Motion and Channel Model
8.2.2 Internal Problem: Data Transmission Optimization (DTO)
8.2.3 External Problem: Surveillance Path Planning (SPP)
8.3 Solution of DTO
8.4 Path Optimization Problem in the Deterministic Case
8.5 Stochastic Problem with Full Information
8.6 Numerical Simulation
8.7 Concluding Remarks
References
9 Second-Order Necessary Optimality Conditions for Abnormal Problems and Their Applications
9.1 Some Examples of Abnormal Problems of Analysis
9.2 Second-Order Necessary Optimality Conditions
9.3 Applications. Inverse Function Theorem
References
10 Alternate Pursuit of Two Targets, One of Which Is a False
10.1 Introduction
10.2 Problem Statement
10.3 Main Result
References
11 Block Jacobi Preconditioning for Solving Dynamic General Equilibrium Models
11.1 Introduction
11.2 Preconditioning
11.3 Model and Method
11.4 Results and Discussion
11.5 Conclusion
References
12 Approximate Feedback Minimum Principle for Suboptimal Processes in Non-smooth Optimal Control Problems
12.1 Introduction
12.2 ε-Feedback Minimum Principle
12.3 Examples
References
13 Solving the Inverse Heat Conduction Boundary Problem for Composite Materials
13.1 Introduction
13.2 Direct Problem Statement and Study
13.3 Statement of the Inverse Boundary Value Problem and Its Reduction to the Problem of Calculating the Values of an Unbounded Operator
13.4 Solution to the Problem (13.14)–(13.17)
References
14 On the Problems of Minmax–Maxmin Type Under Vector-Valued Criteria
14.1 Essential Definitions
14.1.1 Pareto Order
14.1.2 The Notion of Vector-Valued MinMax and MaxMin
14.2 The Linear-Quadratic Problem of Control Under a Vector-Valued Criterion
14.3 The Functional of Type Φ Φ Φ Φ(u) + Ψ Ψ Ψ Ψ(v)
14.4 The Functional of Type F(u,v) = [langleu,A1 V rangle, …, langleu, Ar v rangle]'
14.4.1 Case v inmathbbR1
14.4.2 A Necessary Condition for the Violation of the Minmax Inequality
14.4.3 Case min{ dim(u),dim(v) } 2
14.5 Examples
14.6 Conclusion
Reference
15 One Problem of Statistically Uncertain Estimation
15.1 Introduction and Preliminaries
15.2 Statistically Uncertain Filtering
15.2.1 A Filtering Scheme for Inclusions
15.2.2 Another Filtering Scheme for Inclusions
15.3 Problem Formulation and Its Partial Solution
15.4 Conclusion
References
16 The First Boundary Value Problem for Multidimensional Pseudoparabolic Equation of the Third Order in the Domain with an Arbitrary Boundary
16.1 Introduction
16.2 Statement of the Problem
16.2.1 Locally One-Dimensional Difference Scheme (LOS)
16.2.2 Approximation Error of LOS
16.2.3 Stability of LOS
16.2.4 Uniform Convergence of LOS
References
17 Difference Methods of the Solution of Local and Non-local Boundary Value Problems for Loaded Equation of Thermal Conductivity of Fractional Order
17.1 Introduction
17.2 Statement of the Problem
References
18 A Class of Semilinear Degenerate Equations with Fractional Lower Order Derivatives
18.1 Introduction
18.2 Nondegenerate Equations
18.3 Degenerate Case
18.4 Example
References
19 Program Packages Method for Solution of a Linear Terminal Control Problem with Incomplete Information
19.1 Introduction
19.2 Problem Statement
19.3 Package Terminal Control Problem
19.4 Extended Open-Loop Terminal Control Problem
19.5 Solving Algorithm
19.6 Example
References
20 Application of Correcting Control in the Problem with Unknown Parameter
20.1 Introduction
20.2 The Problem Statement
20.3 The Solving Algorithm
20.4 Estimations
20.5 Conclusions
References
21 On Solving Dynamic Reconstruction Problems with Large Number of Controls
21.1 Introduction
21.2 Dynamics
21.3 Input Data
21.4 Reconstruction Problem
21.5 Algorithm for Constructing Solution of the Reconstruction Problem
21.5.1 Input Data Interpolation
21.5.2 Auxiliary Problem
21.5.3 Solution of the Reconstruction Problem
21.6 Example
21.7 Comparison with Another Approach
21.8 Conclusion
References
22 A Class of Initial Value Problems for Distributed Order Equations with a Bounded Operator
22.1 Introduction
22.2 The Initial Value Problem for the Inhomogeneous Equation
22.3 Applications
22.3.1 A System of Integrodifferential Equations
22.3.2 A Class of Initial Boundary Value Problems
References
23 A Solution Algorithm for Minimax Closed-Loop Propellant Consumption Control Problem of Launch Vehicle
23.1 Introduction
23.2 Mathematical Model of Propellant Consumption Process
23.3 Problem Statement
23.4 The Solution Algorithms for Problems 23.1 and 23.2
23.5 Numerical Example
23.6 Conclusion
References
24 A Problem of Dynamic Optimization in the Presence of Dangerous Factors
24.1 Problem Formulation
24.2 Problem (P) and Problem with State Constraints
24.3 Example
References
25 On Piecewise Linear Minimax Solution of Hamilton–Jacobi Equation with Nonhomogeneous Hamiltonian
25.1 Introduction
25.2 Problem Statement
25.3 Known Facts. Reduced Problem
25.4 Exact Constructing the Solution for Reduced Problem
25.4.1 Splitting of Terminal and Limit Functions
25.4.2 Assumptions on Input Data
25.4.3 Instruments for Formalization of Constructions
25.4.4 Elementary Problems
25.4.5 The Structure of Piecewise Linear Solution
25.5 Problem with Prehomogeneous Hamiltonian
25.6 Conclusion
References
26 Investigation of Stability of Elastic Element of Vibration Device
26.1 Introduction
26.2 Statement of Problem and Solution of Aerohydrodynamic Part
26.3 Investigation of Stability
26.4 Investigation of Dynamics
26.5 Conclusion
References
27 A Geometric Approach to a Class of Optimal Control Problems
27.1 Problem Setting
27.2 Solution Method
27.2.1 Scalar Optimization
27.2.2 Recurrent Relation
27.2.3 Auxiliary Control Problems
27.2.4 Geometric Interpretation
27.3 Model Problem
27.4 Conclusion
References
28 Approximating Systems in the Exponential Stability Problem for the System of Delayed Differential Equations
28.1 Introduction
28.2 Problem Statement
28.3 General Solution of the Approximating System
28.4 Results on Exponential Stability
References
29 On Stability and Stabilization with Permanently Acting Perturbations in Some Critical Cases
29.1 Introduction
29.2 Mathematical Models of the Dynamics of Mechanical Systems with Geometric Constraints
29.3 Application of Malkin's Stability Theorem Under Permanently Acting Perturbations in the Problems of Stability of Steady Motions
29.4 Application of Malkin's Stability Theorem Under Permanently Acting Perturbations for the Stabilization Problems of Steady Motions
References
30 Control Problem with Disturbance and Unknown Moment of Change of Dynamics
30.1 Introduction
30.2 Problem Statement
30.3 Main Result
30.4 Example
30.5 Conclusion
References
31 On an Impulse Differential Game with Mixed Payoff
31.1 Introduction
31.2 Problem Statement
31.3 Main Result
31.4 Example
31.5 Conclusion
References
32 Structure of a Stabilizer for the Hamiltonian Systems
32.1 Introduction
32.2 Optimal Control Problem
32.3 Control Problem Analysis
32.4 Stabilizability Conditions
32.5 Nonlinear Stabilizer Structure
32.6 Conclusion
References
33 Calculus of Variations in Solutions of Dynamic Reconstruction Problems
33.1 Introduction
33.2 Dynamic Reconstruction Problem
33.2.1 Assumptions
33.2.2 Statement of Dynamic Reconstruction Problem
33.3 Solution of Dynamic Reconstruction Problem
33.3.1 Auxiliary Constructions
33.3.2 Calculus of Variations Problem
33.3.3 Necessary Optimality Conditions
33.4 An Algorithm for Constructing DR Solution
33.4.1 Estimates
33.4.2 Constructions of Approximations for Control
33.4.3 Solution of DR
33.5 Comparison with Another Approach
33.6 Conclusion
References
34 Control Problems for Set-Valued Motions of Systems with Uncertainty and Nonlinearity
34.1 Introduction
34.2 Main Assumptions and Problem Formulation
34.2.1 Basic Notations
34.2.2 Description of the Main Problem
34.3 Main Constructions and Problems Solution
34.4 Example and Numerical Simulations
34.5 Conclusion
References

Citation preview

Lecture Notes in Control and Information Sciences Proceedings

Alexander Tarasyev Vyacheslav Maksimov Tatiana Filippova   Editors

Stability, Control and Differential Games Proceedings of the International Conference “Stability, Control, Differential Games” (SCDG2019)

Lecture Notes in Control and Information Sciences - Proceedings Series Editors Frank Allgöwer, Universität Stuttgart, Institute for Systems Theory and Automatic Control, Stuttgart, Germany Manfred Morari, University of Pennsylvania, Department of Electrical and Systems Engineering, Philadelphia, USA

This distinguished conference proceedings series publishes the latest research developments in all areas of control and information sciences – quickly, informally and at a high level. Typically based on material presented at conferences, workshops and similar scientific meetings, volumes published in this series will constitute comprehensive state-of-the-art references on control and information science and technology studies. Please send your proposals to Oliver Jackson, Editor, Springer. e-mail: [email protected]

More information about this series at http://www.springer.com/series/15828

Alexander Tarasyev Vyacheslav Maksimov Tatiana Filippova •

•

Editors

Stability, Control and Differential Games Proceedings of the International Conference “Stability, Control, Differential Games” (SCDG2019)

123

Editors Alexander Tarasyev UrB RAS, IIASA Krasovskii Institute of Mathematics and Mechanics Yekaterinburg, Russia

Vyacheslav Maksimov UrB RAS Krasovskii Institute of Mathematics and Mechanics Yekaterinburg, Russia

Tatiana Filippova UrB RAS Krasovskii Institute of Mathematics and Mechanics Yekaterinburg, Russia

ISSN 2522-5383 ISSN 2522-5391 (electronic) Lecture Notes in Control and Information Sciences - Proceedings ISBN 978-3-030-42830-3 ISBN 978-3-030-42831-0 (eBook) https://doi.org/10.1007/978-3-030-42831-0 Mathematics Subject Classification (2010): 49JXX, 49KXX, 49LXX, 34H05, 34H15, 93CXX, 90CXX © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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, express 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 Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Organization

International Program Committee (IPC) Co-chairs Prof. Academician of RAS Alexander B. Kurzhanski (RU) Prof. Academician of RAS Yurii S. Osipov (RU) Vice-Chairs Prof. Corr. Member of RAS Alexander G. Chentsov (RU) Prof. Corr. Member of RAS Vladimir N. Ushakov (RU) Members Prof. Prof. Prof. Prof. Prof. Prof. Prof. Prof. Prof. Prof. Prof. Prof. Prof.

Tamer Basar (US) Academician of RAS Vitalii I. Berdyshev (RU) Academician of RAS Felix L. Chernousko (RU) Arkadii A. Chikrii (UA) Maurizio Falcone (IT) Faina M. Kirillova (BY) Petar V. Kokotovic (US) Mikhail S. Nikolskii (RU) Leon A. Petrosyan (RU) Marc Quincampoix (FR) Corr. Member of RAS Nina N. Subbotina (RU) Corr. Member of RAS Vladimir E. Tretyakov (RU) Vladimir Veliov (AU)

v

vi

Organization

National Organizing Committee (NOC) Chair Prof. Corr. Member of RAS Nikolay Yu. Lukoyanov (RU) Vice-Chairs Prof. Dr. Alexander Tarasyev (RU) Prof. Dr. Vyacheslav Maksimov (RU) Prof. Dr. Tatiana Filippova (RU) Secretaries Prof. Boris V. Digas (RU) Prof. Anastasiia A. Usova (RU) Members Prof. Prof. Prof. Prof. Prof. Prof. Prof. Prof. Prof.

Mikhail I. Gomoyunov (RU) Mikhail I. Gusev (RU) Igor N. Kandoba (RU) Oxana G. Matviychuk (RU) Vladimir G. Pimenov (RU) Anton R. Plaksin (RU) Alexander N. Sesekin (RU) Platon G. Surkov (RU) Alexander A. Uspenskii (RU)

Organizer Krasovskii Institute of Mathematics and Mechanics of Ural Branch of the Russian Academy of Sciences (IMM UB RAS), Yekaterinburg, Russia, www.imm.uran.ru Sponsors Russian Foundation for Basic Research (RFBR), Grant No.19-01-20057 Ministry of Education and Science of the Russian Federation

Preface

This volume contains selected proceedings of the International Conference Stability, Control and Differential Games (SCDG2019, September 16–20, 2019) organized by the Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences in Yekaterinburg, Russia. The conference was aimed at overviewing the latest advantages in the theory of optimal control, stability theory and differential games and, also, in the field of applications of new techniques and numerical algorithms to solving problems in industry, robotics, mechatronics, power and energy systems, economics and ecology, and other fields of applications. During the conference, specialists in optimal control, differential games, and optimization had the opportunity to share their experience with practitioners to discuss new arising problems, to outline important applications, and to describe new research directions. The volume includes chapters with fundamental results in control theory, stability theory, and differential games presented at the conference and chapters devoted to new and interesting approaches in solving important applied problems in control and optimization. The book gives an overview of the current key problems in control theory, dynamical systems, stability theory, and differential games approaches; evaluates the recent major accomplishments; and forecasts some new areas such as hybrid systems, model predictive control, Hamilton–Jacobi equations, and advanced estimation algorithms. Design of very large distributed systems presents a new challenge to control theory including robust control. Control over the networks becomes an important application area. Distributed hybrid control systems including extremely large number of interacting control loops, coordinating large number of autonomous agents, and handling very large model uncertainties are in the center of contemporary research. All these challenges demand development of new theories, analysis and design methods. The book contributes to this series of problems of the indicated fields of the theory and applications. SCDG2019 continues the series of the International Conferences devoted to Academician Nikolay Nikolayevich Krasovskii and organized by the Institute of Mathematics and Mechanics, Russian Academy of Sciences, Yekaterinburg, Russia: vii

viii

Preface

• “Actual Problems of Stability and Control Theory” (APSCT2009, September 21–26, 2009, Yekaterinburg, Russia) devoted to the Ural scientific school on the theory of stability and control, which was founded and headed by Academician N. N. Krasovskii. • “Systems Dynamics and Control Processes” (SDCP2014, September 15–20, 2014, Yekaterinburg, Russia) dedicated to the 90th Anniversary of Academician Nikolay Nikolayevich Krasovskii. Nikolay Nikolayevich Krasovskii (September 7, 1924—April 4, 2012) was a prominent Russian mathematician who worked in the mathematical theory of control, the theory of dynamical systems, stability theory, and the theory of differential games. He was the Chief of the Ural (Yekaterinburg) scientific school in mathematical theory of control and the theory of differential games. The conference covers the following research topics: • • • • • • •

Stability and stabilization, Optimal control theory and differential games, State estimates, Differential equations, Inverse problems of dynamics, Applications of optimal control theory, and Numerical methods.

The conference featured studies of more than 170 authors from 11 countries (Austria, Armenia, Belarus, Italy, Russia, Uzbekistan, Ukraine, Finland, USA, China, and France). The next international conference SCDG will be organized in 2024. Yekaterinburg, Russia November 2019

Alexander Tarasyev Vyacheslav Maksimov Tatiana Filippova

Contents

1

On Unbounded Limit Sets of Dynamical Systems . . . . . . . . . . . . . . Abdulla A. Azamov and Durdona H. Ruzimuradova

2

About One Problem of Optimal Control of String Oscillations with Non-separated Multipoint Conditions at Intermediate Moments of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. R. Barseghyan

3

4

5

6

Application of Discrete-Time Optimal Control to Forest Management Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrey Krasovskiy and Anton Platov Program and Positional Control Strategies for the Lotka–Volterra Competition Model . . . . . . . . . . . . . . . . . . Nikolai L. Grigorenko, Evgenii N. Khailov, Anna D. Klimenkova, and Andrei Korobeinikov

1

13

27

39

Construction of Dynamically Stable Solutions in Differential Network Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leon Petrosyan and David Yeung

51

On a Differential Game in a System Described by a Functional Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. A. Chikrii, A. G. Rutkas, and L. A. Vlasenko

63

7

The Program Constructions in Abstract Retention Problem . . . . . . Alexander Chentsov

75

8

UAV Path Planning in Search and Rescue Operations . . . . . . . . . . B. M. Miller, G. B. Miller, and K. V. Semenikhin

87

9

Second-Order Necessary Optimality Conditions for Abnormal Problems and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . Aram V. Arutyunov

99

ix

x

Contents

10 Alternate Pursuit of Two Targets, One of Which Is a False . . . . . . 107 Evgeny Ya. Rubinovich 11 Block Jacobi Preconditioning for Solving Dynamic General Equilibrium Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 A. P. Gruzdev and N. B. Melnikov 12 Approximate Feedback Minimum Principle for Suboptimal Processes in Non-smooth Optimal Control Problems . . . . . . . . . . . 127 V. A. Dykhta 13 Solving the Inverse Heat Conduction Boundary Problem for Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 V. P. Tanana and A. I. Sidikova 14 On the Problems of Minmax–Maxmin Type Under Vector-Valued Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Yu. Komarov and Alexander B. Kurzhanski 15 One Problem of Statistically Uncertain Estimation . . . . . . . . . . . . . 157 B. I. Ananyev 16 The First Boundary Value Problem for Multidimensional Pseudoparabolic Equation of the Third Order in the Domain with an Arbitrary Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Murat Beshtokov 17 Difference Methods of the Solution of Local and Non-local Boundary Value Problems for Loaded Equation of Thermal Conductivity of Fractional Order . . . . . . . . . . . . . . . . . . . . . . . . . . 187 M. H. Beshtokov and M. Z. Khudalov 18 A Class of Semilinear Degenerate Equations with Fractional Lower Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Marina V. Plekhanova and Guzel D. Baybulatova 19 Program Packages Method for Solution of a Linear Terminal Control Problem with Incomplete Information . . . . . . . . . . . . . . . . 213 Nikita Strelkovskii and Sergey Orlov 20 Application of Correcting Control in the Problem with Unknown Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Vladimir Ushakov and Aleksandr Ershov 21 On Solving Dynamic Reconstruction Problems with Large Number of Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Evgenii Aleksandrovitch Krupennikov 22 A Class of Initial Value Problems for Distributed Order Equations with a Bounded Operator . . . . . . . . . . . . . . . . . . . . . . . 251 Vladimir E. Fedorov and Aliya A. Abdrakhmanova

Contents

xi

23 A Solution Algorithm for Minimax Closed-Loop Propellant Consumption Control Problem of Launch Vehicle . . . . . . . . . . . . . 263 A. F. Shorikov and V. I. Kalev 24 A Problem of Dynamic Optimization in the Presence of Dangerous Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Sergey M. Aseev 25 On Piecewise Linear Minimax Solution of Hamilton–Jacobi Equation with Nonhomogeneous Hamiltonian . . . . . . . . . . . . . . . . . 283 L. G. Shagalova 26 Investigation of Stability of Elastic Element of Vibration Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Petr A. Velmisov and Andrey V. Ankilov 27 A Geometric Approach to a Class of Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Dmitrii R. Kuvshinov 28 Approximating Systems in the Exponential Stability Problem for the System of Delayed Differential Equations . . . . . . . . . . . . . . 315 R. I. Shevchenko 29 On Stability and Stabilization with Permanently Acting Perturbations in Some Critical Cases . . . . . . . . . . . . . . . . . . . . . . . 325 Aleksandr Ya. Krasinskiy 30 Control Problem with Disturbance and Unknown Moment of Change of Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Viktor I. Ukhobotov and Igor’ V. Izmest’ev 31 On an Impulse Differential Game with Mixed Payoff . . . . . . . . . . . 345 Igor’ V. Izmest’ev 32 Structure of a Stabilizer for the Hamiltonian Systems . . . . . . . . . . 357 Anastasiia A. Usova and Alexander M. Tarasyev 33 Calculus of Variations in Solutions of Dynamic Reconstruction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Nina N. Subbotina 34 Control Problems for Set-Valued Motions of Systems with Uncertainty and Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . 379 Tatiana F. Filippova and Oxana G. Matviychuk

Chapter 1

On Unbounded Limit Sets of Dynamical Systems Abdulla A. Azamov and Durdona H. Ruzimuradova

Abstract This paper is focused on studies of properties of unbounded ω-limit sets of dynamical systems. It is proved that if the ω-limit set  is not connected, then each of its components is unbounded, which clarifies the well-known property confirming that if the ω-limit set is not connected then it is unbounded. It is established that the family of connectivity components of the ω-limit set of a planar analytic system can be finite or countable while nonanalytic systems may admit the ω-limit set with continuum family of components. Examples of an analytic system possessing ω-limit set with infinite number of components and a polynomial system where the ω-limit set has exactly N connectivity components are given. Some open problems are formulated as well.

1.1 Introduction Consider a dynamical system z˙ = f (z) ,

(1.1)

z ∈ Rd , d ≥ 2. Further, we will deal with a fixed trajectory z (t) passing through some initial point ξ , z (0) = ξ and its ω-limit set . The case of  = ∅ will be interesting only. Well-known  is always closed invariant set. If  is bounded, then it is nonempty connected compact set [2, 3, 12].  can be a point or a closed trajectory in simplest A. A. Azamov (B) Institute of Mathematics Named After V. I. Romanovskii, 81, M.ulugbek Str., 100170 Tashkent, Uzbekistan e-mail: [email protected] D. H. Ruzimuradova National University of Uzbekistan, 4, University Str., 100174 Tashkent, Uzbekistan e-mail: [email protected]

© Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_1

1

2

A. A. Azamov and D. H. Ruzimuradova

Fig. 1.1 A quadratic system with the unbounded limit set

cases but in the general case  may have rather complicated structure especially for d ≥ 3. The structure of  was studied rigorously in the case of d = 2 [2, 4, 5, 9, 10, 18]. In this paper properties of a number of connectivity components of  when it is unbounded are studied. First of all, let us bring a sample showing that the ω-limit set may be unbounded even in quadratic systems. It looks x˙ = x + y + x 2 , y˙ = x (y − 1) .

(1.2)

If we take the point (1, 0) as ξ , then  will be the strait y = 1. In order to check this, firstly we note that (1.2) has a unique singular point (0, 0), which is an unstable focus, and the line y = 1 is invariant. In polar coordinates the system (1.2) has the form r˙ = r cos ϕ (cos ϕ + r ) , ϕ˙ = −1 + 21 sin 2ϕ. So that −

3 1 ≤ ϕ˙ (t) ≤ − . 2 2

(1.3)

Consider the behavior of trajectories in the invariant region D = {(x, y) | y < 1}. It turns out that if we put B(x, y) = (1 − y)−3 , then div[B(x, y) f (x, y)] = (1 − y)−3 . Consequently, according to Dulac’s theorem ([13], Sect. 3.9) the system (1.2) cannot have a closed trajectory in the region D. Due to (1.3) the system (1.2) has monodromy mapping ϕ : I → I , where I = {(0, s) | 0 < s < 1}. As ϕ (s) > s for small positive s and due to instability of (0, 0) we may assume ϕ (s) > s for all s ∈ I , otherwise the system (1.2) would have a closed trajectory in D. This implies what was required. One of the trajectories of the system (1.2) is shown in Fig. 1.1.

1 On Unbounded Limit Sets of Dynamical Systems

3

1.2 Unboundedness of the Connectivity Components of a Limit Set According to the property mentioned above, a disconnected ω-limit set is always unbounded. Of course this statement does not exclude possibility that  can have bounded connectivity components together with some unbounded components or it might be a case when all components are bounded, but their union is unbounded. Therefore the following statement clarifies the property mentioned above. Theorem 1.1 If the ω-limit set of the system (1.1) is not connected, then each of its connectivity component is unbounded. Proof Let ω-limit set  of the trajectory z (t) be disconnected and suppose some of its connectivity components are bounded. Denote one of them 0 and put ∗ = \0 . It is clear that ∗ is not empty. If  is bounded then the sets 0 and ∗ are compact that implies (1.4) dist (0 , ∗ ) = inf |x − y| > 0. x∈0 y∈∗

Such a relation lays under proof of connectedness of  in the case when  is bounded [13]. Obviously, the relation (1.4) remains valid in the case when 0 is compact and ∗ is only closed. In the considering case such an argument is not valid, because now 0 is still compact  0 may not be closed (as in the example 0 = {0} × [0, +∞),  but ∗ = \  = 0, 21 , 13 , 41 , . . . × [0, +∞)). Of course  can be represented in the form  = 1 ∪ 2 , where both of sets 1 , 2 are nonempty closed and 1 ∩ 2 = ∅ but it may turn out both of them unbounded that does not allow to yield the relation dist (1 , 2 ) = inf |x − y| = 2δ > 0. x∈1 y∈2

In order to deal with this obstacle we use Alexandroff compactification ([7], Chap. 5, Theorem 21). In other words, we continue the dynamical system (1.1) from the space Rd onto the sphere Sd , Sd ⊂ Rd+1 , by means of the stereographic projection σ : Rd → Sd adding the pole (0, . . . , 0, 1) that will be a singular point for the continued system. In this case the trajectory z (t) goes into the trajectory σ [z (t)] on Sd . The images of components of  will lie in the open subset Sd \ (0, . . . , 0, 1). If α , α ∈ A, is a family of unbounded components of , then their union will be  turned into one component ∗ = α∈A σ (α ) ∪ (0, . . . , 0, 1) of the ω-limit set of the trajectory σ [z (t)]. Of course, ∗ is also compact and the limit set is still disconnected. Thus there are two nonempty disjoint closed sets 1 and 2 such that

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A. A. Azamov and D. H. Ruzimuradova

= 1 ∪ 2 . As 0 is connected we may suppose σ (0 ) ⊂ 1 . Now 1 and 2 are both compact subsets of Sd , that is why dist ( 1 , 2 ) > 0. Therefore 1 possesses a neighborhood W that does not intersect with 2 . Now let us return to the phase space Rd from Sd and denote 1 = σ −1 ( 1 ), 2 = σ −1 ( 2 \ (0, . . . , 0, 1)). Then as 1 is compact, 2 is closed and 1 ∩ 2 = ∅, therefore now the relation (1.4) is true. Further, considerations for the case of bounded ω-limit set can be repeated. For that we consider the open δ-neighborhoods δ1 = 1 + δ D and δ2 = 2 + δ D of the sets 1 and 2 , respectively, where D denotes the open ball |z| < 1. Trajectory z (t) enters each of these sets δ1 , δ2 infinite times. Let z (t2k ) ∈ δ2 , z (t2k+1 ) ∈ δ1 , t1 < t2 < t3 < · · · . Now consider the function ϕ (t)  = dist (z (t) , 1 ). It is continuous t < δ. Thus ϕ and ϕ (t2k ) > δ and ϕ (t2k+1 ) k    = δ for some t k ∈ (t2k , t2k+1 ) , k = 0, 1, . . . . It means z t k ∈ ∂ δ1 . The set ∂ δ1 is compact as the boundary  of t → +∞ such that z t kl the bounded set. Therefore, there exists asubsequence k l  δ z ∈ ∂  z ∈ . On the other , which implies converges as l → ∞ to some point 1    hand, ϕ t kl = δ. The obtained contradiction ends the proof.

1.3 On the Number of Connectivity Components of Limit Sets of Planar Systems Theorem 1.2 Let  be the disconnected ω-limit set of the trajectory z (t) of the dynamical system (1.1). If each connectivity component of  contains at least one non-singular point, then the number of components of  is not more than countable. Proof Due to the disconnectedness of , the trajectory z (t) does not intersect with . Let 0 be an arbitrary connectivity component of  and f (p) = 0 for p ∈ 0 . Let U be a neighborhood of the point p such that f (z) = 0, ∀ z ∈ U and n be the unit vector, which is orthogonal to the vector f (p). We choose ε, ε > 0, in such a way that p + ns ∈ U for |s| ≤ ε. Let zs (t) be the trajectory of (1.1) passing through zs (0) = p + ns. If it is necessary, decreasing positive ε one may guarantee that the inclusion zs (t) ∈ U holds for all t, s, |t| ≤ ε, |s| ≤ ε. Put ε = {zs (t) | |t| ≤ ε, |s| ≤ ε } (see Fig. 1.2). It is easy to see that ε is a neighborhood of the point p. Consequently, − z (t) enters at least one of the subregions + ε = {zs (t) | |t| ≤ ε, 0 < s < ε }, ε = {zs (t) | |t| ≤ ε, −ε < s < 0 } infinite times. For definiteness, + ε will be considered having this property. We show that + ε ∩ (\0 ) = ∅ for some ε, ε > 0. Suppose the contrary: for ∩ (\0 ). We choose a positive δ, such that any positive ε there is q, such that q ∈ + ε δ < ε and q ∈ / ε,δ = {zs (t) | |t| ≤ ε, 0 ≤ s < δ}. It is clear that z (t) enters semineighborhood ε,δ = {zs (t) | |t| ≤ ε, 0 ≤ s < δ} of point p infinite times. Let τ1

1 On Unbounded Limit Sets of Dynamical Systems

5

Fig. 1.2 Construction of “Bendixson’s bag” n

q

z( 1 )

Π ε+ z( 2 )

f ( p)

p

Πε

0

Π ε-

and τ2 be two sequential times when z (t) intersects the transversal p + ns, 0 < s < δ, i.e., z (τi ) = p + nsi , i = 1, 2. Then the segment of the transversal between points z (τ1 ) , z (τ2 ) and the arc of the trajectory z (t) , τ1 ≤ t ≤ τ2 form a closed Jordan curve  that divides R2 into two parts W1 and W2 . By construction, the points p and q lie in the different parts W1 and W2 . Let W1 be a part that is bounded. Then it forms a “Bendixson’s bag”, which implies  ⊂ W1 . This relation contradicts unboundedness of . Thus, the semi-neighborhood + ε of the point p does not intersect \0 . Choosing with rational coordinates, we get the map 0 → ξ0 . As it has been a point ξ0 ∈ + ε proved above, if 0 and 1 are different components of  and ξ0 ∈ 0 , ξ1 ∈ 1 , then ξ0 = ξ1 . Therefore, the family of the connectivity components of  is at most countable.  The proof shows that the following statement is actually true. Corollary 1.1 If each connectivity component of , with the exception of no more than a countable number of them, contains a non-singular point, then the number of connectivity components is finite or countable. Remark Note that the condition of Theorem 1.2 is important. Namely, it is easy to construct a dynamical system of the class C∞ , whose ω-limit set  consists of the set K × [1, +∞), where K is the classic Cantor set. Here  consists of singular points only.

1.4 On the Number of Connectivity Components of the ω-Limit Set in Analytic and Polynomial Systems We construct an analytic system with , consisting of the infinite number of connectivity components. The task belongs to the type of inverse problems, which is similar to one considered in [1, 6, 8, 14–17] and connected with the problem  discussed in [11]. For this purpose, we take the function F (x, y) = ψ (x, y) 1 − y 2 cos x −1  where ψ (x, y) = 1 + x 2 + y 4 . The level line F (x, y) = 0, which is the graph of the functions y = ± (cos x)−1/ 2 , consists of infinite number of branches.

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The function F (x, y) reaches its maximum, equal to 1, only at the point (0, 0). In addition, F (x, y) → 0 as x 2 + y 2 → ∞. Therefore, the level lines F (x, y) = c for 0 < c < 1 are closed curves fulfilling the region 0 < F (x, y) < 1. We denote this region Z and note it is invariant for the systems (1.5) and (1.6) written below. First consider the Hamiltonian system   x˙ = ∂∂Fy = −2y ψ (x, y) cos x − 4y 3 ψ 2 (x, y) 1 − y 2 cos x ,   y˙ = − ∂∂ Fx = −y 2 ψ (x, y) sin x + 2x ψ 2 (x, y) 1 − y 2 cos x ,

(1.5)

for which F (x, y) serves as Hamilton’s function. Therefore, the phase point (x, y), lying in the region Z moves along a level line F (x, y) = c for some c, 0 < c < 1, in the counter-clockwise direction.  Further, we  modify the system (1.5), perturbing in the direction of the vector −Fx , −Fy that is external normal to the level line F (x, y) = c, 0 < c < 1, such that the line F (x, y) = 0 stays an integral: x˙ = Fy (x, y) − λF (x, y) Fx (x, y) , y˙ = −Fx (x, y) − λF (x, y) Fy (x, y) ,

(1.6)

where λ is a positive number (its concrete value will be chosen below). Let (x0 , y0 ) ∈ Z . Further, z (t) denotes the trajectory (x (t) , y (t)) of the system (1.6) with the initial condition x (0) = x0 , y (0) = y0 . Theorem 1.3 The ω-limit set  of the trajectory z (t) consists of the line F (x, y) = 0 and therefore has an infinite number of components. The level line F (x, y) = 0 and the trajectory z (t) are shown in Fig. 1.3.

Fig. 1.3 An analytical system with infinite number of connectivity components

1 On Unbounded Limit Sets of Dynamical Systems

7

Proof The phase point of the system (1.6) still remains in the region Z and moves in the counter-clockwise direction along the trajectory z (t). Since z (t) = 0 for all t, then   d F (z (t)) = −λF (z (t)) Fx2 (z (t)) + Fy2 (z (t)) < 0. dt This implies that z (t) goes away from (0, 0), intersecting each line F (x, y) = c (0 < c < 1) in the direction of decreasing of c. Now we set 



D1+ (n) = (x, y) | π 2 + 2π n < x < 3π 4 + 2π n, y > 1 , 



D2+ (n) = (x, y) | 5π 4 + 2π n < x < 3π 2 + 2π n, y > 1 , and study behavior of z (t) in the region D1+ (n) ∪ D2+ (n), where n is some integer. The derivative y˙ = −Fx (x, y) − λF (x, y) Fy (x, y) is negative in the region D1+ (n) for sufficiently large y. Indeed let us estimate the first summand −Fx (x, y) in region D1+ (n). Since 1 − y 2 cos x < 1 ψ (x, y) and sin x > 2−1/ 2 for (x, y) ∈ D1+ (n), then  

y2 Fx (x, y) = y 2 ψ (x, y) sin x − 2xψ 2 (x, y) 1 − y 2 cos x > ψ (x, y) √ − 2x . 2

(1.7) The inequality (1.7) implies Fx (x, y) > 0 for y > it can be clearly seen that sin x Fx (x, y) = 2 + O y



1 y3

√ 4

8x 2 , (x, y) ∈ D1+ (n) and

 ,

(1.8)

when (x, y) ∈ D1+ (n) and y → ∞. Note that sin x > 0 in the considered region. Further, the estimation     y2 1 + y2 y 2 1 − y 2 cos x  ≤ 1. Therefore (see (1.5))    Fy (x, y) ≤

10y 10 < 3. 2 4 1+x +y y

(1.8) and (1.9) show y˙ < 0 for sufficiently large y, as required.

(1.9)

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A. A. Azamov and D. H. Ruzimuradova

Similarly, it can be proved y˙ > 0 for (x, y) ∈ D2+ (n) for sufficiently large y. Obviously, all discussions can be provided for the regions 



D1− (n) = (x, y) | π 2 + 2π n < x < 3π 4 + 2π n, y < −1 , 



D2− (n) = (x, y) | 5π 4 + 2π n < x < 3π 2 + 2π n, y < −1 as well. Namely, y˙ < 0 for (x, y) ∈ D1− (n) and y˙ > 0 for (x, y) ∈ D2− (n), if |y| is sufficiently great. Further, we show that an arc of the trajectory z (t), lying in the strip 



D + (n) = (x, y) | π 2 + 2π n < x < 3π 2 + 2π n, y > 1 , is convex at least for sufficiently great y, y > 0. To show that it is enough to check d2 y < 0 when y → +∞. Indeed, dx 2 sin x Fx = 2 + O y x˙ =

2 cos x +O y3

x¨ = Therefore, x˙ =





sin 2x +O y6

2 cos x y3

+O

1 y3

1 y4

1 y4









1 y7



2 cos x , Fy = +O y3 , y˙ = −

sin x +O y2

 , y¨ = −

2 +O y5







1 y4 1 y3

1 y6

,  ,

 .

< 0 for (x, y) ∈ D + (n). Finally,

  −y 2 − sin2 x d2 y x˙ y¨ − x¨ y˙ + O (1) = = dx 2 4 cos2 x ˙ 3 (x) for (x, y) ∈ D + (n). Thus, we conclude that y  is negative (positive) for y > 0 (respectively, y < 0) when |y| → +∞. The established properties imply that the trajectory of z (t) cannot go to ∞ in the regions D ± (n) and must return to the strip |y| < 1. Now let us study the behavior of the trajectory of system (1.6) in the region |y| ≤ 1. To do this, we yield upper estimation for the addend λF (x, y) Fx (x, y). It is easy to see 1 + y2 |λF Fx | < λ 1 + y4



   2 |x| 1 + y 2 y2 10λ + . 2 < 2 4 1+x +y 1 + x2 1 + x 2 + y4

1 On Unbounded Limit Sets of Dynamical Systems

9

Thus, even for λ = 1 the larger the value of |x|, the closer the vector fields of (1.5) and (1.6) become to each other. Therefore, the trajectories of the system (1.6) in the strip |y| ≤ 1 also cannot go to ∞ as x → +∞ or x → −∞. This implies existence of monodromy mappings of each of the rays { (x, 0) | x > 0} and {(x, 0) | x < 0} to the other one. The established properties of the trajectory z (t) imply that its ω-limit set coincides with the line F (x, y) = 0.  Theorem 1.4 For every positive integer N there exists a polynomial dynamical system, with a trajectory that its limit set consists of N connectivity components. Proof We begin with the following quadratic function: F0 (x, y) = kx 2 − y + k, where k is a positive parameter. The level line F0 (x, y) = 0 consists of a parabola. The value of the parameter k is equal to the distance between the vertex of the parabola and the origin (0, 0), and at the same time characterizes the curvature of the parabola in its vertex. Let N be an arbitrary positive integer. We set k = coth Nπ and define the function 

2π j 2π j − y sin F j (x, y) = k x cos N N

2



2π j 2π j + x sin − y cos N N

 + k,

for j = 1, 2, ..., N . Obviously, the line F j (x, y) = 0 is obtained by turning the parabola F0 (x, y) = 0. Further, introduce into considerations the fractional rational function N j=1 F 2πN j (x, y) F (x, y) =  2N . 1 + x 2 + y2 Thus, the level line F (x, y) = 0 consists of N lines and each of them does not intersect others. In other words, the line F (x, y) = 0 has N parabolic branches. Now we will construct the Hamiltonian system for which F (x, y) serves as Hamilton’s function and modify it perturbing in the direction of the normal vector −Fx , −Fy saving F (x, y) = 0 as an integral: x˙ = Fy + λFx F, y˙ = −Fx + λFy F.

(1.10)

The level line F (x, y) = 0 and one of the trajectories are shown in Fig. 1.4. It can be easily proved that if x (t) , y (t) is a trajectory of (1.10), passing through a point (x0 , y0 ), F (x0 , y0 ) > 0, then its ω-limit set will be just the line F (x, y) = 0. Now the statement of Theorem 1.4 follows the fact that trajectories of (1.10) coincide with trajectories of the polynomial system, x˙ = P (x, y), y˙ = Q (x, y), where P (respectively, Q) is the numerator of the rational function Fy + λFx F  (−Fx + λFy F).

10 Fig. 1.4 A polynomial system with five connectivity components

A. A. Azamov and D. H. Ruzimuradova

y

x

1.5 Conclusion In this paper, the topological properties of unbounded ω-limit sets of dynamical systems on the plane were studied. The obtained results serve as a qualitative clarification of the phase portrait of such systems. In connection with the results obtained in the article, the following tasks are interesting: 1. Can Theorem 1.2 hold for multidimensional cases? Does analytic systems have no more than a countable number of connectivity components in the space Rd ? 2. It can be established that an ω-limit set of trajectories of quadratic systems is always connected and a number of connectivity components of the ω-limit sets for cubic systems are no more than 2. What is the minimum degree of a polynomial system with the ω-limit set consisting of three connectivity components? 3. Does there exist a function L (n) such that the ω-limit set of trajectory for any polynomial system of a degree n has at most L (n) connectivity components? Acknowledgements The authors thank the Reviewer for his comments that helped to improve the exposition.

References 1. Al’mukhamedov, M.I.: On the construction of a differential equation having given curves as limit cycles. Izv. Vysš. Uˇcebn. Zaved. Mat. 44, 12–16 (1965) [in Russian] 2. Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.G.: Qualitative Theory of SecondOrder Dynamic Systems. Halsted Press, New York-Toronto (1973) 3. Birkhoff, G.D.: Dynamical Systems. Colloquium Publications, No 9, American Math. Soc., New York (1927)

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4. Buendía, J.E., Lopez, V.J.: A topological characterization of the ω-limit sets of analytic vector fields on open subsets of the sphere (2017). arXiv:1711.00567V2 [math. CA] 5. Buendía, J.E., Lopez, V.J.: Some remarks on the ω-limit sets for plane, sphere and projective plane analytic flows. Qual. Theory Dyn. Syst. 16, 93–298 (2017) 6. Ganiev, Kh: Synthesis of second-order structurally stable systems with prescribed structure. Din. Sist. 7, 35–46 (1975) [in Russian] 7. Kelly, J.L.: General Topology. D. Van Nostrand Company Inc., Canada (1955) 8. Llibre, J., Ramírez, R., Sadovskaia, N.: Planar vector fields with a given set of orbits. J. Dyn. Differ. Equ. 23(4), 885–902 (2011) 9. Lopez, V.J., Llibre, J.S.: A topological characterization of the ω-limit sets for analytic flows on the plane, the sphere and the projective plane. Adv. Math. 216(2), 677–710 (2007) 10. Lopez, V.J., Lopez, G.S.: A characterization of ω-limit sets for continuous flows on surfaces. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 9(8), 515–521 (2006) 11. Markus, L.: Global structure of ordinary differential equations in the plane. Trans. Am. Math. Soc. 76, 127–148 (1954) 12. Nemytskii, V.V., Stepanov, V.V.: Qualitative Theory of Differential Equations. Princeton University Press, Princeton (1960) 13. Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (2001) 14. Ruzimuradova, D.H., Tilavov, A.M.: A rational dynamical system with ω-limit set consisting of 4n components. Dokl. Acad. Scien. Uzb. 1, 3–5 (2018) [in Russian] 15. Sverdlove, R.: Inverse problems for dynamical systems in the plane. In: Dynamical Systems: Proceedings of a University of Florida International Symposium, pp. 499–502. Academic Press, New York (1977) 16. Valeeva, R.T.: Construction of a differential equation with given limit cycles and critical points. Volž. Mat. Sb. 5, 83–85 (1966) [in Russian] 17. Vdovina, E.: Second inverse problem in the qualitative theory of differential equations. Differ. Equ. 14, 1249–1253 (1979) 18. Vinograd, R.: On the limit behavior of an unbounded integral curve, Moskov. Gos. Univ. Uch. Zap. Mat. 155(5), 94–136 (1952) [in Russian]

Chapter 2

About One Problem of Optimal Control of String Oscillations with Non-separated Multipoint Conditions at Intermediate Moments of Time V. R. Barseghyan Abstract The problem of optimal control of string vibrations with given initial, final conditions and with non-separated values of the deflection function and velocities at intermediate moments of time with a quality criterion specified for the entire time interval is considered. The problem is solved using the methods of separation of variables and theories of optimal control of finite-dimensional systems with non-separated multipoint intermediate conditions. As an application of the proposed approach, an optimal control action is constructed for the string vibration with given non-separated conditions on the values of the deflection function and string velocities at two intermediate moments of time.

2.1 Introduction One of the most common processes in nature and technology is vibrationary which is modeled by the wave equations [1–4]. At the same time, in practice, control problems often arise when it is necessary to generate the desired vibration form that satisfies intermediate conditions. Many control processes from various fields of science and technology lead to the necessity to investigate multipoint boundary value problems of control and of optimal control, a characteristic feature of which is the presence of non-separated (non-local) conditions at several intermediate points of the interval of investigations. Due to numerous applications, the attention of researchers was attracted to multipoint boundary value problems in which, along with the classical boundary (initial and final) conditions, non-separated (non-local) multipoint intermediate conditions are also given [5–16]. Non-separated multipoint boundary value problems, on the one hand, arise as mathematical models of real processes, and, on the other hand, for many processes, the correct formulation of local boundary value problems is impossible. The non-separability of multipoint conditions is also, V. R. Barseghyan (B) Institute of Mechanics of NAS of RA, Marshal Baghramyan Ave., 24B, 0019 Yerevan, Armenia e-mail: [email protected] Mathematics and Mechanics Department of YSU, St. Alec Manukyan 1, 0025 Yerevan, Armenia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_2

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in particular, because of the impossibility in practice to measure instantaneously or in separate points the measured parameters of the state of an object. Numerous examples of technological processes leading to control and optimal control problems in systems with distributed parameters are considered in [1–3], and various methods of solving are proposed. The problems of control and of optimal control of vibratory processes, with both external and boundary control actions under various types of boundary conditions, are considered in [1–4, 8–19], and various methods for solving control problems are proposed. In this paper, the problem of optimal control of string vibrations with given initial, final conditions and non-separated values of the deflection and velocities of the points of the string at intermediate moments of time with a quality criterion given for the entire time interval is considered. By the method of separation of variables, the initial problem is reduced to an optimal control problem with a countable number of ordinary differential equations with given initial, final, and non-separated multipoint intermediate conditions. An optimal control action is constructed, which is obtained using the methods of the theory of optimal control of finite-dimensional systems with multipoint intermediate conditions. As an application of the proposed constructive approach, an optimal control action is constructed for the string vibration with given non-separated values of the deflection function and velocities of the string points at two intermediate moments of time, as well as in the case when at one moment of time only the deflection value is specified and at another moment of time values of velocities are specified.

2.2 The Formulation of the Problem Consider a homogeneous, elastic, stretched string with the length l, the edges of which are fixed. Let distributed forces act on a string in a vertical plane with a density u(x, t), which is a control action. Let the state of the distributed vibratory system, i.e., deviations from the equilibrium state, is described by the function Q (x, t), 0 ≤ x ≤ l, 0 < t < T , which for 0 < x < l and 0 < t < T follows the wave equation ∂2 Q ∂2 Q = a 2 2 + u(x, t) 2 ∂t ∂x

(2.1)

with homogeneous boundary conditions Q(0, t) = 0,

Q(l, t) = 0, 0 ≤ t ≤ T

and satisfies the initial and final conditions  ∂ Q  = ψ0 (x), 0 ≤ x ≤ l, Q(x, 0) = ϕ0 (x), ∂t t=0

(2.2)

(2.3)

2 About One Problem of Optimal Control of String Oscillations …

15

 ∂ Q  = ψT (x) ≡ ψm+1 (x), 0 ≤ x ≤ l. ∂t t=T (2.4)

Q(x, T ) = ϕT (x) ≡ ϕm+1 (x),

In Eq. (2.1) a 2 = Tρ0 , where T0 is the tension of the string and ρ is the density of the homogeneous string. The function Q (x, t), satisfying Eq. (2.1), is twice continuously differentiable up to the boundary of the region. It is assumed that the function u(x, t) ∈ L 2 (), where  = {(x, t) : x ∈ [0, l], t ∈ [0, T ]}. Let at some intermediate moments of time 0 = t0 < t1 < · · · < tm < tm+1 = T non-separated (non-local) conditions on the values of the deflection function are given in the form: m 

f k Q(x, tk ) = α(x),

k=1

m  k=1

ek

 ∂ Q(x, t)  = β(x),  ∂t t=tk

(2.5)

where f k and ek are given values (k = 1, . . . , m), α(x) and β(x) are some known functions. It is assumed that ϕ0 (x), ψ0 (x), ϕT (x), ψT (x), α(x), and β(x) are given smooth functions satisfying the compatibility conditions. In general, it may be that at some moments of time tk (k = 1, . . . , m) under the conditions (2.5) there exists either the value of the deflection function or the value of the derivative of this function, i.e., it is not necessary that at each moment   of time tk (k = 1, . . . , m) under these conditions functions Q(x, tk ) i ∂ Q(x,t)  ∂t t=tk

were simultaneously present. In such cases, we will assume that the corresponding coefficients f k or ek are equal to zero. The problem of optimal control of string vibrations with given non-separated values of the deflection function and velocities at intermediate moments of time tk (k = 1, . . . , m) can be formulated as follows: among the possible controls u(x, t), 0 ≤ x ≤ l, 0 ≤ t ≤ T , it is required to find optimal control action u 0 (x, t) that takes the string vibrations (2.1) with boundary conditions (2.2) from the given initial state (2.3) to the given final state (2.4), ensuring satisfaction of non-separated multipoint intermediate conditions (2.5) and minimizing the functional: ⎤ 21 ⎡ T l   J [u] = ⎣ (u(x, t))2 dxdt ⎦ . 0

(2.6)

0

It is assumed that the system (2.1) with constraints (2.2)–(2.5) on the time interval [0, T ] is completely controllable [6, 20].

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V. R. Barseghyan

2.3 The Problem Solution To construct a solution to the formulated problem, we look for a solution to Eq. (2.1) with boundary conditions (2.2) in the following form: Q(x, t) =

∞ 

πn x. l

Q n (t) sin

n=1

(2.7)

We represent the function u(x, t) as a Fourier series u(x, t) =

∞ 

πn x. l

u n (t) sin

n=1

(2.8)

We substitute the expansions (2.7), (2.8) into relations (2.1)–(2.5). Due to the orthogonality of the system of eigenfunctions, it follows that the Fourier coefficients Q n (t) satisfy a countable system of ordinary differential equations aπ n 2 Q¨ n (t) + λ2n Q n (t) = u n (t), λ2n = n = 1, 2, . . . l

(2.9)

and the following initial, non-separated multipoint intermediate and final conditions: Q n (0) = ϕn(0) , m 

f k Q n (tk ) = αn ,

k=1

Q n (T ) = ϕn(T ) = ϕn(m+1) ,

Q˙ n (0) = ψn(0) , m 

ek Q˙ n (tk ) = βn ,

(2.10) (2.11)

k=1

Q˙ n (T ) = ψn(T ) = ψn(m+1) ,

(2.12)

where Q n (t), ϕn(0) , ψn(0) , ϕn(m+1) , ψn(m+1) , u n (t), αn , and βn denote the Fourier coefficients corresponding to the functions Q (x, t), ϕ0 (x), ψ0 (x), ϕm+1 (x), ψm+1 (x), u(x, t), α(x), and β(x). The general solution of Eq. (2.9) with initial conditions (2.10) and its time derivative has the following form: Q n (t) =

ϕn(0)

1 1 cos λn t + ψn(0) sin λn t + λn λn

Q˙ n (t) = −λn ϕn(0) sin λn t + ψn(0) cos λn t +

t u n (τ ) sin λn (t − τ )dτ, 0

t u n (τ ) cos λn (t − τ )dτ. 0

(2.13)

2 About One Problem of Optimal Control of String Oscillations …

17

Now, taking into account the intermediate non-separated (2.11) and final (2.12) conditions and using the approaches given in [6, 7], from Eq. (2.13), we obtain that the functions u n (τ ) for each n must satisfy the following system of equalities: T

T u n (τ ) sin λn (T − τ )dτ = C1n (T ),

0

u n (τ ) cos λn (T − τ )dτ = C2n (T ), 0

m 

tk fk

k=1 m  k=1

(m) u n (τ ) sin λn (tk − τ )dτ = C1n (t1 , . . . , tm ),

0

tk ek

(m) u n (τ ) cos λn (tk − τ )dτ = C2n (t1 , . . . , tm ),

(2.14)

0

where C1n (T ) = λn ϕn(m+1) − λn ϕn(0) cos λn T − ψn(0) sin λn T, C2n (T ) = ψn(m+1) + λn ϕn(0) sin λn T − ψn(0) cos λn T, 

 m  1 (0) (m) (0) f k ϕn cos λn tk + ψn sin λn tk , C1n (t1 , . . . , tm ) = λn αn − λn k=1 (m) C2n (t1 , . . . , tm ) = βn −

m 

  ek −λn ϕn(0) sin λn tk + ψn(0) cos λn tk .

(2.15)

k=1

We introduce the following functions: h 1n (τ ) = sin λn (T − τ ), h 2n (τ ) = cos λn (T − τ ),  m  sin λn (tk − τ ) (m) (k) (k) f k h 1n (τ ), h 1n (τ ) = h 1n (τ ) = 0 k=1  m  cos λn (tk − τ ) (k) (k) h (m) (τ ) = e h (τ ), h (τ ) = k 2n 2n 2n 0 k=1

0≤τ ≤T f or 0 ≤ τ ≤ tk , f or tk < τ ≤ tm+1 = T f or 0 ≤ τ ≤ tk . f or tk < τ ≤ tm+1 = T (2.16)

Then, the integral relations (2.14) with the help of the function (2.16) are written as follows: T

T u n (τ )h 1n (τ )dτ = C1n (T ),

0

u n (τ )h 2n (τ )dτ = C2n (T ), 0

18

T

V. R. Barseghyan

u n (τ )h (m) 1n (τ )dτ

=

(m) C1n (t1 , . . . , tm ),

0

T

(m) u n (τ )h (m) 2n (τ )dτ = C 2n (t1 , . . . , tm ),

0

n = 1, 2, . . .

(2.17)

From relation (2.17), it follows that for each harmonic, the motion described by Eq. (2.9) with conditions (2.10)–(2.12) is completely controllable if and only if for (m) (m) (t1 , . . . , tm ), C2n (t1 , . . . , tm ) any given value of the constants C1n (T ), C2n (T ), C1n (2.15) a control u n (t) t ∈ [0, T ], satisfying condition (2.17), can be found. Taking into account the decomposition (2.8) and the orthogonality of the system of eigenfunctions, the minimized functional (2.6) can be written as follows: T  l

∞

l  [u(x, t)] dxdt = 2 n=1

T u 2n (τ )dτ .

2

0

0

0

T As for each n = 1, 2, . . . 0 u 2n (τ )dτ ≥ 0, hence, minimization of the functional (2.6) is equivalent to minimizing the functionals T u 2n (τ )dτ (n = 1, 2, . . .).

(2.18)

0

Thus, the solution of the stated optimal control problem (2.1)–(2.6) for each n = 1, 2, . . . is reduced to finding an optimal control u 0n (t) t ∈ [0, T ] that satisfies the integral relations (2.17) and minimizes the functional (2.18). The problem of optimal control with the functional (2.18) and with integral conditions (2.17) can be considered as a problem of conditional extremum from the calculus of variations. However, as can be seen from notation (2.16), the integrand in relation (2.17) is discontinuous, and therefore classical methods of the calculus of variations are not applicable to the study of this problem [6, 20]. Note that conditions (2.17) are linear operations generated by a function u n (t) on the time interval [0, T ], and functional (2.18) is the norm of a linear normed space. Therefore, the solution of the obtained optimal control problem (2.17)–(2.18) is advisable to search using the algorithm for solving the problem of moments [6, 20]. Following [6, 20], to solve the finite-dimensional problem of moments (2.17)– (2.18), it is necessary to find some values p1n , p2n , q1n , q2n , n = 1, 2, . . . related by conditions (m) (m) + q2n C2n = 1, p1n C1n (T ) + p2n C2n (T ) + q1n C1n

for which

(2.19)

2 About One Problem of Optimal Control of String Oscillations …

19

T (ρn0 )2

= min

h 2n (t)dt,

(2.19)

(2.20)

0

where (m) h n (t) = p1n h 1n (t) + p2n h 2n (t) + q1n h (m) 1n (t) + q2n h 2n (t).

(2.21)

0 0 0 0 , p2n , q1n , q2n , n = 1, 2, . . ., minimizing (2.20) with To determine the values p1n conditions (2.19), we apply the Lagrange indefinite multipliers method. We introduce the function

f ( p1n , p2n , q1n , q2n ) =

T 

2 (m) p1n h 1n (t) + p2n h 2n (t) + q1n h (m) 1n (t) + q2n h 2n (t) dt+

0

  (m) (m) + γn p1n C1n (T ) + p2n C2n (T ) + q1n C1n + q2n C2n −1 , where γn is Lagrange multiplier. Based on this method, calculating the derivatives of the function f ( p1n , p2n , q1n , q2n ) with respect to p1n , p2n , q1n , q2n , n = 1, 2, . . . and setting them equal to zero, we get the following system of algebraic equations: γn C1n (T ), 2 γn an(2) p1n + bn(1) p2n + dn(2) q1n + en(1) q2n = − C2n (T ), 2 γn (m) (2) (2) (1) (2) bn p1n + dn p2n + dn q1n + en q2n = − C1n , 2 γn (m) (2) (1) (2) (1) cn p1n + en p2n + en q1n + cn q2n = − C2n , n = 1, 2, . . . 2 an(1) p1n + an(2) p2n + bn(2) q1n + cn(2) q2n = −

(2.22)

where the following notations are introduced: an(1)

T =

(h 1n (τ )) dτ , 2

bn(1)

0

dn(1)

=

T

cn(1)

=

0

=

(h 2n (τ ))2 dτ , 0

h (m) 1n (τ )

2

dτ =

0

T

T

T  m 0

h (m) 2n (τ )

2

dτ =

dτ ,

k=1

T  m 0

2 f k h (k) 1n (τ )

k=1

2 ek h (k) 2n (τ )

dτ ,

an(2)

T =

h 1n (τ )h 2n (τ )dτ , 0

20

V. R. Barseghyan

bn(2)

dn(2)

cn(2)

en(1)

T =

h 1n (τ )h (m) 1n (τ )dτ

=

h 1n (τ )

0

0

T

T

=

h 2n (τ )h (m) 1n (τ )dτ

=

0

0

T

h 1n (τ )h (m) 2n (τ )dτ

=

0

0

T

= 0

T

=

0

=

 m 

 f k h (k) 1n (τ )

dτ ,

 m 

 ek h (k) 2n (τ )

dτ ,

k=1

h 2n (τ )

 m 

 ek h (k) 2n (τ )

dτ ,

k=1

T  m 0

dτ ,

k=1

0

(m) h (m) 1n h 2n (τ )dτ

 f k h (k) 1n (τ )

k=1

h 1n (τ )

T

h 2n (τ )h (m) 2n (τ )dτ

 m 

h 2n (τ )

T =

en(2) =

T

k=1

f k h (k) 1n (τ )

 m 

 ek h (k) 2n (τ )

dτ .

(2.23)

k=1

Adding condition (2.19) to Eq. (2.22), we obtain a closed system of algebraic equations with the unknown values p1n , p2n , q1n , q2n , γn , n = 1, 2, . . .. We introduce the following notations:  an(2) bn(2) cn(2)  bn(1) dn(2) en(1)  , dn(2) dn(1) en(2)  en(1) en(2) cn(1)      (1)  C1n (T ) an(2) bn(2) cn(2)   an C1n (T ) bn(2) cn(2)       C (T ) b(1) d (2) e(1)   a (2) C (T ) d (2) e(1) 

n ( p1n ) =  2n(m) n(2) n(1) n(2) , n ( p2n ) =  n(2) 2n(m) n(1) n(2) ,  C1n dn dn en   bn C1n dn en   C (m) e(1) e(2) c(1)   c(2) C (m) e(2) c(1)  n n n n n n 2n 2n    (1) (2)  (1) (2) (2)  an an C1n (T ) cn(2)   an an bn C1n (T )     (2) (1)   a b C (T ) e(1)   a (2) b(1) d (2) C (T ) 

n (q1n ) =  n(2) n(2) 2n(m) n(2) , n (q2n ) =  n(2) n(2) n(1) 2n(m) ,  bn dn C1n en   bn dn dn C1n   c(2) e(1) C (m) c(1)   c(2) e(1) e(2) C (m)  n n n n n n 2n 2n  (1)  an  (2) a

n =  n(2)  bn  c(2) n

and assume that n = 0.

2 About One Problem of Optimal Control of String Oscillations …

21

Then, the solution of system (2.22) with condition (2.19) can be represented as follows:

n ( p1n ) 0

n ( p2n ) 0

n (q1n ) , p2n = , q1n = , An An An

n (q2n )

n = , γn = −2 , n = 1, 2, . . . . An An

0 p1n = 0 q2n

(2.24)

Here, the following notation is made: (m) (m) + n (q2n )C2n . An = n ( p1n )C1n (T ) + n ( p2n )C2n (T ) + n (q1n )C1n 0 0 0 0 , p2n , q1n , q2n into (2.21), we obtain Substituting from (2.24) the values for p1n

h 0n (t) =

h˜ 0n (t) , An

where (m) h˜ 0n (t) = n ( p1n )h 1n (t) + n ( p2n )h 2n (t) + n (q1n )h (m) 1n (t) + n (q2n )h 2n (t). (2.25)

Having the optimal functions h 0n (t) from (2.20) and taking into account (2.25), we get T 0 2 h˜ n (t) dt. (ρn0 )2 = ABn2 , gde Bn = n

0

Thus, according to [6, 20], the desired optimal control u 0n (t) is determined by the expression: u 0n (t) =

1 An ˜ 0 h 0 (t) = h (t). (ρn0 )2 n Bn n

Note that with the notation (2.16) we will have ⎧ m ⎪ ⎪ f k sin λn (tk − t), 0 ≤ t ≤ t1 ⎪ ⎪ ⎪ k=1 ⎪ m ⎪ ⎪ ⎪ ⎪ f k sin λn (tk − t), t1 < t ≤ t2 ⎪ ⎪ ⎨ k=2 ... h (m) , 1n (t) = ⎪ m  ⎪ ⎪ ⎪ f k sin λn (tk − t), tm−2 < t ≤ tm−1 ⎪ ⎪ ⎪ k=m−1 ⎪ ⎪ ⎪ ⎪ f sin λn (tm − t), tm−1 < t ≤ tm ⎪ ⎩ m 0, tm < t ≤ tm+1 = T

(2.26)

22

V. R. Barseghyan

⎧ m ⎪ ⎪ ek cos λn (tk − t), 0 ≤ t ≤ t1 ⎪ ⎪ ⎪ k=1 ⎪ m ⎪  ⎪ ⎪ ⎪ e cos λn (tk − t), t1 < t ≤ t2 ⎪ ⎪ k=2 k ⎨ ... h (m) . 2n (t) = ⎪ m  ⎪ ⎪ ⎪ ⎪ ⎪ k=m−1 ek cos λn (tk − t), tm−2 < t ≤ tm−1 ⎪ ⎪ ⎪ ⎪ ⎪ e cos λn (tm − t), tm−1 < t ≤ tm ⎪ ⎩ m 0, tm < t ≤ tm+1 = T (m) Substituting the values of the functions h 1n (t), h 2n (t), h (m) 1n (t), h 2n (t) into (2.25), we obtain ⎧ ⎪ h˜ (1)0 0 ≤ t ≤ t1 ⎪ n (t), ⎪ ⎪ ˜ (2)0 ⎪ (t), t 1 < t ≤ t2 h ⎪ ⎪ ⎨ n . . . , (2.27) h˜ 0n (t) = ˜ (m−1)0 ⎪ (t), tm−2 < t ≤ tm−1 hn ⎪ ⎪ ⎪ ⎪ (t), tm−1 < t ≤ tm h˜ (m)0 ⎪ n ⎪ ⎩ ˜ (m+1)0 (t), tm < t ≤ tm+1 = T hn

where h˜ (1)0 n (t) = n ( p1n )h 1n (t) + n ( p2n )h 2n (t) + n (q1n )

m 

f k sin λn (tk − t)

k=1

+ n (q2n )

m 

ek cos λn (tk − t),

k=1

h˜ (2)0 n (t) = n ( p1n )h 1n (t) + n ( p2n )h 2n (t) + n (q1n )

m 

f k sin λn (tk − t)

k=2

+ n (q2n )

m 

ek cos λn (tk − t),

k=2

... (t) = n ( p1n )h 1n (t) + n ( p2n )h 2n (t) + n (q1n ) h˜ (m−1)0 n

m 

f k sin λn (tk − t)

k=m−1

+ n (q2n )

m 

ek cos λn (tk − t)

k=m−1

(t) = n ( p1n )h 1n (t) + n ( p2n )h 2n (t) + n (q1n ) f m sin λn (tm − t) h˜ (m)0 n + n (q2n )em cos λn (tm − t), h˜ (m+1)0 (t) n

= n ( p1n )h 1n (t) + n ( p2n )h 2n (t).

2 About One Problem of Optimal Control of String Oscillations …

23

Thus, having an explicit form of the function h˜ 0n (t), from (2.26), we obtain the optimal function u 0n (t) for each n = 1, 2, . . .. Next, substituting the optimal function u 0n (t) into (2.13), we obtain Q 0n (t) on the time interval t ∈ [0, T ]. Therefore, from formulas (2.7) and (2.8), we obtain the optimal function of string deflection Q 0 (x, t) and optimal control u 0 (x, t). Thus, for optimal control, we will have ⎧ ∞  ⎪ ⎪ ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ⎪... ⎨ ∞ u 0 (x, t) =  ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ∞ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ∞ ⎪  ⎪ ⎪ ⎩ n=1

An ˜ (1)0 x, h (t) sin πn Bn n l

0 ≤ t ≤ t1

An ˜ (2)0 x, h (t) sin πn Bn n l

t 1 < t ≤ t2

An ˜ (m−1)0 (t) sin πn x, tm−2 h Bn n l An ˜ (m)0 (t) sin πn x, h Bn n l

< t ≤ tm−1

.

tm−1 < t ≤ tm

An ˜ (m+1)0 (t) sin πn x, tm h Bn n l

< t ≤ tm+1 = T

From this expression it is clear that the optimal control action u 0 (x, t) is a piecewise continuous function.

2.4 Example Assume that m = 2 (0 < t1 < t2 < t3 = T ). Assuming that t1 = al , t2 = 2 al , T = 4 al , we obtain that t1 λn = π n, t2 λn = 2π n, T λn = 4π n, λn (T − t1 ) = 3π n, λn (T − t2 ) = 2π n, λn (t1 − t2 ) = −π n. Assuming that f 2 = e1 = 0 and f 1 = e2 = 1, the conditions (2.5) have the following form: Q(x, t1 ) = α(x),

 ∂ Q(x, t)  = β(x).  ∂t t=t2

Taking into account the formulas of optimal solution derived in the previous sections, for the considered case, we will have 3  3 l 4 l λn  (3) ϕn − (−1)n αn , , n ( p1n ) = 4 a a 2 3  3 l  (3) ψn − βn ,

n ( p2n ) = 4 a

n =

24

V. R. Barseghyan

3   l λn  4αn − (−1)n ϕn(3) + 3ϕn(0) ,

n (q1n ) = a 2 3  3 l 

n (q2n ) = 2βn − ψn(3) − ψn(0) . 4 a Calculating the values of An and Bn , explicit forms of the function of the optimal control will be obtained in the form: for 0 ≤ t ≤ al u 0 (x, t) =

for

l a

  

∞  (3) (3) A n λn 3 n × Bn 2 ϕn − (−1) αn sin λn (T − t) + 4 ψn − βn cos λn (T − t)  n=1

 (3) (0) + λ2n 4αn − (−1)n ϕn + 3ϕn sin λn (t1 − t)

 (3) (0) 3 + 4 2βn − ψn − ψn cos λn (t2 − t) sin πn l x;

 l 3 a

< t ≤ 2 al 3  ∞ 

An λn  (3) l 3 (3) ϕn − (−1)n αn sin λn (T − t) + ψn − βn cos λn (T − t) a B 2 4 n=1 n !

3 πn + 2βn − ψn(3) − ψn(0) cos λn (t2 − t) sin x; 4 l

u 0 (x, t) =

for 2 al < t ≤ 4 al u 0 (x, t) =

3  ! ∞ 

An λn  (3) l 3 (3) πn ϕn − (−1)n αn sin λn (T − t) + ψn − βn cos λn (T − t) sin x. × a Bn 2 4 l n=1

Thus, having explicit forms of the functions of optimal control using the above formulas, one can find the corresponding expression for the string deflection function.

2.5 Conclusion The problem of optimal control of string vibrations with given non-separated values of the function of deflection and velocities at intermediate moments of time by the method of separation of variables is reduced to a control problem with a countable number of ordinary differential equations with given initial, final, and non-separated multipoint intermediate conditions. The quality criterion is given for the entire time interval. The problem is solved using the methods of the theory of optimal control of finite-dimensional systems with multipoint intermediate conditions. As an application of the proposed approach, an optimal control action is constructed for the string vibration with given non-separated values of deflection and velocities of the string points at two intermediate moments of time.

2 About One Problem of Optimal Control of String Oscillations …

25

References 1. Butkovskiy, A.G.: Methods of the Systems Control with Distributed Parameters, 568 pp. Nauka (1975) (in Russian) 2. Sirazetdinov, T.K.: Systems Optimization with Distributed Parameters, 480 pp. Nauka (1977) (in Russian) 3. Degtyarev, G.L., Sirazetdinov, T.K.: Theoretical Foundations of Optimal Control of Elastic Spacecraft, 216 pp. Nauka (1986) (in Russian) 4. Znamenskaya, L.N.: Control of Elastic Vibrations, 176 pp. FIZMATLIT (2004) (in Russian) 5. Aschepkov, L.T.: Optimal control of the system with intermediate conditions. PMM 45(2), 215–222 (1981). (in Russian) 6. Barseghyan, V.R.: Control of Compound Dynamic Systems and of Systems with Multipoint Intermediate Conditions, 230 pp. Nauka (2016) (in Russian) 7. Barseghyan, V.R., Barseghyan, T.V.: On an approach to the problems of control of dynamic system with nonseparated multipoint intermediate conditions. Autom. Remote Control 76(4), 549–559 (2015) 8. Barseghyan, V.R., Saakyan, M.A.: The optimal control of wire vibration in the states of the given intermediate periods of time. Proc. of NAS RA: Mech. 61(2), 52–60 (2008). (in Russian) 9. Barseghyan, V.R.: Optimal control of a membrane vibration with fixed intermediate states. Proc. YSU 188(1), 24–29 (1998). (in Russian) 10. Barseghyan, V.R.: About one problem of optimal boundary control of string vibrations with restrictions in the intermediate moment of time. In: Proceedings of the 11th International Chetaev Conference. Analytical Mechanics, Stability and Control, Kazan, 14–18 June 2017, vol. 3, part 1, pp. 119–125 (2017) (in Russian) 11. Barseghyan, V.R., Movsisyan, L.A.: Optimal control of the vibration of elastic systems described by the wave equation. Int. Appl. Mech. 48(2), 234–239 (2012) 12. Korzyuk, V.I., Kozlovskia, I.S.: Two-point boundary problem for the equation of string vibration with the given velocity at the certain moment of time. Proc. Inst. Math. NAS Belarus 18(2), 22–35 (2010) (in Russian) 13. Korzyuk, V.I., Kozlovskia, I.S.: Two-point boundary problem for the equation of string vibration with the given velocity at the certain moment of time. Proc. Inst. Math. NAS Belarus 19(1), 62–70 (2010) (in Russian) 14. Makarov, A.A., Levkin, D.A.: Multipoint boundary value problem for pseudodierential equations in multilayer. Vistnyk of V.N. Karazin Kharkiv Natil. Univ. Ser. Math. Appl. Math. Mech. 69(1120), 64–74 (2014) (in Ukrainian) 15. Assanova, A.T., Imanchiev, A.E.: On the solvability of a nonlocal boundary value problem for a loaded hyperbolic equations with multi-point conditions. Bull. Karaganda Univ. Ser.: Math. 1(81), 15–20 (2016) (in Russian) 16. Bakirova, E.A., Kadirbayeva, Zh.M.: On a solvability of linear multipoint boundary value problem for the loaded differential equations. Izvestiya NAS RK. Ser. fiz.-mat. 5(309), 168–175 (2016) (in Russian) 17. Il’in, V.A., Moiseev, E.I.: Optimization of boundary controls of string vibrations. Uspekhi Mat. Nauk 60(6), 89–114 (2005) 18. Manita, L.A.: Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends. J. Appl. Math. Mech. 74(5), 611–616 (2010) 19. Kopets, M.M.: Optimal control of vibrations of a rectangular membrane. Cybern. Comput. Eng. 177, 28–42 (2014) (in Russian) 20. Krasovsky, N.N.: The Theory of Motion Control, 476 pp. Nauka (1968) (in Russian)

Chapter 3

Application of Discrete-Time Optimal Control to Forest Management Problems Andrey Krasovskiy and Anton Platov

Abstract Forests provide multiple ecosystem services, including woody and nonwoody biomass products, contribution to carbon budget, and provision of public goods. Therefore, forest management problems are important in the context of optimizing those services. In this study, we formalize the spatial dynamic forest management model as the discrete-time optimal control problem with discrete time and age dynamics. The objective of the forest owner is to maximize the sum of discounted profits over time. Control variables stand for ratios of the forest harvested at every time period, every cell (forest type) and for each age class. In this setup, we deal with a high-dimensional bilinear control problem. The problem is solved using the discrete Pontryagin’s maximum principle. This approach allows us to derive an optimal solution in a constructive manner and reduce computational costs which may arise in the linear programming and recursive dynamics optimization methods. Results are illustrated by the model example with sample age-dependent cost functions, biomass factors, and price projection. Future research will deal with including mortality factors and disturbances into the model dynamics.

3.1 Introduction We consider an optimal control problem contained in bioeconomic models [1], known as the optimal rotation, forest management, or Faustmann model. Our modeling deals with economic aspect of forestry (see, e.g., [2]). In this study, an optimal control problem is formulated for a discrete system. The dynamics of the system A. Krasovskiy (B) Ecosystems Services and Management (ESM) Program, International Institute for Applied Systems Analysis (IIASA), Schlossplatz 1, 2361 Laxenburg, Austria e-mail: [email protected] A. Platov Lomonosov Moscow State University (MSU), Leninskie Gory, Moscow 119991, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_3

27

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describes the changes in an age sequence of forest stands. The optimization criterion is represented by the discounted profit, calculated on the finite time horizon. To derive a solution to the problem, we apply Pontryagin’s maximum principle [3], developed for discrete systems by Boltyanskii [4]. Let us note that for solving optimization problems arising in similar models, researchers traditionally apply methods of linear programming (see, e.g., [5]) or dynamic programming (see, e.g., [6]). In practice, various software solvers are used; an extensive review of mathematical programming methods and corresponding solvers can be found in [7] and heuristic methods applied in forest planning problems are discussed in [8]. The use of solvers may lead to high computational costs due to high dimensionality of problems, as well as to the lack of transparency in the solving routines. Although applied forest management models are often complex, their basic constructions can be formulated mathematically in a well compact form [9]. Nevertheless, the optimal control theory in the framework of the Pontryagin maximum principle is not used with rare exceptions (see, e.g., [10]). In this paper, we use the discrete-time maximum principle as a method for solving the optimal forest management problem. We propose a model that satisfies the basic forestry requirements, while adhering to mathematical rigor. A solution to the bilinear optimal control problem, formulated in the paper, is constructive. This allows us to avoid the high computational costs mentioned above. The study presents analytical results proving the uniqueness of the solution under typical assumptions, as well as the numerical example for illustrative case study.

3.2 Problem Statement Let us allocate every type of an individual forest stand into a cell with number i, i = 1, . . . , N . In practice, the cell corresponds to a separate forest area with the stand of one forest type (species). In every cell the trees are divided into age classes, i.e., by age a ∈ [0, A]. Let us denote by symbol xi (a, t) the forest area in cell i, age class a, at time t ∈ [0, T ]. We consider a discrete-time model; the time step t and age class step a are determined according to the following relations: a j+1 = a j + a, tk+1 = tk + t, t1 = t, t K = T, j = 1, . . . , M − 1, k = 1, . . . , K − 1, t = a, where M is the number of age classes and K is the number of time periods. For every forest type there is a factor that transforms the forest area into the biomass depending on the age class. We denote the biomass factors by symbols βi (a j ) ≥ 0. Symbol u i (a j , tk ) ∈ [0, 1] denotes control variable, that is, the ratio of forest stand of type i, age a j , that is harvested at time tk .

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For convenience, we introduce the following vector notations:  T xi (tk ) := xi (a1 , tk ), . . . , xi (a M , tk ) ,  T ui (tk ) := u i (a1 , tk ), . . . , u i (a M , tk ) , and further: xi (k) := xi (tk ) ∈ R M , ui (k) := ui (tk ) ∈ R M ,   x(k) := x1 (k), . . . , x N (k) ∈ Mat M×N ,   u(k) := u1 (k), . . . , u N (k) ∈ Mat M×N , T  β(k) := βi (a1 ), . . . , βi (a M ) ∈ R M .

3.2.1 Discrete Dynamics of Forest Areas Let us define the following set:    j j U := {u i } ∈ Mat M×N 0 ≤ u i ≤ 1 . If control u(k) ∈ U for every k = 1, . . . , K , then u is called the admissible control. Discrete-time dynamics of the state variable are subject to the equation: xi (k + 1) = (L + MDg(ui (k)))xi (k),

(3.1)

where matrices L, M ∈ Mat M×M are determined in the following way: ⎡

0 0 ⎢ 1 0 ⎢ ⎢ 0 1 L := ⎢ ⎢... ... ⎢ ⎣ 0 0 0 0

0 0 0 ... 0 0

... ... ... ... ... ...

0 0 0 ... 1 0

0 0 0 ... 0 1

⎤ ⎡ ⎤ 1 1 1 ... 1 1 1 0 ⎢ −1 0 0 . . . 0 0 0 ⎥ 0⎥ ⎥ ⎢ ⎥ ⎢ 0 −1 0 . . . 0 0 0 ⎥ ⎥ 0⎥ ⎥ ⎢ , M := ⎢ ... ... ... ... ... ... ...⎥. ...⎥ ⎥ ⎢ ⎥ ⎣ 0 0 0 . . . −1 0 0 ⎦ 0⎦ 0 0 0 . . . 0 −1 −1 1

Here operator Dg transforms vector y = (y1 , y2 , . . . , yn )T into the diagonal matrix: ⎡

y1 ⎢ 0 ⎢ Dg(y) = ⎢ ⎢... ⎣ 0 0

0 y2 ... 0 0

... ... ... ... ...

0 0 ... yn−1 0

⎤ 0 0 ⎥ ⎥ ...⎥ ⎥. 0 ⎦ yn

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One can give the following interpretation to dynamics (3.1). The forest area, left after harvesting in the age class a j in the time period k, in the next time period k + 1 moves to the age class a j+1 ; the area is accumulated in the last age class; all cut areas are planted. The biomass (e.g., timber) harvested at the time period k is calculated as follows: H (k) =

N

βiT Dg(ui (k))xi (k).

(3.2)

i=1

3.2.2 Modeling Economic Decision-Making Let us assume that harvesting decisions in the forestry sector are determined by the profit (net present value) maximizing behavior [7]. A one-period benefit from selling the harvested biomass (3.2) is calculated as follows: B(k) = p(k)H (k),

(3.3)

where symbol p denotes the biomass price. Below we solve the problem for a given price projection p(k). After substituting (3.2) to (3.3), the benefit function takes the form: N

βiT Dg(ui (k))xi (k). B(u(k), x(k)) := p(k) i=1

The cost function is generated by several components: cutting (logging) cost C L = C L (a), extracting (incl. transportation) cost C E = C E (a), and planting cost C P = C P (a). All costs can be represented in conventional units per hectare. The total cost of control actions u is calculated as follows: C(u(k), x(k)) =

N M

 [CiL (a j ) + CiE (a j ) + CiP (a j ) xi (a j , k)u i (a j , k). (3.4)

i=1 j=1

The profit function is determined by the following relation: (u(k), x(k)) := B(u(k), x(k)) − C(u(k), x(k)).

(3.5)

3.2.3 Optimal Control Problem Given the forest growth dynamics xi (k + 1) (3.1), the initial age distribution xi1 := xi (1), i = 1, . . . , N ,

(3.6)

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the biomass factors β i , and the profit function (k) (3.5), the problem is to find among ˆ maximizing the net present value: admissible controls ui (k), the optimal control u(k),  max J := u∈U

K

 ρk (k) ,

(3.7)

k=1

where U is the set of all admissible controls and ρk is a given discount factor.

3.3 Solution to the Bilinear Optimal Control Problem Note the validity of the following statement, the proof of which follows directly from the form of matrices L, M. Assertion 3.1 If all components of vectors xi1 are non-negative, then for any admissible control u all components of the solution to (3.1), (3.6) are non-negative as well.  Remark 3.1 In the problem statement, the components of vectors xi1 correspond to forest areas. Therefore, they are non-negative by definition. Remark 3.2 In the text below, when we use operators max and sign with respect to a row vector, we mean that they are applied to every component of the row vector. According to the Pontryagin maximum principle [4], components uˆ i of optimal control uˆ in each k-th time period maximize every component of the row vector: uˆ i (k) = arg max

u(k)∈U



 ρk ( p(k)βiT − CiT ) + λi (k + 1)M ×

 × Dg(ˆxi (k))Dg(ui (k)) . (3.8)

Here λi (k + 1) is the solution to the adjoint equation   λi (k) = ρk ( p(k)βiT −CiT )Dg(uˆ i (k)) + λi (k + 1) LT + Dg(uˆ i (k))MT ,

(3.9)

which is calculated in the inverse time, starting from the transversality condition: λi (K + 1) = 0.

(3.10)

  Here dynamics xˆ i (k) are determined by Eq. (3.1): xˆ i (k + 1) = L + Dg(uˆ i (k))M xˆ i (k) with the initial condition: xˆ i1 := xi (1).

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It is important to note that condition (3.8) does not allow to explicitly determine the component of vector uˆ i (k), when its multiplier is equal to zero. While solving applied problems, the occurrence of the zero values in vectors   ρk ( p(k)βiT −CiT )+λi (k+1)M

(3.11)

is extremely rare for the bang–bang controls. At the same time, the zero components often appear in vector xˆ i (k). Indeed, component uˆ i (a j , k) of control uˆ i (k), that is derived from (3.8) takes one of two values: either 0 or 1. Let uˆ i (a j , k) = 1, 1 < j < M − 1, then according to Eq. (3.1) in the next time period, k + 1, we get xˆi (a j , k) = 0. Below we formulate a condition corresponding to a typical situation, i.e., when the components of vector (3.11) do not vanish. As noted above, even in this case, condition (3.8) does not always uniquely determine components of the optimal control. Nevertheless, the optimal control can be found by simplifying (3.8), i.e., by eliminating multiplier Dg(ˆxi (k)). Let us justify this simplification. Instead of condition (3.8), we consider new condition: u˜ i (k) = arg max

u(k)∈U

   ρk ( p(k)βiT −CiT )+λi (k+1)M Dg(ui (k)) .

(3.12)

Let us formulate the typicality assumption. ˜ ˜ to problem (3.9), (3.10), (3.12), such Assumption   A. There exists solution {λ, u} ˜ i (k+1)M do not contain zero component for that vectors ρk ( p(k)βiT −CiT )+λ all i, k. Lemma If assumption A is satisfied, then problem (3.9), (3.10), (3.12) has a unique solution and all components of u˜ take the values from set {0, 1}. Proof According to condition (3.10) and assumption A, vector ρk ( p(k)βiT −CiT ) does not contain zero components. Therefore, u˜ i (K ) is uniquely determined, and ˜ i (K ) its components take the values from {0, 1}. According to u˜ i (K ) and (3.9), λ ˜ i (k) continues to is uniquely determined. The process of constructing u˜ i (k) and λ k = 1.  The following theorem holds. ˜ u} ˜ be a solution to (3.9), (3.10), Theorem 3.1 Let assumption A be satisfied, {λ, ˜ Then (3.12), and x˜ be a solution to problem (3.1), (3.6), corresponding to control u. ˜ x˜ } is the solution to problem (3.1), (3.6), (3.7). {u, Proof Without loss of generality, we consider the case when N = 1, and omit index i. Let us consider new auxiliary optimal control problems, which are generated by a small parameter ε > 0. The dynamics of state variable and the set of admissible controls are the same, but the optimality criterion takes the following form: Jε (u, x) := J (u, x) + ε J¯(u),

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where J¯(u) :=

N

  ˜ + 1)M , (3.13) σ(k)u(k), σ(k) = sign ρk ( p(k)β T − CT ) + λ(k

k=1

i.e., max Jε (u, x). u∈U

(3.14)

˜ x˜ } is the solution to problem (3.1), (3.6), (3.14) for Let us show that pair {u, any ε > 0. ¯ x¯ } be a solution to the auxiliary problem (3.14), satisfying the maximum Let {u, principle:    ¯ + 1)M Dg(¯x(k))+ ¯ u(k) = arg max ρk p(k)β T − CT ) + λ(k u(k)∈U  + εσ(k) Dg(u(k)). (3.15) ¯ is subject to the adjoint equation: Here λ   T ¯ + 1) LT + Dg(u(k))M ¯ ¯ ¯ , + λ(k λ(k) = ρk ( p(k)β T − CT )Dg(u(k)) which is solved in the inverse time, starting from the transversality condition: ¯ λ(K + 1) = 0. ¯ and corresponding optimal control u. ¯ In Let us construct the adjoint variable λ the time period K , according to (3.11), (3.13), we get   σ(K ) = sign ρk ( p(K )β T − CT ) . ¯ Substituting σ(K ) and λ(K + 1) = 0 to condition (3.15), we get ¯ ) = arg max u(K

u(K )∈U



  ρ K p(K )β T −CT Dg(¯x(K ))+    + ε sign ρ K p(K )β T − CT Dg(u(k)). (3.16)

¯ ) coincides with condition (3.12) for derivCondition (3.16) for deriving control u(K ˜ ¯ ˜ ). Indeed, λ(K ing control u(K + 1) = λ(K + 1) = 0 and according to Assertion 3.1, all components of vector x(K ) are non-negative. Hence, all components of the multiplier of Dg(u(K )) in (3.16) have the same signs as all components of the

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multiplier in (3.12) (independent of the concrete values of vector x¯ (k)). Therefore, ¯ ) = u(K ˜ ). u(K By induction we get the complete coincidence of controls and adjoint variables: ¯ ˜ ¯ ˜ ¯ ˜ u(k) = u(k), λ(k) = λ(k), k = 1, . . . , K . Thus, we have shown that u(k) = u(k) ˜ ˜ and x¯ (k) = x˜ (k), i.e., {u(k), x˜ (k), λ(k)} is the unique solution to the system of the Pontryagin maximum principle for the auxiliary problem and, hence, it is the solution to the auxiliary problem for any ε > 0. ˆ xˆ } be an arbitrary admissible process in the initial problem (3.1), (3.6). Let {u, Then, according to the previous part of the proof, for any ε > 0, one gets ˜ x˜ ) ≥ Jε (u, ˆ xˆ ) = J (u, ˆ xˆ ) + ε J¯(u). ˆ ˜ x˜ ) + ε J¯(u) ˜ = Jε (u, J (u, ˜ x˜ ) ≥ J (u, ˆ xˆ ). Passing to the limit in the last inequality, when ε → 0+, we get J (u, ˜ x˜ } is the solution to problem (3.1), (3.6), (3.7). Therefore, {u, Remark 3.3 There is a nonemptyset of problems  for which assumption A is satisfied. For example, if for all k, j, ρk p(k)β T − CT = (2, 2, . . . , 2), then assumption j A is automatically satisfied and u i (k) = 1 for all i, j. Remark 3.4 Suppose functional J in (3.7) has a linear terminal term, i.e., J (u, x) :=

K

ρk (k) +

k=1

N

βiT Dg(S)xi (K + 1),

i=1

 T where S := S1 , . . . , S M , with constants S j ≥ 0 for all j = 1, . . . , M. Then Theorem 3.1 remains valid if condition (3.10) is replaced by the following condition: λi (K + 1) = βiT Dg(S).

3.4 Illustrative Example Here we present modeling results based on a hypothetical case study. The software, including visualization block, is coded using the Python libraries Matplotlib and Toolkits. The age class step was set to 20 years. The last age class contains all trees older than 140 years. We consider one cell on the time interval of 160 years. Therefore, in this example, N = 1, K = 8, and M = 8. Input parameters are shown in Figs. 3.1 and 3.2, and modeling results in Figs. 3.3, 3.4, 3.5, and 3.6. Discount factor has the form ρk = 1/(1 + r )qk , where parameters  p = 8/991. Price trend p (in  q = 1, sin(t)  3 −t/100  · 1 − 2  . Euro/m ) is taken as p(t) = 35 · e The optimal profit is shown in Fig. 3.6. Technically, the forest manager can consider several scenarios of (generally unknown) price projections. Using optimal

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Fig. 3.1 The biomass factor β (in m3 /ha) with respect to age a

Fig. 3.2 Aggregated cost C (in Euro/m3 ) with respect to age a

Fig. 3.3 Optimal control u

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Fig. 3.4 Distribution of area X (in ha) with respect to age a and time period t Fig. 3.5 Dynamics of harvested biomass H (in thousands of m3 ) with respect to time period t

Fig. 3.6 Dynamics of instantaneous profit with discount ρt (t) (in millions of Euro) with respect to time period t

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response in terms of profits, they can make various decisions, e.g., depending on their risk preferences. Although modeling results are presented for one cell, the consideration of multiple cells with individual cost and biomass profiles is straightforward due to analytical formulations.

3.5 Conclusions The solution, justified in Theorem 3.1, was derived using the Pontryagin maximum principle in the form of the constructive rule. This rule determines the optimal time and locations for harvesting depending on the spatially explicit forest growth dynamics and price projections. The model structure, including the cost functions depending on the age class for a certain species, as well as the biomass factors, allows for flexibility in model formulations. In the future, natural mortality along with the risk functions would be implemented in the model. Acknowledgements The research was supported by the RESTORE+ project (www.restoreplus. org), which is part of the International Climate Initiative (IKI), supported by the Federal Ministry for the Environment, Nature Conservation and Nuclear Safety (BMU) based on a decision adopted by the German Bundestag. The second author was also supported by the Russian Science Foundation (RSF) grant (Project No. 19-11-00223).

References 1. Clark, C.: Mathematical Bioeconomics. Wiley (1990) 2. Rämö, J., Tahvonen, O.: Optimizing the harvest timing in continuous cover forestry. Environ. Resour. Econ. 67(4), 853–868 (2017) 3. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience, New York (1962) 4. Boltyanskii, V.G.: Optimal Control of Discrete Systems. Wiley (1978) 5. Cerdá, E., Martín-Barroso, D.: Optimal control for forest management and conservation analysis in dehesa ecosystems. Eur. J. Oper. Res. 227(3), 515–526 (2013) 6. Ferreira, L., Constantino, M., Borges, J.G., Garcia-Gonzalo, J., Barreiro, S.: A climate change adaptive dynamic programming approach to optimize eucalypt stand management scheduling: a Portuguese application. Can. J. For. Res. 46(8), 1000–1008 (2016) 7. Segura, M., Ray, D., Maroto, C.: Decision support systems for forest management: a comparative analysis and assessment. Comput. Electron. Agric. 101, 55–67 (1970) 8. Pukkala, T., Kurttila, M.: Examining the performance of six heuristic optimisation techniques in different forest planning problems. Silva Fenn. 39(1), 67–80 (2005) 9. Usher, M.B.: A matrix model for forest management. Biometrics 25(2), 309–315 (1969) 10. Chikumbo, O., Mareels, I.M.Y.: Optimal control and parameter selection problems in forest stand management. WIT Trans. Ecol. Environ. 51 (1970)

Chapter 4

Program and Positional Control Strategies for the Lotka–Volterra Competition Model Nikolai L. Grigorenko, Evgenii N. Khailov, Anna D. Klimenkova, and Andrei Korobeinikov Abstract In this notice, we consider a control model of cancer dynamics that describes the interaction between normal and cancer cells and is based on the Lotka– Volterra competition model. For this control model, we consider a minimization problem of a terminal functional, which is a weighted difference of the concentrations of cancer and normal cells at the final moment of treatment. For this model, we consider three types of treatment. For each type, using the Pontryagin maximum principle, we analyze and establish properties of the optimal controls and illustrate the analytical findings by numerical calculations. Moreover, apart from the optimal controls, we also construct the positional controls for two variants of control constraints.

4.1 Introduction Cancer is one of the most common death causes. For example, more than 18 million cancer cases were diagnosed in 2018; about 9.6 million of these were fatal. The design of cancer control strategies and anti-cancer therapies involves mathematical modeling as its essential part. Thus, in order to describe the interaction of cancer cells and normal cells in the process of treating cancers such as lymphoma, myeloma, and leukemia, a competition model based upon the Lotka–Volterra competition model N. L. Grigorenko (B) · E. N. Khailov · A. D. Klimenkova Lomonosov Moscow State University, Moscow, Russia e-mail: [email protected] E. N. Khailov e-mail: [email protected] A. D. Klimenkova e-mail: [email protected] A. Korobeinikov Centre de Recerca Matematica, Barcelona, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_4

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is commonly used. Moreover, to find more effective therapies, the optimal control theory can be employed. Initial advance in this direction was reported in [1].

4.2 Model We consider the interaction of two types of cells, namely, the normal cells of concentration x(t) and the cancer cells of concentration y(t). We assume that these two cell types compete for a limited resource and that their dynamics is described by the logistic growth differential equations. Furthermore, we assume that the populations are objects of two types of therapies, namely, a cytotoxic therapy (that kills the cells) of intensity u(t), and a cytostatic therapy (that inhibits the cells proliferation) of intensity w(t). Controls u(t) and w(t) can represent drug concentrations or the radiotherapy intensity at time t. The population dynamics is described by the nonlinear control system of differential equations: ⎧ ˙ = r (1 − κ1 w(t))(1 − x(t) − a12 y(t))x(t) − m 1 u(t)x(t), ⎨ x(t) y˙ (t) = (1 − κ2 w(t))(1 − y(t) − a21 x(t))y(t) − m 2 u(t)y(t), (4.1) ⎩ x(0) = x0 , y(0) = y0 ; x0 , y0 > 0. Here, r is the intrinsic growth rate of the normal cells; a12 and a21 represent the relative compatibility of the cancer and normal cells, respectively. We assume that a12 · a21 = 1 (that is, cancer is more competitive than the normal cells). Parameters m 1 and m 2 represent the efficacy (the killing rates) of the cytotoxic therapy with respect to the normal and cancer cells, and κ1 and κ2 are the efficacies of inhibiting the normal and cancer cells proliferation by the cytostatic therapy; it is supposed that the inequalities m 2 > m 1 and κ2 > κ1 hold. Both controls, u(t) and w(t), are bounded, that is,   0 ≤ u(t) ≤ u max ≤ 1, 0 ≤ w(t) ≤ wmax < min κ1−1 , κ2−1 .

(4.2)

The system (4.1) is defined on a given time interval [0, T ]. When the controls u(t) and w(t) are absent, model (4.1) is the classical Lotka– Volterra model of two competing populations. The qualitative behavior of this model is determined by a mutual location of lines x + a12 y = 1 and y + a21 x = 1. Figure 4.1 shows four robust scenarios that are possible for the Lotka–Voltera competing model. For the cases in Fig. 4.1, the system can have up to four nonnegative equilibrium states, namely, (0, 0), (0, 1), (1, 0), and (b1 , b2 ), where b1 = (1 − a12 )/(1 − a12 a21 ) and b2 = (1 − a21 )/(1 − a12 a21 ). For this model, the origin is always an unstable node, whereas the other equilibrium states can be either saddles (marked by circles in Fig. 4.1) or attracting nodes (marked by dots in Fig. 4.1).

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Fig. 4.1 Robust scenarios that are possible for the Lotka–Volterra model of two competing populations

For model (4.1), the set of admissible controls (T ) is formed by all Lebesguemeasurable couples of functions (u(t), w(t)), which for almost all t ∈ [0, T ] satisfy constraints (4.2). The boundedness, positiveness, and continuation of solution (x(t), y(t)) for system (4.1) can be established by standard arguments. For system (4.1), on the set of admissible controls (T ), we consider the problem of minimization of a terminal functional J (u, w) = y(T ) − αx(T ),

(4.3)

where α > 0 is a given weight. The functional is a weighted difference of the cancerous and normal cell concentrations at the final moment of the therapy. The boundedness of solutions of system (4.1) guarantees the existence of the optimal solution for minimization problem (4.3).

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4.3 Optimal Controls For this model, we consider three scenarios, namely, (a) the cytotoxic therapy only (w(t) ≡ 0), (b) the cytostatic therapy only (u(t) ≡ 0), and (c) the therapy has the cytotoxic and cytostatic action, (w(t) ≡ u(t)). For each scenario, we conduct an analytical study and find the types of optimal controls. We illustrate these analytical findings by calculations using BOCOP-2.1.0 software package.

4.3.1 Killing Cancer Cells Initially, we consider the following system: ⎧ x(t) ˙ = r (1 − x(t) − a12 y(t))x(t) − m 1 u(t)x(t), t ∈ [0, T ], ⎪ ⎪ ⎪ ⎨ y˙ (t) = (1 − y(t) − a21 x(t))y(t) − m 2 u(t)y(t), x(0) = x0 , y(0) = y0 ; x0 , y0 > 0, ⎪ ⎪ ⎪ min . ⎩ J1 (u) = y(T ) − αx(T ) → (u(·),0)∈(T )

To analyze the optimal control u ∗ (t) and the corresponding optimal solution (x∗ (t), y∗ (t)), we apply the Pontryagin maximum principle. Analyzing the switching function, we come to the following conclusions: • If a21 m 1 − r m 2 ≥ 0 and m 1 − ra12 m 2 ≥ 0 hold, then the optimal control u ∗ (t) is a piecewise-constant function with the only one switching of the type  u ∗ (t) =

0 , if 0 ≤ t ≤ θ∗ , u max , if θ∗ < t ≤ T.

• If a21 m 1 − r m 2 ≤ 0 and m 1 − ra12 m 2 ≤ 0 hold, then the optimal control u ∗ (t) is a piecewise-constant function with the only one switching of the type  u ∗ (t) =

u max , if 0 ≤ t ≤ θ∗ , 0 , if θ∗ < t ≤ T.

• If a21 m 1 − r m 2 < 0 and m 1 − ra12 m 2 > 0 hold, then on entire interval [0, T ] the optimal control u ∗ (t) is a bang–bang control taking the values 0 or u max with a finite number of switchings. • If a21 m 1 − r m 2 > 0 and m 1 − ra12 m 2 < 0 hold, then on a subinterval  ⊂ [0, T ] the optimal control u ∗ (t) can have a singular arc of the type u sing (t) =

1−r (m 1 + m 2 )ra12 a21 − (r 2 a12 m 2 + a21 m 1 ) x∗ (t). + m2 − m1 m 2 (m 1 − ra12 m 2 )

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If the inclusion u sing (t) ∈ (0, u max ) holds for all t ∈ , then it is possible to concatenate the singular arc u sing (t) with bang–bang portions of the control u ∗ (t). These results were previously presented in [1].

4.3.2 Inhibiting Cancer Cells Proliferation For the second scenario, we consider the following system: ⎧ x(t) ˙ = r (1 − κ1 w(t))(1 − x(t) − a12 y(t))x(t), t ∈ [0, T ], ⎪ ⎪ ⎪ ⎨ y˙ (t) = (1 − κ2 w(t))(1 − y(t) − a21 x(t))y(t), x(0) = x0 , y(0) = y0 , (x0 , y0 ) = (b1 , b2 ); x0 , y0 > 0, ⎪ ⎪ ⎪ min . ⎩ J2 (w) = y(T ) − αx(T ) → (0,w(·))∈(T )

The Pontryagin maximum principle leads to the following conclusions: • For the parameters a12 and a21 that correspond to Scenarios 1, 3, and 4 in Fig. 4.1, the optimal control w∗ (t) is a piecewise-constant function with one switching either of the type  0 , if 0 ≤ t ≤ τ∗ , w∗ (t) = (4.4) wmax , if τ∗ < t ≤ T, or of the type

 w∗ (t) =

wmax , if 0 ≤ t ≤ τ∗ , 0 , if τ∗ < t ≤ T.

• For the parameters a12 and a21 that correspond to Scenario 2 in Fig. 4.1, the optimal control w∗ (t) is a piecewise-constant function with the only one switching of the type (4.4). Thus, if the cytostatic therapy is only used, the optimal control on entire interval [0, T ] has no more than one switching.

4.3.3 Cytotoxic and Cytostatic Actions Are Simultaneously Applied In this case, we consider the following system: ⎧ x(t) ˙ = r (1 − κ1 u(t))(1 − x(t) − a12 y(t))x(t) − m 1 u(t)x(t), t ∈ [0, T ], ⎪ ⎪ ⎪ ⎨ y˙ (t) = (1 − κ2 u(t))(1 − y(t) − a21 x(t))y(t) − m 2 u(t)y(t), x(0) = x0 , y(0) = y0 ; x0 , y0 > 0, ⎪ ⎪ ⎪ min . ⎩ J3 (u) = y(T ) − αx(T ) → (u(·),u(·))∈(T )

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Our numerical study shows that in this case the optimal control u ∗ (t) can be a bang– bang control or it can have a singular arc. All types of the optimal control u ∗ (t) confirmed the switching function analysis.

4.3.4 Numerical Results Figure 4.2 presents results of computations for the scenarios considered in Sects. 4.3.1–4.3.3. In this figure, the diagrams in the upper row depict the optimal controls for the three scenarios, namely, the cytotoxic therapy only (scenario (a)), the cytostatic therapy only (scenario (b)), and the therapy producing cytotoxic and cytostatic effects (scenario (c)). In this figure, the end of interval [0, T ] is shown by the vertical dot-dashed lines. The diagrams in the lower row depict the corresponding phase portraits. On these phase portraits, the solid black lines represent the optimal solutions, whereas the gray dashed lines represent the solutions in the absence of the control. The gray solid lines show the continuations of the optimal solutions for a longer time interval (in this case, for [T, 2T ]). We would like to stress that in these examples the initial conditions are located in the domain of attraction of the point (0, 1) that corresponds to extinction of the normal cells. The numerical results confirm that at least in some conditions the optimal control is able to move the state of the system into the domain of attraction of the point (1, 0), where cancer cell population is dying out due to the competition.

Fig. 4.2 The optimal controls and phase trajectories for cytotoxic therapy (a), cytostatic therapy (b), and a therapy producing both effects (c). In scenario (a) x(0) = 0.6, y(0) = 0.4, r = 0.3, a12 = 1.5, a21 = 1.2, m 1 = 0.2, m 2 = 0.7, α = 1, T = 30 and u max = 1. In (b) x(0) = 0.4, y(0) = 0.3, r = 0.5, a12 = 1.5, a21 = 1.4, κ1 = 0.1, κ2 = 0.5, α = 1, T = 30 and wmax = 1. In (c) x(0) = 0.5, y(0) = 0.5, r = 0.6, a12 = 1.5, a21 = 1.4, κ1 = 0.2, κ2 = 0.5, m 1 = 0.3, m 2 = 0.4, α = 1, T = 30, and u max = 1

4 Program and Positional Control Strategies …

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4.4 Positional Control Strategies In this section, we consider the problem of constructing the positional control u(x, y) and constant control w ∈ [0, wmax ] capable of transferring a solution of system (4.1) to a small neighborhood of the point (1, 0). The positional control and the corresponding solution of system (4.1) are understood according to [2]. Such a control, considered as a controlled therapy, can be applied as an initial approximation for an optimal control problem and employed for adjusting the process with respect to current values of the state variables. To construct such a positional control, we apply the method of analytical design of aggregated regulators [3]. In the absence of restrictions on the control u and with the constant control w, a version of this method is to find a macro-variable ϕ(x, y), which is a differentiable function and for which the following hold: • the differential equation T1 (ϕ) t + ϕ = 0 generates the positional control u(x, y) t→+∞

such that y(t) −→ 0 by the second equation of system (4.1). Here T1 is a positive parameter and (ϕ) t is a derivative by virtue of system (4.1). • the function y(x) obtained from the equality ϕ(x, y) = 0 and substituted into the first equation of system (4.1) at u = u(x, y) ensures for the solution of the first t→+∞ t→+∞ equation of the system at y(t) −→ 0 the limit relationship x(t) −→ 1. If there are restrictions on the control u, this approach is applicable for the expansion of the original system and the use of several macro-variables.

4.4.1 Case 1: Unrestricted Control u and Constant Control w Let us assume that in control system (4.1) there are no restrictions on control u and that control w is constant, w ∈ [0, wmax ] on a non-fixed time interval [0, ]. Let β1 and T1 be given positive parameters. Let us define the macro-variable ψ(x, y) = β1 y as an asymptotically stable for t → +∞ solution of the differential equation T1 (ψ) t + ψ = 0. Here (ψ) t is a derivative by virtue of system (4.1). Let us denote f (x, y) = 1 − x − a12 y and g(x, y) = 1 − y − a21 x. Then, by virtue of the second equation of system (4.1), the corresponding positional control is

−1 (1 − κ . w)g(x, y) + T u = m −1 2 2 1

(4.5)

 Indeed, for t > T1 ln −1 β1 y0 , where is a small positive value, we have |ψ| < and, respectively, |y| < β1−1 . Moreover, using (4.5), for all such t the first equation of system (4.1) has the following form:  (1 − κ2 w)(1 − y) + T1−1 x˙ = x r (1 − κ1 w)(1 − a12 y) − m 1 m −1 2 

−x r (1 − κ1 w) − m 1 m −1 2 (1 − κ2 w)a21 .

(4.6)

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Fig. 4.3 Graphs of the functions of y(x), u(t) and p(w)

If inequalities  (1 − κ2 w)(1 − y) + T1−1 > 0, r (1 − κ1 w)(1 − a12 y) − m 1 m −1 2 r (1 − κ1 w) − m 1 m −1 2 (1 − κ2 w)a21 > 0 hold, then the logistic equation (4.6) has the solution x(t) such that t→+∞

x(t) −→ p(w) =

 (1 − κ2 w) + T1−1 r (1 − κ1 w) − m 1 m −1 2 r (1 − κ1 w) − m 1 m −1 2 (1 − κ2 w)a21

.

Finally, for given β1 and T1 , one can take the value w ∈ [0, wmax ] such that ensuring that the value | p(w) − 1| is minimal. Figure 4.3 shows the phase trajectories of system (4.1) for x0 = 1.0 and y0 ∈ {0.1; 0.4; 1.0} for the control u given by (4.5), the graphs of the control u(t) = u(x(t), y(t)) and the graph of p(w) for x0 = 1.0, y0 = 1.0, and for the following parameters: r = 0.9, a12 = 1.5, a21 = 1.0, κ1 = 0.2, κ2 = 0.45, m 1 = 0.3, m 2 = 0.4, β1 = 1.0, T1 = 6.0.

(4.7)

We can conclude that, for an arbitrary initial point (x0 , y0 ), there exist values of β1 , T1 , and w which ensure that the control (4.5) transfers the corresponding phase trajectory of system (4.1) into a small neighborhood of the terminal point (1, 0) for a finite time. To find the best values for the parameters β1 , T1 and w, we can solve numerically the finite-dimensional minimization problem: I1 (β1 , T1 , w) = (x( ) − 1)2 + y 2 ( ) → min .

4 Program and Positional Control Strategies …

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4.4.2 Case 2: Bounded Control u ∈ [0, umax ] and Constant Control w Let us consider the control system (4.1) with the bounded control u ∈ [0, u max ] and constant control w, w ∈ [0, wmax ]. We also assume that function u(t) is differentiable. Let us introduce additional variables v and z, such that u = v + 0.5u max , and consider the extended four-dimensional control system: ⎧ x˙ = r (1 − κ1 w)x f (x, y) − m 1 x(v + 0.5u max ), ⎪ ⎪ ⎪ ⎪ ⎨ y˙ = (1 − κ2 w)yg(x,  y) − m 2 y(v + 0.5u max ), v˙ = h 1 z − h 1 A1 tan π u −1 max v , ⎪ ⎪ u , z ˙ = h ⎪ 2 1 ⎪ ⎩ x(0) = x0 , y(0) = y0 , v(0) = 0, z(0) = 0.

(4.8)

Here u 1 (t) is a new Lebesgue-measurable function, and A1 and h i , i = 1, 2 are given positive parameters. System (4.8) is considered on a non-fixed time interval [0, ]. Analyzing system (4.8), it is easy to verify that variable v(t) satisfies the inequality |v(t)| ≤ 0.5u max . Let us denote F(x, y, w) = r (1 − κ1 w)x f (x, y) and G(x, y, w) = (1 − κ2 w) yg(x, y). We introduce the first macro-variable ψ1 (x, y, v, z) = z + u 2 (x, y, v) requiring that it satisfies the differential equation T1 (ψ1 ) t + ψ1 = 0. Here T1 is a given positive parameter and (ψ1 ) t is the derivative by virtue of system (4.8). Substituting ψ1 in the last equation of system (4.8), we find (u 2 ) x [F(x, y, w)x − m 1 x(v + 0.5u max )] u 1 = −h −1 2 + (u 2 ) y [G(x, y, w)y − m 2 y(v + 0.5u max )]

(4.9)

+ (u 2 ) v v˙ + T1−1 (z + u 2 (x, y, v)) .

Control u 1 (x, y, v, z) transfers system (4.8) into a small neighborhood of the set {(x, y, v, z) : ψ1 = 0}. By these arguments, system (4.8) can be reduced to the three-dimensional control system: ⎧ x˙ = x F(x, y, w) − m 1 x(v + 0.5u max ), ⎪ ⎪ ⎨ y˙ = yG(x, y, w) − m2 y(v + 0.5u max ), −1 v ˙ = h u − h A tan π u v ⎪ 1 2 1 1 max , ⎪ ⎩ x(0) = x0 , y(0) = y0 , v(0) = 0.

(4.10)

One can find the control u 2 from the condition that the second macro-variable ψ2 (x, y, v) = G(x, y, w) − m 2 (v + 0.5u max ) + β2 satisfies the differential equation T2 (ψ2 ) t + ψ2 = 0. Here β2 and T2 are given positive parameters, and (ψ2 ) t is a derivative by virtue of system (4.10).

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Substituting ψ2 into the last equation of system (4.10), we obtain u 2 = −(m 2 h 1 )−1 (1 − κ1 w)a21 x [F(x, y, w) − m 1 (v + 0.5u max )]

 −1 + (1 − κ2 w)y [G(x, y, w) − m 2 (v + 0.5u max )]

− m 2 h 1 A1 tan π u max v − T2−1 (G(x, y, w) − m 2 (v + 0.5u max ) + β2 ) .

(4.11) Control u 2 (x, y, v) transfer system (4.10) into a small neighborhood of the set {(x, y, v) : ψ2 = 0}. In order to find a specific formula for control u 1 given by (4.9), it is necessary to calculate the partial derivatives of function u 2 with respect to variables x, y, and v. The expressions of these partial derivatives, as well as this of control u 1 , are cumbersome and, therefore, they are omitted here. We only note that this control is admissible when maxt∈[0, ] |v(t)| < 0.5u max . Motion along ψ2 = 0 is described by the following system: ⎧ ⎪ ⎪ x˙ = x r (1 − κ1 w)(1 − a12 y) − m 1 m −1 2 ((1 − κ2 w)(1 − y) + β2 ) ⎨ 

−1 − x r (1 − κ , w) − m m (1 − κ w)a 1 1 2 21 2 ⎪ ⎪ ⎩ y˙ = −β2 y. If inequalities

(4.12)

r (1 − κ1 w) − m 1 m −1 2 ((1 − κ2 w) + β2 ) > 0, r (1 − κ1 w) − m 1 m −1 2 (1 − κ2 w)a21 > 0

hold, then the logistic system (4.12) has a solution x(t) such that t→+∞

x(t) −→ q(w) =

r (1 − κ1 w) − m 1 m −1 2 ((1 − κ2 w) + β2 ) r (1 − κ1 w) − m 1 m −1 2 (1 − κ2 w)a21

.

Finally, for given β1 , h 1 , h 2 , T1 , T2 , and A1 , one can choose such a value for w ∈ [0, wmax ] that minimizes the value |q(w) − 1|. Figure 4.4 shows a phase trajectory (x(t), y(t)) of system (4.1) for the control u 1 given by (4.9) and the graphs of corresponding u(t) = u(x(t), y(t)) and q(w) for some parameters. From the arguments above, it can be seen that for some (but not for all) initial points (x0 , y0 ) there exist the values of β2 , h 1 , h 2 , T1 , T2 , A1 , and w such that the control (4.9) transfers the corresponding trajectory of system (4.1) into a small neighborhood of the terminal point (1, 0) for a finite time interval. To find the best values of the parameters β2 , h 1 , h 2 , T1 , T2 , A1 , and w for which the considered problem is solvable, it is proposed to solve numerically the finite-dimensional minimization problem: I2 (β2 , h 1 , h 2 , T1 , T2 , A1 , w) = (x( ) − 1)2 + y 2 ( ) → min .

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Fig. 4.4 A phase trajectory y(x) and the graphs of the functions of u(t) and q(w) for x0 = 1.0 and y0 = 0.4. The values of parameters are β2 x = 0.1, h 1 = 1.0, h 2 = 0.0001, T1 = 1.0, T2 = 2.0, A1 = 5.0, u max = 0.2; the other parameters are from (4.7)

References 1. Khailov, E.N., Klimenkova, A.D., Korobeinikov, A.: Optimal control for anti-cancer therapy. In: Korobeinikov, A., Caubergh, M., Lázaro, T., Sardanyés, J. (eds.) Extended Abstracts Spring 2018. Trends in Mathematics, vol. 11, pp. 35–43. Birkhäuser, Basel (2019) 2. Krasovskii, N.N., Subbotin, A.I.: Positional Differential Games. Nauka, Moscow (1974) (in Russian) 3. Kolesnikov, A., Veselov, G., Kolesnikov, Al., Monti, A., Ponci, F., Santi, E., Dougal, R.: Synergetic synthesis of Dc-Dc boost converter controllers: theory and experimental analysis. In: Conference Proceedings 17th Annual IEEE Applied Power Electronics Conference and Exposition, Dallas, TX, USA, pp. 409–415 (2002)

Chapter 5

Construction of Dynamically Stable Solutions in Differential Network Games Leon Petrosyan and David Yeung

Abstract This paper presents a novel measure of the worth of coalitions—named as a cooperative-trajectory characteristic function—to generate a time-consistent Shapley value solution in a class of network differential games. This new class of characteristic function is evaluated along the cooperative trajectory. It measures the worth of coalitions under the process of cooperation instead of under minmax confrontation or Nash non-cooperative stance. The resultant time-consistent Shapley value calibrates the marginal contributions of individual players to the grand coalition payoff based on their cooperative actions/strategy. The cooperative-trajectory characteristic function is also time consistent and yields a new cooperative solution in network differential games.

5.1 Introduction A fast-growing branch of game theory is network games or games on networks. Mazalov and Chirkova [4] provided a comprehensive disquisition on the theory and applications of networking games. Given that most real-life game situations are dynamic (intertemporal) rather than static, network differential games have become a field that attracts theoretical and technical developments. The impacts of Krasovskii’s [3] classic book on the fundamentals of differential games are profound. Wie [12, 13] developed differential game models for studying traffic network. Zhang et al. [17] presented a differential game of network defense. Pai [6] provided a differential game formulation of a controlled network. Meza and Lopez-Barrientos [5] examined a differential game of a duopoly with network externalities. Cao and Ertin [1] studied the minimax equilibrium of network differential games. Petrosyan [9] develL. Petrosyan (B) Saint Petersburg State University, Universitetskaya Emb., 7/9, Saint Petersburg, Russia e-mail: [email protected] D. Yeung Hong Kong Shue Yan University, Hong Kong, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_5

51

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oped cooperative differential games on networks. Coordinating players in a network to maximize their joint gain and distribute the cooperative gains in a dynamically stable scheme is a topic of ongoing research. Designing solutions for cooperative network differential games is a classic problem that attracts research efforts. The Shapley [11] value appears to be one of the best solutions in attributing a fair gain to each player in a complex situation like a network. However, the determination of the worth of the subsets of players (characteristic function) in the Shapley value is not indisputably unique. In addition, in a differential game, the worth of the coalitions of players changes as the game evolves. In this paper, we present a novel characteristic function—named as cooperative-trajectory characteristic function—to generate a time-consistent Shapley value solution in a class of network differential games. In computing the values of characteristic function for coalitions, we maintain the cooperative strategies for all players and evaluate the worth of the coalitions along the cooperative trajectory. Thus, the worth of coalition S includes its cooperative payoff with the exclusion of the gains through network connection from players outside the coalition. The rationale for such formulation is to attribute the contributions of the players in the process of cooperation. Worth-noting is that this new set of characteristic function is time consistent. The time consistency property of the characteristic function has not been shared by existing characteristic functions in differential games (see [2, 8, 10, 14–16]). It is the first time that the worth of coalitions is measured under the process of cooperation instead of under minmax confrontation or Nash non-cooperative stance. This paper is organized as follows. Section 5.2 presents a class of network differential games and derives the worth of coalition in the form of cooperative-trajectory characteristic function. Section 5.3 shows that using this type of characteristic function, time-consistent Shapley value imputation can be obtained. Conclusions are given in Sect. 5.4.

5.2 Game Formulation and Worth of Coalition Consider a class of differential games on network. Let N = {1, 2, . . . , n} be the set of players which are connected in the network. The nodes of this network are players from the set N . We use the nodes of the network in the same manner as players in corresponding nodes. We denote the set of nodes by Nˆ and denote the set of all arcs in network Nˆ by L. The arcs in L are the ar c (i, j) ∈ L for players i, j ∈ N . For notational convenience, we denote the set of players  connected to player i as K˜ (i) = { j : ar c(i, j) ∈ L}, and the set K (i) = K˜ (i) {i}, for i ∈ N . Let x i ∈ R m be the state variable of player i ∈ N , and u i ∈ U i ⊂ R k the control variable player i ∈ N . Every player i ∈ N can cut the connection with any other players from the set N at any instant of time.

5 Construction of Dynamically Stable Solutions …

53

The state dynamics of the game is x˙ i = f i (x i , u i ), x i (t0 ) = x0i for t ∈ [t0 , T ] and i ∈ N .

(5.1)

The payoff function of player i depends upon his state variable, the state variables of players from the set K (i), and his own control variable: Hi (x0i , x0K (i) , u 1 , . . . , u n )

=

T  

j

h i (x i (τ ), x j (τ ), u i (τ ))dτ h i ≥ 0.

(5.2)

j∈K (i) t

0

Now, consider the case where the players cooperate to maximize their joint payoff. Let the cooperative trajectory be denoted by x(t) ¯ = (x¯ 1 (t), x¯ 2 (t), . . . , x¯ n (t)), and the optimal cooperative strategies of player i by u¯ i (t), for t ∈ [t0 , T ] and i ∈ N . The joint cooperative payoff can be expressed as T   

j

h i (x¯ i (τ )x¯ j (τ )u¯ i (τ ))dτ

i∈N j∈K (i) t

0

 

T j

(5.3)

x˙ i = f i (x i , u i ), x i (t0 ) = x0i for t ∈ [t0 , T ] and i ∈ N .

(5.4)

=

max

u 1 ,u 2 ,...,u n

h i (x i (τ ), x j (τ ), u i (τ ))dτ

i∈N j∈K (i) t

0

subject to

Then we consider allotting the cooperative gains to individual players according to the Shapley value. One of the issues in using the Shapley value is the determination of the worth of coalitions (which is often called characteristic function). In cooperative differential games, the characterization of the worth of a coalition of players S is not indisputably unique except for the case of zero-sum games in which coalition S seeks to maximize its payoff while coalition N \S seeks to minimize the payoff. In this section, we present a new formulation of the worth of coalition S ⊂ N as V (S; x0 , T − t0 ) =





T j

h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ ))dτ.

(5.5)

i∈S j∈K (i)∩S t 0

Note that the worth of coalition S is measured by the sum of the payoffs of the players in the coalition in the cooperation process with the exclusion of the gains from players outside coalition S. Thus, the characteristic function reflecting the worth of coalition S in (5.5) is formulated along the cooperative trajectory x(t). ¯ This is a novel feature and we name it as cooperative-trajectory characteristic function.

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For simplicity in notation, we denote T j

αi j (x0 , T − t0 ) =

h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ ))dτ t0

and

T j

αi j (x(t), ¯ T − t) =

h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ ))dτ,

(5.6)

t

for t ∈ [t0 , T ]. Using the notations in (5.6), we can express (5.5) as V (S; x0 , T − t0 ) =





i∈S j∈K (i)



αi j (x0 , T − t0 ),

(5.7)

S

and similarly, along the cooperative trajectory x(t) ¯ V (S; x(t), ¯ T − t) =





i∈S j∈K (i)



αi j (x(t), ¯ T − t), for t ∈ [t0 , T ]. S

Proposition 5.1 The following inequalities hold for cooperative-trajectory characteristic function: V (S 1 ∪ S2 ; x0 , T − t0 ) ≥ V (S 1 ; x0 , T − t0 ) + V (S 2 ; x0 , T − t0 ) −V (S 1 ∩ S2 ; x0 , T − t0 ). Proof See Appendix.

(5.8) 

Similarly, along the cooperative trajectory x(t). ¯ Proposition 5.2 The following inequalities hold for cooperative-trajectory characteristic function along the cooperative trajectory: ¯ T − t) ≥ V (S 1 ; x(t), ¯ T − t)) + V (S 2 ; x(t), ¯ T − t)) V (S 1 ∪ S2 ; x(t), −V (S 1 ∩ S2 ; x(t), ¯ T − t)), for t ∈ [t0 , T ]. Proof Follow the proof of Proposition 5.1.

(5.9) 

The inequalities (5.8) and (5.9) imply that the game is convex and so are the subgames along the cooperative trajectory. This also means that the core of the game is not void and the Shapley value belongs to the core. In the case where S1 and S2 are distinct

5 Construction of Dynamically Stable Solutions …

55

subsets, we obtain the super-additivity of the cooperative-trajectory characteristic function as ¯ T − t) ≥ V (S 1 ; x(t), ¯ T − t)) + V (S 2 ; x(t), ¯ T − t)). V (S 1 ∪ S2 ; x(t), The cooperative-trajectory characteristic function in (5.5) can be expressed as

V (S; x0 , T − t0 ) =





i∈S j∈K (i)

=





i∈S j∈K (i)





T j

h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ ))dτ S t0

t j

h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ ))dτ + V (S; x(t), ¯ T − t), for S ⊂ N . S t0

(5.10) Condition (5.10) exhibits the time consistency property of the cooperative-trajectory characteristic function V (S; x0 , T − t0 ). This property has not been shared by any existing characteristic functions in differential games. Finally, any individual player attempting to act independently will have the links to other players in the network being cut off.

5.3 Imputation of Cooperative Network Gains In this section, we consider using the Shapley value to distribute the cooperative network gain. We first derive the Shapley value using the cooperative-trajectory characteristic function V (S; x(t), ¯ T − t). Then we formulate an imputation distribution procedure (IDP) such that the Shapley imputation can be realized.

5.3.1 Time-Consistent Dynamic Shapley Value Imputation Now consider allocating the grand coalition cooperative network gain V (N ; x0 , T − t0 ) according to the Shapley value. Player i’s payoff under cooperation would be Sh i (x0 , T − t0 ) =  S⊂N , i∈S

(|S| − 1)!(n − |S|)! [V (S; x0 , T − t0 ) − V (S\{i}; x0 , T − t0 )], (5.11) n!

for i ∈ N .

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Invoking (5.7), the difference V (S; x0 , T − t0 ) − V (S\{i}; x0 , T − t0 ) can be expressed as 



αi j (x0 , T − t0 ) −

i∈S j∈K (i)∩S

=







αl j (x0 , T − t0 ) =

l∈S\{i} j∈K (l)∩S\{i}

αi j (x0 , T − t0 ) +

j∈K (i)∩S



α ji (x0 , T − t0 ).

(5.12)

j∈K (i)∩S

Therefore, we can obtain the cooperative payoff of player i as 

Sh i (x0 , T − t0 ) = ⎡ ⎣

(|S| − 1)!(n − |S|)! × n!

S⊂N i∈S 



αi j (x0 , T − t0 ) +

j∈K (i)∩S

⎤ α ji (x0 , T − t0 )⎦ .

(5.13)

j∈K (i)∩S

For subgames along the cooperative trajectory, the cooperative payoff of player i can be obtained as follows:

Sh i (x(t), ¯ T − t) = ⎡ ⎣

 S⊂N i∈S

 j∈K (i)∩S

αi j (x(t), ¯ T − t) +

(|S| − 1)!(n − |S|)! × n!



⎤ α ji (x(t), ¯ T − t)⎦ ,

(5.14)

j∈K (i)∩S

for t ∈ [t0 , T ]. In addition, we show that using the cooperative-trajectory characteristic function, the Shapley value imputation is time consistent. Proposition 5.3 The Shapley value imputation in (5.13)–(5.14) is time consistent. Proof Substituting the values of αi j (x0 , T − t0 ) into (5.13), we obtain

5 Construction of Dynamically Stable Solutions …

57



Sh i (x0 , T − t0 ) =

S⊂N i∈S T

 j∈K (i)

j



[h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ )) + h ij (x¯ j (τ ), x¯ i (τ ), u¯ j (τ ))dτ ] S t0



=

S⊂N , i∈S j



[h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ ))dτ + h ij (x¯ j (τ ), x¯ i (τ ), u¯ j (τ ))] S t0

+

 S⊂N i∈S

(|S| − 1)!(n − |S|)! × n!

T

 j∈K (i)

(|S| − 1)!(n − |S|)! × n!

t

 j∈K (i)

(|S| − 1)!(n − |S|)! × n!

j



[h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ )) + h ij (x¯ j (τ ), x¯ i (τ ), u¯ j (τ ))]dτ.

(5.15)

S t

Invoking (5.14), we have Sh i (x0 , T − t0 ) =

 S⊂N i∈S



t

j∈K (i)∩S t0

(|S| − 1)!(n − |S|)! × n!

j

[h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ )) + h ij (x¯ j (τ ), x¯ i (τ ), u¯ j (τ ))]dτ + Sh i (x(t), ¯ T − t),

which exhibits the time consistency property of the Shapley value imputation ¯ T − t), for t ∈ [t0 , T ].  Sh i (x(t), Crucial to the analysis is the design of an IDP such that the Shapley imputation (5.13)–(5.14) can be realized. This will be done in the next section.

5.3.2 Instantaneous Characteristic Function and IDP Differentiating the characteristic function (5.10) with respect to t yields 



i∈S j∈K (i)



j

h i (x¯ i (t), x¯ j (t), u¯ i (t)) = − S

d V (S; x(t), ¯ T − t). dt

(5.16)

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L. Petrosyan and D. Yeung

Following Petrosian et al. [7], we define W (S; x(t), ¯ T − t) =





i∈S j∈K (i)



j

h i (x¯ i (t), x¯ j (t), u¯ i (t)), for S ⊂ N ,

(5.17)

S

as the “instantaneous characteristic function” of the game at time t ∈ [t0 , T ]. Following the proof of Proposition 5.1, we can obtain the inequality ¯ T − t) ≥ W (S 1 ; x(t), ¯ T − t)) + W (S 2 ; x(t), ¯ T − t)) W (S 1 ∪ S2 ; x(t), −W (S 1 ∩ S2 ; x(t), ¯ T − t)), for t ∈ [t0 , T ]. (5.18) We introduce the instantaneous core C(x(t), ¯ T − t) as the set of imputations ξ(t) = (ξ1 (t), ξ2 (t), . . . , ξn (t)) such that n 

ξi (t) = W (N ; x(t), ¯ T − t) and

i=1

for S ⊂ N , S = N and ∈ [t0 , T ].



ξi (t) ≥ W (S; x(t), ¯ T − t),

i∈S

(5.19)

Given the properties of W (S; x(t), ¯ T − t) in (5.18), the set C(x(t), ¯ T − t) is nonempty. An IDP leading to the realization of the Shapley value imputation (5.13)– (5.14) has to satisfy T

T ξi (τ )dτ = Sh i (x0 , T − t0 ) and

t0

ξi (τ )dτ = Sh i (x(t), ¯ T − t). t

Theorem 5.1 An imputation distribution procedure (IDP) prescribing player i ∈ N at time t ∈ [t0 , T ] an allotment βi (t) =



(|S| − 1)!(n − |S|)! n!

S⊂N i∈S + h ij (x¯ j (τ ), x¯ i (τ ), u¯ j (τ ))], βi (t) ≥ 0,

 j∈K (i)



j

[h i (x¯ i (t), x¯ j (t), u¯ i (t)) + S

(5.20)

¯ T − t) in would lead to the realization of the Shapley value imputation Sh i (x(t), (5.13)–(5.14). Proof Using the IDP [8, 10] in (5.20), we obtain the Shapley value

5 Construction of Dynamically Stable Solutions …

59

T βi (τ )dτ = t0

=



(|S| − 1)!(n − |S|)! n!

T



j

[h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ ))

(5.21)

j∈K (i)∩S t 0 S⊂N i∈S + h ij (x¯ j (τ ), x¯ i (τ ), u¯ j (τ ))]dτ = Sh i (x0 , T − t0 );

T βi (τ )dτ = t

=



(|S| − 1)!(n − |S|)! n!



T j

h i (x¯ i (τ ), x¯ j (τ ), u¯ i (τ )) +

(5.22)

j∈K (i)∩S t S⊂N i∈S + h ij (x¯ j (τ ), x¯ i (τ ), u¯ j (τ ))]dτ = Sh i (x(t), T − t)

and t Sh i (x0 , T − t0 ) =

βi (τ )dτ + Sh i (x(t), ¯ T − t), for t ∈ [t0 , T ].

(5.23)

t0

One can readily verify that the imputation distribution procedure β(t) = (β1 (t), β2 (t), . . . , βn (t)) satisfies n 

βi (t) = W (N ; x(t), ¯ T − t) and

i=1



βi (t) ≥ W (S; x(t), ¯ T − t),

(5.24)

i∈S

for S ⊂ N , S = N and t ∈ [t0 , T ]. Hence, β(t) belongs to the instantaneous core C(x(t), ¯ T − t).

5.4 Conclusions This paper presents a novel form for measuring the worth of coalitions—named as cooperative-trajectory characteristic function—for a class of network differential games. Using this type of characteristic function, we generate a time-consistent dynamic Shapley value solution in a class of network differential games.

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The salient features of the cooperative-trajectory characteristic function include • measuring the worth of coalitions in the process of cooperation along the cooperative trajectory, • generating time-consistent Shapley value imputation in a dynamic framework, • the marginal contributions of individual players are evaluated based on their cooperative actions/strategy, • the time consistency of the cooperative-trajectory characteristic function itself along the cooperative trajectory, and • the simplicity in evaluating the cooperative-trajectory characteristic function. This is the first time that cooperative-trajectory characteristic function is used in network differential games, further fruitful research along this line is expected. Acknowledgements The work was supported by Russian Science Foundation grant Optimal Behavior in Conflict-Controlled Systems (N 17-11-01079).

Appendix: Proof of Proposition 5.1. Using (5.7), we have V (S 1 ∪ S2 ; x0 , T − t0 ) =

i∈S1

=







 

αi j (x0 , T − t0 ) +

i∈S1 j∈K (i)∩S1





αi j (x0 , T − t0 )

i∈S2 j∈K (i)∩S2



−

αi j (x0 , T − t0 )

S2 j∈K (i)∩(S1 ∪S2 )



αi j (x0 , T − t0 )

i∈S1 ∩S2 j∈K (i)∩(S1 ∩S2 )

+





αi j (x0 , T − t0 ) +

i∈S1 j∈K (i)∩S2

≥







αi j (x0 , T − t0 )

i∈S2 j∈K (i)∩S1

αi j (x0 , T − t0 ) +

i∈S1 j∈K (i)∩S1

−







αi j (x0 , T − t0 )

i∈S2 j∈K (i)∩S2





αi j (x0 , T − t0 )

i∈S1 ∩S2 j∈K (i)∩(S1 ∩S2 )

= V (S 1 ; x0 , T − t0 ) + V (S 2 ; x0 , T − t0 ) − V (S 1 ∩ S2 ; x0 , T − t0 ). Hence Proposition 5.1 follows.

(5.25)

5 Construction of Dynamically Stable Solutions …

61

References 1. Cao, H., Ertin, E.: MiniMax equilibrium of networked differential games. ACM Trans. Auton. Adapt. Syst. 3(4) (2018). https://doi.org/10.1145/1452001.1452004 2. Gromova, E.: The Shapley Value as a Sustainable Cooperative Solution in Differential Games of Three Players. Recent Advances in Game Theory and Applications, Static & Dynamic Game Theory: Foundations & Applications. Springer (2016). https://doi.org/10.1007/978-3319-43838-2_4 3. Krasovskii, N.N.: Control of Dynamic System. Nauka (1985) 4. Mazalov, V., Chirkova, J.V.: Networking Games: Network Forming Games and Games on Networks. Academic Press (2019) 5. Meza, M.A.G., Lopez-Barrientos, J.D.: A differential game of a duopoly with network externalities. In: Petrosyan, L.A., Mazalov, V.V. (eds.) Recent Advances in Game Theory and Applications. Springer, Birkhäuser (2016). https://doi.org/10.1007/978-3-319-43838-2 6. Pai, H.M.: A differential game formulation of a controlled network. Queueing Syst.: Theory Appl. Arch. 64(4), 325–358 (2010) 7. Petrosian, O.L., Gromova, E.V., Pogozhev, S.V.: Strong time-consistent subset of core in cooperative differential games with finite time horizon. Math. Theory Games Appl. 8(4), 79–106 (2016) 8. Petrosjan, L.A.: The Shapley value for differential games. In: Olsder G.J. (ed.) New Trends in Dynamic Games and Applications. Annals of the International Society of Dynamic Games, vol. 3, pp. 409–417. Birkhäuser Boston (1995) 9. Petrosyan, L.A.: Cooperative differential games on networks. Trudy Inst. Mat. i Mekh. UrO RAN 16(5), 143–150 (2010) 10. Petrosyan, L., Zaccour, G.: Time-consistent Shapley value allocation of pollution cost reduction. J. Econ. Dyn. Control. 27, 381–398 (2003) 11. Shapley, L.S.: A value for N-person games. In: Kuhn, H., Tucker, A. (eds.) Contributions to the Theory of Games, pp. 307–317. Princeton University Press, Princeton (1953) 12. Wie, B.W.: A differential game model of Nash equilibrium on a congested traffic network. Networks 23, 557–565 (1993) 13. Wie, B.W.: A differential game approach to the dynamic mixed behavior traffic network equilibrium problem. Eur. J. Oper. Res. 83(1), 117–136 (1995) 14. Yeung, D.W.K., Petrosyan, L.A.: Subgame Consistent Cooperation—A Comprehensive Treatise. Springer (2016) 15. Yeung, D.W.K.: Subgame consistent Shapley value imputation for cost-saving joint ventures. Math. Game Theory Appl. 2(3), 137–149 (2010) 16. Yeung, D.W.K., Petrosyan, L.A.: Dynamic Shapley Value and Dynamic Nash Bargaining. Nova Science, New York (2018) 17. Zhang, H., Jiang, L.V., Huang, S., Wang, J., Zhang, Y.: Attack-defense differential game model for network defense strategy selection. IEEE Access (2018). https://doi.org/10.1109/ACCESS. 2018.2880214

Chapter 6

On a Differential Game in a System Described by a Functional Differential Equation A. A. Chikrii, A. G. Rutkas, and L. A. Vlasenko

Abstract We study a differential game of approach in a system whose dynamics is described by a functional differential equation in a Hilbert space. The main assumption on the equation is that the operator multiplying the system state at the current time is the generator of a strongly continuous semigroup of bounded linear operators. Weak solutions of the equation are represented by the variation of constant formula. To obtain solvability conditions for the approach of the system state to a cylindrical terminal set, we use the technique of set-valued mappings and their selections and also constraints on support functionals of sets defined by the behaviors of pursuer and evader. The paper contains an example to illustrate the differential game in a system described by a partial functional differential equation with time delay. In particular, we investigate the heat equation with heat loss and with controlled distributed heat source and leak.

6.1 Introduction Numerous applications to physics, biology, radio engineering, gas dynamics, etc. stimulated the development of the theory of functional differential equations [1]. We study a differential game between a pursuer and an evader in a dynamical system with the following linear functional differential equation in a real separable Hilbert space Y (6.1) y  (t) = Ay(t) + F(yt ) + u(t) − v(t), t ∈ [0, T ]. A. A. Chikrii (B) Glushkov Institute of Cybernetics, Kiev, Ukraine e-mail: [email protected] A. G. Rutkas · L. A. Vlasenko Kharkov National University of Radio Electronics, Kharkov, Ukraine e-mail: [email protected] L. A. Vlasenko e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_6

63

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As customary in the theory of functional differential equations, the symbol yt denotes the function yt (τ ) = y(t + τ ), −ω ≤ τ ≤ 0 (ω > 0), belonging to a class B(−ω, 0; Y ) of Y -valued functions of the argument τ ∈ [−ω, 0]. A. D. Myshkis drew our attention to the fact that for the first time similar classes of functions were introduced by N. N. Krasovskii in studying time delay equations in a general functional differential form [2, 3]. He considered continuous or piecewise continuous functions as the class B(−ω, 0; Rn ). The problems considered here are related to studies [4–7], which are devoted to differential–difference games in finitedimensional spaces. In paper [8], the theory of functional differential games for neutral-type systems with atomic difference operator (according to the terminology in [1]) is developed. Degenerate delay systems from [9] belong to class of neutral-type systems with non-atomic difference operator. More general classes of neutral-type systems in Banach spaces are considered in paper [10], which also contains a physical example of high-frequency electrical circuit whose evolution is described by an equation of this class. To study differential games for these systems, it is required to bring spectral analytical methods from [11]. With respect to Eq. (6.1), we assume that a closed linear operator A in Y with dense domain D A generates a strongly continuous semigroup {S(t), t ≥ 0} on Y ; F is a mapping from B(−ω, 0; Y ) into Y ; controls u(t) and v(t) of the pursuer and evader are measurable vector functions (the notations of strong and weak measurability in separable spaces are equivalent) with values in control domains U and V , which are closed convex bounded sets in the space Y . Examples of mappings F can be found in [1, 12]. We shall define the operator F in Sect.  6.2. We use the following notation:  ·  and ·, · are the norm and the inner product in corresponding spaces; A∗ is the adjoint operator of A with domain D A∗ ; L(Y, Z ) is the space of bounded linear operators from Y to Z , L(Y ) = L(Y, Y ); L 2 (0, T ; Y ) is the space of Y -valued square-norm integrable functions on [0, T ]; and C([0, T ]; Y ) is the space of Y -valued continuous functions on [0, T ].

6.2 Statement of the Game Problem Let the initial conditions for Eq. (6.1) be given as follows: y(t) = g(t), a.e. t ∈ [−ω, 0],

y(+0) = y0 ,

(6.2)

where g(t) is a vector function with values in Y , y0 ∈ Y . For given admissible controls u(t) and v(t), we consider a solution y(t) = y(t; u, v) of problem (6.1), (6.2) or a state of the system. Solutions are understood in the weak sense, but not in the weakened sense as in [13]. We say that a function y(t), t ∈ [−ω, T ] is said to be a solution of problem (6.1), (6.2), if y(t) ∈ C([0, T ]; Y ), y(t) satisfies initial conditions (6.2), for each h ∈ D A∗ the function is absolutely continuous and satisfies the equation

6 On a Differential Game in a System Described …

65

  d = y(t), A∗ h + F(yt ) + u(t) − v(t), h , a.e. t ∈ [0, T ]. dt The goal of the game in system (6.1), (6.2) is to bring the state y(t) to the cylindrical terminal set M = M0 ⊕ M1 at a time T0 ∈ (0, T ] in the class of admissible controls of the pursuer for any admissible control of the evader. The terminal set M is the orthogonal sum of a closed linear subspace M0 in Y and a convex closed set M1 from the orthogonal complement M⊥ 0 to M0 in Y . Let  be the orthogonal projector in Y to M⊥ 0 , U0 and V0 denote the sets of admissible controls of the pursuer and evader. Throughout the paper we assume that the mapping F in (6.1) is defined by the Stieltjes measure η(τ ), τ ∈ [−ω, 0]: 0 dη(τ )ϕ(τ ), η(τ ) = −

F(ϕ) =

n 

0 χ(−∞,−ωr ] (τ )Ar −

r =1

−ω

B(s)ds, τ

where 0 < ω1 < · · · < ωn = ω are delays, χ(−∞,−ωr ] (τ ) is the characteristic function of the semi-infinite interval (−∞, −ωr ], Ar ∈ L(Y ), B(s) ∈ L 2 (−ω, 0; L(Y )). Then the delay term F(yt ) in (6.1) is written by 0 F(yt ) =

dη(τ )y(t + τ ) =

n 

0 Ar y(t − ωr ) +

r =1

−ω

B(τ )y(t + τ )dτ.

(6.3)

−ω

Regarding the initial function in (6.2), we assume that g(t) ∈ L 2 (−ω, 0; Y ). According to Corollary 2.1 from [14], there exists a unique solution of the initial value problem (6.1)–(6.3) given by the variation of constant formula 0 y(t) = W (t)y0 +

t G t (τ )g(τ )dτ +

−ω

W (t − s)[u(s) − v(s)]ds,

(6.4)

0

where the fundamental solution W (t) is a unique solution of W (t) =

⎧ ⎨ ⎩

S(t) +

t 0

S(t − s)

0, t < 0

0 −ω

dη(τ )W (s + τ )ds, t ≥ 0

and for each t ∈ [0, T ], the operator valued function G t (τ ) for almost all τ ∈ [−ω, 0] is given by G t (τ ) =

n  r =1

τ W (t − τ − ωr )Ar χ[−ωr ,0] (τ ) + −ω

W (t − τ + ξ )B(ξ )dξ.

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The operator function W (t) is strongly continuous on [0, ∞) and, for some M, δ > 0, satisfies W (t) ≤ Meδt . In the particular case, when there is no delay, Formula (6.4) takes the form t y(t) = S(t)y0 +

S(t − s)[u(s) − v(s)]ds, 0

which is the definition of the weak solution or the motion from paper [15].

6.3 Solvability Conditions of the Game Problem Consider conditions for the solvability of the game problem in system (6.1)–(6.3). First, using the technique of set-valued mappings and their selections, we derive sufficient conditions for the game termination in a finite time, following the schemes in [16, 17] in finite-dimensional case and in [18, 19] in infinite-dimensional case. Put = {(t, s) : 0 ≤ s ≤ t ≤ T }. Consider the set-valued mappings Q(t,s, v) = W (t − s)[U − v], Q : × V  Y,

Q(t, s, v), Q 0 :  Y. Q 0 (t, s) =

(6.5)

v∈V

We suppose that the set-valued mapping Q 0 (t, s) takes nonempty values on the set

(Pontryagin’s condition). Let γ (t, s) be a measurable in s ∈ [0, t] selection of the set-valued mapping Q 0 (t, s). With the help of the vector functions, 0 ζ (t) = W (t)y0 +

t G t (τ )g(τ )dτ, ζγ (t) = ζ (t) +

−ω

γ (t, s)ds, 0

introduce the set-valued mapping from × V into R1

(t, s, v) = { α ≥ 0 : [Q(t, s, v) − γ (t, s)] ∩  α [M1 − ζγ (t)] = ∅}. The mapping has nonempty closed images, it is measurable in (s, v) ∈ [0, t] × V . Consider the support functional of (t, s, v) in the direction +1: α(t, s, v) = sup{ α≥0 :  α ∈ (t, s, v), (t, s, v) ∈ × V }.

(6.6)

This support functional is the resolving function in finite-dimensional case [16, 17] or the resolving functional in infinite-dimensional case [18, 19]. If ζγ (t) ∈ M1 , / M1 , then the supremum in (6.6) is attained and the then α(t, s, v) = ∞. If ζγ (t) ∈

6 On a Differential Game in a System Described …

67

resolving functional is measurable in (s, v) ∈ [0, t] × V . If v(s) ∈ V0 and ζγ (t) ∈ / M1 , then the set-valued mapping (t, s, v(s)) and its support function α(t, s, v(s)) are measurable in s ∈ [0, t]. Introduce the set ϒ = ϒ1 ∪ ϒ2 , ϒ1 = {t ∈ [0, T ] : ζγ (t) ∈ M1 }, t / M1 ∧ inf α(t, s, v(s))ds ≥ 1}. ϒ2 = {t ∈ [0, T ] : ζγ (t) ∈

(6.7)

v∈V0

0

Let there exist T0 ∈ ϒ. Consider the set-valued mappings from [0, T0 ] × V into Y U1 (s, v) = {u ∈ U : W (T0 − s)[u − v] = γ (T0 , s)}, U2 (s, v) = {u ∈ U : W (T0 − s)[u − v] − γ (T0 , s) ∈ α(T0 , s, v)[M1 − ζγ (T0 )]. / M1 . Clearly, the mapping U2 (s, v) is meaningful if ζγ (t) ∈ If ζγ (T0 ) ∈ M1 , then, for an arbitrary admissible evader control v(s), we set the pursuer control u(s) on the interval [0, T0 ] equal to an measurable selection u 1 (s) of the set-valued mapping U1 (s, v(s)). For this choice of the admissible pursuer control, we obtain y(T0 ) = ζγ (T0 ) ∈ M1 and the state y(t) (6.4) will be brought to the terminal set at time T0 . / M1 and v(s) ∈ V0 . Then we search a time t∗ ∈ (0, T0 ] such that Now, let ζγ (T0 ) ∈ t∗ α(T0 , s, v(s))ds = 1.

(6.8)

0

We set a pursuer control u(s) on the interval [0, t∗ ) equal to a measurable selection u 2 (s) of the set-valued mapping U2 (s, v(s)) and on the interval [t∗ , T0 ] equal to a measurable selection u 1 (s) of the set-valued mapping U1 (s, v(s)). In this case, the following relations are fulfilled: t∗ y(T0 ) = ζγ (T0 ) +

{W (T0 − s)[u 2 (s) − v(s)] − γ (T0 , s)}ds 0

t∗ ∈ ζγ (T0 ) +

t∗ α(T0 , s, v(s))[M1 − ζγ (T0 )]ds =

0

α(T0 , s, v(s))M1 ds ⊂ M1 , 0

where the integral of the set-valued mapping is understood in the Aumann sense as the set of integrals of integrable selections. This, again, means that the state y(t) (6.4) will be brought to the terminal set at time T0 . Thus, we have the following result. Theorem 6.1 Suppose that, for conflict-controlled system (6.1)–(6.3), the following constraints are satisfied: the operator A generates a strongly continuous semigroup

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of bounded linear operators; g(t) ∈ L 2 (−ω, 0; Y ); B(s) ∈ L 2 (−ω, 0; L(Y )); the set-valued mapping Q 0 (t, s) in (6.5) takes nonempty values on the set ; for some selection γ (t, s) ∈ Q 0 (t, s), the set ϒ (6.7) is not empty. Then, the state of system (6.1)–(6.3) can be brought to the cylindrical terminal set M = M0 ⊕ M1 at any time T0 ∈ ϒ. Now, to obtain solvability conditions of the game problem in system (6.1)–(6.3), we use support functionals of two sets defined by the behaviors of pursuer and evader, as in [20, 21]. Consider the following operator  ∈ L(L 2 (0, T ; Y ), Y ) and its images T0 z = 

W (T0 − s)z(s)ds, U = U0 , V = −V0 . 0

For the sets U , V , introduce the support functionals     ϕU (h) = sup h, z , ϕV (h) = sup h, z . z∈U

(6.9)

z∈V

Let the terminal set have the form M = M0 ⊕ Bd (d ≥ 0), where Bd is the dneighborhood of zero in M⊥ 0 . In this case, the inequality sup inf y(T0 ; u, v) ≤ d

v∈V0 u∈U0

(6.10)

is necessary and sufficient to terminate the game in system (6.1)–(6.3) in time T0 . Theorem 6.2 Suppose that the following assumptions for conflict-controlled system (6.1)–(6.3) with the terminal set M = M0 ⊕ Bd (d ≥ 0) are valid: the operator A generates a strongly continuous semigroup of bounded linear operators; g(t) ∈ L 2 (−ω, 0; Y ); B(s) ∈ L 2 (−ω, 0; L(Y )). In order that T0 be a game completion time, it is necessary and sufficient that, for every vector h ∈ Y with unit norm, the relation holds   (6.11) ϕV (h) − ϕU (−h) ≤ d − h, ζ (T0 ) .

Proof For arbitrary admissible controls u ∈ U0 , v ∈ V0 , there exists a unique solution y(t) (6.4) of problem (6.1)–(6.3). For all h ∈ Y , the relation holds 

       h, y(T0 ; u, v) = h, u − h, v + h, ζ (T0 ) .

Introduce the function     p(h) = sup inf h, y(T0 ; u, v) = inf sup h, y(T0 ; u, v) . v∈V0 u∈U0

u∈U0 v∈V0

6 On a Differential Game in a System Described …

69

By (6.9), we obtain   p(h) = ϕV (h) − ϕU (−h) + h, ζ (T0 ) . Hence, we can rewrite (6.11) as p(h) ≤ d.

(6.12)

Let us prove the necessity in inequality (6.11) or (6.12) for every vector h ∈ Y with unit norm to terminate the game by time T0 . Suppose the contrary, i.e., there exists a vector h ∈ Y with h = 1 such that p(h) > d. The convex closed bounded set V0 is weakly compact in the Hilbert space L 2 (0, T ; Y ) [22]. Therefore, the least upper bound in the definition of ϕv (h) (6.9) is attained for v0 ∈ V0 and we have the relations   d < p(h) ≤ h, y(T0 ; u, v0 ) , ∀u ∈ U0 . This implies that y(T0 ; u, v0 ) > d

(6.13)

for every admissible control u ∈ U0 , which is a contradiction of the game completion in time T0 . Thus, the necessary in inequality (6.11) is proved. Now we prove the sufficiency. Show, if inequality (6.11) or (6.12) holds for all h ∈ Y with h = 1, then the game can be terminated in time T0 . Suppose the contrary, i.e., there exists a control v0 ∈ V0 such that inequality (6.13) is true for any control u ∈ U0 . The continuous convex functional f (u) = y(T0 ; u, v0 ) defined on the Hilbert space L 2 (0, T ; Y ) attains its minimum on the convex closed bounded set U0 . Hence, there exists an ε > 0 such that minu∈U0 f (u) > d + ε. It means that the convex set U − (v0 ) + ζ (T0 ) and the closed ball Bd+ε with center as the origin and radius d + ε are disjoint in the space Y . It follows from the separation theorem [22] that there exists a vector h ∈ Y of unit norm such that     inf h, y(T0 ; u, v0 ) ≥ sup h, b ≥ d + ε > d.

u∈U0

b∈Bd+ε

Then p(h) > d, which contradicts the assumption (6.12). The theorem is proved. 

6.4 Application to Partial Functional Differential Equations Partial differential equations with time delay arise in problems of mathematical physics, in which the delay effect is taken into account. In the domain [0, T ] × [0, l], the heat equation in the case of heat exchange has the form

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∂ y(t, x) a(x) + a0 (x)y(t, x) ∂x 0 + a1 (x)y(t − ω, x) + b(τ, x)y(s + τ, x)dτ = u(t, x) − v(t, x).

∂ ∂ y(t, x) = ∂t ∂x

−ω

(6.14) 0 The terms a0 (x)y(t, x), a1 (x)y(t − ω, x) and −ω b(τ, x)y(s + τ, x)dτ correspond to the heat loss; the functions u(t, x), v(t, x) correspond to the controlled distributed heat source and leak. We consider the boundary and initial conditions y(t, 0) = y(t, l) = 0, t ∈ [0, T ] y(+0, x) = y0 (x), y(t, x) = g(t, x), a.e. (t, x) ∈ [−ω, 0] × [0, l].

(6.15)

Assume that the functions in (6.14), (6.15) are real and satisfy the following conditions: a(x) ∈ C 1 [0, l] is a positive function, a0 (x) ∈ C[0, l], a1 (x) ∈ L 2 [0, l], b(τ, x) ∈ L 2 ([−ω, 0] × [0, l]), y0 (x) ∈ L 2 [0, l], g(τ, x) ∈ L 2 ([−ω, 0] × [0, l]). Admissible controls of the pursuer (source) u(t, x) ∈ L 2 ([0, T ] × [0, l]) and the evader (leak) v(t, x) ∈ L 2 ([0, T ] × [0, l]) satisfy the constraints: u(t) ∈ U and v(t) ∈ V , where U and V closed balls in L 2 (0, l) with the center at zero and the radii 1 > 0 and 2 > 0. The goal of the game in system (6.14), (6.15) is to bring the state y(t, x) to zero in a time not exceeding T in the class of admissible controls of the pursuer for any admissible control of the evader. We exclude the trivial case of the game and assume that the function y0 (x) is nonzero. In the real space Y = L 2 [0, l], mixed problem (6.14), (6.15) can be represented in abstract form (6.1)–(6.3), where the operator A is generated in L 2 (0, l) by the Sturm–Liouville differential expression d/dx(a(x)d/dx) + a0 (x) with the Dirichlet boundary conditions, A1 q = a(x)q(x), B(τ )q = b(τ, x)q(x). It is well known (see, for example, [23]) that the operator A is self-adjoint with compact resolvent in L 2 (0, l); its spectrum consists of a countable set of real eigenvalues {λm }∞ m=1 of finite multiplicity with accumulation point −∞; the corresponding eigenfunctions {em (x)}∞ m=1 form an orthonormal basis  l). The operator A generates an  in Lλm2t(0, q, em em (x). The solution y(t, x) of e analytic subgroup given by S(t)q = ∞ m=1 the mixed problem (6.14), (6.15) is regarded in the sense of the solution of the abstract problem (6.1)–(6.3). There exists a unique solution y(t, x) ∈ C([0, T ], L 2 (0, l)) of the mixed problem (6.14), (6.15), which admits the representation (6.4). The terminal set M = {0}, M0 = {0}, M1 = B0 = {0},  is the identity operator. To apply Theorem 6.2, we find T0 ϕU (h) = 1 0

W ∗ (T0 − s)hds, ϕV (h) =

2 ϕU (h). 1

Here and below, we consider the norm and the inner product in the space L 2 (0, l). By Theorem 6.2, in order that T0 be a game completion time in system (6.14), (6.15),

6 On a Differential Game in a System Described …

71

it is necessary and sufficient that the following relation holds for all h ∈ L 2 (0, l) and h = 1: T0 (1 − 2 )



∗

0

W (T0 − s)hds ≥ h, W (T0 )y0 +

 G t (τ )g(τ )dτ .

−ω

0

Further, to be definite, we choose a1 (x) = 1, b(τ, x) = 0 in (6.14) and y0 (x) = em (x), g(t, x) = 0 in (6.15). Using the representation W (t) =

N −1 

S(t − kω)

k=0

(t − kω)k , (N − 1)ω ≤ t ≤ N ω, k!

N = 1, 2, . . . ,

we obtain, in order that T0 be a game completion time, it is necessary and sufficient that T0 (1 − 2 ) wm (T0 − s)ds ≥ wm (T0 ), 0

(6.16)

N −1 

(t − kω)k , (N − 1)ω ≤ t ≤ N ω. wm (t) = eλm (t−kω) k! k=0 Let us verify that the conditions of Theorem 6.1 hold for the differential game in system (6.14), (6.15). Since 1 > 2 , choose the selection γ (t, s) = 0 ∈ Q 0 (t, s) = ∅. Define the function (6.6): α(t, s, v) = sup{ α≥0 :  α w(t, s) − v ∈ U }, w(t, s, x) =

wm (t) em (x). wm (t − s)

It is not difficult to see that   2 w(t, s), v + w(t, s), v + w(t, s)2 (12 − v2 )

 α(t, s, v) =

w(t, s)2

.

The set ϒ in (6.7) coincides with the set of T0 satisfying (6.16). It follows from the proof of Theorem 6.1 that, for any admissible evader control v(s) = v(s, x), the admissible pursuer control u(s) = u(s, x) to terminate the game at time T0 is the following:  u(s, x) =

  v(s, x) − v(s), em em (x) − α(T0 , s, v(s))wm (T0 − s)]em (x), s ∈ [0, t∗ ) , v(s, x), s ∈ [t∗ , T0 ]

where the time t∗ is defined in (6.8).

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References 1. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) 2. Krasovskii, N.N.: The stability of quasilinear systems with after-effect. Dokl. Akad. Nauk SSSR 119, 435–438 (1958) (in Russian) 3. Krasovskii, N.N.: Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford, California (1963) 4. Krasovskii, N.N., Osipov, Y.S: Linear differential-difference games. Dokl. Akad. Nauk SSSR 197, 777–780 (1971) (in Russian) 5. Kurzhanskii, A.B.: Differential approach games in systems with lag. Differencial’nye Uravneniya 7, 1398–1409 (1971) (in Russian) 6. Nikol’skii, M.S.: Linear differential pursuit games in the presence of lags. Differencial’nye Uravneniya 8, 260–267 (1972) (in Russian) 7. Chikrii, A.A., Chikrii, G.T.: Group pursuit in differential-difference games. Differencial’nye Uravneniya 20, 802–810 (1984) (in Russian) 8. Lukoyanov, N.Y., Plaksin, A.R.: Differential games for neutral-type systems: an approximation model. Proc. Steklov Inst. Math. 291, 190–202 (2015). https://doi.org/10.1134/ S0081543815080155 9. Vlasenko, L.A., Rutkas, A.G.: On a class of impulsive functional-differential equations with nonatomic difference operator. Math. Notes 95, 32–42 (2014). https://doi.org/10.1134/ S0001434614010040 10. Vlasenko, L.A.: Existence and uniqueness theorems for an implicit delay differential equation. Differ. Equ. 36, 689–694 (2000). https://doi.org/10.1007/BF02754227 11. Rutkas, A.G.: Spectral methods for studying degenerate differential-operator equations. I. J. Math. Sci. 144, 4246–4263 (2007). https://doi.org/10.1007/s10958-007-0267-2 12. Kunisch, K., Schappacher, W.: Necessary conditions for partial differential equations with delay to generate C0 -semigroups. J. Differ. Equ. 50, 49–79 (1983). https://doi.org/10.1016/ 0022-0396(83)90084-0 13. Vlasenko, L.A., Myshkis, A.D., Rutkas, A.G.: On a class of differential equations of parabolic type with impulsive action. Differ. Equ. 44, 231–240 (2008). https://doi.org/10.1134/ S0012266108020110 14. Nakagiri, S.: Optimal control of linear retarded systems in Banach spaces. J. Math. Anal. Appl. 120, 169–210 (1986). https://doi.org/10.1016/0022-247X(86)90210-6 15. Osipov, Y.S., Kryazhimskii, A.V., Maksimov V.I.: N.N. Krasovskii’s extremal shift method and problems of boundary control. Autom. Remote. Control. 70, 577–588 (2009). https://doi. org/10.1134/S0005117909040043 16. Chikrii, A.A.: Conflict-Controlled Processes. Kluwer, Boston, London, Dordrecht (1997). https://doi.org/10.1007/978-94-017-1135-7 17. Chikrii, A.A.: An analytical method in dynamic pursuit games. Proc. Steklov Inst. Math. 271, 69–85 (2010). https://doi.org/10.1134/S0081543810040073 18. Vlasenko, L.A., Chikrii, A.A.: The method of resolving functionals for a dynamic game in a Sobolev system. J. Autom. Inf. Sci. 46, 1–11 (2014). https://doi.org/10.1615/ JAutomatInfScien.v46.i7.10 19. Vlasenko, L.A., Rutkas, A.G., Chikrii, A.A.: On a differential game in an abstract parabolic system. Proc. Steklov Inst. Math. 293(Suppl. 1), 254–269 (2016). https://doi.org/10.1134/ S0081543816050229 20. Vlasenko, L.A., Rutkas, A.G.: On a differential game in a system described by an implicit differential-operator equation. Differ. Equ. 51, 798–807 (2015). https://doi.org/10.1134/ S0012266115060117

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21. Rutkas, A., Vlasenko, L.: On a differential game in a nondamped distributed system. Math. Methods Appl. Sci. (2019). https://doi.org/10.1002/mma.5712 22. Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. American Mathematical Society, Providence (1957) 23. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin, Heidelberg (1995)

Chapter 7

The Program Constructions in Abstract Retention Problem Alexander Chentsov

Abstract For axiomatically determined dynamic system, the game retention problem with intermittent time is considered. The admissible control procedures of the player I interested in the retention implementing are defined as set-valued quasistrategies (non-anticipating set-valued reactions on the noise operation). The solvability set and the structure of solving quasistrategies are established. For investigation, the known method of program iterations is used. Two variants of this method (direct and indirect variants) are used. For these variants, duality relations are obtained.

7.1 Introduction The control problems with noise are considered in the differential games (DG) theory. The fundamental position of theory of DG is theorem about alternative established by Krasovskii and Subbotin (see [1, 2]). This theorem defined the solvability conditions for DG pursuit-evasion. By this theorem corresponding solvability conditions in the sense of saddle point were obtained for other DG (we recall known monograph [3] in which many concrete game problems of control are considered). A variant of the pursuit-evasion DG for finite time is the retention game. In this game, one player tends to keep the trajectory in the given set which defines state constraints; the aim of another player is opposite. We note that the game can be also considered for an infinite time horizon although, in this case, the alternative theorem does not hold. For investigation of the game retention problem, we use constructions of [4] connected with the known method of program iterations (see [4–8]). This approach corresponds to employment of methods of program control for the DG solution in the class of quasistrategies (see [9, 10]) and positional strategies (see [1, 2]). Now,

A. Chentsov (B) Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg, Russia e-mail: [email protected] Ural Federal University, Yekaterinburg, Russia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_7

75

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we consider the axiomatically defined variant of the retention problem; we follow to approach [4] (in particular, see [4, Sect. 1]). We note the remark of the conclusion of [4, Sect. 1] about topological nature of the used investigation scheme.

7.2 General Notions and Designations The standard set-theoretical symbolics is used (we use quantors and propositional 

connectives; ∅ is the empty set). By = we denote the equality by definition. A family is a set of all elements which are sets too. For every object x, by {x} we denote singleton containing x : x ∈ {x}. If H is a set, then by P(H ) (by P  (H )) we denote the family of all (all nonempty) subsets of H . By B A we define the set of all mappings from a set A into a set B. If A and B are nonempty sets, f ∈ B A , and C ∈ P  (A), then  ( f | C) = ( f (x))x∈C ∈ B C . 





By R we denote real numbers, R+ = {ξ ∈ R | 0  ξ }, N = {1; 2; . . .}, and N0 = {0; 1; 2; . . .} = {0} ∪ N. If H is a nonempty set and h ∈ H H , then (h k )k∈N0 : N0 → H H 

(7.2.1)



is defined traditionally: (h 0 (x) = x ∀x ∈ H )&(h k = h ◦ h k−1 ∀k ∈ N), where ◦ is the superposition symbol. Moreover, we use two variants for the infinite degree of an operator. ∞

(1) If E is a set and ζ ∈ P(E)P(E) , then ζ ∈ P(E)P(E) is defined by the rule ∞



ζ (Σ) =



ζ k (Σ) ∀Σ ∈ P(E).

(7.2.2)

k∈N0 

(2) For sets A and B, suppose that M(A, B) = P(B) A . The second variant corresponds to the case when (in (7.2.1)) H = M(A, B). Namely, for β ∈ M(A, B)M(A,B) , the mapping β ∞ ∈ M(A, B)M(A,B) is defined by the rule 

β ∞ (C)(a) =



β k (C)(a) ∀C ∈ M(A, B) ∀a ∈ A.

k∈N0

(7.2.3)

In (7.2.2) and (7.2.3), we have two variants of the infinite degree of an operator. Of course, for a set E, the family P(E) is equipped with the ordering by inclusion; moreover, for every sequence (Σk )k∈N ∈ P(E)N and Σ ∈ P(E) def

((Σk )k∈N ↓ Σ) ⇐⇒ ((Σ =

 k∈N

Σk )&(Σs+1 ⊂ Σs ∀s ∈ N)).

7 The Program Constructions in Abstract Retention Problem

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We have usual monotone sequential convergence of sets. On this basis, we introduce the monotone sequential convergence of multifunctions: if A and B are sets, (u i )i∈N ∈ M(A, B)N and u ∈ M(A, B), then def

((u i )i∈N ⇓ u) ⇐⇒ ((u i (x))i∈N ↓ u(x) ∀x ∈ A). Finally, for u ∈ M(A, B) and v ∈ M(A, B) def

(u  v) ⇐⇒ (u(x) ⊂ v(x) ∀x ∈ A).

7.3 The Program Operators and Quasistrategies We fix T ∈ P  (R) for which T = {t} ∀t ∈ R. Later, T plays the role of an analogue of control time interval (we note two important particular cases: T = R+ (the case of the problem with continuous time) and T = N0 (the case of the problem with discrete time); in general, we have to deal with intermittent time). Fix a TS (X, τ ), 

X = ∅, and suppose that F = {X \ G : G ∈ τ }. We use X as phase spase. By ⊗T (τ ) we denote the standard Tychonoff product of samples of (X, τ ) with the index set T ; see [11, 2.3]. So, (X T , ⊗T (τ )) is a TS; this TS is the Tychonoff degree of (X, τ ). We fix C ∈ P  (X T ) and by I denote topology on C induced from (X T , ⊗T (τ )). Of course, we obtain the topology of pointwise convergence in C; (C, I) is a subspace of (X T , ⊗T (τ )). By F and K we denote the families of all closed and all compact 

(in (C, I)) subsets of C, respectively. We use D = T × X as the position space; for H ∈ P(D) and t ∈ T 

H t = {x ∈ X | (t, x) ∈ H } ∈ P(X ). 

Then, F = {F ∈ P(D) | Ft ∈ F ∀t ∈ T } is the family of all subsets of D closed in the natural topology P(T ) ⊗ τ of the product (T, P(T )) (this is the discrete of T ) and (X, τ ). We fix nonempty sets Υ and Ω ∈ P  (Υ T ); elements of Ω play the role of indeterminate factors of a process. Consider the mapping S : D × Ω → P  (C)

(7.3.1)

(S is the fixed multifunction from D × Ω in C) as an abstract analogue of a control system. If z ∈ D and ω ∈ Ω, then S(z, ω) is considered as the bundle of trajectories of (7.3.1) corresponding to the initial position z and ω ∈ Ω.

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Fix N ∈ F and consider the sets N t, t ∈ T , as state constraints. If t ∈ T then   Tt = T ∩ ] − ∞, t] and Tt = T ∩ [t, ∞[; with Tt , we connect the Past and with Tt , the Future of our process. If (t, x) ∈ D, ω ∈ Ω, and h ∈ S((t, x), ω), then situation h(ξ ) ∈ N ξ  ∀ξ ∈ Tt

(7.3.2)

is considered as the retention for trajectory h. We consider questions connected with guaranteed realization of (7.3.2) in the class of quasistrategies. In this connection, we introduce germs of mappings from Ω and C: under t ∈ T , we suppose that 



(Ωt (ω) = {ν ∈ Ω | (ω | Tt ) = (ν | Tt )} ∀ω ∈ Ω) 



&(Z t (u) = {v ∈ C | (u | Tt ) = (v | Tt )} ∀u ∈ C).

(7.3.3)

Now, we introduce the heredity operators, namely, for t ∈ T Γt : M(Ω, C) → M(Ω, C)

(7.3.4)

is defined by the rule: ∀C ∈ M(Ω, C) ∀ω ∈ Ω 





Γt (C)(ω) = {h ∈ C(ω) | Z ξ (h) ∩ C(ν) = ∅ ∀ξ ∈ Tt ∀ν ∈ Ωξ (ω)}.

(7.3.5)

Then non-anticipating multifunctions can be defined in terms of (7.3.2)–(7.3.5): under t ∈ T 

Nt = {α ∈ M(Ω, C) | α = Γt (α)} = {α ∈ M(Ω, C) | ∀ω1 ∈ Ω ∀ω2 ∈ Ω ∀ξ ∈ Tt ((ω1 | Tξ ) = (ω2 | Tξ )) ⇒ ({(h | Tξ ) : h ∈ α(ω1 )} = {(h | Tξ ) : h ∈ α(ω2 )})}. (7.3.6) So, non-anticipating multifunctions are fixed points of Γt (7.3.4) and only they have this property. Now, for every position (t, x) ∈ D, we introduce the corresponding set of quasistrategies: 

Q (t,x) = {α ∈ Nt | α(ω)∈ P  (S((t, x), ω)) ∀ω ∈ Ω} P  (S((t, x), ω))). = Nt ∩ (

(7.3.7)

ω∈Ω

We note that quasistrategies of the set (7.3.7) are multifunctions with nonempty values. These values are sets of trajectories of our system (7.3.1). Now, we introduce a special type of multifunctions. Namely, for H ∈ P(D), (t, x) ∈ D, and ω ∈ Ω 

Π (ω | (t, x), H ) = {h ∈ S((t, x), ω) | (ξ, h(ξ )) ∈ H ∀ξ ∈ Tt }.

(7.3.8)

7 The Program Constructions in Abstract Retention Problem

79

We consider Π (· | z, H ) under H ∈ P(D) and z ∈ D; this mapping associates the set (7.3.8) for every “point” ω ∈ Ω. Of course, 

Cz0 = Π (· | z, N ) ∈ M(Ω, C);

(7.3.9)

in (7.3.9), we obtain the aim multifunction. Then, under z ∈ N , the set 

Q 0z = {α ∈ Q z | α  Cz0 }

(7.3.10)

is considered as the set of quasistrategies solving our basic problem. With (7.3.10), the next question is connected: for which z ∈ N , the property Q 0z = ∅ holds? So, the natural problem about the successful solvability of the retention problem (see (7.3.8)–(7.3.10)) arises. Of course, we keep in mind the set 

N 0 = {z ∈ N | Q 0z = ∅}.

(7.3.11)

Moreover, for z 0 ∈ N 0 , we consider the question about construction of quasistrategy α 0 ∈ Q 0z |z=z 0 . For investigation of these questions, along with (7.3.4), we introduce A ∈ P(D)P(D) by the rule 

A(H ) = {z ∈ H | Π (ω | z, H ) = ∅ ∀ω ∈ Ω}.

(7.3.12)

We call A the operator of program absorption. Up to the end of this section, we suppose that following Conditions 7.1 and 7.2 hold. Condition 7.1 For every t ∈ T and ω ∈ Ω, the set {(x, h) ∈ X × C | h ∈ S((t, x), ω)} is closed in the natural topology of product of TS (X, τ ) and (C, I). Condition 7.2 For every t ∈ T , x ∈ X , and ω ∈ Ω, there exist a τ -neighborhood H of the point x and a (compact in (C, I)) set K ∈ K for which S((t, y), ω) ⊂ K ∀y ∈ H. In connection with these conditions, see [4, Conditions 4.2 and 4.3]. We note following property [4, Corollary 6.2]: A(F) ∈ F ∀F ∈ F.

(7.3.13)

Moreover, by [4, Proposition 6.2], we have the next analogue of the sequential continuity: for every sequence (Fi )i∈N ∈ FN and a set F ∈ F ((Fi )i∈N ↓ F) ⇒ ((A(Fi ))i∈N ↓ A(F)).

(7.3.14)

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By [4, Theorem 6.1], the following suppositions realize: ∞

∞

∞

(1) A (N ) ∈ F and A(A (N )) =A (N ); ∞ (2) ∀H ∈ P(D) ((H = A(H ))&(H ⊂ N )) ⇒ (H ⊂A (N )); ∞ ∞ ∞ (3) ∀U ∈ P(N ) (A (N ) ⊂ U ) ⇒ (A (N ) =A (U )). ∞

So, (under Conditions 7.1 and 7.2) A (N ) is the fixed point of the operator A (see property 1); in (2) and (3), we obtain useful additional properties). Of course, we have the sequence (Ak (N ))k∈N0 in the family F for which A0 (N ) = N , Al+1 (N ) = ∞

A(Al (N )) under l ∈ N0 and A (N ) is the intersection of all sets of the given sequence. So, we obtain the “indirect” iterated procedure. Proposition 7.1 If (Ni )i∈N is a sequence in F for which (Ni )i∈N ↓ N , then (Ak (Ni ))i∈N ↓ Ak (N ) ∀k ∈ N0 . Proof Since A0 (Ni ) = Ni under i ∈ N and A0 (N ) = N , we have convergence (A0 (Ni ))i∈N ↓ A0 (N ). Then, 

N = {k ∈ N0 | (Ak (Ni ))i∈N ↓ Ak (N )} ∈ P  (N0 ) and 0 ∈ N. Let r ∈ N. Then r ∈ N0 and (Ar (Ni ))i∈N ↓ Ar (N ).

(7.3.15)

Then, Ar +1 (N j ) = A(Ar (N j )) under j ∈ N and Ar +1 (N ) = A(Ar (N )). Therefore, from (7.3.14) and (7.3.15) (Ar +1 (Ni ))i∈N ↓ Ar +1 (N ). Therefore, r + 1 ∈ N. So, (0 ∈ N)&(k + 1 ∈ N ∀k ∈ N). By induction the equality N = N0 is established. So, we obtain the required property.  Corollary 7.1 If (Ni )i∈N is a sequence in F with the property (Ni )i∈N ↓ N , then ∞

∞

(A (Ni ))i∈N ↓A (N ).

(7.3.16)

Proof From Proposition 7.1, we obtain that ∞

A (N ) =

 k∈N0

Ak (N ) =

  k   k  ∞ ( A (Ni )) = ( A (Ni )) = A (Ni ). k∈N0 i∈N

i∈N k∈N0

i∈N

(7.3.17) Later, from (7.3.8), we have that Π (ω | z, H1 ) ⊂ Π (ω | z, H2 ) under ω ∈ Ω, z ∈ D, H1 ∈ P(D), and H2 ∈ P(D) with H1 ⊂ H2 . By (7.3.12) A(H1 ) ⊂ A(H2 ) under H1 ∈ P(D) and H2 ∈ P(D) with H1 ⊂ H2 . If j ∈ N, then N j+1 ⊂ N j by suppo

sition. Then, A0 (N j+1 ) ⊂ A0 (N j ). So, for N = {k ∈ N0 | Ak (N j+1 ) ⊂ Ak (N j )}, we obtain that N ∈ P  (N0 ) and 0 ∈ N. Let r ∈ N. Then, Ar (N j+1 ) ⊂ Ar (N j ) and

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by the abovementioned property of A the situation Ar +1 (N j+1 ) = A(Ar (N j+1 )) ⊂ A(Ar (N j )) = Ar +1 (N j ) holds. So, r + 1 ∈ N. We obtain that (0 ∈ N)&(k + 1 ∈ N ∀k ∈ N). Then, by induction N = N0 . Therefore, Ak (N j+1 ) ⊂ Ak (N j ) ∀k ∈ N0 . As a corollary, we obtain that   ∞ ∞ Ak (N j+1 ) ⊂ Ak (N j ) =A (N j ). A (N j+1 ) = k∈N0

k∈N0 ∞

∞

Since the choice of j was arbitrary, A (Ns+1 ) ⊂ A (Ns ) ∀s ∈ N. From (7.3.17), we obtain (7.3.16). 

7.4 The Direct Iterated Procedure Consider non-anticipating multifunctions and quasistrategies. But, at first, we consider the “direct” iterated procedure. For any (t, x) ∈ N , we introduce the set 

0 0 N(t,x) = {α ∈ Nt | α  C(t,x) } with -greatest element 



0 ]= (na)[C(t,x)





C(ω)

0 C∈N(t,x)

ω∈Ω

0 ∈ N(t,x) .

(7.4.1)

Of course, by [4, (7.2)] Q 0z = {α ∈ N0z | α(ω) = ∅ ∀ω ∈ Ω} ∀z ∈ N . Now, for 0 ))k∈N0 in the set M(Ω, C): (t, x) ∈ N , we introduce the sequence (Γtk (C(t,x) 0 0 0 0 ) = C(t,x) )&(Γtk (C(t,x) ) = Γt (Γtk−1 (C(t,x) )) ∀k ∈ N); (Γt0 (C(t,x)

(7.4.2)

0 of course, we obtain multifunction Γt∞ (C(t,x) ) ∈ M(Ω, C) for which 0 )(ω) = Γt∞ (C(t,x)

 k∈N0

0 Γtk (C(t,x) )(ω) ∀ω ∈ Ω.

(7.4.3)

From (7.4.2) and (7.4.3), we have the next property: for (t, x) ∈ N 0 0 ))k∈N ⇓ Γt∞ (C(t,x) ). (Γtk (C(t,x)

(7.4.4)

In the following, we use the natural definition of pasting for mappings. Namely, for a set , u ∈ T , v ∈ T , and t ∈ T , we suppose that (u  v)t ∈ T is defined by the rule 



((u  v)t (ξ ) = u(ξ ) ∀ξ ∈ Tt )&((u  v)t (ξ ) = v(ξ ) ∀ξ ∈ Tt \ {t}). In this section, we suppose that the following Condition 7.3 holds.

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Condition 7.3 (ω1  ω2 )t ∈ Ω ∀ω1 ∈ Ω ∀ω2 ∈ Ω ∀t ∈ T . So, S(z, (ω1  ω2 )t ) ∈ P  (C) is defined for z ∈ D ω1 ∈ Ω ω2 ∈ Ω t ∈ T . Moreover, in the following, we suppose that the next Conditions 7.4 and 7.5 hold. Condition 7.4 ∀(t∗ , x∗ ) ∈ N ∀ω1 ∈ Ω ∀ω2 ∈ Ω ∀t ∈ Tt∗ ∀h ∈ S((t∗ , x∗ ), (ω1  ω2 )t ) ∃ht ∈ S((t, h(t)), ω2 ) : (h | Tt ) = (ht | Tt ). Condition 7.5 ∀(t∗ , x∗ ) ∈ N ∀ω1 ∈ Ω ∀ω2 ∈ Ω ∀h 1 ∈ S((t∗ , x∗ ), ω1 ) ∀t ∈ Tt∗ ∀h 2 ∈ S((t, h 1 (t)), ω2 ) (h 1  h 2 )t ∈ S((t∗ , x∗ ), (ω1  ω2 )t ). So, in the following, Conditions 7.3–7.5 are assumed to be fulfilled. Theorem 7.1 If H ∈ P(N ) and (t∗ , x∗ ) ∈ N , then Γt∗ (Π (· | (t∗ , x∗ ), H )) = Π (· | (t∗ , x∗ ), A(H )). In connection with Theorem 7.1, see [4, 12, 13]. From Theorem 7.1, we obtain the following property: if (t∗ , x∗ ) ∈ N and k ∈ N0 , then Γt∗k (C(t0 ∗ ,x∗ ) ) = Π (· | (t∗ , x∗ ), Ak (N )). ∞

Theorem 7.2 If (t∗ , x∗ ) ∈ N , then Γt∗∞ (C(t0 ∗ ,x∗ ) ) = Π (· | (t∗ , x∗ ), A (N )). We note that the proof of Theorems 7.1 and 7.2 is similar to [13, Proposition 6.1 and Theorem 6.2] for the case of DG of pursuit-evasion.

7.5 Main Problem In this section, we consider the question about construction of N 0 and quasistrategies solving the retention problem. By [4, Proposition 8.1] we obtain that, for z ∈ D and ω ∈ Ω, the set Cz0 (ω) is closed in S(z, ω) with topology induced from (X T , ⊗T (τ )). Under Conditions 7.1 and 7.2, we have the inclusion Cz0 (ω) ∈ K for z ∈ D and ω ∈ Ω. In the following, we suppose that (X, τ ) is a T2 -space (see [11, I.5]). Proposition 7.2 If (t∗ , x∗ ) ∈ N and C(t0 ∗ ,x∗ ) ∈ KΩ , then the following relations hold: Γt∗∞ (C(t0 ∗ ,x∗ ) ) = (na)[C(t0 ∗ ,x∗ ) ] ∈ KΩ . Corollary 7.2 Under Conditions 7.1 and 7.2, for (t∗ , x∗ ) ∈ N , the following relations hold: Γt∗∞ (C(t0 ∗ ,x∗ ) ) = (na)[C(t0 ∗ ,x∗ ) ] ∈ KΩ .

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So, we obtain conditions sufficient for convergence of the “direct” iterated procedure to multifunction of the type (7.4.1). Proposition 7.3 For z ∈ N and Cz0 ∈ KΩ ∞

(na)[Cz0 ] = Π (· | z, A (N )).

(7.5.1)

Corollary 7.3 Under Conditions 7.1 and 7.2, for every z ∈ N , equality (7.5.1) holds. By K we denote the family of all sequentially compact in (C, I) subsets of C. Then, we obtain (see [4, Proposition 8.3]) a “sequentially compact” version of Proposition 7.3: if (t∗ , x∗ ) ∈ N and C(t0 ∗ ,x∗ ) ∈ KΩ , then ∞

Γt∗∞ (C(t0 ∗ ,x∗ ) ) = (na)[C(t0 ∗ ,x∗ ) ] = Π (· | (t∗ , x∗ ), A (N )) ∈ KΩ . In this connection, we recall [14, p. 348]. Let 

(DOM)[α] = {ω ∈ Ω | α(ω) = ∅} ∀α ∈ M(Ω, C).

(7.5.2)

We note that by (7.3.7) (DOM)[α] = Ω ∀(t, x) ∈ N ∀α ∈ Q (t,x) . Moreover, by [4, Proposition 8.4] and (7.5.2), we obtain that ∀(t, x) ∈ N ∞

0 )] = Ω) ⇒ ((t, x) ∈A (N )). ((DOM)[Γt∞ (C(t,x)

(7.5.3)

As a corollary, for every z ∈ N , the next implication holds ∞

((DOM)[(na)[Cz0 ]] = Ω) ⇒ (z ∈A (N )).

(7.5.4) ∞

From [4, Proposition 8.5], we obtain that, under Conditions 7.1 and 7.2, A (N ) = {z ∈ N | (DOM)[(na)[Cz0 ]] = Ω} = {z ∈ N | (na)[Cz0 ] ∈ Q 0z }. As a result, the next basic proposition (see [4, Theorem 8.1]) takes place: under Conditions 7.1 and 7.2 ∞

N 0 =A (N );

(7.5.5)

∞

in this case, for (t∗ , x∗ ) ∈A (N ), by (7.4.1) ∞

(na)[C(t0 ∗ ,x∗ ) ] = Γt∗∞ (C(t0 ∗ ,x∗ ) ) = Π (· | (t∗ , x∗ ), A (N )) ∈ Q 0(t∗ ,x∗ ) .

(7.5.6)

So, in (7.5.5) and (7.5.6), we have the solution of our main problem and concrete quasistrategy solving the retention problem. Now, we consider the case, when z ∗ ∈ ∞

N \ A (N ). Then, by (7.5.4)

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Ω \ (DOM)[(na)[Cz0∗ ]] = ∅.

(7.5.7)

Let ω∗ ∈ Ω \ (DOM)[(na)[Cz0∗ ]]. So, (na)[Cz0∗ ](ω∗ ) = ∅. Proposition 7.4 For every α ∈ N0z∗ , the equality α(ω∗ ) = ∅ holds. Proof By (7.4.1) α  (na)[Cz0∗ ] and, as a corollary, α(ω∗ ) ⊂ (na)[Cz0∗ ](ω∗ ). So,  α(ω∗ ) = ∅.

7.6 Conclusion In article, the game retention problem is considered in axiomatic setting; we follow to [4]. We consider this problem for a topological phase space. Two variants of the program iteration methods are used. The natural duality of these variants (direct and indirect) is established. The solvability set and the structure of solving quasistrategies are established. In connection with the considered problem, we note investigations [15, 16]. Finally, we recall the original approach of Serkov [17], connected with application of transfinite variant of program iteration method.

References 1. Krasovskii, N.N., Subbotin, A.I.: An alternative for the game problem of convergence. J. Appl. Math. Mech. 34, 948–965 (1970). https://doi.org/10.1016/0021-8928(70)90158-9 2. Krasovsky, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York Inc. (1988) 3. Isaacs, R.: Differential Games. Wiley, New York (1965) 4. Chentsov, A.G.: An abstract confinement problem: a programmed iterations method of solution. Avtomat. i Telemekh. 65, 157–169 (2004) 5. Chentsov, A.G.: On a game problem of guidance, Dokl. Akad. Nauk SSSR 226, 73–76 (1976) 6. Chistyakov, S.V.: On solutions for game problems of pursuit. Prikl. Mat. Mekh. 41, 825–832 (1977) 7. Ukhobotov, V.I.: Construction of a stable bridge for a class of linear games. J. Appl. Math. Mech. 41, 350–354 (1977). https://doi.org/10.1016/0021-8928(77)90021-1 8. Chentsov, A.G.: On the game problem of convergence at a given moment of time. Math. USSR-Izv. 12, 426–437 (1978). https://doi.org/10.1070/IM1978v012n02ABEH001985 9. Roxin, E.: Axiomatic approach in differential games. J. Optim. Theory Appl. 3, 153 (1969). https://doi.org/10.1007/BF00929440 10. Elliott, R.J., Kalton, N.J.: Values in differential games. Bull. Amer. Math. Soc. 78, 427–431 (1972) 11. Engelking, R.: General Topology. PWN, Warszawa (1977) 12. Chentsov, A.G.: On the duality of various versions of the programmed iteration method. Russian Math. (Iz. VUZ) 45(12), 74–85 (2001) 13. Chentsov, A.G.: On interrelations between different versions of the method of program iterations: a positional version. Cybern. Syst. Anal. 38, 422–438 (2002). https://doi.org/10.1023/ A:102036882866

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14. Chentsov, A.G., Morina, S.I.: Extension and Relaxations. Kluwer Academic Publishers, Dordrecht, Boston, London (2002) 15. Serkov, D.A., Chentsov, A.G.: Programmed iteration method in packages of spaces. Dokl. Math. 94, 583–586 (2016) 16. Serkov, D.A., Chentsov, A.G.: Implementation of the programmed iterations method in packages of spaces. Izv. IMI UdGU 48, 42–67 (2016) 17. Serkov, D.A.: Transfinite sequences in the programmed iteration method. Proc. Steklov Inst. Math. 300, 153–164 (2018)

Chapter 8

UAV Path Planning in Search and Rescue Operations B. M. Miller, G. B. Miller, and K. V. Semenikhin

Abstract Unmanned aerial vehicle (UAV) search operations usually involve two consequent steps. The first one is the area of interest observation and data capturing with the aid of optoelectronic cameras, and the second one is the data transmission to the flight control center. Very often the position of the observed object is rather far from the area covered by a communication network formed by stationary general use base stations or a group of mobile base stations temporarily deployed for the search mission. This, as well as the constraints caused by surface features, may prevent the immediate transmission of the collected data. Therefore, after capturing the data, the UAV must find an appropriate position to perform a successful transfer of the information flow. Moreover, in case of manifold search mission, the UAV must be ready for an unplanned task correction, and therefore a landing and commencing a new flight are not acceptable. In this paper, UAV search mission planning is considered as a path optimization problem. The optimization goal is to set up the best conditions for data capturing and transmission under velocity and timing mission constraints. The optimization parameters are the UAV path and the data transmission plan; thus, the optimization problem is naturally decomposed into the internal problem of data transmission optimization and the external surveillance path planning. B. M. Miller (B) Monash University, Melbourne, Australia e-mail: [email protected] A.A Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia G. B. Miller Institute of Informatics Problems of Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] K. V. Semenikhin Moscow Aviation Institute, Moscow, Russia e-mail: [email protected] Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, Russia

© Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_8

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The first one is solved in an explicit way for a given path, and for the second one, the numerical solution in the deterministic case is found with the aid of the maximum principle. For the stochastic case, the approach is also outlined. The findings are illustrated by simulation results.

8.1 Introduction Nowadays, UAVs search and rescue missions become more and more important in various areas including the search of missing persons in different incidents such as forest fires, earthquakes, shipwrecks, etc. [1, 2]. The typical approach to the operation planning is based on the usage of a swarm of UAVs which provides maximum coverage of the operation area [3, 4]. In this case, the optimization of the search mission may be formulated as a scheduling problem with a constraint for each UAV to spend enough time in the assigned research area [5]. Though this approach takes into account the limited UAV’s power resources and overall mission time limits, it ignores the quality of the gathered and transmitted information [3]. Meanwhile, the collected data quality may be crucial for the whole search and rescue operation, since those kind of missions are usually multi-step and each successive step uses the information gathered on previous ones. Therefore, in the planning of the search, besides the time and power constraints, one needs to take into account the quality of observations and transmission that depends on chosen UAV’s trajectory. One possible approach which minimizes the average distances to the point of interest is given in [5]. The observation-transmission quality is usually decreasing with distance; however, it remains acceptable even rather far from the observationtransmission site, so it is necessary to incorporate this dependence into the criterion for the UAV path planning. Generally, the search and rescue operations try to collect evidence about the position of a missing person. The search algorithms must take into account fundamental parameters such as – – – –

quality of sensory data collected by the UAVs; UAV’s energy constraints; environmental hazards (e.g., surface, winds, trees); quality of information exchange/coordination between UAVs.

Most of these parameters depend on the chosen UAV path. We assume that the best way to deal with all these factors is to use the information criterion as the capacity of the communication channel, which depends both on the distance between the UAV and the observation area and the distance between UAV and the base stations. The typical dependence is the inverse polynomial which is adequate from the physical point of view and is convenient as a corresponding term in the optimal condition equations. The optimization problem here is twofold. First optimization parameter is the optimal data transmission policy, which affects the limited power resources and should deal with channel perturbations whose level depends on the distance

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to the base station. The second optimization parameter is the UAV path, which naturally depends on the velocity and the mission timing. In the next section, we give the problem statement and its decomposition on two, namely, internal and external optimization problems. The first one is solved in Sect. 8.3 as a simple minimization problem and the second one requires the application of the optimal control theory. The numerical solution of the second problem in the deterministic case is provided in Sect. 8.4, and for the stochastic case, the approach is outlined in Sect. 8.5. The results of numerical experiments are presented in Sect. 8.6. The last contains conclusions and outlines further research directions.

8.2 Problem Statement 8.2.1 UAV Motion and Channel Model In this research, we use a rather simple motion model, like in [6] assuming the planar motion at constant altitude with constant velocity  X (t) =

x1 (t) x2 (t)



x˙1 (t) = V cos γ (t), x˙2 (t) = V sin γ (t),

(8.1)

where V ∈ [Vmin , Vmax ] is a given constant velocity, t ∈ [0, T ] with a fixed mission time limit T , and the control parameter is the yaw angle γ (t) ∈ [−π/2, π/2]. The initial X 0 and the terminal X T points of the flight are known. During its mission, the UAV must visit the vicinity of the set observation points, which are localized at m (t)} and may change their positions with time. S = {X ob The position of a base station X bs (t) is also known. The wireless channel between the UAV and the base station is assumed to be stochastic. Its state Xt at time instant t may take values from a discrete set of unit vectors Xt ∈ {ek , k = 1, . . . , M} as in standard model of a Markov chain in continuous time [11]. Each vector ek corresponds to a certain level of data transmission quality. The number of possible states M could be arbitrary and the simplest model could be limited with just two states: “good”—stable connection, low loss rate and “bad”—unstable connection, high loss rate caused by the wireless signal fading with distance or obstacles preventing its propagation. The channel model is defined as a controlled Markov chain with M possible states and a transition rate matrix , which naturally depends on the distance between the UAV and the base station:  = (r ), where r = X (t) − X bs (t). Each channel state corresponds to a certain noise level 1/k , so if the power level is u t , then the product k u t is equal to the signal-to-noise ratio and thus log(1 + k u t ) is the instantaneous channel capacity in the corresponding state ek [8, 9].

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8.2.2 Internal Problem: Data Transmission Optimization (DTO) DTO problem is to maximize the amount of useful information transferred along the fixed path of the UAV minus power consumed for this transmission. An optimization criterion, which corresponds to this optimization goal may be defined in the following form: T  u ν log(1 + , Xt  u t ) − κ u t dt → max , (8.2) J0 [γ , u] = E u={u(·)}

0

where  = col(1 , . . . ,  M ); ·, · is the scalar product; coefficient ν > 0 determines the utility of the information transmitted via the channel; the linear term κu t stands for the power consumed for the transmission at level u t , and hence the coefficient κ > 0 determines the strictness of overall power consumption constraint. Parameters k and ν, as well as the channel state Xt , depend on the UAV position X (t); therefore, the functional in the whole depends on the control γ (·). The utility function ν should depend on the distance between the UAV position X (t) and the area S assigned for observation. In the simplest case of a single observation point S = X ob (t), the utility function may be based on various decreasing dependencies on the current distance d = X (t) − X ob (t), for example, exponen2 tial ν = e−(d/d0 ) or inverse proportional ν = a/(b + cd 2 ), where d0 , a, b, and c are some coefficients, which are selected following the mission specification. When the observation has to be done over a set of points or some area (i.e., when S is not a singleton), then various metrics may be engaged to define the distance between the UAV and the set S. Since the precise specification of the utility coefficient ν is not necessary for this research, in further considerations it is simply assumed a function of the UAV position X (t).

8.2.3 External Problem: Surveillance Path Planning (SPP) This part of the problem deals with UAV path optimization. For a fixed UAV path, the DTO problem solution (8.2) provides the optimal data transmission plan u t and the maximum criterion value maxu={u t } J0 [γ , u]. In SPP this value is again maximized with respect to the UAV path, i.e., via the yaw angle γ = γ (t) which, in turn, defines the UAV path through the dynamics equation (8.1): J¯[γ ] = max J0 [γ , u] − κT X (T ) − X T 2 → max , u={u t }

γ ={γ (·)}

(8.3)

where κT is responsible for the accuracy of the terminal point X T achievement. Thus, the goal of the SPP problem is to find the best path by means of controlling the yaw angle from (3) given the transmission policy u t that is already found in the

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DTO problem. Once the optimal value of J¯ is obtained, an additional numerical procedure is needed to maximize it in one scalar parameter—velocity V . This completes the scheme of solving the combined SPP-DTO problem.

8.3 Solution of DTO Consider the maximization problem for integrand in criterion (8.2). For the sake of simplicity, we first address the case of a single base station with one possible channel state:   (8.4) μ0 = max ν log(1 +  u) − κ u , u≥0

where ν, , andκ are positive constants described above. Thus, μ0 may be considered as useful information minus power inputs for the optimal value of the power level u0. The derivative of (8.4) equals ν/(1/ + u) − κ. It is positive for ν > κ (1/ + u), that is, for u < ν/κ − 1/, and it is negative for u > ν/κ − 1/. Since μ0 is concave, u 0 = max{0, ν/κ − 1/} delivers the maximum in the problem (8.4). If u 0 = 0 we get μ0 = 0. Otherwise, if u 0 > 0, then μ0 = ν log(ν/κ) − ν + κ/. Therefore, the maximum in the problem (8.4) equals μ0 = ν g 0 (ρ), where ρ = κ/(ν), − log ρ − 1 + ρ, if ρ < 1, g0 (ρ) = 0, if ρ ≥ 1.

(8.5)

Underline that the optimal value is zero if ρ ≥ 1. This condition implies the inequality κ ≥ ν, which means that the power waste is more significant than the absence of the transmitted information. For the data flow optimization problems with incomplete information, it is essential to maintain a minimal data exchange over a communication channel in order to determine its state [7]. Though this fact does not matter much in the problem at hand, it should be taken into consideration in favor of the applicability of the results in future works. Hence, we have to reformulate the problem (8.4) so that the power level would be limited from below with some nonzero value ε: με = max (ν log(1 +  u) − κ u) , where ε > 0. u≥ε

(8.6)

The corresponding maximum point is defined by relation: u ε = max{ε, ν/κ − 1/}.

(8.7)

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Condition u ε > ε, that is, ν/κ − 1/ > ε, may be rewritten as ν/κ > 1 + ε or, using the notation of (8.5), as ρ < 1/(1 + ε). Thus, the maximum in the problem (8.6) equals με = ν gε (ρ), where ρε = 1 +1 ε , (8.8) − log ρ − 1 + ρ, ρ < ρε , gε (ρ) = − log ρε − 1 + ρε , ρ ≥ ρε .

8.4 Path Optimization Problem in the Deterministic Case In this section, we address the external optimization problem (8.3). As in Sect. 8.3, we assume single communication channel with one possible state, so that M = 1 and , X t u t = u t and imply that the scalar noise level coefficient  smoothly depends on the distance between the UAV and the base station:  = (r ), r = X (t) − X bs (t). The utility coefficient ν = ν(d) is also assumed to be a smooth function of the distance d between the UAV and the observation set S. Since the base station position, observation area S, and the distance d(X (t), S) are known, we further assume the noise and utility coefficients to be smooth functions of the UAV position:  = (X (t)), ν = ν(X (t)). Consider the maximization problem for expression under the expectation in criterion (8.2), taking into account the terminal term from (8.3): J¯[γ , u] =

T

 ν(X (t)) log(1 + (X (t)) u t ) − κ u t dt − κT X (T ) − X T 2 → max

u t ≥ε, γt

0

X˙ = V (t),

 X (0) = X 0 , (t) =

 sin γ (t) , |γ (t)| ≤ π/2. cos γ (t)

(8.9)

Given the yaw angle γ (t), maximization on u t yields the optimal power level u ε = max{ε, ν(X (t))/κ − 1/((X (t))} and the corresponding functional value J¯[γ ] = max J¯[γ , u] =

T f (X (t)) dt − κT X (T ) − X T 2 ,

u t ≥ε

(8.10)

0

where the function under the integral equals f (X ) = ν(X ) gε



 κ (X )ν(X )

(8.11)

with gε (·) defined in (8.8). The optimization problem for the criterion (8.10) on the paths of the controlled system (8.9) is solved with the aid of the maximum principle as in [6].

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Let t = col(ψ1 (t), ψ2 (t)) be the conjugate variables. Introduce the Hamiltonian H (X (t), γt , t ) = t , V t  + f (X (t)).

(8.12)

According to the necessary optimality condition in the form of the Pontryagin maximum principle [10], if the pair (X ∗ (t), γt∗ ) is the optimal path and control, then there exists the conjugate variable t∗ = col(ψ1∗ (t), ψ2∗ (t)), such that ˙ s∗ = − 

∂ H (X, γ , ) ∂ f (X ) = − X =X ∗ (s) ∂X ∂ X X =X ∗ (s)

(8.13)

∗ and the optimal control γt delivers the maximum to the Hamiltonian (8.12), ∂ H (X,γ ,) i.e., ∗ = 0 for any s ∈ [0, T ], hence ∂γ γ =γ (s)

sin γ ∗ (s) =

ψ1 (s) ψ12 (s) + ψ22 (s)

, cos γ ∗ (s) =

ψ2 (s) ψ12 (s) + ψ22 (s)

.

(8.14)

Adding the optimal control representation (8.14) to the differential equations for system path (8.9) and the conjugate variables (8.13), we get a boundary value problem, which in scalar form may be represented as follows: V ψ1 (s) V ψ2 (s) x˙1 (s) = , x˙2 (s) = , 2 2 2 ψ1 (s) + ψ2 (s) ψ1 (s) + ψ22 (s) ∂ f (x1 , x2 ) ∂ f (x1 , x2 ) , ψ˙ 2 (s) = − ψ˙ 1 (s) = − ∂ x1 ∂ x2

(8.15)

with initial and terminal conditions X (0) = X 0 , (T ) = col(ψ1 (T ), ψ2 (T )) = −κT (X (T ) − X T ).

(8.16)

Finally, the boundary value problem (8.15)–(8.16) with the instantaneous utility function f (X ) = f (x1 , x2 ) (8.11) and the optimal control u t = u ε (X ∗ (t)) (8.7) yields the solution to the path planning/power level optimization problem (8.2), (8.3) for the case of a single possible channel state. It should be noted that the proposed boundary value problem does not necessarily have a solution for any combination of its parameters. In order to determine reasonable values of κ and κT , it suffices to solve an auxiliary optimization problem for system (8.9) T ν(X (t)) log(1 + (X (t)) u t ) dt → max u t ≥ε, γt

0

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T subject to constraints 0 u t dt ≤ UT and X (T ) − X T  ≤ r T , where UT is the limit for the integral power consumption of the UAV transmitter and r T is the radius of the circle around the initial point, where the UAV is expected to return after the mission.

8.5 Stochastic Problem with Full Information Now let us return to the original problem, where the channel state is governed by a controlled Markov chain Xt with transition rate matrix dependent on the distance between the UAV and the base station:  = (X (t)). It is assumed that the power level u t is a stochastic process in a form of a Markov  M control, defined by a set of υk (t)I {Xt = ek }. deterministic functions υk (t), k = 1, . . . , M: u t = k=1 Then the criterion (8.2) can be represented as follows: T J0 [γ , u] =

ν(X (t))

M 

 log(1 + k (X (t))υk (t)) − κ υk (t) pk (t) dt,

(8.17)

k=1

0

where pk (t) = P{Xt = ek } is the probability of the process Xt to be in the state ek at the time instant t and k are the noise coefficients which correspond to the channel states ek . Note that these coefficients may depend on the distance to the base station, like in the deterministic problem of Sect. 8.4: k = k (X (t)). After the maximization of the criterion (8.17) with respect to the set of variables υk (t) ≥ ε, we get the following optimal control problem: J¯[γ ] =

T ν(X (t)) 0

M  k=1

 gε

κ k (X (t))ν(X (t))

 pk (t) dt − κT X (T ) − X T 2 → max . γ

This problem is considered for a deterministic differential system with the combined state col(x1 (t), x2 (t), p1 (t), . . . , p M (t)) given by the motion model (8.9) and Kolmogorov equations P˙t =  (X (t))Pt , Pt = col( p1 (t), . . . , p M (t)) with initial conditions X (0) = X 0 and P0 = col( p1 (0), . . . , p M (0)). A solution to this problem can also be obtained using the maximum principle using the same reasoning as in Sect. 8.4. In the case when the states of the base stations are observed via some stochastic observation process, one can estimate the conditional probabilities of the base station states and use them instead of probabilities Pt . Of course, this does not yield an optimal solution, but if the state observations are valid, this suboptimal control may be considered as a good approximation to the optimal one.

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8.6 Numerical Simulation In this section, we provide a numerical simulation of the model (8.1)–(8.3) with the following specifications: – observation interval is t ∈ [0, 60] min; – observable object is a single point which moves from X ob (0) = (−1.31, −0.52) to X ob (T ) = (3.77, 7.62) along the path shown in Fig. 8.1 (left, white dash line) with constant speed; – UAV starts at X 0 = (2.0, −1.0) and is aimed to finish around X T = (7.5, −4.0); – UAV speed is subject to optimization within the bounds V ∈ [0, 50] km/h; – coefficient of the return accuracy significance is κT = 0.05 1/km2 ; – observation utility function is given by ν(X (t)) = 1/(1 + d 2 /d02 ), where d = d(X (t)) = X (t) − X ob (t) is the distance between the UAV and the observable object and d0 = 2.5 km is the distance, where the utility halves; – position of the base station is X bs = (7.0, 7.0) (red triangle in Fig. 8.1, right); – noise coefficient (X (t)) = 1/(1 + r 2 /r02 ) depends on the distance from the base station r = r (X (t)) = X (t) − X bs , where r0 = 2.0km is a distance where the noise coefficient is twice as high as above the base station; – coefficient of the energy constraint strictness in (8.2) is κ = 0.05; – minimum power level ε = 0.01. In the presented setting, the observations are made close to the observable object, but far from the base station are less valuable, since the collected data may not be properly transferred over a noisy channel. At the same time, the integral criterion encourages the continuous data collection, so the UAV paths, which allow tracking of the observable object on its way, should be more beneficial than those, which tend to stay closer to the best view or less noise areas. x∗2 (t)[km]

u ∗ (t)

6 4 5 2 0 −2 −2

0 0

2

4

6

x∗1 (t)[km]

0 min

30 min

60 min

Fig. 8.1 Left: observation point path (white dash line), base station position (red triangle), and UAV optimal path (blue curve). Right: optimal power level u ∗ (green line)

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An additional optimization procedure was performed in order to obtain the UAV velocity V ∗ , which delivers a maximum to the criterion (8.3). To that end, the set of possible speed values [Vmin , Vmax ] was transformed into the uniform grid with a step equal to h = 0.01, and for each of the speed values Vi = Vmin + i h, i ∈ [0, (Vmax − Vmin )/ h], an optimal trajectory was found as a solution to the maximum principle equations (8.15)–(8.16). The best-found speed value is V ∗ = 12.36 km/h, with the criterion value equal to J¯(V ∗ ) = 3.93. The optimal path for the UAV speed V ∗ is shown in Fig. 8.1 (left) and the corresponding optimal power level u ∗ = u ε (8.7) in Fig. 8.1 (right).

8.7 Concluding Remarks Usually, the problem of search and rescue UAV mission is threefold. These are – planning the schedule of visiting various search areas and data transmission; – optimal trajectory design, which must take into account the positions of the observed object and the base stations, timing and velocity constraints; – optimal data acquisition and transmission planning, based on given UAV trajectory. These tasks depend on one another, and they must be solved taking into account all the information available online, such as the state of the transmission channel with each particular base station and the position of probably moving objects of observation. Here, we present approaches to the solution of the second and the third tasks in a deterministic setting. Although a scheme for solving the stochastic problem is also outlined, a more realistic statement should include first an estimation step and then a solution of the stochastic control problem with incomplete information. For the estimation step, there are well-developed methods, such as presented in [11]; however, the optimal control problem implies a solution of a functional partial differential equation for which there are no effective numerical methods. Meanwhile, for practical purposes, a suboptimal solution is often sufficient. These suboptimal algorithms are commonly designed under the assumption that the observations are informative, and the estimate provides more or less adequate information about the state. Of course, even in these suboptimal settings, obtaining the exact solutions is still impossible; however, numerical approaches are rather useful for engineering practice, especially if they take into account the principal features of the problem. Acknowledgements The work of G. B. Miller is supported by the Russian Foundation of Basic Research (RFBR Grant No. 19-07-00187-A).

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References 1. Waharte, S., Trigoni, N.: Supporting search and rescue operations with UAVs. In: International Conference on Emerging Security Technologies, EST 2010, Canterbury, UK, 6–7 Sept 2010 2. Schumacher, C., Chandler, P., Pachter, M., Pachter, L.: Constrained optimization for UAV task assignment. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, Rhode Island, USA, 16–19 Aug 2004 3. Hoang, V.T., Phung, M.D., Dinh, T.H., Ha, Q.P.: Angle-encoded swarm optimization for UAV formation path planning. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2018, pp. 5239–5244, Madrid, Spain, 1–5 Oct. 2018 4. Barkley, B.E., Paley, D.A.: Multi-target detection, tracking, and data association on road networks using unmanned aerial vehicles. In: IEEE Aerospace Conference, Big Sky, MT, USA, 4–11 March 2017 5. Rasmussen, S.J., Shima, T.: Tree search algorithm for assigning cooperating UAVs to multiple tasks. Int. J. Robust Nonlinear Control. 18, 135–153 (2008) 6. Andreev, M.A., Miller, A.B., Miller, B.M., Stepanyan, K.V.: Path planning for unmanned aerial vehicle under complicated conditions and hazards. J. Comput. Syst. Sci. Int. 51(2), 328–338 (2012) 7. Miller, B.M., Miller, G.B., Semenikhin, K.V.: Optimization of the data transmission flow from moving object to nonhomogeneous network of base stations. In: Preprints of the 20th World Congress of the International Federation of Automatic Control, Toulouse, France, July 2017. IFAC-PapersOnLine 50(1), 6160–6165 (2017) 8. Golomb, S.W., Peile, R.E., Scholtz, R.A.: Basic Concepts in Information Theory and Coding. Springer Science+Business Media, LLC, New York (1994) 9. Kuznetsov, N.A., Myasnikov, D.V., Semenikhin, K.V.: Optimal control of data transmission over a fluctuating channel with unknown state. J. Commun. Technol. Electron. 63(12), 1506– 1517 (2018) 10. Bryson, A.E., Ho, Y.-C.: Applied Optimal Control: Optimization, Estimation and Control. Hemisphere, Washington (1975) 11. Elliott, R., Aggoun, L., Moore, J.: Hidden Markov Models: Estimation and Control. Springer, New York (2008)

Chapter 9

Second-Order Necessary Optimality Conditions for Abnormal Problems and Their Applications Aram V. Arutyunov

Abstract In this paper, we discuss necessary optimality conditions applicable to constrained optimization problems degenerating at the point of minimum. Some applications of these results to inverse function existence problem are studied.

9.1 Some Examples of Abnormal Problems of Analysis We will explain what abnormality is about on the example of two related problems (for more details see [1]). Let us start from the optimization theory. Let X be a linear space. Consider the minimization problem with constraints: f 0 (x) → min,

F(x) = 0.

(9.1)

Here F : X → Y = Rk is a given map, and the minimum of a given function f 0 : X → R is sought on the admissible set M := {x ∈ X : F(x) = 0}. For simplicity, assume that X is a Banach space (we may even assume that X = Rn ), and f 0 , F are twice continuously differentiable in some neighborhood of the point x0 , which we assume to be a local minimizer for the problem (9.1). Then, in the point x0 , the Lagrange multipliers rule holds true. In order to formulate it let us introduce the Lagrange function L(x, λ) = λ0 f 0 (x) + y ∗ , F(x); λ = (λ0 , y ∗ ), λ0 ∈ R1 , y ∗ ∈ Y, where (k + 1)-dimensional vector λ = (λ0 , y ∗ ) and its components are called Lagrange multipliers.

A. V. Arutyunov (B) Institute of Control Sciences of RAS, Profsoyuznaya st., 65, 117997 Moscow, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_9

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Lagrange Principle. Let x0 be a local minimizer in the problem (9.1). Then there exists a Lagrange multiplier λ such that ∂L (x0 , λ) = 0, λ0 ≥ 0, |λ| = 1. ∂x

(9.2)

Lagrange principle is well known as it provides first-order necessary optimality conditions. The set of Lagrange multipliers, satisfying (9.2), is denoted by (x0 ). Consider the two following cases. First, let the point x0 be normal, that is, imF  (x0 ) = Y . In the Russian literature, this condition is also known as “Lyusternik’s condition”. In addition, along with the term “normal” point it also often used the term “nondegenerate” or “regular” point. So, if the point of minimum x0 is normal, then in view of (9.2) λ0 > 0 and hence, taking into account the positive homogeneity of the relations (9.2) in the variable λ, without loss of generality, we can assume that λ0 = 1. At the same time, there exists unique Lagrange multiplier λ = (1, y ∗ ). Besides, when the minimizer x0 is normal, classic second-order necessary conditions hold: ∃ λ ∈ (x0 ) :

∂ 2L (x0 , λ)[h, h] ≥ 0 ∀ h ∈ X : F  (x0 )h = 0. ∂x2

(9.3)

Consider the second case: let the point x0 be abnormal, i.e., imF  (x0 ) = Y. Then, Lagrange principle (9.2) is true with λ0 = 0 and arbitrary y ∗ = 0 from the kernel of the conjugate operator ker F  (x0 )∗ (which is not trivial due to imF  (x0 ) = Y ). Thus, in any abnormal point, Lagrange principle holds automatically regardless of the functional f 0 being minimized, and so this principle appears only as a paraphrase of abnormality definition. Therefore, Lagrange principle is useless if we try to investigate whether an abnormal point is local minimizer or not. As for the classical second-order necessary conditions (9.3), at abnormal points of extremum they can be violated. The following two-dimensional example clearly shows what has been said X = R2 , f 0 (x) = −|x|2 → min, f 1 (x) = x12 − x22 = 0, f 2 (x) = x1 x2 = 0, where x = (x1 , x2 ) ∈ R2 . In this problem, there is a unique point x0 = 0 satisfying constraints, which naturally appears to be local minimizer here. However, this is easy to see that the conditions (9.3) do not hold. Thus, in abnormal points, Lagrange principle is not meaningful, whereas the classic second-order necessary conditions are not applicable (they might not be true at such points).

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So the following problem arises: what are the meaningful necessary conditions for minimum in the problem (9.1) without a priori assumptions of normality of the point being under investigation? We now turn to the related problem. Let the mapping F : X → Y be continuously differentiable in a neighborhood of the point x0 ∈ X and y0 = F(x0 ). The question is, whether there exists a neighborhood V of the point y0 , such that for all y ∈ V the equation F(x) = y (9.4) has such a solution x(y) that x(y0 ) = x0 and the map x(·) is continuous at y0 , or even more, continuous throughout the neighborhood V ? If the point x0 is normal, then the classical theorem on the inverse function gives a positive answer to this question, and the mapping x(·) can be chosen continuously differentiable. If the point x0 is abnormal, it is no longer the case. For example, the scalar equation, x12 + x22 = y, considered in a neighborhood of x = 0, for y < 0, has no solutions, but the equation x12 − x22 = y has infinitely many continuous solutions for which x(0) = 0, although all of them are not differentiable at the origin. This raises the question of obtaining conditions, which are weaker than normality, and which would guarantee the existence of the solution x(·) to Eq. (9.4) with the desired properties.

9.2 Second-Order Necessary Optimality Conditions Consider the extremum problem with equality and inequality constraints f 0 (x) → min,

f i (x) ≤ 0, i = 1, l,

f (x) = 0.

(9.5)

For non-negative integers s, we introduce the sets (it may so happen that some of them will be empty) s (x0 ), consisting of those Lagrange multipliers λ ∈ (x0 ), for which there exists (depending on λ) a subspace  ⊆ X , such that codim ≤ s;  ⊆ ker F  (x0 ); ∂ 2L (x0 , λ)[x, x]2 ≥ 0 ∀ x ∈ , ∂x2

f 0 (x0 ) ∈ ⊥ .

Here, the map F = ( f 1 , . . . , fl , f ) acts from X into Rm , and m = l + k.

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Consider the cone K(x0 ) = {h ∈ X :  f i (x0 ), h ≤ 0 ∀ i, f  (x0 )h = 0}, called “cone of critical directions”. Theorem 9.1 (second-order necessary conditions) Let the point x0 be a local minimizer in the problem (9.5). Then, as m = k + l the set m = m (x0 ) is not empty and, moreover, ∂ 2L (x0 , λ)[h, h] ≥ 0 ∀ h ∈ K(x0 ). max λ∈m ∂ x 2 This theorem was obtained in [2, 3]. Its proof is based on a perturbation method and some functional analysis results from [5]. For the case of abnormal minimizers x0 , Theorem 9.1 can be slightly strengthened. Theorem 9.2 Let the point x0 be abnormal local minimizer in the problem (9.5). Then, as m = k + l the set m−1 = m−1 (x0 ) is not empty and, moreover, max

λ∈m−1

∂ 2L (x0 , λ)[h, h] ≥ 0 ∀ h ∈ K(x0 ). ∂x2

This theorem was obtained in [6]. Its proof uses some facts from real algebraic geometry. The result of this theorem was generalized to the problem f 0 (x) → min,

x ∈ C,

F(x) ∈ D,

where F : X → Y = Rm . Here C, D defined closed subsets of X, Y (but not necessarily convex!). The formulation of these second-order necessary conditions can be found in [1, Sect. 2]. Denote by F2 (x0 ) the cone consisting of all y ∈ Y , y = 0, such that ∂∂ Fx (x0 , y0 )∗ y = 0 and there exist a subspace of X  ⊆ K er such that codim ≤ k;

∂F (x0 ), ∂x

∂2 y, F(x0 )[ξ, ξ ] ≥ 0 ∀ ξ ∈ . ∂x2

Note that the cone F2 (x0 ) might be empty. It, for example, certainly empties when the point x0 is normal, since, in this case, ∂∂ Fx (x0 )∗ y = 0 ∀y = 0. Another observation: if we add y = 0 to F2 (x0 ), then this cone becomes closed, but not necessarily convex.

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Definition 9.1 Mapping F : X → Y is said to be 2-normal in the point x0 , provided the cone convF2 (x0 ) is acute, i.e., it does not contain any non-trivial subspaces (the case F2 (x0 ) = ∅ is not excluded since empty cone is acute by definition). When X = Rn and n  m, the set of 2-normal mappings is dense in Cs3 (Rn , Rm ). Here Cs3 (Rn , Rm ) is the space of three times differentiable mappings from Rn to Rm equipped with Whitney topology (see, for example, [8, Chap. 2, Sect. 1]). Let us, for simplicity, assume that X is a Hilbert space. Theorem 9.3 Assume the map F be 2-normal at x0 and the second-order necessary conditions stated in Theorem 9.1 hold. Then there exists a vector y ∈ Rm such that for any ε > 0, in the perturbed problem: f 0 (x) + εx − x0 2 → min, f i (x) + εyi x − x0 2 ≤ 0, i = 1, l, f i (x) + εyi x − x0 2 = 0, i = l + 1, m, the point x0 is a strict local minimizer. The proof of this theorem can be found in [2, 3]. Second-order necessary condition in Theorems 9.1 and 9.2 was adopted on various classes of optimal control problems, namely, control problems with terminal, mixed constraints, the problems with impulsive controls, the control problems with delay, etc. Exact formulations can be found in [1, Sect. 7] and [7]. Second-order sufficient conditions for infinite-dimensional constrained optimization problems can be found in [1, Sect. 4].

9.3 Applications. Inverse Function Theorem We now turn to the inverse function theorem. Suppose again that we are given a mapping F : X → Y = Rk , and F(x0 ) = y0 . For all y of neighborhood of y0 , we consider the nonlinear equation F(x) = y. The question is, when this equation for all y close to y0 has a solution? If the point x0 is normal, then the answer is the classical inverse function theorem. But here, let us formulate an inverse function theorem which is true even without a priori assumptions of normality. Let Y1 = I m ∂∂ Fx (x0 ), Y2 = (I m ∂∂ Fx (x0 ))⊥ . Denote by P1 : Y → Y1 , P2 : Y → Y2 projection operators of Y onto subspaces Y1 and Y2 , respectively.

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Definition 9.2 Say that the mapping F : X → Y satisfies solvability condition at point x0 provided there exist positive constants κ1 , κ2 , such that ∀ y : |y − y0 | < κ1 ∃ x = x(y) : F(x(y)) = y;   1 x(y) − x0  ≤ κ2 |P1 (F(x0 ) − y)| + |P2 (F(x0 ) − y)| 2 .

Theorem 9.4 (inverse function theorem) Assume the map F be 2-normal in the point x0 . Then, in order for F to satisfy solvability condition at x0 , it is necessary and sufficient that ∃h ∈ X :

∂F ∂2 F (x0 )h = 0; y, 2 (x0 )[h, h] < 0 ∀ y ∈ F2 (x0 ). ∂x ∂x

This result was obtained in [1, 3, 4]. The proof is based on the above-presented second-order necessary conditions. If the point x0 is normal, then this theorem becomes classic inverse function theorem. Later, this result was extended up to the implicit function theorem, to the inverse function theorem on the cone, to nonlinear systems containing inequalities, etc. All the formulated results were published in [1, 3, 4]. It was also applied to the study of controllability in degenerate dynamical systems. With the help of these theorems on inverse and implicit functions, there were obtained new theorems on the existence of bifurcation points; for nonlinear systems, we constructed a new theory of sensitivity for extremum problems without a priori assumptions of normality. Acknowledgements The investigation is supported by Russian Science Foundation (Project No. 17-11-01168).

References 1. Arutyunov, A.V.: Smooth abnormal problems in extremum theory and analysis. Russ. Math. Surveys. 67(3), 403–457 (2012) 2. Arutyunov, A.V.: Second-order conditions in extremal problems. The abnormal points. Trans. Am. Math. Soc. 350(11), 4341–4365 (1998) 3. Arutyunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Mathematics and Its Application. Kluwer Academic Publisher, Dordrecht (2000) 4. Arutyunov, A.V.: Necessary Extremum Conditions and an Inverse Function Theorem without a priori Normality Assumptions, Differential equations and dynamical systems, Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 236, Nauka, Moscow, pp. 33–44 (2002)

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5. Arutyunov,A.V.: Perturbations of extremum problems with constraints and necessary conditions of optimality // Itogi nauki i tekhniki. VINITI. Math. Anal. 27, 147–235 (1989) 6. Arutyunov, A., Karamzin, D., Pereira, F.: Necessary optimality conditions for problems with equality and inequality constraints: the abnormal case. J. Optim. Theory Appl. 140(1), 391–408 (2009) 7. Arutyunov, A., Karamzin, D., Pereira, F.: Optimal Impulsive Control: The Extension Approach. Springer Nature Switzerland AG, Cham, Switzerland (2019) 8. Hirsch, M.: Differential Topology. Springer, New York, Heidelberg, Berlin (1976)

Chapter 10

Alternate Pursuit of Two Targets, One of Which Is a False Evgeny Ya. Rubinovich

Abstract The differential game of alternating pursuit of two targets, one of which is a false target with a given classification probability, is considered on the plane. The players have simple motions. The criterion is the average time to meet the true target.

10.1 Introduction Differential games of pursuit-evasion with a group target on the plane, when the pursuers are less than the targets, have long attracted the attention of researchers. The fact is that the flat case, on the one hand, allows, as a rule, to obtain visual analytical solutions, and on the other hand, it is quite acceptable for modeling the conflict interaction of real dynamic objects. The criteria for the effectiveness of such interaction can be traditional payment functionals in the game theory of pursuitevasion such as “Time” or “Miss”. Mathematical formalizations of such games allow taking into account information discrimination of players, in particular, the presence of false targets, incomplete information about the initial conditions, etc. Games of pursuit of a group target in the presence of false targets are divided into differential games of alternate and joint pursuit. In the first class, as a rule, the task of the pursuer is to catch all the targets in turn, or all the true targets, if the pursuer has the ability to classify the targets from a certain distance or in direct contact. In the case of joint pursuit, the task of the pursuer is partial convergence directly with the group target. The tasks of joint pursuit of two targets were considered in [1–3]. Problems of alternate pursuit of two targets were studied in [4–9]. In [7], it has solved the problem of using a mobile false target to distract the pursuer in the assumption of the known laws of guidance and circular capture zone (detection zone) of the pursuer. Works [10–21] belong to a new and rapidly developing direcE. Ya. Rubinovich (B) V.A. Trapeznikov Institute of Control Sciences, 65, Profsoyuznaya Str., Moscow 117997, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_10

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tion of differential games of three players of the type Attacker–Target–Defender or Missile–Target–Defender, respectively, ATD or MTD games. In these productions, the attacking player seeks to catch (hit) the fleeing target, while the task of the mobile defender, whose role can be played by a false target, is to catch the attacking player. The principal difference between the problem statement in [21] and [10–20] is the incompleteness of a priori information about the pursuer. The target only knows the initial bearing on the pursuer and the magnitude of his speed. The direction of the velocity vector of the pursuer purpose is not known. In this paper, the differential game of alternating pursuit on the plane by the pursuer P of two consistently evading targets E A , E B (one of which is false) and with simple movements of the players differs from the previously solved problem statements [4–6, 9] in which minimization of the total time pursuit of two targets is considered (the game Γtime ). In our statement, it is fundamentally the presence of a false target, and what kind of target a false pursuer does not know. So, if the first target in the order of persecution is false, the pursuer is forced to organize the pursuit of the second target. Therefore, the task of the pursuer is to minimize the mathematical expectation of time to catch the true target, taking into account the given probability classification.

10.2 Problem Statement A conflict–cooperative differential game on the plane of one pursuer P against a coalition of two consistently evading targets E A and E B one of which is false is considered. The movements of the players are simple (inertialess). The velocity vectors of the players are the controls, which are imposed geometric (modular) restrictions: the speed of the pursuer |u(t)| ≤ 1, the speed of the targets |vi (t)| ≤ β < 1, i = A, B; here β is a given constant. The pursuer does not know which target is true, but he knows the target classification probabilities p A and p B , ( p A + p B = 1), where pi is the probability that i-th target is true. The targets E A and E B are pursued alternately until a point meeting with the true target. The payoff functional (criterion) has the form (10.1) G = p A T A + p B TB → min max , P

E A ,E B

where Ti is the time to meet the target E i . The criterion (10.1) makes sense of the mathematical expectation of the time before meeting the true target. It is minimized by the pursuer. The targets consistently maximize this criterion. Such a statement with the criterion of “full time pursuit of two targets” (game Γtime ) was considered in [4] and with the criterion of “miss on the true target” (game Γmiss ) was considered in [2]. In these works, it was noted that the priority choice of meetings with the targets is, in fact, additional control of the pursuer, which can be chosen:

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(a) programmatically (as a function of time) at the start of the game; (b) positionally, in the pursuit process, as a function of the current position of the players. In [2, 4], both these possibilities are considered and it is shown that at certain initial positions in the case (b) the stage of alternate pursuit is preceded by the stage of joint pursuit of some duration [0, θ] of the group of two targets and during which the upcoming order of meetings with the targets is not defined. It is determined directly at instant θ of the end of the joint pursuit phase. The considered statement is ideologically close to [2, 4]; however, as noted above, differs from it by the payment and also by the fact that the probabilities of targets classification are a priori known.

10.3 Main Result Case (a)—programmatic choice of the meetings order. The index 1 will be assigned to the first in the order of pursuit. Then the payoff (10.1) takes the form G = p1 T1 + p2 (T1 + T12 ) = T1 + p2 T12 ,

(10.2)

where T12 is the time in pursuit of E 2 after a meeting with E 1 . The players’ motion equations: after the time T1 prior toT1 ⎧ ⎧ ⎨ z˙ 1 (t) = v1 (t) − u(t), ⎨ z˙ 1 (t) = 0, z˙ 2 (t) = v2 (t) − u(t), z˙ 2 (t) = v2 (t) − u(t), ⎩ ⎩ z˙ 3 (t) = 1, z˙ 3 (t) = p2 .

(10.3)

Here z i (t), i = 1, 2, is a two-dimensional vector directed from P t to E it , where P t and E it are current players’ positions; z 3 (t) is a variable that has the meaning of model time and allows us to determine the payoff (10.2) in the terminal form G = z 3 (T ),

(10.4)

where T is the duration of the game in the worst case for the pursuer, when the target E 1 is false. Initial conditions are specified as z 10 , z 20 , z 3 (0) = 0. In addition, the equation of the discontinuity surface of the right parts of the equation (10.3) (meeting condition P and E 1 ) (10.5) z 1 (T1 ) = 0,

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Fig. 10.1 The optimal trajectories of the players in the Case (a) (β = 0.5)

terminal condition (meeting P and E 2 ) z 2 (T ) = 0.

(10.6)

The game solution in Case (a) is based on the maximum principle for problems with the game situation [2, 4]. The following proposition takes place. Proposition 10.1 Optimal movements of the players are carried out along the straight lines with the maximum modulo speeds. It is known that then the point P T1 = E 1T1 of the meeting of players P and E 1 lies on the Apollonius circle (Fig. 10.1, O  is the center of the circle) [2, 4]: 

 −1 2 −1 2   + y 2 = |z 10 |β 1 − β 2 . x − |z 10 | 1 − β 2

(10.7)

To localize this point, we have rewritten the payoff (10.2) as

l2 1 − p1 β l 1 l1 l2 + l1 , p2 = + G= + β 1−β 1−β β (1 − p1 β) p2−1 where li  P T1 E i0 and consider the auxiliary problem of reflection from the circle (10.7) of a light beam emitted from the point E 10 at a speed β (since the beam of light minimizes the time of its movement in the given medium between the given points). After the reflection from the circle (10.7) at point P T1 , the speed of light increases abruptly and the beam of light begins to move at a speed (1 − p1 β) p2−1 ≥ 1 > β toward the point E 20 . At point P T1 , the law of reflection must be fulfilled sin ν/ sin μ = β/[(1 − p1 β) p2−1 ],

(10.8)

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which localizes this point on the circle of Apollonius (Fig. 10.1). By virtue of (10.6) the angle of incidence ν is always less than the reflection angle μ. Successive application of the Pontryagin maximum principle to the problem (10.3)–(10.6) leads to a system of algebraic equations ⎧ 0 = z 10 + T1 (βh 1 − u 0 ), ⎪ ⎪ ⎨ 0 = z 20 + T1 (βh 2 − u 0 ) − T12 (1 − β)h 2 , u 0 = λ1 h 1 + λ2 h 2 , ⎪ ⎪ ⎩ 1 = βλ1 + (1 − β p1 ) p2−1 λ2 ,

(10.9)

with respect to the unknown λ1 , λ2 , T1 , T12 and three direction angles of unit vectors h 1 , h 2 , and u 0 . Here βh 1 = v10 , βh 2 = v20 , u 0 are optimal controls of the players. At instant t = 0, the sequence of meetings is determined as follows. Let  A (P 0 , E 0A , E 0B ) and  B (P 0 , E 0A , E 0B ) be payoffs (as functions of initial positions) corresponding to the program choice of the sequence of meetings E A → E B or E B → E A , respectively. The optimal sequence of meetings is determined at instant t = 0 from the following conditions: E A → E B , if min k (P 0 , E 0A , E 0B ) = k=A,B

 A (P 0 , E 0A , E 0B ), E B → E A , if min k (P 0 , E 0A , E 0B ) =  B (P 0 , E 0A , E 0B ). k=A,B

In the case of identical targets ( p A = p B = 0.5), the target closest to the pursuer is initially pursued. This target is assigned an index of 1. Case (b)—positional choice of the meetings order. As in the Case (a) index 1 will be assigned to the first in the order of pursuit. In the Case (a), the decision on the order of pursuit the player P takes at instant θ = 0 of the pursuit beginning. Due to the assumption P does not change the order of pursuit, i.e., at any behavior of the opponent the pre-planned purpose is initially pursued. In this formulation, under some initial conditions, untenable, from a practical point of view, solutions are obtained (although optimal in the sense of setting the problem). For example, for z 10 = z 20  z 0 , the players move in a straight line (along the vector z 0 ), with the target E 1 moving from P and the target E 2 moving to P. Since the target E 1 is initially pursued, the P encounters with the target E 2 until T1 is not fixed, even if the target E 2 is true. In order to avoid such paradoxes and make the problem statements practically meaningful, we allow the pursuer to carry out re-targeting, which physically means that P can choose the order of pursuit in the process of movement. In the process of players’ movement, a sign of the difference between payoffs (as a function of the current player’s positions {z 1 (t), z 2 (t), z 3 (t)}) Φ(z 1 (t), z 2 (t), z 3 (t))  2 (z 1 (t), z 2 (t), z 3 (t)) − 1 (z 1 (t), z 2 (t), z 3 (t)), generally speaking, can change in the interval [0, θ]. The positional choice of the sequence of meetings allows us to take into account the fact that whenever at some point in the game equality Φ(z 1 (t), z 2 (t), z 3 (t)) = 0

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is realized, the targets E 1 and E 2 become identical for the pursuer in the sense of the payoff value. This means that at this point t, the payoff (as a function of the current state {z 1 (t), z 2 (t), z 3 (t)}) does not depend on the choice of the sequence of the meeting and the player P can re-index the targets, i.e., change the order of the upcoming sequence of meetings. The possibility of such reindexing allows to guarantee the preservation of the sign of inequality Φ(z 1 (t), z 2 (t), z 3 (t)) ≥ 0

(10.10)

in interval [0, θ] of joint pursuit, θ  sup{t : Φ(z 1 (t), z 2 (t), z 3 (t)) = 0}. Inequality (10.10) plays the role of phase constraints and defines the relationship between the equations of motion (10.3) and the positional control of the sequence of meetings. In our case, we propose to use the inequality (10.11) instead of (10.10), which is simpler for analytical studies. This inequality coincides with (10.10) for the initial arrangement of players on the same row. p1 |z 2 (t)| ≥ p2 |z 1 (t)|,

t ∈ [0, θ].

(10.11)

Now let us find the set L 0 of possible initial positions E 20 of the target E 2 , at which in the case of a fixed (i.e., programmatic) pursuit order (E 1 → E 2 ) the inequality (10.11) is satisfied and, in addition, there is a moment ϑ = inf{t ∈ [0, T1 ] : p1 |z 2 (t)| − p2 |z 1 (t)| = 0}.

(10.12)

To do this, consider the time function R(t)  ( p1 |z 2 (t)|)2 − ( p2 |z 1 (t)|)2 . On optimal trajectories corresponding to the programmatically selected sequence of encounters and the initial conditions z 10 = {1, 0}; z 20 = { cos δ,  sin δ} (i.e., on the scale 0  |z 10 | = 1 and   |z 20 |), R(t) is a parabola R(t) = [ cos δ + at]2 p12 + [ sin δ + bt]2 p12 − [1 + td]2 p22 , where a  β cos ψ2 − cos ϕ, b  β sin ψ2 + sin ϕ, d  β cos ψ1 − cos ϕ, and δ is the angle of target sight by player P at t = 0. The angles ϕ, ψ1 , ψ2 are the direction angles of the unit vectors u 0 , h 1 , h 2 of (10.9) (Fig. 10.1). When performing (10.11) and (10.12) we have R(t) ≥ 0, R(ϑ) = 0, ϑ ∈ [0, T1 ]. Therefore, for t = ϑ, the function R(t) has a minimum and therefore the discriminant Δ of the square equation R(t) = 0 is equal to zero Δ = [(a cos δ + b sin δ) p12 − dp22 ]2 − [(a 2 + b2 ) p12 − d 2 p22 ](2 p12 − p22 ) = 0. (10.13) A simple analytical solution is obtained in the case of identical targets ( p1 = p2 = 0.5). In this case, from (10.13), it follows T1 (β sin σ + sin(ϕ + δ)) = l2 + βT1 ,

(10.14)

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where from P 0 P T1 E 20 by theorem of sines we get l2 / sin(ϕ + δ) = T1 / sin σ = / sin(2ψ1 ).

(10.15)

Solving together (10.14) and (10.15), we obtain  sin σ = 1 (= 0 ),

(10.16)

or in coordinate form x0 sin ψ2 − y0 cos ψ2 = 1,

where z 20 = {x0 , y0 } ∈ L 0 .

(10.17)

The projection of the first vector equality of (10.9) on the axes O X and OY gives 1 + βT1 cos ψ1 = T1 cos ϕ,

β sin ψ1 = sin ϕ.

(10.18)

From (10.17) and (10.18), we get cos ϕ − β cos ψ1 = sin(2ψ1 ).

(10.19)

The relation (10.17) indicates that the set L 0 of points {x0 , y0 } = z 20 is tangent to the circle of unite radius with the center O = P 0 , drawn at an angle ψ2 of the axis O X (i.e., to the vector z 10 ), where the angles ϕ and ψ1 are determined by the system (10.18), (10.19). Next, the solutions of the system (10.18), (10.19) will be marked with a symbol *: ϕ∗ , ψ1∗ and ψ2∗ = π − ϕ∗ − 2ψ1∗ . From the construction, it follows that when E 20 ∈ L 0 is actually L 0 (the tangent is M 0 N 0 , Fig. 10.2) is extremal, i.e., the optimal trajectory of the avoidance target E 2 is in L 0 .

Fig. 10.2 Division into zones of the area of possible initial positions of the far target in the Case (b) (β = 0.5)

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The tangent L 0 = M 0 N 0 divides the set of initial possible positions of the far target (the appearance of the circle O of radius 0 = 1 in the half-plane Y ≥ 0) into two zones C and D (Fig. 10.2). In the C zone (the appearance of the curvilinear E 10 N 0 M 0 ), the pursuit order is fixed from the beginning of the game and a strict inequality (10.11) is performed in the entire pursuit process. Let E 20 ∈ D are inside E 10 N 0 M 0 . We introduce a movable coordinate system t t X Y related to the current position P t of the player P and the axis X t directed along the vector z 1t . The scale along the axes X t and Y t will be chosen so that |z 1 (t)| = 1. In this coordinate system, for each t, a curvilinear E 10 N t M t with a boundary L t = M t N t = {(xt , yt ) : xt sin ψ2∗ − yt cos ψ2∗ = 1} is constructed. The condition (10.5) becomes equivalent to the following: |z 1 (T1 )| = ∞. This means that the point E 2t (representing the position of the far target) at some instant will go beyond the zone D, (E 10 N t M t ). According to the accepted indexing rule, this output is possible only through the boundary L t = M t N t . Let τ  inf{t : E 2t ∈ L t }. Then, taking the instant τ for the initial and fixing the pursuit order, we obtain that the depicting point E 2t from the instant τ will move along the boundary L t = M t N t toward the point M t , and at instant (10.20) θ  sup{t : p1 |z 2 (t)| − p2 |z 1 (t)| = 0} the position E 2θ = M θ is implemented. The angular coordinate δ ∗ of the point M θ is equal to the angle of sight of the targets at the time θ δ ∗  ∠{z 1 (θ), z 2 (θ)} = π/2 − 2ψ1∗ − ϕ∗ ,

(10.21)

where ψ1∗ and ϕ∗ are from (10.18) and (10.19). The instant θ splits the game into two stages: [0, θ] and [θ, T ]. Let E 2θ = M θ . Taking the instant θ for the initial (i.e., assuming θ = t0 = 0 and E 2θ = E 20 ) we consider the game at the stage [θ, T ] = [t0 , T ] in the original fixed coordinate system. Since according to (10.20) from the moment of θ the pursuit order is fixed, the average time of target capture in this case will be equal to T θ = 0.5(βT1∗ + l2∗ )/(1 − β), where T1∗ = P 0 P T1 , l2∗ = P T1 M 0 (Fig. 10.2). But T1∗ = 1/ sin(2ψ1∗ ) and l2∗ = coth(2ψ1∗ ), hence T θ = 0.5 coth ψ1∗ /(1 − β)  k (in scale 0 = |z 1 (θ)| = 1). If 0 > 1, then T ∗ = 0 k. Therefore, the total (two-step) average time from t0 = 0 to capture both targets is equal to (10.22) T = θ + k|z 1 (θ)|. Consequently, the output of the depicting point E 2t in the position E 2θ = M θ must be carried out by players so that the criterion (10.22) maximizes v1 , v2 and minimizes u. Thus, at the [0, θ] stage, players play an auxiliary differential game with dynamics (10.7), the above restrictions, criterion (10.22), and terminal conditions for θ p1 |z 2 (θ)| − p2 |z 1 (θ)| = 0, |z 1 (θ)||z 2 (θ)| cos δ ∗ = z 1 (θ), z 2 (θ),

(10.23)

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Fig. 10.3 The optimal trajectories of the players in the Case (b) with E 10 = E 20 and β = 0.5

where δ ∗ is from (10.21) and ·, · means scalar product. In step [0, θ], a joint prosecution by the player P and the targets E i takes place. The physical meaning of the present phase of a joint prosecution is that during this stage, the pursuer keeps the target in the face of uncertainty regarding the future of order capture, since the final index (i.e., the order of the catch), according to (10.20), is set at the instant θ (Fig. 10.3).

References 1. Ol’shansky, V.K., Rubinovich, E.Ya.: Simplest differential games of pursuieing a system of two plants. Autom. Remote. Control. 35(1), 19–28 (1974) 2. Rubinovich, E.Ya.: Elementary differential game of alternate pursuit with miss-type criterion. In: Proceedings of the 8th International Symposium on Dynamic Games and Applications, pp. 189–202. The Netherlands, Maastricht (1998) 3. Rubinovich, E.Ya.: Two targets pursuit-evasion differential game with a restriction on the targets turning. In: Preprints, 17th IFAC Workshop on Control Applications of Optimization, Yekaterinburg, Russia, 15–19 October 2018. IFAC-PapersOnLine 51–32, 503–508 (2018) 4. Abramyants, T.G., Maslov, E.P., Rubinovich, E.Ya.: A simplest differential game of alternate pursuit. Autom. Remote. Control. 41(8), 1043–1052 (1981) 5. Breakwell, J.V., Hagedorn, P.: Point capture of two evaders in succession. J. Opt. Theory Appl. 27(1), 89–97 (1979) 6. Shevchenko, I.I.: On successive pursuit. Autom. Remote. Control. 42(11), 1472–1477, 54–59 (1981) 7. Maslov, E.P., Ivanov, M.N.: On comparison of two pursuit methods in the problem of successive rendezvous. Autom. Remote. Control. 44(7), 856–861 (1983)

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8. Abramyants, T.G., Maslov, E.P., Rubinovich, E.Ya.: Mobile objects control in conditions of artificially organized information incompleteness. Problemy Upravleniya 4, 75–81 (2005) (in Russian) 9. Petrosyan L.A., Shiryaev V.D.: Group pursuit of some evaders by a certain pursuer. Vest. Leningr. Univ., Mat. Mekh. Astron. 13, 50–57 (1980) (Russian, English summary) 10. Ivanov, M.N., Maslov, E.P.: On one problem of deviation. Autom. Remote. Control. 45(8), 1008–1014 (1984) 11. Boyell, R.L.: Defending a moving target against missile or torpedo attack. IEEE Trans. Aerosp. Electron. Syst. AES 12, 582–586 (1976) 12. Boyell, R.L. Counterweapon aiming for defence of a moving target. IEEE Trans. Aerosp. Electron. Syst. AES 16, 402–408 (1980) 13. Garcia, E., Casbeer, D.W., Pham, K., Pachter, M.: Cooperative aircraft defense from an attacking missile. In: Proceedings of 53th IEEE Conference Decision and Control (CDC), pp. 2926–2931, Los Angeles, USA, 15–17 December 2014 14. Pachter, M., Garcia, E., Casbeer, D.W.: Active target defense differential game. In: 52nd Annual Allerton Conference on Communication, Control, and Computing, pp. 46–53. IEEE (2014) 15. Perelman, A., Shima, T., Rusnak, I.: Cooperative differential games strategies for active aircraft protection from a homing missile. J. Guid. Control. Dyn. 34(3), 761–773 (2011) 16. Shima, T.: Optimal cooperative pursuit and evasion strategies against a homing missile. AIAA J. Guid. Control. Dyn. 34(2), 414–425 (2011) 17. Takeshi, Y., Sivasubramanya, B.N., Hi-royuki, T.: Modified command to line-of-sight intercept guidance for aircraft defense. J. Guid. Control. Dyn. 36(3), 898–902 (2013) 18. Naiming, Q.I., Qilong, S.U.N., Jun, Z.H.A.O.: Evasion and pursuit guidance law against defended target. Chin. J. Aeronaut. 30(6), 1958–1973 (2017) 19. Martin, W., Tal, S., Ilan, R.: Minimum effort intercept and evasion guidance algorithms for active aircraft defense. J. Guid. Control. Dyn. 39(10), 2297–2311 (2016) 20. Garcia, E., Casbeer, D.W., Pachter, M.: Active target defense differential game with a fast defender. IET Control. Theory Appl. 11(17), 2985–2993 (2017) 21. Rubinovich, E.Ja.: Missile-target-defender problem with incomplete a priori information. In: Dynamic Games and Applications (Special Issue) (2019). On open access: https://rdcu.be/ bhvyh. https://doi.org/10.1007/s13235-019-00297-0

Chapter 11

Block Jacobi Preconditioning for Solving Dynamic General Equilibrium Models A. P. Gruzdev and N. B. Melnikov

Abstract We apply preconditioning to improve computation of perfect foresight competitive equilibrium for the global economy with multiple production sectors and regions. The model is solved by an iterative method of the Gauss–Seidel type. At each iteration, first intertemporal variables are updated, and then equations for intratemporal variables are solved using the Newton–Krylov method. The Jacobian matrix of the system of equations for intra-temporal variables is close to the block diagonal part over regions. The inverse of this diagonal part is used as the preconditioner to speed up convergence of the Krylov method at each nonlinear iteration. The application of the block Jacobi preconditioning is illustrated in the example of the integrated Population-Economy-Technology-Science (iPETS) model.

11.1 Introduction Dynamic general equilibrium models are widely used for estimating effects of demographic and technological changes on energy use and CO2 emissions. The equilibrium is described in the framework of the Arrow–Debreu theory, which leads to a system of nonlinear equations. The number of equations can reach hundreds of thousands and millions, and solving such systems can be time-consuming. Parallel computing and reordering techniques can be used to reduce the computing time (see, e.g., [4, 6]). An alternative is to transform the linearized system of equations into one that is easier to solve, which is called preconditioning (see, e.g., [13, 14]). The two most popular options of preconditioning are performing the incomplete LU decomposition (ILU) and using one of the stationary methods, such as Jacobi, A. P. Gruzdev (B) Lomonosov Moscow State University, Moscow, Russia e-mail: [email protected] N. B. Melnikov Central Economics and Mathematics Institute, Lomonosov Moscow State University, Moscow, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_11

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Gauss–Seidel, or successive overrelaxation (SOR), as the preconditioner (for details, see, e.g., [5]). A block Jacobi preconditioner with respect to time was applied to speed up the Newton–Krylov method for the stacked system [3, 10]. An application of Jacobi and ILU preconditioners for small-scale stochastic growth models was studied by [8]. However, application of stacked Newton-type methods can be impractical or infeasible for large systems with multiple regions and long time horizons. We present and analyze a block Jacobi preconditioner with respect to regions in the extended path method [1] (for use of the block Jacobi method with regional blocks for solving the model, see, e.g., [2]). The method is illustrated in the example of the integrated Population-Economy-Technology-Science (iPETS) model, a general equilibrium growth model with multiple regions, goods, sectors, households, and factors of production [9]. The paper is organized as follows. In Sect. 11.2, we give a brief summary of preconditioning for iterative linear system solvers. In Sect. 11.3, we first describe the model and solution method, and then present our preconditioning approach and its implementation. In Sect. 11.4, we analyze the computational results. Section 11.5 provides some discussion, concluding remarks, and topics for future research.

11.2 Preconditioning We consider a nondegenerate linear system Ax = b with the solution x ∗ = A−1 b. A usual stopping rule for an iterative method is r /r0  ≤ η,

(11.1)

where r = b − Ax is the residual, r0 is the residual at the initial iterate, and η is a given tolerance. Norms of the error e = x − x ∗ and residual r = −Ae are related by A−1 r  r  e ≤ = κ(A) , e0  A−1 r0  r0 

(11.2)

where κ(A) ≡ A−1 A is the condition number. In particular, using the spectral norm A2 = |λmax (A T A)|, we have  κ2 (A) =

|λmax (A T A)| , |λmin (A T A)|

(11.3)

where λmin and λmax are the minimal and maximal eigenvalues of the matrix A T A. The idea of preconditioning the linear system of equations Ax = b consists in replacing it by another linear system AM y = b, which is easier to solve, then setting x = M y. The matrix M is called the (right) preconditioner. To be efficient, M must be relatively inexpensive to compute and approximate A−1 . As we see from

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inequality (11.2), if AM has a smaller condition number than A, then the relative residual of the preconditioned system is a better indicator of the relative error than the relative residual of the original system. We use the GMRES (Generalized Minimum RESidual) method, which is a Krylov subspace method for nonsymmetric systems (for details, see, e.g., [5]). The kth (k ≥ 1) iteration of GMRES is the solution to the least squares problem b − Ax2 → min , x∈x0 +Kk

where x0 is the initial iterate and Kk = span {r0 , Ar0 , . . . , Ak−1 r0 } is the Krylov subspace. Since x ∈ x0 + Kk , we have xk = x0 +

k−1 

γl Al r0 ,

l=0

where γl are coefficients of the linear combination. Then the residual rk = b − Axk is determined by k  γl−1 Al r0 ≡ p(A)r0 , rk = r0 − l=1

where p(A) is a polynomial of degree k such that p(0) = 1. Hence rk 2 ≤  p(A)r0 2 . Let p(A) = (I − A)k and assume I − A2 = ρ < 1, then rk 2 ≤ ρ k r0 2 ,

(11.4)

and it takes k ≥ log η/ log ρ iterations to meet the stopping criterion (11.1). If, however, I − A2 > 1, we can try to find M such that I − AM2 < 1 and apply GMRES to the preconditioned system AM y = b.

11.3 Model and Method First, we describe the structure of the global general equilibrium model we employ in this paper. The iPETS model has three types of agents: consumers, producers, and government. Consumers maximize the lifetime utility function over an infinite time horizon under budget constraints taking prices as given. As is typical of dynamic

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general equilibrium models, the infinite time horizon is replaced by a sufficiently large finite time interval assuming some sort of boundary conditions at the terminal moment T . The first-order optimality condition gives a discrete boundary value problem. Producers choose the level of production to maximize profits given technology (i.e., production functions) and prices (this is equivalent to cost minimization given the level of production) at each time step. Prices are determined by the market clearing conditions for production factors, intermediate and final goods. The government sector in each region purchases intermediate goods to produce a final good. The government sector is “neutral” in the sense that the real value of government purchases is constant over time, and the government budget constraint is satisfied in each period through lump-sum transfers to households. Moreover, the government collects taxes at rates determined from the benchmark data. The first-order optimality conditions for agents and market clearance conditions for goods give the system of nonlinear equations that describe the intertemporal equilibrium F(K , P) = 0, (11.5) where K stands for intertemporal variables, such as capital, consumption, investment, etc. for all regions r = 1, N R and all time steps t = 0, T , and P stands for intratemporal variables, such as prices and input–output ratios of final and intermediate goods, international trade, etc., for all goods, regions, and time steps. Use of constant elasticity of substitution (CES) utility functions and nested CES production functions ensures that there is no corner solution. Hence F(K , P) is a smooth function in a neighborhood of the equilibrium. System (11.5) is solved by the following iterative method of the Gauss–Seidel type (for details, see [7]). Let the ith iterate of prices P i and capital K i be determined. First, we solve the system F(K i , P) = 0 with respect to prices, and thus obtain the next iterate P i+1 . This part of the algorithm is implemented as the inner loop. Next, the new iterate of capital K i+1 is determined from the system F(K , P i+1 ) = 0 by updating K i using the successive overrelaxation (SOR) method. Updating consists of replacing values of the intertemporal variables K i , namely, aggregate capital stocks and current expectations for the return on investment, by a convex combination of the current value and a new value which is available after solving all time periods at each iteration. Separate weights (relaxation factors) are applied to determine the convex combination for capital stocks and expected returns.1 This part of the algorithm is implemented as the outer loop (Fig. 11.1). Equations for the prices Pti at different time steps t = 0, T can be solved independently: (11.6) Ft (K i , Pt ) = 0.

1 In

practice, one can choose the relaxation factors small enough to ensure convergence, but a large number of iterations can be required to achieve prescribed tolerance level.

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For better convergence, the system of equations at time t is broken down into two subsystems and solved with nested A-loop and B-loop, similarly to the above outer and inner loops. The A-loop solves the system for the factor prices (prices of capital, labor, and land), and a lump-sum transfer to (from) households to balance the government’s budget constraint for each region and time step. This system of equations has a relatively small dimension, and therefore Newton’s method is applied. Inside the A-loop, at each Newton’s step, the B-loop applies to a much larger, separable system of prices for intermediate goods. This system of nonlinear equations equates vector of prices for intermediate goods with marginal cost of production that is derived from the dual cost function for each industry which produces intermediate goods. In general, these cost functions depend on all intermediate goods prices. The larger dimension of the B-loop is solved using the Krylov subspace method instead of LU factorization of the Jacobian, which is slower. For this, we employ the Newton Iterative Solver—NITSOL, which implements a version of the Newton–Krylov method with backtracking [11]. NITSOL supports an interface for right preconditioning of the linearized system. For brevity, we denote the system of equations solved in the B-loop as Fi,t, j (u) = 0,

(11.7)

where i is the outer loop iteration number, t is the time step, j is the Newton iteration number of the A-loop, and u = Pt . At the kth Newton–Krylov iteration of the B-loop, one solves the linearized system Fi,t, j (u k )s = −Fi,t, j (u k ),

(11.8)

where u k is the kth iterate of the solution and s is the step to the next iterate u k+1 = u k + s. If the Jacobian matrix A = Fi,t, j (u k ) is close to its block diagonal part over −1 regions diag(A1 , . . . , A N R ), then the inverse matrix M = diag(A−1 1 , . . . , A N R ) can be used as a preconditioner. To make preconditioning effective, one should not calculate the preconditioner too often. In the following, we describe our implementation based on using the preconditioner calculated at (t, j, k) = (1, 1, 1) for all (t, j, k). NITSOL has two parameters jacv and ipar (Fig. 11.1). The first one is the name of the user-supplied subroutine that calculates the Jacobian-vector product or preconditioner-vector product. The second one ipar passes integer parameters to the solver. We use ipar for checking whether it is necessary to recalculate the preconditioner or not. The parameter ipar is a global vector that stores the current iteration numbers (t, j, k) for each call of the NITSOL.2 Subroutine jacv contains two options depending on the value of the parameter ijob (Fig. 11.1). If ijob is equal to one, we calculate the preconditioner-vector product. If all components of ipar are equal to one, we update the preconditioner M and save it to the global variable Msvr . Otherwise, we use the existing pre2 Strictly speaking, the last component is not the iteration number k but rather a flag, which takes two values: zero if k = 1 and one if k > 1.

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Fig. 11.1 Block Jacobi preconditioning with NITSOL

conditioner Msvr . After calculating the matrix-vector product x = M y, we save statistics of function evaluations and matrix-vector products. This step is necessary because NITSOL can save statistics on calculations inside the solver but not in the user-defined subroutine. There are two ways of calculating the Jacobian matrix Fi,t, j (u k ) for the problem (11.8) in NITSOL. The first one is to compute the entire Jacobian matrix A = Fi,t, j (u k ) and the second one is to implement only the matrix-vector product Ax. In the first case, one needs n function evaluations to obtain the Jacobian matrix by the difference approximation (assuming that Fi,t, j (u k ) has been already evaluated). Then the Jacobian matrix is stored and used for each matrix-vector product. In the second case, one function evaluation is required for calculating each matrix-vector product of finite-difference approximation

Fi,t, j (u k )s =

 1 Fi,t, j (u k + δs) − Fi,t, j (u k ) , δ

(11.9)

where δ is the difference step. This is one of the built-in options in NITSOL. −1 For calculating the preconditioner M = diag (A−1 1 , . . . , A N R ), we first calculate the Jacobian. Then we loop over regions, extract r th diagonal block Ar of the Jacobian, find the inverse matrix Ar−1 , and stack the blocks together. The LAPACK subroutine DGETRF computes the LU factorization using partial pivoting with row interchanges and subroutine DGETRI computes the inverse of the LU factorization.

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11.4 Results and Discussion The code for the iPETS model is written in Fortran and compiled using Intel Fortran compiler. We calibrate the iPETS model to reproduce major outcomes for shared socioeconomic pathways (SSPs) as implemented by [12]. As the initial guess for prices P 0 we take the exact solution for SSPs but perturb the exact solution for capital by taking the balanced growth path as the initial guess K 0 . There are N R = 9 regions in the model and T = 105 time steps, 1 year each. Our goal is to compare the version with and without preconditioning in the B-loop. Therefore, we limit the maximum number of iterations for the outer loop to MAXIT = 10 and relax the A-loop accuracy to acc_a = 0.1, which ensures convergence of the A-loop in one iteration. The Jacobian matrix A = Fi,t, j (u k ) for (i, t, j, k) = (1, 1, 1, 1) is presented in Fig. 11.2. Each small square in the figure corresponds to a value of the matrix. The color gradient of the values in the matrix is shown on the right part of the figure. Blocks of the matrix are indicated by the green lines. The size of each block is 16 × 16. The non-diagonal blocks correspond to the trade between regions in the model. As one can see, outside its block diagonal part the Jacobian matrix is sparse. Therefore, use of the block Jacobi preconditioner M makes the system easier to solve. Indeed, we have I − A2 = 490 for the initial system and I − AM2 = 0.87 for the preconditioned system. According to formula (11.4), this ensures convergence of GMRES for the preconditioned system. The block Jacobi preconditioning not only decreases the number of the Krylov iterations for the linear system (11.8) with the stopping rule (11.1) but also improves convergence to the solution. Indeed, the spectrum of the symmetric matrix A T A is spread out over a wide interval (Fig. 11.3), and hence the condition number (11.3)

Fig. 11.2 Colormap of the Jacobian matrix (u ) of the A = Fi,t, j k nonlinear system (11.7) computed with (i, t, j, k) = (1, 1, 1, 1)

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Fig. 11.3 Eigenvalues of the matrices A T A and (AM)T (AM)

Table 11.1 Numbers of function evaluations, preconditioner-vector products, linear and nonlinear iterations, and speedup in solving the nonlinear system (11.6) with or without computing the Jacobian and with or without the (right) preconditioner Mt, j,k

Function evals M y matvecs Linear iters Nonlinear iters Speedup

Store Jacobian w/o precond

Precond Mt, j,k

Jacobian-free w/o precond

Precond Mt, j,k

21615 0 150 4 1

624 7 3 3 26.4

155 0 150 4 59.8

908 7 3 3 19.2

for the initial linear system is much greater than one, κ2 (A) = 4 · 105 . For the preconditioned system, the spectrum of (AM)T AM is clustered near one (Fig. 11.3), and the condition number becomes five orders of magnitude lower, κ2 (AM) = 2.4. The Newton–Krylov methods do not require computing and storing the Jacobian but rather computing Jacobian-vector products. However, the Jacobian matrix is still required for preconditioning. Therefore, first we compare the performance of the two methods for the B-loop: storing the Jacobian matrix computed by the finite-difference approximation versus computing only Jacobian-vector products by formula (11.9). Table 11.1 shows various statistics for solving the nonlinear system (11.6) with or without computing the Jacobian and with or without the preconditioner Mt, j,k . We present the numbers for the first outer loop iteration i = 1 and first A-loop iteration j = 1. Since system (11.6) is solved independently for different time steps t = 0, T , we present the average numbers over t. As we see, the Jacobian-free method clearly outperforms the method with storing the Jacobian both in terms of function evaluations (11.7) and speedup. Use of the preconditioner does not change the number of nonlinear iterations much but decreases the number of linear iterations 50 times. Preconditioning requires less function evaluations when we store the Jacobian than in the Jacobian-free case. But the Jacobian-free method gives 60 times speedup compared to the method when we store the Jacobian. We do not obtain a greater speedup when we use the Jacobian-free method with preconditioning. The reason is that the preconditioner is computed too often. To improve the speedup with preconditioning, we compute it only once for (t, j, k) = (1, 1, 1), and then reuse M1,1,1 for all other (t, j, k). Table 11.2 shows

11 Block Jacobi Preconditioning for Solving Dynamic General Equilibrium Models

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Table 11.2 Numbers of function evaluations, preconditioner-vector products, linear and nonlinear iterations, and speedup in solving the nonlinear system (11.6) without computing the Jacobian matrix (1) without a preconditioner, (2) with the preconditioner Mt, j,k , and (3) with the preconditioner M1,1,1 w/o precond Precond Mt, j,k Precond M1,1,1 Function evals M y matvecs Linear iters Nonlinear iters Speedup

155 0 150 4 1

908 7 3 3 0.3

16 14 10 4 1.6

statistics for different preconditioning methods in the Jacobian-free model runs. As we see, the matrix Mt, j,k is closer to the Jacobian matrix At, j,k than M1,1,1 . Therefore, the preconditioner Mt, j,k takes 3 linear iterations and preconditioner M1,1,1 takes 10 linear iterations (which is still 15 times less than without a preconditioner). Use of the preconditioner Mt, j,k also requires two times less preconditioner-vector products than use of the preconditioner M1,1,1 . However, use of the preconditioner Mt, j,k requires much more function evaluations than use of the preconditioner M1,1,1 . Hence, the computing time with the preconditioner M1,1,1 is less than with the preconditioner Mt, j,k . Compared to the model that runs without a preconditioner, use of the preconditioner M1,1,1 gives 1.6 times speedup and produces results within the same tolerance.

11.5 Conclusion To summarize, we have presented and analyzed the block Jacobi preconditioning with respect to regions for solving dynamic multi-region general equilibrium models. We demonstrated the use of the preconditioner in the example of the iPETS model. By analyzing the spectrum of the Jacobian, we showed that the block Jacobi preconditioning improves the convergence of GMRES method to the solution. We demonstrated that the Jacobian-free method outperforms the method with storing the Jacobian both in terms of function evaluations and speedup. In the model that runs with the Jacobian-free method, the block Jacobi preconditioning gives 1.6 times speedup and reduces the number of function evaluations by an order of magnitude. As future work, we plan to apply our preconditioning approach to the iPETS model with more complex economical scenarios and higher accuracy. The advantage of our approach is that preconditioning with respect to regions can be combined with the parallelization over time steps [6, 7]. Acknowledgements This research was carried out using the equipment of the shared research facilities of high-performance computing (HPC) resources at Lomonosov Moscow State University.

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References 1. Fair, R., Taylor, J.: Solution and maximum likelihood estimation of dynamic nonlinear rational expectations models. Econometrica 51, 1169–1185 (1983) 2. Faust, J., Tryon, R.: Block distributed methods for solving multi-country econometric models. In: Gilli, M. (ed.) Computational Economic Systems: Models, Methods & Econometrics, pp. 229–242. Springer, Dordrecht (1996) 3. Gilli, M., Pauletto, G.: Krylov methods for solving models with forward-looking variables. J. Econ. Dyn. Control. 22, 1275–1289 (1998) 4. Ha, P.V., Kompas, T.: Solving intertemporal CGE models in parallel using a singly bordered block diagonal ordering technique. Econ. Model. 52, 3–12 (2016) 5. Kelley, C.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, PA (1995) 6. Melnikov, N., Gruzdev, A., Dalton, M., O’Neill, B.C.: Parallel algorithm for solving largescale dynamic general equilibrium models. In: Proceedings of the 1st Russian Conference on Supercomputing—RuSCDays 2015, Moscow, pp. 84–95 (2015) 7. Melnikov, N., Gruzdev, A., Dalton, M., Weitzel, M., O’Neill, B.C.: Parallel extended path method for solving perfect foresight models. Computational Economics (2019). Submitted 8. Mrkaic, M., Pauletto, G.: Preconditioning in economic stochastic growth models. IFAC Proc. Vol. 34(20), 349–353 (2001) 9. O’Neill, B.C., Dalton, M., Fuchs, R., Jiang, L., Pachauri, S., Zigova, K.: Global demographic trends and future carbon emissions. Proc. U.S.A. Natl. Acad. Sci. 107, 17521–17526 (2010) 10. Pauletto, G., Gilli, M.: Parallel Krylov methods for econometric model simulation. Comput. Econ. 16, 173–186 (2000) 11. Pernice, M., Walker, H.: NITSOL: A Newton iterative solver for nonlinear systems. SIAM J. Sci. Comput. 19, 302–318 (1998) 12. Ren, X., Weitzel, M., O’Neill, B.C., Lawrence, P., Meiyappan, P., Levis, S., Balistreri, E.J., Dalton, M.: Avoided economic impacts of climate change on agriculture: Integrating a land surface model (CLM) with a global economic model (iPETS). Clim. Change 146, 517–531 (2018) 13. Saad, Y., van der Vorst, H.A.: Iterative solution of linear systems in the 20th century. J. Comput. Appl. Math. 123(1), 1–33 (2000) 14. Wathen, A.J.: Preconditioning. Acta Numer. 24, 329–376 (2015)

Chapter 12

Approximate Feedback Minimum Principle for Suboptimal Processes in Non-smooth Optimal Control Problems V. A. Dykhta

Abstract We consider a non-smooth optimal control problem with Lipschitz dynamics with respect to state variables and a terminal functional, which is defined by a semiconcave function (the difference between smooth and continuous convex functions). For suboptimal processes of this problem, the variational type necessary optimality condition is obtained. This condition, on one hand, is a generalization of the so-called Feedback Minimum Principle obtained by the author in previous publications, and on the other hand, it significantly strengthens ε-Maximum Principle for suboptimal processes obtained by I. Ekeland. An important feature of our result is that the obtained condition of suboptimality is formulated using a family of auxiliary problems of dynamic optimization, due to the multiplicity of solutions of the adjoint inclusion and the plurality of subgradients of the terminal function.

12.1 Introduction Consider the following optimal control problem (P): x˙ = f (t, x, u), x(t0 ) = x0 , u(t) ∈ U, t ∈ T = [t0 , t1 ], J (σ ) = l(x(t1 )) → inf . Here, σ denotes pairs (x, u) ∈ AC(T, R n ) × L ∞ (T, U ), U is a compact set in R m , vector function f (t, x, u) is continuous, and Lipschitz continuous with respect to (w.r.t.) x. Furthermore, we suppose f satisfying the sublinearity condition, so the set of trajectories is precompact in space C. The terminal function l can be written as follows: l(x) = l1 (x) − l2 (x),

V. A. Dykhta (B) Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk,, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_12

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where function l1 is smooth , and l2 is convex and continuous on R n (this means that l is semiconcave). In [1, 2], the necessary optimality conditions for an admissible pair of functions σ¯ = (x, ¯ u), ¯ using feedback controls of the functional descent, were obtained. These conditions, called the Feedback Minimum Principle, are ideologically related to the most important concepts of the Krasovskii control theory school [3–5] in the Ural region (in particular, to the concept of u-stable functions). Notice that the Feedback Minimum Principle turned out to be a rather constructive and effective method for solving optimal control problems of various types. However, the scope of applicability of this result is not wide enough; this is why we appeal to its extension to suboptimal processes. Thus, we cover problems without the existence of optimal control in the class U = L ∞ (T, U ) and make progress to generalize the Feedback Minimum Principle to problems with terminal constraints. Actually, if such constraints are “removed” by a penalty method, then in the approximating problems, one needs the suboptimality conditions instead of the exact optimality conditions.

12.2 ε-Feedback Minimum Principle Let  be the set of all admissible processes in problem (P). The process σ¯ = (x, ¯ u) ¯ is called ε-optimal, if J (σ¯ ) ≤ inf J () + ε (here, ε > 0). Note that ε-optimal processes exist for any ε > 0. For the first time, the necessary ε-optimality conditions were proposed by Ekeland [6] in the form of weakened Pontryagin maximum principle (ε-maximum principle). In our paper, the result of I. Ekeland (and of his followers) is strengthened in several directions, for problem (P). The semiconcavity of the function l(x) implies that, for any subgradient s ∈ ¯ 1 )), the smooth function ∂l2 (x(t Ls (x) = l1 (x) − s, x is an upper support function for l at the point x(t ¯ 1 ). More precisely, it means that, for all x ∈ R n , the following inequality holds: ¯ 1 )). l(x) − l(x(t ¯ 1 )) ≤ Ls (x) − Ls (x(t Thus, ε-optimality of the process σ¯ in problem (P) implies its ε-optimality for all ¯ 1 )): comparison problems from the family {C P s }, s ∈ ∂l2 (x(t I s (σ ) := Ls (x(t1 )) → inf, σ ∈ .

(C P s )

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Introduce the function H (t, x, ψ, u) = ψ · f (t, x, u), the Clarke adjoint inclusion [7] ˙ ¯ ψ(t), u(t)), ¯ − ψ(t) ∈ ∂x H (t, x(t), ¯ 1 )) − ∂l2 (x(t ¯ 1 )), ψ(t1 ) ∈ l1x (x(t and the set of its solutions . Note that the transversality condition (for ψ(t1 )) corresponds to the minimum condition of Pontryagin function H w.r.t. controls u ∈ U instead of the maximum condition. Given any ψ ∈ , let ¯ p ψ (t, x) = ψ(t) + l1x (x) − l1x (x(t)),   ψ Uψ,ε (t, x) = u ∈ U | H (t, x, p (t, x), u) ≤ h(t, x, p ψ (t, x)) + ε ,

(12.1)

where h(t, x, ψ) = min{H (t, x, ψ, u) | u ∈ U }. For any ψ ∈ , denote by Vψ,ε the set of all selectors of set-valued map (12.1) (we call them “feedback controls”). By Xψ,ε we denote the set of solutions for the original dynamic systems driven by controls v(t, x) from Vψ,ε (the union of constructive ¯ 1 )) will motions and Carathéodory solutions). All sets corresponding to s ∈ ∂l2 (x(t be marked with a superscript s. Let us make some necessary explanations of the introduced notation. Due to the ¯ 1 )) non-smoothness of the original control system, any fixed subgradient s ∈ ∂l2 (x(t defines (in general) some set of cotrajectories  s for comparison problem (C P s ) (and, of course, for problem (P)). Any cotrajectory ψ ∈  s generates map (12.1) and other corresponding sets, in particular, a set of comparison trajectories Xψ,ε . The first statement of the Approximate Feedback Principle is given by Condition A If process σ¯ = (x, ¯ u) ¯ is ε-optimal in problem (P) then ∀s ∈ ¯ 1 )) ∂l2 (x(t  Ls (x(t ¯ 1 )) ≤ Ls (x(t1 )) + ε ∀ x ∈ {Xψ,ε | ψ ∈  s }. This necessary optimality condition means that, in the sets of comparison trajectories there is no trajectory which violates the ε-optimality of x¯ (or pair σ¯ ). Note that such a condition does not appear in the well-known versions of the Ekeland ε- maximum principle. ¯ on T . We call σ¯ an ε-extremal of problem (P), if ∃ ψ ∈  : u(t) ¯ ∈ Uψ,ε (t, x(t)) (The case ε = 0 corresponds to the Clarke extremal.) By εs we denote the set of cotrajectories from  s that ensure ε-extremality of σ¯ . The final variational statement is as follows: ¯ 1 )) : εs = ∅ Condition B If σ¯ is an ε-extremal of problem (P), then ∀ s ∈ ∂l2 (x(t trajectory x¯ is ε-optimal in the following problem: Ls (x(t1 )) → inf, x ∈

 {Xψ,ε | ψ ∈ εs }.

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Note that the ε-feedback principle for non-smooth problems essentially uses the multiplicity of trajectories and the sets of comparison feedback controls.

12.3 Examples Example 1 x˙ = g(x)u, x(0) = 0, |u| ≤ 1, J (x, u) = − cos x(1) − |x(1)| → min, where x and u are scalars, g(x) is a Lipschitz continuous functions with the property g(0) = 0. For the process σ¯ = (x¯ = u¯ ≡ 0), we have  = {ψ ≡ −s | |s| ≤ 1},

p ψ (x) = −s + sin x ∀ψ.

Clarke extremal condition [7] for σ¯ takes the following form: ∃ ψ = −s :

min (−sg(0)u) = 0.

|u|≤1

Obviously, this condition holds only for s = 0, i.e., for ψ0 = 0. Therefore, σ¯ is a Clarke extremal. However, due to Condition A with ε = 0, a similar minimum condition must hold ∀ s, |s| ≤ 1, but it is impossible. So, the pair σ¯ is not optimal. Now, we use Condition A for feedback improvement of σ¯ . According to formula (12.1) with ε = 0, the controls of possible descent from Uψ,0 (t, x) can be found from the problem u(−s + sin x) → min, |u| ≤ 1 (for all |s| ≤ 1). It is convenient to take s = 1. It gives the feedback control v(x) = u ∗ ≡ 1. Suppose that the corresponding trajectory x∗ (t) satisfies the inequality L1 (x∗ (1)) < ¯ = −1. Then, the pair (x∗ , u ∗ ) improves the process σ¯ . L1 (x(1)) Notice that for discarding σ¯ and its improvement, we used trajectories that were not included in the extremality conditions. Example 2 Consider a modified version of an example from the fundamental article [8, p. 209]:   x˙1 = u u 2 − 41 , x1 (0) = 0, x˙2 = |x1 | − x12 , x2 (0) = 0, 1 |u| ≤ 1, J (x, u) = x2 (1) + (x12 − u 2 )dt → inf . 0

It is clear that this problem reduces to a problem of type (P) with a linear objective function, but we avoid this formalism.

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Note that for all trajectories |x˙1 (t)| ≤ 34 for all t, and, therefore, |x1 (t)| ≤ 1. Hence, x˙2 (t) ≥ 0 ∀ t and x2 (1) ≥ 0. Moreover, the equality x2 (1) = 0 is possible only for x1 (t) ≡ 0 and for three controls u = 0, 21 , − 21 (of course, then x2 (t) ≡ 0). Starting from these controls, for which the trajectory x(t) ¯ ≡ 0 and the adjoint inclusion coincide, we begin to solve the problem. The set of cotrajectories for these processes has the following description:  = {ψ(t) = (ψ1 (t), ψ2 (t)) | ψ1 (t) = c(t − 1), ψ2 ≡ 1, |c| ≤ 1} (it is parameterized by c ∈ R). It is easy to verify that, for any ψ ∈ , the weakened minimum condition for the function H w.r.t. control is not satisfied on the selected processes (there is a maximum on them). Thus, they are not ε-extremals, and the necessary Condition B does not make sense for them. We use Condition A to improve this triple of processes. In general, one can try any ψ ∈ , but we take the simplest of them, associated to c = 0. Then H = x12 − u 2 and map (12.1) with ε = 0 takes the form: Uψ,0 = {1, −1}. Elementary selectors v 1,2 = ±1 do not lead to the goal because the functional increases on the corresponding processes. But even with the constant map Uψ,0 one can construct its feedback selectors, and we choose  −1, x1 > 0, v(x1 ) = +1, x1 ≤ 0. (Of course, such a choice looks artificial, but it can be justified through the transformation of the functional using the Gabasov matrix function.) The discontinuous feedback control v(x1 ) generates a sequence of processes {x k , u k } ⊂  with corresponding state Euler polygons and the following piecewise-constant controls:

i i +1 , , i = 0, . . . , k − 1. t∈ k k

u (t) = (−1) k

k

for

For this sequence, J (x k , u k ) → −1

as

k → ∞.

It is clear that the sequence is minimizing (although, x2k (1) > 0 ∀ k). One can verify this by checking Condition B. Note that the optimal process in this example does not exist, and the sequence {x k , u k } in the limit gives a sliding mode that is optimal in a convexified problem. In [8], this example was considered with the terminal constraints x1 (1) = x2 (1) = 0, which are “almost” ignored in our consideration (we only added the term x2 (1) to J ). Despite this, the above-described sequence {x k , u k } is also minimizing for the problem [8], and the limiting sliding mode is optimal in its extension. This somewhat unusual situation is related to the violation of the conditions of normality

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(correctness) in the example from [8]: the trajectory of the optimal sliding mode cannot be approximated by trajectories of the original system, which exactly satisfy the terminal constraints. Conclusion The presented necessary suboptimality conditions with feedback controls of the descent of the cost functional provide a useful tool for the qualitative and numerical analysis of optimal control problems. At the same time, they harmoniously combine with such fundamental results of dynamic optimization as the Pontryagin maximum principle, dynamic programming, stability and stabilization of dynamic systems. Acknowledgements This work was partially supported by the Russian Foundation for Basic Research (Project No. 17-01-00733).

References 1. Dykhta, V.A.: Variational necessary optimality conditions with feedback descent controls for optimal control problems. Dokl. Math. 91(3), 394–396 (2015) 2. Dykhta, V.A.: Positional strengthenings of the maximum principle and sufficient optimality conditions. Proc. Steklov Inst. Math. 293(1), S43–S57 (2016) 3. Krasovskii, N.N., Subbotin, A.I.: Game-theoretical Control Problems. Springer, New York (1988) 4. Krasovskii, N.N.: Control of a Dynamical System. Nauka, Moscow (1985). [in Russian] 5. Subbotin, A.I.: Generalized Solutions of First-Order Partial Differential Equations: The Dynamical Optimization Perspective. Birkhäuser, Boston (1995) 6. Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. (N.S.). 1(3), 443–474 (1979) 7. Clarke, F.H.: Functional Analysis. Calculus of Variations and Optimal Control. Graduate Texts in Mathematics. Springer, London (2013) 8. Ioffe, A.D., Tikhomirov, V.M.: Extension of variational problems. Tr. Mosk. Mat. Obs. 18, 187–246 (1968)

Chapter 13

Solving the Inverse Heat Conduction Boundary Problem for Composite Materials V. P. Tanana and A. I. Sidikova

Abstract The paper deals with the problem of determining the boundary condition in the heat equation consisting of homogeneous parts with different thermal properties. As boundary conditions, the Dirichlet condition at the left end of the rod (at x = 0) corresponding to the heating of this end and the linear condition of the third kind at the right end (at x = 1) corresponding to the cooling when interacting with the environment are considered. In a point of discontinuity of heat transfer properties (at x = x0 ) conditions of continuity for temperature and heat flow are set. In the inverse problem, the boundary condition at the left is considered unknown over the entire infinite time interval. To find it, the value of the direct problem solution at the point of x0 , that is, the point of the rod division into two homogeneous sections, is specified. In this paper, an analytical study of the direct problem was carried out, which allowed us to apply the time Fourier transform to the inverse boundary value problem. The inverse heat conduction boundary problem was solved using the projection-regularization method and order-accurate error estimates of this solution were obtained.

13.1 Introduction Composite materials are widely adopted in various branches of modern technology. Further progress in the development of many areas of instrument engineering is largely associated with an increase in application of such materials, and when creating new aerospace and special equipment, their role becomes crucial. The development of instrument engineering follows the path of complication of the models being studied and problem statements [1]. Based on the model representations of mechanics, a composite material can be defined as a nonuniform medium, described by functions V. P. Tanana (B) · A. I. Sidikova South Ural State University, Chelyabinsk 454080, Russia e-mail: [email protected] A. I. Sidikova e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_13

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that are discontinuous in coordinates. The paper investigates and solves the inverse problem of determining the temperature at the left end of the rod for a system of two heat conduction equations with different thermal diffusivity coefficients. Since the solution of such problems requires high accuracy, it is necessary to obtain guaranteed estimates, which significantly increase the reliability of the numerical results. Questions related to error estimates remain insufficiently studied. In this regard, the paper presents an analytical study of the direct problem, which allowed us to apply the time Fourier transform to the inverse boundary value problem, after which the method of projection regularization [2] was used to obtain an approximate solution and an order-accurate error estimate of this solution. It should be noted that a fairly wide class of inverse boundary value problems is presented in [3–6].

13.2 Direct Problem Statement and Study Let a thermal process be described by a system of equations ∂u 1 (x, t)/∂t = a12 ∂ 2 u 1 (x, t)/∂ x 2 , x ∈ (0, x0 ], t ∈ (0, ∞),

(13.1)

∂u 2 (x, t)/∂t = a22 ∂ 2 u 2 (x, t)/∂ x 2 , x ∈ [x0 , 1), t ∈ (0, ∞),

(13.2)

u 1 (x, 0) = 0, x ∈ [0; x0 ];

u 2 (x, 0) = 0, x ∈ [x0 ; 1],

(13.3)

u 1 (0, t) = q(t), t > 0,

(13.4)

∂u 2 (1, t)/∂ x + ku 2 (1, t) = 0, k > 0, t > 0,

(13.5)

u 1 (x0 , t) = u 2 (x0 , t), a1 ∂u 1 (x0 , t)/∂ x = a2 ∂u 2 (x0 , t)/∂ x, t > 0, (13.6) where q(t) ∈ C 2 [0, +∞), q(0) = q  (0) = 0, and the number t0 > 0 exists, such that, for any t ≥ t0 q(t) = 0. In the direct problem (13.1)–(13.6), we need to find the following function:  u(x, t) =

u 1 (x, t), u 2 (x, t),

0 ≤ x ≤ x0 , t ≥ 0, x0 < x ≤ 1, t ≥ 0.

(13.7)

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Let t1 = t0 + 2. Having done the substitution 

 h 1 (x) = 1 −

u 1 (x, t) = v1 (x, t) + q(t)h 1 (x), x ∈ [0, x0 ], u 2 (x, t) = v2 (x, t) + q(t)h 2 (x), x ∈ [x0 , 1],

   ka2 x ka1 (x − x0 ) + a2 kx0 , h 2 (x) = 1 − , a1 + ka1 (1 − x0 ) + ka2 x0 a1 + ka1 (1 − x0 ) + ka2 x0

we pass to the problem ∂v1 (x, t)/∂t = a12 ∂ 2 v1 (x, t)/∂ x 2 − q  (t)h 1 (x), x ∈ (0, x0 ], t ∈ (0, t1 ], ∂v2 (x, t)/∂t = a22 ∂ 2 v2 (x, t)/∂ x 2 − q  (t)h 2 (x), x ∈ [x0 , 1), t ∈ (0, t1 ], v1 (x, 0) = 0, x ∈ (0; x0 ), v2 (x, 0) = 0, x ∈ (x0 ; 1), v1 (0, t) = 0, ∂v2 (1, t)/∂ x + κv2 (1, t) = 0, κ > 0, t ∈ [0, t1 ], v1 (x0 , t) = v2 (x0 , t), a1 ∂v1 (x0 , t)/∂ x = a2 ∂v2 (x0 , t)/∂ x, t ∈ [0, t1 ],

(13.8)



h 1 (x), x ∈ [0, x0 ], h 1 (0) = 0, h 2 (1) + kh 2 (1) = 0, h(x) ∈ C[0, 1]. h 2 (x), x ∈ [x0 , 1], Solving the Sturm–Liouville problem

h(x) =

a12 v1 (x) + σ 2 v1 (x) = 0, a22 v2 (x) + σ 2 v2 (x) = 0, 

v1 (0) = 0, v2 (1) + κv2 (1) = 0, v1 (x0 ) = v2 (x0 ), a1 v1 (x0 ) = a2 v2 (x0 ), 

where v(x) =

v1 (x), x ∈ [0, x0 ], v2 (x), x ∈ [x0 , 1],

v(x) ∈ C[0, 1],

we will find the eigenvalues and the corresponding eigenfunctions of this problem:

ψn (x) =

⎧ ⎪ ⎪ ⎪ ⎨

σn x a1 sin σan 1x0 sin σn (1−x) ⎪ a2 ⎪ ⎪ ⎩ σn (1−x0 ) sin a2

sin

, + +

0 < x < x0 , σn κa2 σn κa2

cos σn (1−x) a2 0) cos σn (1−x a2

(13.9) , x0 < x < 1,

where σn are positive roots of a transcendental equation  tan

 σ x0 1 − x0 σ =− + . a2 a1 ka2

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Let us transform the previous equation    σ 1 − x0 x0 1 − x0 x0 σ+ σ = 0. sin + cos + a2 a1 a2 a1 k 2 a22 + σ 2 k 2 a22 + σ 2 

ka2

Denoting sin ξ = ka2 / k 2 a22 + σ 2 , cos ξ = σ/ k 2 a22 + σ 2 , we obtain  cos

  1 − x0 x0 + σ − ξ = 0. a2 a1

Since σ > 0 and tan σ < 0, according to the previous equation,  σn =

 π a1 a2 a1 a2 + ξn + πn 2 a1 (1 − x0 ) + a2 x0 a1 (1 − x0 ) + a2 x0

and we obtain that, for any n |ξn | ≤ ka2 /σn . Therefore, v(x, t) =

∞  t

n=1

gn (θ ) =

 2 gn (θ ) · e−λn (t−θ) dθ ψn (x),

0



sin(σn x0 /a1 ) −2a1 a2 q  (θ ) ka1 a2 sin(σn x0 /a1 ) . − 2 (1 − x0 )a1 + x0 a2 σn σn (a1 + ka1 (1 − x0 ) + ka2 x0 )

Using orthogonality of the system of eigenfunctions {ψn (x)}, defined by (13.9), we obtain the following lemma. Lemma 13.1 There exists a unique solution v(x, (13.8) that satis t)1,0of the problem   fies the condition v(x, t) ∈ C([0, 1] × [0, t ]) C ([0, x ) (x , 1)) × (0, t1 ]) 1 0 0  C 2,1 ((0, x0 ) (x0 , 1) × (0, t1 ]) as well as for any t ∈ [0, t1 ] v(x, t) H 2 [0, 1]. For continuation of the solution v(x, t) with respect to t on the half-line [t0 , ∞), we consider the problem ∂v1 (x, t)/∂t = a12 ∂ 2 v1 (x, t)/∂ x 2 , x ∈ (0, x0 ), t ≥ t0 , ∂v2 (x, t)/∂t = a22 ∂ 2 v2 (x, t)/∂ x 2 , x ∈ (x0 , 1), t ≥ t0 , v(x, t0 ) = v0 (x), x ∈ [0; 1], v1 (0, t) = 0, ∂v2 (1, t)/∂ x + kv2 (1, t) = 0, k > 0, t ≥ t0 , v1 (x0 , t) = v2 (x0 , t), a1 ∂v1 (x0 , t)/∂ x = a2 ∂v2 (x0 , t)/∂ x, t ≥ t0 . The solution to this problem takes a form as given below:

13 Solving the Inverse Heat Conduction Boundary Problem …

v(x, t) =

∞

137

vn e−λn (t−t0 ) ψn (x), 2

n=1

2a1 a2 sin2 (σn x0 /a1 ) vn = (1 − x0 )a1 + x0 a2

x0

dx v0 (x) · ψn (x) + a1

0

1 x0

dx . v0 (x) · ψn (x) a2

Lemma 13.1 implies that v0 (x) = v(x, t0 ) ∈ H 2 [0, 1] (definition of the space H 2 [0, 1] is given, for example, in [7]), v0 (0) = 0, v0 (1) + kv0 (1) = 0. Since 2 2 2 e−σn (t−t0 ) = e−σn · e−σn (t−t0 −1) , we obtain the existence of the number d1 > 0 such that ∀t ≥ t0 + 2 2 sup{|v(x, t)|, |vx (x, t)|} ≤ d1 e−σn (t−t0 −1) and ∀ε > 0 ∃d2 (ε) ∀[ε, x0 − ε]



[x0 + ε, 1 − ε] ∀t ≥ t0 + 2

vxx (x, t) ≤ d2 (ε) · e−(t−t0 −1) . Then, it follows from the theorem proven in [8], and Lemma 13.1. Theorem 13.1 Let (t) ∈ C(−∞, ∞) and be bounded on this line. Then, the following relations are valid: ∞

u x (x, t)(t)dt

−∞

∞

u x x (x, t)(t)dt

−∞

∂ = ∂x

∞

∂2 = 2 ∂x

u(x, t)(t)dt ,

−∞

∞

u(x, t)(t)dt .

−∞

13.3 Statement of the Inverse Boundary Value Problem and Its Reduction to the Problem of Calculating the Values of an Unbounded Operator Let Mr ⊂ L 2 [0; ∞),   +∞ +∞ 2  2 2 Mr = q(t) : q(t) ∈ L 2 [0; ∞), |q(t)| dt + |q (t)| dt ≤ r , 0

where r is a known positive number.

0

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Suppose that in the condition (13.4), the function q(t) is unknown and the function f (t) is given instead, which is defined by the formula u(x0 , t) = f (t).

(13.10)

It is required, by using f (t), to define the function q(t) such that, when substituting it in the condition (13.4), the solution u(x, t) to the problem (13.1)–(13.6) satisfies the relation (13.10). Suppose that, at f (t) = f 0 (t) ∈ C[0, ∞), there exists the solution q0 (t) to the inverse problem (13.1)–(13.3), (13.5), (13.6), (13.10), which satisfies the condition (13.7), and when the function q0 (t) is substituted into the condition (13.4) of the direct problem (13.1)–(13.6), the solution to this problem will satisfy the condition (13.10), but, instead of the function f 0 (t), we are given some approximation f δ (t) ∈ C[0, ∞) and the error level δ > 0 such that

f δ (t) − f 0 (t) L 2 [0,∞) ≤ δ.

(13.11)

It is required, using f δ , δ, and Mr , to determine the approximate solution qδ to the problem (13.1)–(13.3), (13.5), (13.6), (13.10) and to estimate the value of qδ − q0 L 2 [0,∞) .  To solve this problem, let us introduce an operator F, mapping L 1 [0; ∞) L 2 [0; ∞) into L 2 (−∞; ∞) and defined by the formula 1 F[q(t)] = √ 2π

∞

q(t)e−itτ dt, |τ | ≥ 0, q(t) ∈ L 2 [0; ∞).

0

The previous formula implies that q(t) = 0 at t ≤ 0. From Theorem 13.1, it follows that the F transform can be applied to the solution for the problem (13.1)–(13.3), (13.5), (13.6), (13.10). iτ uˆ 1 (x, τ ) = a12 ∂ 2 uˆ 1 (x, τ )/∂ x 2 , x ∈ (0, x0 ), |τ | ≥ 0, iτ uˆ 2 (x, τ ) = a22 ∂ uˆ 2 (x, τ )/∂ x 2 , x ∈ (x0 , 1), |τ | ≥ 0,

(13.12) uˆ 1 (x0 , τ ) = fˆ(τ ), ∂ uˆ 2 (1, τ )/∂ x + k uˆ 2 (1, τ ) = 0, |τ | ≥ 0, uˆ 1 (x0 , τ ) = uˆ 2 (x0 , τ ), a1 ∂ uˆ1 (x0 , τ )/∂ x = a2 ∂ uˆ2 (x0 , τ )/∂ x, |τ | ≥ 0, where uˆ 1 (x, τ ) = F[u 1 (x, t)], uˆ 2 (x, τ ) = F[u 2 (x, t)],F is the Fourier transform with respect to t. √ Let β(τ ) be defined by the formula sinh β(τ ) = ka2 /μ0 τ − ik 2 a 2 . It follows from the previous equation that β(τ ) → 0

at

τ → ∞.

(13.13)

13 Solving the Inverse Heat Conduction Boundary Problem …

139

The problem (13.12) reduces to the problem of evaluating the unbounded operator T , T fˆ(τ ) = q(τ ˆ ),

(13.14)

√ √ cosh[μ0 (1 − x0 ) τ /a2 + μ0 x0 τ /a1 + β(τ )] , T (τ ) = √ cosh[μ0 (1 − x0 ) τ /a2 + β(τ )]

(13.15)

where domain of definition of T , D(T ) = { fˆ(τ ) : fˆ(τ ) ∈ L 2 [0, ∞), T fˆ(τ ) ∈ L 2 [0, ∞)}.

(13.16)

It follows from (13.15) and (13.16) that the operator T is linear, unbounded, closed, and injective. Let qˆ0 (τ ) = T fˆ0 (τ ), fˆ0 (τ ) = F[ f 0 (t)], fˆδ (τ ) = F[ f δ (t)]. From (13.11), it follows that fˆδ − fˆ0 ≤ δ. The set Mr under the F transform r ⊃ F[Mr ], defined by the formula passes into the set M   ∞ 2 2 2  ˆ ) : q(τ ˆ ) ∈ L 2 (−∞, ∞), (1 + τ )|q(τ ˆ )| r mdτ ≤ 2r . Mr = q(τ −∞

(13.17) r . From the fact that q0 (t) ∈ Mr , we obtain qˆ0 (τ ) ∈ M

13.4 Solution to the Problem (13.14)–(13.17) Lemma 13.2 For any ε > 0, there exists the number τε > 0 such that, for any τ ≥ τε ,  1−

√ √  √ x0 τ/2 ε | cosh(μ0 (1 − x0 ) τ /a2 + μ0 x0 τ /a1 + β(τ ))| a1 e ≤ √ 4 + 2ε | cosh(μ0 (1 − x0 ) τ /a2 + β(τ ))|  ≤ 1+

 √ x0 τ/2 ε e a1 . 4+ε

Proof Since β(τ ) = β1 (τ ) + iβ2 (τ), then from relations | cosh z 1 | =  cosh2 x1 − sin2 y1 and | cosh z 2 | = sinh2 x2 + cos2 y2 , z 1 = x1 + i y1 , z 2 = x2 + i y2 , we obtain √ √ | cosh[μ0 (1 − x0 ) τ /a2 + μ0 x0 τ /a1 + β(τ )]| √ | cosh[μ0 (1 − x0 ) τ /a2 + β(τ )]|

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√ √ cosh[(1 − x0 ) τ/2/a2 + x0 τ/2/a1 + β1 (τ )] , ≤ √ sinh[(1 − x0 ) τ/2/a2 + β1 (τ )] √ √ cosh[(1 − x0 ) τ/2/a2 + x0 τ/2/a1 + β1 (τ )] √ sinh[(1 − x0 ) τ/2/a2 + β1 (τ )] √

√

1 − e−2(1−x0 ) τ/2/a2 −2x0 τ/2/a1 −2β1 (τ ) √ ≤ −x √τ/2/a . 1 · (1 − e −2(1−x 0 ) τ/2/a2 −2β1 (τ ) ) e 0 Since from (13.13) and the above, it follows that, for any η > 0 there will be found 1 a22 ln2 such that, for any |τ | ≥ τ1 , τ1 > 0, τ1 ≥ τε , for example, τ1 = 2(1 − x0 )2 η √ √ τ/2/a2 −2x0 τ/2/a1 −2β1 (τ )

sup{e−2(1−x0 )

√

, e−2(1−x0 )

τ/2/a2 −2β1 (τ )

} < η,

then, for any τ, |τ | ≥ τ1 , the following inequality is valid: √ √ √ | cosh[μ0 (1 − x0 ) τ /a2 + μ0 x0 τ /a1 + β(τ )]| 1 + η x0 a τ/2 ≤ e 1 . (13.18) √ 1−η | cosh[μ0 (1 − x0 ) τ /a2 + β(τ )]| Similarly, it can be shown that √ √ √ | cosh[μ0 (1 − x0 ) τ /a2 + μ0 x0 τ /a1 + β(τ )]| 1 − η x0 a τ/2 e 1 . (13.19) ≥ √ 1+η | cosh[μ0 (1 − x0 ) τ /a2 + β(τ )]| It is easy to verify that, if we assume η = ε/(8 + 3ε), then the statement of the lemma will follow from (13.18) and (13.19).  To solve the problem (13.14), we use the projection-regularization method [9]. This method is based on a regularizing family of operators {Tα : α > τε }, which is defined by the formula  Tα fˆ(τ ) =

T fˆ(τ ); 0 ;

−α ≤ τ ≤ −τε , τε ≤ τ ≤ α, |τ | > α.

(13.20)

Let us define the approximate value qˆδα (τ ) of the problem (13.14) by the formula qˆδα (τ ) = Tα fˆδ (τ ); |τ | ≥ τε .

(13.21)

To select the regularization parameter α = α(δ, r ), let us consider in (13.21) the estimate

qˆδα (τ ) − qˆ0 (τ ) ≤ qˆδα (τ ) − qˆ0α (τ ) + qˆ0α (τ ) − qˆ0 (τ ) ,

(13.22)

13 Solving the Inverse Heat Conduction Boundary Problem …

141

where qˆ0α (τ ) = Tα fˆ0 (τ ). √ Since it follows from (13.20) and (13.21) that qˆδα (τ ) − qˆ0α (τ ) ≤ 2 Tα δ, we pass to the estimate of Tα . Lemma 13.3 Under the above conditions, the following relations are valid: 

  √  √ x0 α/2 x0 α/2 ε ε a1 1− ≤ Tα ≤ 1 + e e a1 , α ≥ τε . 4 + 2ε 4+ε

The statement of the lemma follows from the definition of the norm of an operator and Lemma 13.2. √ √ x0 α/2 x0 α/2 ε ε , b2 = 1 − , then b2 e a1 ≤ Tα ≤ b1 e a1 , α ≥ Let b1 = 1 + 4+ε 4 + 2ε τε , and  −α ω (α) = sup

∞ |qˆ0 (τ )| dτ +

2

2

−∞

 |qˆ0 (τ )|2 dτ : qˆ0 (τ ) ∈ Mˆ r .

(13.23)

α

Then, sup{ qˆ0α − qˆ0 : qˆ0 ∈ Mˆ r } = ω(α).

(13.24)

From (13.17), we obtain that, under the condition of qˆ0 (τ ) ∈ Mˆ r −τε ∞ 2 2 2 (1 + τ )|qˆ0 (τ )| + (1 + τ 2 )|qˆ02 (τ )|2 dτ ≤ 2r 2 . −∞

τε

From (13.23) and the previous relation, we obtain ω2 (α) = 2r 2 /(1 + α 2 ). Thus, from (13.22), (13.24), the previous equation, and Lemma 13.3, we get

qˆδα (τ ) − qˆ0 (τ ) ≤

√ √  x0 α/2 2r/ 1 + α 2 + b1 e a1 δ.

Let us select the regularization parameter α = α(δ) in the formula (13.21) from the condition √  x0 α/2 r/ 1 + α 2 = b1 e a1 δ.

(13.25)

Considering the above, we obtain √ 

qˆδα(δ) (τ ) − qˆ0 (τ ) ≤ 2 2r/ 1 + α(δ).

(13.26)

√ x0 α/2 √ Since the function 1 + α 2 e a1 strictly increases in α and varies from 1 to ∞, there exists the unique solution α(δ) to equation (13.25).

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To present the estimate (13.26) in the class of elementary functions, we consider two equations: e

√ x0 α/2 a1

= r/b1 δ, e

√ 2x0 α/2 a1

= r/b1 δ.

(13.27)

We denote the solutions to equation (13.27) as α 1 (δ) and α 2 (δ), relatively. Then, for sufficiently small values of δ, the following relations are valid: α 2 (δ) ≤ α(δ) ≤ α 1 (δ). It follows from (13.27) that α 1 (δ) = and from (13.28) α(δ) ∼ ln2 δ

(13.28)

2a12 2 r a12 2 r and , ln α (δ) = ln 2 b1 δ b1 δ x02 2x02 at δ → 0.

(13.29)

Let us denote the solution to the problem (13.14)–(13.17) by the formula qˆδ (τ ) = qˆδα(δ) (τ ).

(13.30)

Then, it follows from (13.26) and (13.30) that √

qˆδ (τ ) − qˆ0 (τ ) ≤ 2 2r/ 1 + α 21 (δ, r ). Finally, we define the solution qδ (t) to the inverse problem (13.1)–(13.3), (13.5), (13.6), (13.10) by the formula  qδ (t) =

ReF −1 [qˆδ (τ )] ; t ∈ [0, t0 ], 0 ; 0 < t , t > t0 ,

where F −1 is the inverse operator of F. Given the above, the following estimate holds for qδ (t): √

qδ (t) − q0 (t) ≤ 2 2r/ 1 + α 21 (δ, r ).

(13.31)

From (13.29) and (13.31), it follows that there exists the number d > 0 such that, for any δ ∈ (0, δ0 ), the formula is valid:

qδ (t) − q0 (t) ≤ d · r ln−2 δ.

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143

References 1. Alifanov, O.M.: Inverse Heat Transfer Problems. Machinostroyeniye Publ., Moscow (1988) 2. Tanana, V.P., Danilin, A.R.: The optimality of regularizing algorithms in the solution of illposed problems. Differ. Equat. 12, 1323–1326 (1976) 3. Kabanikhin, S.I.: Inverse and Ill-Posed Problems. Siberian Academic Press, Novosibirsk (2009) 4. Ivanov, V.K., Vasin, V.V., Tanana, V.P.: Theory of Linear Ill-Posed Problems and its Applications. VSP, Nitherlands (2002) 5. Tikhonov, A.N., Glasko, V.B.: Methods of determining the surfaces temperature of a body. USSR. Comput. Math. Math. Phys. 7, 910–914 (1967) 6. Lavrent’ev, M.M., Romanov, V.G., Shishatsky, S.P.: Ill-Posed Problems of Mathematical Physics and Analyses. Nauka Publ., Moscow (1980) 7. Osipov, Y.S., Vasiliev, F.P., Potapov, M.M.: Bases of the Dynamical Regularization Method. MSU Publ., Moscow (1999) 8. Zorich, V.A.: Mathematical Analysis. Nauka Publ., Moscow (1984) 9. Tanana, V.P., Sidikova, A.I.: Optimal Methods for Ill-Posed Problems With Applications to Heat Conduction. De Gruyter, Berlin (2018)

Chapter 14

On the Problems of Minmax–Maxmin Type Under Vector-Valued Criteria Yu. Komarov and Alexander B. Kurzhanski

Abstract This paper is devoted to methods of solving problems of dynamic optimization under multivalued criteria. Such problems require a full description of the related Pareto boundary for the set of all values of the vector criteria and also an investigation of the dynamics of such set. Of special interest are problems for systems that also include a bounded disturbance in the system equation. Hence, it appears useful to develop methods of calculating guaranteed estimates for possible realizations of related solution dynamics. Such estimates are introduced in this paper. Introduced here are the notions of vector values for minmax and maxmin with basic properties of such items. In the second part of this work, there given are some sufficient conditions for the fulfillment of an analogy of classical scalar inequalities that involve relations between minmax and maxmin. An illustrative example is worked out for a linear-quadratic type of vector-valued optimization with bounded disturbance in the system equations.

14.1 Essential Definitions 14.1.1 Pareto Order Consider a mapping F(x) : Rn → R p . Definition 14.1 Vector x ∈ R p is told to be dominated by the vector y ∈ R p in the sense of Pareto if  x = y, yi  xi , i = 1, . . . , p. Yu. Komarov (B) · A. B. Kurzhanski Lomonosov MSU, Moscow, Russia e-mail: [email protected] A. B. Kurzhanski e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_14

145

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We will denote this relation as y  x. Definition 14.2 The vector-valued minimum for the set of values of this mapping is further defined as MinF(X ) = { f ∗ ∈ F(X ) | x ∈ X : F(x)  f ∗ } . Definition 14.3 The vector-valued maximum for this set is here defined as   MaxF(X ) = f ∗ ∈ F(X ) | x ∈ X : f ∗  F(x) .

14.1.2 The Notion of Vector-Valued MinMax and MaxMin Consider the mapping F(u, v) : U × V → R p . We further assume that set F(U, V ) satisfies certain conditions that ensure for this set the existence of both Pareto boundaries, namely, for minimum and for maximum. An example of such conditions is given for lower and upper bounds of F(U, V ) under the given ordering as ∃M∗ , M ∗ : M∗  F(u, v)  M ∗

∀u ∈ U, v ∈ V.

Besides this the set is supposed to be closed. Definition 14.4 The vector-valued minmax for the elements of F(·, ·) over the set U × V will be taken as    Minu Maxv F(u, v) = Min MaxF(u, V ) . u∈U

Each maximum inside the brackets is taken over values of u ∈ U while the value v fixed. A similar definition for vector-valued maxmin will be as follows. Definition 14.5 The vector-valued maxmin for the elements of F(·, ·) over U × V will be taken as the mapping  Maxv Minu F(u, v) = Max

 v∈V

 MinF(U, v) .

14 On the Problems of Minmax–Maxmin Type Under Vector-Valued Criteria

147

Definition 14.6 We will assume that arbitrary sets A, B ⊂ R p satisfy conditions A  B,

(14.1)

if the next property is true: ∀b ∈ B \ A ⇒ ∃a ∈ A : a  b. Some trivial relations for the mappings of the above are as follows. Proposition 14.1 The next relations are true: • MinF(X, Y )  Min x Max y F(x, y); • MaxF(X, Y )  Max y Min x F(x, y). In addition to that, for validating the proofs of further propositions we also indicate the correctness of following relations. Proposition 14.2 The Pareto boundary satisfies the next properties: • ∀X ⊂ R p ⇒ Max{MinX } = MinX, Min{MaxX } = MaxX ; • ∀A ⊆ B ⊂ R p ⇒ MinB  Min A  Max A  MaxB; • if ∃Min(A + B), then Min(A + B) = Min(A + MinB). Remark 14.1 The last equation is described in detail in [1].

14.2 The Linear-Quadratic Problem of Control Under a Vector-Valued Criterion Consider a dynamic control system under disturbance of next type: x˙ = B(t)u + C(t)v, t ∈ [t0 , ϑ], x0 = x 0, u(t) ∈ P(t), v(t) ∈ R(t), with vector criterion ⎡

⎤ ⎡ ⎤ x, N1 x + u, M1 u − v, P1 v J1 (ϑ, x, u, v) ⎢ ⎥ ⎢ ⎥ .. .. J (ϑ, x, u, v) = ⎣ ⎦=⎣ ⎦, . . J p (ϑ, x, u, v)

x, Nr x + u, Mr u − v, Pr v

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where a, b denotes the scalar production of a and b, x ∈ Rn , u ∈ Rm , v ∈ R p , and Ni ∈ Rn×n , Mi ∈ Rm×m , Pi ∈ R p× p . We will now investigate the properties of minmax and maxmin solutions to this problem. Using the Cauchy formula for the given equation of our system dynamics, we get

t x(t) = x +

t B(τ )u(τ )dτ +

0

t0

C(τ )v(τ )dτ, t0

we further write down the components of the vector functional J [ϑ] as follows: Ji (ϑ, x, u, v) = x, Ni x + u, Mi u − v, Pi v =  ϑ =

B(τ )u(τ )dτ, (Ni +

Ni )

t0

 + x , (Ni + 0

Ni )



ϑ

C(τ )v(τ )dτ + t0

ϑ





B(τ )u(τ )dτ + x , (Ni + 0

Ni )

t0

 ϑ +

B(τ )u(τ )dτ, (Ni + Ni )



ϑ

B(τ )u(τ )dτ + t0

 ϑ C(τ )v(τ )dτ, (Ni + t0

C(τ )v(τ )dτ + t0

t0

+



ϑ

Ni )



ϑ

C(τ )v(τ )dτ + t0

  + u, Mi u − v, Pi v + x 0 , (Ni + Ni )x 0 . We further regroup these components in such way that each component of the vector functional would be presented in the form of a sum of the next type: Ji = Fi (u, v) + ϕi (u) + ψi (v), where

14 On the Problems of Minmax–Maxmin Type Under Vector-Valued Criteria

u˜ ∈

v˜ ∈

⎧ ϑ ⎨ ⎩

t0

⎧ ϑ ⎨ ⎩

t0

149

Fi (u, ˜ v˜ ) = u, ˜ Ai v˜ ,

⎫ ⎬ B(τ )u(τ )dτ | u(τ ) ∈ P(τ ) , ⎭ ⎫ ⎬ C(τ )v(τ )dτ | v(τ ) ∈ R(τ ) . ⎭

In vector form, the last relation will be J [ϑ] = F(u, v) + (u) + (v). We now give a separate description for the behavior of the minmax and maxmin for each term of the last relation.

14.3 The Functional of Type (u) + (v) Consider functional S(u, v) = (u) + (v). Moving along a chain of inequalities, we have Minu Maxv S(u, v) = Minu Maxv {(u) + (v)} = = Min {Max {(u) ˜ + (v)|v ∈ V } |u˜ ∈ U } = = Min {(u) ˜ + Max(V )|u˜ ∈ U } = Min {Min(U ) + Max(V )} . And similarly for the maxmin: Maxv Minu S(u, v) = Maxv Minu {(u) + (v)} = = Max {Min {(u) + (˜v)|u ∈ U } |˜v ∈ V } = = Max {Min(U ) + (˜v )|˜v ∈ V } = Max {Min(U ) + Max(V )} . Since a randomly selected nonempty set A satisfies relation Max A  Min A, if both Min A and Max A exist, we apply this relation to S(u, v) = (u) + (v), which yields the inequality Maxv Minu S(u, v)  Minu Maxv S(u, v).

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14.4 The Functional of Type F(u, v) = [u, A1 V  , . . . , u, Ar v] 14.4.1 Case v ∈ R1 Consider a functional of type    u, Av A u, v   , F(u, v) = = u, Bv B u, v 

where v ∈ V ⊆ R1 , u ∈ U ⊆ Rm , a, b is scalar production of a and b, and A, B ∈ Rm . Given scalar v, we introduce an alternative representation for this functional, which is a   Au F(u, v) = v . Bu Then F(U, v) = v

  A U = vU˜ , B

  A U. B Assume vmax = max v, vmin = min(v), then

where U˜ =

  Minu Maxv vU˜ = Minu vmax · u|u ∈ U˜ = vmax · MinU˜ and

which means

  Maxv Minu vU˜ = Maxv v · MinU˜ = vmax · MinU˜ , Minu Maxv vU˜ = Maxv Minu vU˜ .

For describing the cases of higher dimensions we will need some additional propositions.

14.4.2 A Necessary Condition for the Violation of the Minmax Inequality Consider functional F(u, v) : U × V → Rr , where U ⊆ Rm , V ⊆ R p . The main minmax inequality for the Pareto ordering will be named as

14 On the Problems of Minmax–Maxmin Type Under Vector-Valued Criteria

Minu Maxv F(u, v)  Maxv Minu F(u, v),

151

(14.2)

where the order  is understood to be in the sense of (14.1). Proposition 14.3 Suppose there exist F(u ∗ , v∗ ) ∈ Maxv Minu F such that

f ∗ = F(u ∗ , v∗ ) ∈ Minu Maxv F,

f∗ =

f ∗  f∗. Then fˆ = F(u ∗ , v∗ ) will be such that 

fˆ ≷ f ∗ , fˆ ≷ f ∗ .

Proof The proof will be given by indicating that other cases are impossible. Since f ∗ ∈ Minu Maxv F, we have f ∗ ∈ MaxF(u ∗ , V ), which means f ∗ ⊀ fˆ. Similarly f ∗ ∈ MinF(U, v∗ ), which means fˆ ⊀ f ∗ . 1. Suppose fˆ  f ∗ . Then



fˆ  f ∗  f ∗ , fˆ  f ∗ ,

which leads to a contradiction. 2. A similar reasoning is true for cases f ∗ = fˆ, f ∗  fˆ and f ∗ = fˆ. 3. Now suppose fˆ ≷ f ∗ . Then if f ∗  fˆ and f ∗ = fˆ, we come to a contradiction. 4. Case fˆ ≷ f ∗ is treated similarly to the previous one. Hence, condition f ∗  f ∗ automatically yields 

fˆ ≷ f ∗ , fˆ ≷ f ∗ .

Corollary 14.1 Suppose F = (F1 , . . . , Fr ) and ∃ f ∗ = F(u ∗ , v∗) ∈ Minu Maxv F, f ∗ = F(u ∗ , v∗ ) ∈ Maxu Minv F : f ∗  f ∗ . Then ∃i = j, k = l; i, j, k, l = 1, . . . , r : 

 F j (u ∗ , v∗ ) − F j (u ∗ , v∗ ) < 0,    Fk (u ∗ , v∗ ) − Fk (u ∗ , v∗ ) Fl (u ∗ , v∗ ) − Fl (u ∗ , v∗ ) < 0. Fi (u ∗ , v∗ ) − Fi (u ∗ , v∗ )



14.4.3 Case min {dim(u), dim(v)}  2 Suppose criterion F = (F1 , . . . , Fr ) is given by relations Fi = u, Ai v ,

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where u ∈ U ⊆ Rm , v ∈ V ⊆ R p . We will now look for sufficient conditions for the validity of the minmax inequality for the Pareto ordering. Suppose the last criterion is not true. Then for some i = j, k = l it follows that:  

    ∗    u ∗ , Ai v∗ − u ∗ , Ai v∗ u , A j v∗ − u ∗ , A j v∗ < 0,       u ∗ , Ak v∗ − u ∗ , Ak v∗ u ∗ , Al v∗ − u ∗ , Al v∗ < 0,

or



  u ∗ , Ai (v∗ − v∗ ) u ∗ , A j (v∗ − v∗ ) < 0,  ∗   (u − u ∗ ), Ak v∗ (u ∗ − u ∗ ), Al v∗ < 0.

If we now manage to find conditions which for all nonzero u ∈ Rn , v ∈ R m yield   u, Ai v u, A j v  0, then we come to a contradiction with the earlier supposition and these conditions will hence be sufficient for the validity of the minmax inequality (14.2). Rewrite   u, Ai v u, A j v = u Ai vv A j u  0. We will now look for the restrictions which ensure that matrix Q = Ai vv A j would be positive semi-definite simultaneously for all v ∈ Rm . For this situation, we further use the next propositions. Theorem 14.1 (The criterion of Silvester) A symmetric matrix A = A ∈ Rm×m is positively semi-definite if and only if all its main minor matrices are non-negative. Since, in general, the considered matrix, Q, is not symmetrical, the application of Silvester criterion requires the next additional proposition. Lemma 14.1 For an arbitrary matrix A ∈ Rm×m , the relation ∀v = 0 ⇒ v, Av  0 is true if and only if the next inequality is true: A + A  0. 2 Since for two arbitrary matrices the next relation is true rank AB  min{rank A, rank B}, rank(A + B)  rank A + rank B, then, applying equality

(14.3) (14.4)

14 On the Problems of Minmax–Maxmin Type Under Vector-Valued Criteria

153

  rank vv = 1, we come to rank

Q + Q 2

!

   2 rank Q  2 min rank Ai , rank(vv ), rank A j  2.

Hence, all the minors of matrix S = (Q + Q ) of order 3 and above are equal to zero. Proposition 14.4 Suppose for all i, j = 1, . . . , r, i = j all the angular minors Mk [S] of matrix S = Ai vv A j + A j vv Ai , of order k  2 are non-negative simultaneously for all v ∈ (V − V ). Then the inequality (14.2) is true. Corollary 14.2 Suppose conditions of Proposition 14.4 are true. Then ∀i = j = 1, . . . , r ⇒ [Ai ]kl [A j ]kl  0, ∀k = 1, . . . , n, l = 1, . . . , m. Proof We now find the explicit relation for the diagonal elements of matrix S = Q + Q . For getting more suitable notations, we rename the system variables, namely, Ai = A = [ai j ], A j = B = [bi j ], vv = X = [xi j ]. Then, due to definition of matrix products, we find # $ " " " ""

Q ii = [AX B ]ii = [AX ]ik [B ]ki = air xr k bik = air (vr vk )bik .

k

k

r

k

r

Denote αi = (ai1 , ai2 , . . . , aim ), βi = (bi1 , bi2 , . . . , bim ) to be the vectors of matrix rows for A and B, respectively. Then " "    bik vk air vr = 2 αi , v βi , v = 2v αi βi v  0. Sii = 2Q ii = 2 k

r

Thus, for all k = 1, . . . , m, we have ([Ai ]k ) [A j ]k  0.   Note that for rank (Ak ) Bk  1, this relation allows to get a simpler formula of the last condition, having

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Akl Bkl  0, ∀k = 1, . . . , n, l = 1, . . . , m. Remark 14.2 Note that the given reasoning is true not only for the case of square matrices Ai .

14.5 Examples Example 14.1 Let U = V = [−1; 1],  A1 =

   1 0 1/2 0 , A2 = − . 0 1/2 0 1

Realizing a numerical calculation of maxmin and minmax for the given conditions, we obtain the graph shown in Fig. 14.1. Under given values of matrices Ai we have an inverse relation Minu Maxv F(u, v)  Maxv Minu F(u, v). Example 14.2 We first indicate that Corollary 14.2 is not a sufficient condition for the validity of the main inequality (14.2). Let U = V = [−1; 1], 

   12 10 A1 = , A2 = − . 51 21

Fig. 14.1 The result of numerical calculations for the boundaries in Example 14.1

1.5 1 MaxMin MinMax

0.5 0 -0.5 -1 -1.5 -1.5

-1

-0.5

0

0.5

1

1.5

14 On the Problems of Minmax–Maxmin Type Under Vector-Valued Criteria Fig. 14.2 The result of numerical calculation of the boundaries in Example 14.2

155

0.8 0.6 MaxMin MinMax

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

-3

-2

-1

0

1

2

3

After a numerical calculation, we get a graph Fig. 14.2. It thus occurs that Corollary 14.2 is not a sufficient condition for fulfilling inequality (14.2), and hence an additional analysis of second-order minors for matrix S = Q + Q cannot be omitted.

14.6 Conclusion In the course of this paper, we formulated a vector-valued analogy of minmax under Pareto ordering and a similar analogy for maxmin. It was determined that an analogy of the classical scalar inequality relations between minmax and maxmin may not always be true for the vector-valued case. It thus required to investigate conditions that would ensure the correctness of such analogy through an example for a linearquadratic vector-valued problem under bounded disturbance. This research is supported by Russian Foundation for Basic Research (research project 16-29-04191 ofi_m).

Reference 1. Kurzhanski, A.B., Komarov, Y.A.: Hamiltonian formalism for the problem of optimal motion control under multiple criteria. Dokl. Math. 97(3), 291–294 (2018)

Chapter 15

One Problem of Statistically Uncertain Estimation B. I. Ananyev

Abstract An abstract filtering problem for multistage stochastic inclusions is considered. For its solution, three various schemes for the building of the sets consisting of conditional distributions are proposed. We are interested in the coincidence of these sets. Some sufficient conditions for this are given. Two theorems on the coincidence are proved for finite sets and for two multidimensional Euclidean case with special multivalued mappings and special selections.

15.1 Introduction and Preliminaries In this paper, we deal with an estimation problem under uncertainty. A classical formulation of the estimation problem is the following. There are two random variables x and y, which are related through their joint probability distribution so that the value of one provides information about the value of the other. We get to know the value of y in order to estimate the value of x. It is known that the conditional distribution of x on the basis of knowledge of y gives the most complete such information. Consider a general situation for the multistage filtering problem. Below we complicate the problem and call it statistically uncertain. Given a probability space (, F, P), two Borel spaces X and Y , and multistage equations z i = h i (z i−1 , ω), z i = [xi , yi ] ∈ X Y, i ∈ 1 : N ,

(15.1)

one needs to obtain a conditional distribution P(x N ∈ A|y N ) for all sets A ∈ B X . Here xi is the non-observable projection of z i on X , yi is the observable projection on Y . Hereafter, y N = {y1 , . . . , y N } is a set of observable projections and B X is the Borel σ -algebra on topological space X . B. I. Ananyev (B) N.N. Krasovski Institute of Mathematics and Mechanics, Kovalevskaya 16, 620990 Yekaterinburg, Russia e-mail: [email protected] Ural Federal University, Mira 19, 620002 Yekaterinburg, Russia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_15

157

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Let us remind a notation. A topological space is said to be Borel if it homeomorphic to a Borel set of some Polish (metrizable, complete, and separable) space. If P is a class of subsets of a set X , then σ (P) is the smallest σ -algebra containing P. If X 1 , X 2 , . . . is a sequence of sets, the Cartesian product of X 1 , X 2 , . . . is denoted by X 1 X 2 . . . . For two σ -algebras A and G on X and Y , respectively, their product on X Y is defined as A · G = σ ({AG : A ∈ A, G ∈ G}). We have B X Y = B X · BY . If f is a random real function on (, F, P) with the finite expectation E| f | < ∞ and A is a σ -subalgebra of F, then E( f |A) is the conditional expectation. For the sequence y N , the σ -algebra σ (y N ) equals σ (y1−1 (BY ) ∪ · · · ∪ y N−1 (BY )). For brevity, E( f |σ (y N )) = E( f |y N ). As a rule, along with equations (15.1) there is a flow of increasing σ -algebras F1 ⊂ · · · ⊂ F N ,

(15.2)

such that every functions h i are B X Y · Fi |B X Y -measurable. The conditional distribution for the projection x N may be obtained recursively as follows. Let h i (z, ω) = [ f i (z, ω), gi (z, ω)] ∈ X Y and h i−1 (z, Z )= {ω : h i (z, ω) ∈ Z } ∀Z ∈ B X Y . First, measurable stochastic kernels are introduced: qi (Z |z) = P(h i−1 (z, Z )), Z ∈ B X Y ,

(15.3)

on X Y given z. These kernels represent transition probabilities from (i − 1) stage to i stage. Denote by P(X ) the set of all probability Borel measures on a Polish space X . It is known [2] that if X is a Borel space, then P(X ) is also a Borel space in the weak topology. Each of the stochastic kernel qi (·|z) is a collection of probability measures in P(X Y ) parameterized by z ∈ X Y . The measurability of kernels (15.3) means the following. Let γ : X Y → P(X Y ) be any mapping of the form γ (z) = qi (·|z), then γ −1 (BP(X Y ) ) ⊂ B X Y . The notion of measurability of kernels is equivalent to the Borel measurability of functions qi (Z |z) in z for all Z ∈ B X Y . If X, Y, Z are three Borel spaces then according to [2, Proposition 7.27] for any stochastic measurable kernel q(·|z) on X Y given z there exists a stochastic measurable kernel r (·|z, y) on X given [z, y] and a stochastic measurable kernel s(·|z) on Y given z such that  r (A|z, y)s(dy|z) ∀A ∈ B X , ∀B ∈ BY .

q(AB|z) =

(15.4)

B

Here s(B|z) = q(X B|z). Using (15.4) on the first stage, one can write  q1 (AB|z 0 ) =

r1 (A|z 0 , y1 )s1 (dy1 |z 0 ) ∀A ∈ B X , ∀B ∈ BY , B

where r1 (·|z 0 , y1 ) is a stochastic measurable kernel. Therefore, the conditional distribution P(x1 ∈ A|y 1 ) = p1 (A|z 0 , y 1 ) of the state x1 equals r1 (A|z 0 , y1 ). Further, we act by induction. Let the distribution pi−1 (·|z 0 , y i−1 ) be already constructed. Then on the stage i, we have

15 One Problem of Statistically Uncertain Estimation

159



 qi (AB|[xi−1 , yi−1 ]) pi−1 (d xi−1 |z 0 , y i−1 ) = X

ri (A|z 0 , y i )si (dyi |z 0 , y i−1 ), B

(15.5)



where si (B|z 0 , y i−1 ) =

qi (X B|[xi−1 , yi−1 ]) pi−1 (d xi−1 |z 0 , y i−1 ). X

So, we have pi (A|z 0 , y i ) = ri (A|z 0 , y i ), ∀A ∈ B X . As a matter of fact, stochastic kernels (15.3) play the main role in above procedure and they can be given a priori instead of Eqs. (15.1). Particularly, let X = R n , Y =  R m , and qi (AB|z i−1 ) = AB u i (x, y|z i−1 )d xd y, where u i (x, y|z i−1 ) is a Lebesguemeasurable non-negative probability density. Then, on the first stage, one has the conditional density  uˆ 1 (x|z 0 , y1 ) = u 1 (x, y1 |z 0 )

Rn

u 1 (x, y1 |z 0 )d x

for the conditional measure r1 (·|z 0 , y1 ). Let uˆ i−1 (x|z 0 , y i−1 ) be the conditional density for the conditional measure ri−1 (·|z 0 , y i−1 ). Then, on the stage i, one has the conditional density  i i−1 uˆ i (x|z 0 , y ) = R n u i (x, yi |[xi−1 , yi−1 ])uˆ i−1 (xi−1 |z 0 , y )d xi−1   ˆ i−1 (xi−1 |z 0 , y i−1 )d xi−1 d x, i > 1, R n R n u i (x, yi |[x i−1 , yi−1 ])u

(15.6)

for the conditional measure ri (·|z 0 , y i ). Knowing the conditional measure, one can calculate the conditional expectation or dispersion of any real measurable function of x N .

15.2 Statistically Uncertain Filtering Suppose we have multistage inclusions z i ∈ Q i (z i−1 , ω), z i ∈ X Y, i ∈ 1 : N ,

(15.7)

instead of Eq. (15.1) on the same probability space. Then the reasonings of Sect. 15.1 are inapplicable because stochastic kernels like (15.3) are unknown. This is a situation of statistically uncertain filtering. Introduce some new notation concerning the uncertain case. For any set X denote by P(X ) the collection of all subsets of X . Let (, F), (X, A) be two measurable spaces and  :  → P(X ) be a multivalued mapping. Given A ∈ P(X ), the upper inverse for  is defined as  ∗ (A) = {ω ∈  : (ω) ∩ A = ∅}. Sometimes, if  is fixed, we write A∗ =  ∗ (A). The lower inverse for  is ∗ (A) = {ω ∈

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 : (ω) ⊂ A}. In the same way, we write A∗ = ∗ (A). The multivalued mapping  is said to be strongly measurable if  ∗ (A) ∈ F for all A ∈ A. In this case, we shortly say that  is F|A-measurable. As (∗ (Ac ))c =  ∗ (A), the notion of strong measurability can be formulated with the help of the lower inverse ∗ and vice versa. The probabilities of the sets A∗ and A∗ are denoted by P∗ (A) and P∗ (A), respectively, or simply P ∗ (A) and P∗ (A) if it doesn’t lead to ambiguity. For multivalued mapping , denote by S() the set of all measurable selections: S() = {h :  → X, F|A − measurable and h(ω) ∈ (ω) ∀ω}. Introduce the distribution Ph (A) = P(h −1 (A)) ∀A ∈ A of the selection h and the set P = {Ph : h ∈ S()} ⊂ P(X ) of all such distributions. Define also the set P (A) = {Ph (A) : h ∈ S()} ⊂ [0, 1] ∀A ∈ A. In order to exclude the case S() = ∅, the multivalued mapping  have to satisfy some conditions (see [3, Theorem 5.1]). For example, let X be separable metric with A = B X and  :  → X be F|B X -measurable with complete values, then  has a measurable selection. We need also the set  = { p ∈ P(X ) : p(A) ≤ P (A) ∀A ∈ B X } and the set M =  p ∈ P(X ) : p(A) ≤ P∗ (A) ∀A ∈ B X . As to inclusions (15.7), we suppose that the multivalued mappings Q i are B X Y · Fi |B X Y -measurable where flow (15.2) is given. It is always supposed that S(Q i ) = ∅.

15.2.1 A Filtering Scheme for Inclusions Consider the sets S(Q i ) of selections for inclusions (15.7). For each of the selection h i (z, ω) = [ f i (z, ω), gi (z, ω)] ∈ Q i (z, ω), we construct the conditional measures ri (·|z 0 , y i ), i ≥ 1, according to (15.5). Therefore, we can get the set R1 =



  r1 (·|z 0 , y 1 )

h 1 ∈S(Q 1 )

of conditional distributions on the first stage. Let the set Ri−1 of distributions be already constructed. Then, on the stage i, we have 

 qi (AB|z i−1 )ri−1 (dxi−1 |z 0 , y i−1 ) = X

ri (A|z 0 , y i )si (dyi |z 0 , y i−1 ), B

 where si (B|z 0 , y i−1 ) = X qi (X B|z i−1 )ri−1 (dxi−1 |x0 , y i−1 ), for all ri−1 ∈ Ri−1 , h i ∈ S(Q i ). So, we obtain the set Ri =



  ri (·|z 0 , y i )

(15.8)

h i ∈S(Q i ), ri−1 ∈Ri−1

and use it for the calculation of the set of conditional expectations or dispersions for any real measurable function of x N .

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As in the classical case, stochastic kernels (15.3) play the main role in above procedure. Each selection h i (z, ω) ∈ S(Q i ) uniquely generates the measurable stochastic kernel qi (·|z) on X Y given z. Denote by Pi,z the set of all such kernels given z. The sets Pi,z can be given a priori instead inclusions (15.7). For example, in the case X = R n , Y = R m , we may have the set Ui of probability densities u i (x, y|z i−1 ) on the stage i. Then, using (15.6), one can recursively obtain the set Uˆ i of conditional densities uˆ i (x|z 0 , y i ) for the set Ri of conditional distributions in (15.8). Example 15.1 Suppose that i ∈ 1 : 2, X = Y = R, and the sets Ui on each stage consist of two densities:  1, if (x − x) ¯ 2 + (y − y¯ )2 ≤ 1; u(x, y|¯z ) = 0, otherwize;  √ 1, if |x − x| ¯ + |y − y¯ | ≤ 1/ 2; v(x, y|¯z ) = 0, otherwize. Then stochastic kernels have two forms: q(Z |¯z ) = λ(Z ∩ B1 (¯z )) or q(Z |¯z ) = λ(Z ∩ B1/√2 (¯z )) on each stage where λ is the Lebesgue measure on R 2 and Br (z), Br (z) are circles for corresponding norms with the center z. Therefore, the conditional density uˆ 2 (x2 |z 0 , y 2 ) is defined by four formulas. One of them (on each stage u(x, y|¯z ) is used) is



  uˆ 2 (x2 |z 0 , y 2 ) = x2 + 1 − (y2 − y1 )2 ∧ x0 + 1 − (y1 − y0 )2



  − x2 − 1 − (y2 − y1 )2 ∨ x0 − 1 − (y1 − y0 )2    4 1 − (y2 − y1 )2 1 − (y1 − y0 )2 ,   if |x2 − x0 | ≤ 1 − (y2 − y1 )2 + 1 − (y1 − y0 )2 . Here a ∧ b = min{a, b} and a ∨ b = max{a, b}.

15.2.2 Another Filtering Scheme for Inclusions In Sect. 15.2.1, the sets Ri in (15.8) were built with the help of measurable selections h i (z, ω) = [ f i (z, ω), gi (z, ω)] ∈ Q i (z, ω), i.e., with the help of the sets S(Q i ), consisting of all B X Y · Fi |B X Y -measurable selections. Each selection h i (z, ω) uniquely generates the measurable stochastic kernel qi (·|z) on X Y given z. We denote by P(X Y |z) the set of all B X Y |BP(X Y ) -measurable stochastic kernels q(·|z) on X Y given z. Define also the set Pi,z (Z ) = {qi (Z |z) : qi ∈ Pi,z } ⊂ [0, 1] for every z. The sets Pi,z and Pi,z (Z ) are the most precise pieces of information that inclusions (15.7) give about the probability distribution of unknown selection

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h i (z i−1 , ω), and about the values qi (Z |z i−1 ), respectively. After that we introduce the set   i,z = q ∈ P(X Y |z) : q(Z |z) ∈ Pi,z (Z ) ∀Z ∈ B X Y ∀z ∈ X Y .

(15.9)

It is clear that Pi,z ⊂ i,z . These sets are not convex in general. To define one more set, introduce upper probability Pi∗ (Z |z) = P({ω ∈  : Q i (z, ω) ∩ Z = ∅}) ∀Z ∈ B X Y given z. Then define the set   Mi,z = q ∈ P(X Y |z) : q(Z |z) ≤ Pi∗ (Z |z) ∀Z ∈ B X Y ∀z ∈ X Y .

(15.10)

From various points of view, it is preferable to work with set (15.10) because it is convex, but some information with this set may be lost. We have the relation between defined sets: (15.11) Pi,z ⊂ i,z ⊂ Mi,z . Thus, we can build recursive procedures like (15.8) using stochastic kernels qi (·|z) ∈ i,z or qi (·|z) ∈ Mi,z . We call the corresponding sets of conditional means as Di and Mi , respectively. According to (15.11), we get Ri ⊂ Di ⊂ Mi ∀i ∈ 1 : N .

(15.12)

15.3 Problem Formulation and Its Partial Solution Our problem is to give some sufficient conditions for coincidence of the sets in (15.12). In the general case, these sets may be considerably different. Consider the example that we have built on the base of Example 3.1 in [5]. Example 15.2 Let X = Y = {0, 1} be the discrete spaces of two elements, the probability space be equal to (, F, P), where  = {ω1 , ω2 }, F = P(), P(ω1 ) = 1/3. The product X Y is four vertexes of the square. For simplicity, number the vertexes: [1, 0] = 1, [0, 0] = 2, [0, 1] = 3, [1, 1] = 4. The mapping Q(z, ω) does not depend on the stage i and is defined as follows: Q(2, ω1 ) = {1, 2, 3}, Q(2, ω2 ) = {1, 2}, Q(3, ω1 ) = {2, 3, 4}, Q(3, ω2 ) = {2, 3}, Q(4, ω1 ) = {3, 4, 1}, Q(4, ω2 ) = {3, 4}, Q(1, ω1 ) = {4, 1, 2}, Q(1, ω2 ) = {4, 1}. On each step, the set S(Q), given z, has six selections h(z, ω) and, therefore, the set Pi,z has six stochastic kernels (distributions). Let z = 2, then Pi,z = {(1, 0, 0, 0), (2/3, 1/3, 0, 0), (2/3, 0, 1/3, 0), (1/3, 2/3, 0, 0), (0, 1, 0, 0), (0, 2/3, 1/3, 0)}. Here ( p1 , p2 , p3 , p4 ) = (P{q = 1}, P{q = 2}, P{q = 3}, P{q = 4}). We see that the distribution (1/3, 1/3, 1/3, 0) ∈ i,z given z = 2, but it does not belong to Pi,z .

15 One Problem of Statistically Uncertain Estimation

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On the other hand, the distribution (0.5, 0.3, 0.2, 0) ∈ Mi,z given z = 2, but it ∈ / i,z . The same is true for other z, whence inclusions (15.11) are strict. From this it follows that inclusions (15.12) are strict as well. Recall that the sets Mi,z are convex as opposed to other sets Pi,z and (15.9). Nevertheless, some cases are described in literature when the sets like in (15.11), (15.12) given z are coincided. But it is done only for finite sets X or for onedimensional intervals like [A(ω), B(ω)] (see [5]). As in the Euclidean case we have n + m ≥ 2, then we are forced to consider at least the sets of the form Q i (z, ω) = [A(ω), B(ω)][C(ω), D(ω)], i.e., products of two closed intervals. It is supposed that A(ω) ≤ B(ω) and C(ω) ≤ D(ω). But, as far as we know, the problem of coincidence of sets in (15.11), (15.12) in that case still is open in the class of all measurable selections. In this paper, we establish two theorems about coincidence of the sets in (15.11) and (15.12) for special cases. First, suppose that the product X Y is a finite space with σ -algebra P(X Y ). A set A ∈ F in a probability space (, F, P) is called an atom if for all subsets F  B ⊂ A one has P(B) = P(A) or P(B) = 0. The probability space is called non-atomic if it contains no atoms. Theorem 15.1 Let the probability space be non-atomic, X and Y be finite. Then the sets in (15.11), (15.12) coincide. As a matter of fact, Theorem 1 follows from [6, Theorem 2]. The non-atomic property is used here to construct a function f :  → [0, 1] with uniform distribution. From the other hand, for any strongly measured multifunction Q i (z, ω) and any stochastic is a selection h i : X Y [0, 1] ∈ S(Q i ) on the probkernel q(·|z) ∈   Mi,z given z there ability space [0, 1], B[0,1] , λ given z such that it generates the kernel q(·|z). The selection is built with the help of quantile functions. At last, the composition proves the theorem. Now, let X = R, Y = R, and let Q i (z, ω) = (A(ω), B(ω))(C(ω), D(ω)) be an open rectangle that does not depend on i and z. Further, we simply write Q(ω). It is supposed that A(ω) < B(ω) and C(ω) < D(ω). Collect some known general results about mappings with compact and open values. Theorem 15.2 (Theorems 3.2, 3.3 in [5]) Let (, F, P) be a probability space, X be a Polish space, and  :  → P(X ) be F|B X -measurable mapping with compact values. Then 1. The set function (upper probability) P ∗ (A) is continuous for decreasing sequences of compact sets. 2. P ∗ (A) = sup K ⊂A compact P ∗ (K ) = inf A⊂G open P ∗ (G) ∀A ∈ B X . 3. The set M = {q ∈ P(X ) : q(A) ≤ P ∗ (A) ∀A ∈ B X } is convex and weakly compact. 4. P ∗ (A) = max P (A) ∀A ∈ B X . 5. M = clw (convP ). If the F|B X -measurable mapping  has open values, then

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6. P ∗ (A) = sup J ⊂A finite P ∗ (J ) ∀A ∈ B X . 7. P ∗ (A) = max P (A) ∀A ∈ B X . The results of Theorem 15.2 are applicable for our sets in (15.11) given z. In ∗ ∗ accordance with Theorem 15.2,  (Z )), PQ =  we have2Q(ω) = (ω),∗ P (Z ) = P(Q {Ph : h ∈ S(Q)}, and M Q = q ∈ P(R ) : q(Z ) ≤ P (Z ) ∀Z ∈ B R 2 . A mapping Q :  → P(X Y ) is simple if it has a finite number of set values. For our open simple rectangle Q(ω), we prove the coincidence PQ =  Q = M Q in the spe˜ 2 ) ⊂ P(R 2 ) of distributions. Namely, the class P(R ˜ 2 ) consists of cial class P(R 2 all probability measures q ∈ P(R ) possessing the properties: the cumulative distribution function F(x, y) = q((−∞, x](−∞, y]) is continuous and such that the inequality F(x,   y) ≤ F(u, v) implies that x ≤ u and y ≤ v. Besides, the support set supp(q)= z ∈ R 2 : for every open neighborhood Nz  z one has q(Nz ) > 0 is ˜ 2 ) are given in the following lemma. compact. The properties of P(R ˜ 2 ) is convex and nonempty. For any open rectangle Lemma 15.1 The class P(R 2 ˜ ) such that = (a, b)(c, d) and for any measure q ∈ P((R  q( ) = 1 there is a selection h :  → on probability space [0, 1], B[0,1] , λ the distribution Ph of which coincides with q. ˜ 2 ) is obvious. Further, we write z 1 = [x1 , y1 ] ≤ z 2 = Proof The convexity of P(R ¯ = [a, b][c, d] one can build a [x2 , y2 ] iff x1 ≤ x2 and y1 ≤ y2 . For any rectangle continuous function of the form ⎧ ⎪ ((x − a)2 + u(x)2 )/((d − c)2 + (b − a)2 ) if ⎪ ⎪ ⎪ ⎪ ⎪ y ≥ u(x) = c + (d − c)(x − a)/(b − a); ⎨ F(x, y) = ((x¯ − a)2 + u(x) ¯ 2 )/((d − c)2 + (b − a)2 ) if ⎪ ⎪ ⎪ y < u(x) = c + (d − c)(x − a)/(b − a), where ⎪ ⎪ ⎪ ⎩x¯ = a + (b − a)(y − c)/(d − c). This function possesses all necessary properties and F(x + α, y + β) + F(x, y) − F(x + α, y) − F(x, y + β) ≥ 0 for all α ≥ 0, β ≥ 0, and for all x, y. Therefore, the ˜ 2 ). If q( ) = 1, then its compact support function F generates a measure q ∈ P(R ¯  ⊂ which strictly consupp(q) ⊂ . Therefore, there exists a closed rectangle ¯  such that ω = tains supp(q). We can choose a continuous function h : [0, 1] → F(h(ω)), where F is the continuous cumulative probability distribution for q. Using properties of F, we obtain the equality of sets: {ω : h(ω) ≤ z} = {ω : ω ≤ F(z)}, whence the Lebesgue measures of these sets coincide. Thus, the required selection is found.  Now, we can prove the following theorem. Theorem 15.3 Let the probability space (, F, P) be non-atomic, then for strongly measurable, simple, and open rectangle mapping Q :  → P(R 2 ) we have the coin˜ 2 ). cidence PQ =  Q = M Q of the sets in the class P(R

15 One Problem of Statistically Uncertain Estimation

165

Proof As the probability space is non-atomic, there is a measurable mapping α :  → [0, 1] with uniform distribution [6]. At first, let Q(ω) ≡ on . We obtain that Q ∗ (Z ) = if Z ∩ = ∅ and Q ∗ (Z ) = ∅ if Z ∩ = ∅. So, M Q =  ˜ 2 ) : q( ) = 1 . By Lemma 1 choose a continuous selection h : [0, 1] → q ∈ P(R such that Ph = q. Then the distribution of composition h(α(ω)) equals q. Thus, PQ =M Q . Consider the general case. Then Q can be expressed in the form n ˜ be the Q = i=1 i I Bi , where {B1 , . . . , Bn } is a partition of , Bi ∈ F. Let minimal open rectangle containing all i . Define the class   ˜ \ i = {E 1 , . . . , E m }. This is a partiD = H1 ∩ · · · ∩ Hn : Hi = i or Hi = ˜ and any i is a union of elements from D. Let f : D → {1, . . . , m} be the tion of bijection. Consider the mapping Q  = f (Q) :  → P({1, . . . , m}). It is a strongly measurable mapping between non-atomic space and a finite set. We have 

∗

Q (I ) = Q

∗

 

 =

Ei

 

B j ∈ F.

(15.13)

i∈I j ⊃E i

i∈I

By [6, Theorem 2], we get M Q  = PQ  . ˜ 2 ) that belongs to M Q and define the finite distribution Consider a measure q ∈ P(R q  = q( f −1 ) : P({1, . . . , m}) → [0, 1]. Given I ⊂ {1, . . . , m}, ⎛ q  (I ) = q( f −1 (I )) = q ⎝



⎞

⎛

∗ ⎝ E i ⎠ ≤ PQ

i∈I



⎞ Ei ⎠ =

 j ∩(∪i∈I E i ) =∅

i∈I

∗ (I ), P(B j ) = PQ 

taking into account (15.13). Therefore, q  ∈ M Q  = PQ  . From this fact, we deduce the existence of the measurable selection of Q  , u :  → {1, . . . , m}, such that Pu = q  . Introduce Fi = u −1 (i) ∈ F for each i and define the multivalued mapping  Q i :  → P(R ), as ω → 2

E i if ω ∈ Fi , ∅ otherwise.

Consider the measure qi : B R 2 → [0, 1] given by qi (A) = q(A ∩ E i ) for all A ∈ B R 2 . Then, qi (A) ≤ P(Q i∗ (A)) ∀A ∈ B Ei , because q(E i ) = P(Fi ) for all i. In the same way as above there is a continuous selection wi ∈ S(Q i ) such that Pwi = qi . Now define w :  → R 2 , as ω → wi (ω) if ω ∈ Fi . The function w is a measurable selection of Q such that Pw = q. Indeed, the class {F1 , . . . , Fm } = {u −1 (1), . . . , u −1 (m)} is a partition of . Besides, Q i (ω) = ∅ ∀ω ∈ Fi , whence w is well defined. If ω ∈ Fi , then w(ω) = wi (ω) ∈ Q i (ω) = E i = f −1 ({i}) = f −1 ({u(ω)}) ∈ f −1 (Q  (ω)) = {E j : E j ⊂ Q(ω)}, therefore w(ω) ∈ Q(ω).

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 m  −1 wi (G) ∩ Fi ∈ F. Hence, w is a measurable Given G ∈ B R 2 , w−1 (G) = i=1 selection of Q. Given A ∈ B R 2 , Pw (A) =

m 

P(w−1 (A) ∩ Fi ) =

i=1

m 

P(wi−1 (A) ∩ Fi ) =

i=1

=

m 

m 

qi (A)

i=1

q(A ∩ E i ) = Q(A).

i=1

Hence, Pw = q and M Q = PQ .



Theorem 15.3 can be distributed to the case X Y = R n+m for multivalued simple mappings n+m Q i : X Y  → X Y whose values represent open boxes of the (ai (z, ω), bi (z, ω)), where functions ai (·, ω), bi (·, ω) are continform i=1 ˜ n+m ) of probability measures is uous for every ω. Whereupon, the class P(R used. Note that other statistically uncertain filtering schemes are considered in [1].

15.4 Conclusion This paper considers an abstract filtering problem for multistage stochastic inclusions. To solve the problem, three various schemes are proposed for the building of the sets consisting of conditional distributions. Some sufficient conditions for the coincidence of these sets are given. Two theorems on the coincidence are proved for finite sets and for two multidimensional Euclidean case with special multivalued mappings and special selections. Further, it is supposed to extend results to more general classes of selections and mappings. Also, distribution of some results from [4] to the case of random trajectory tubes of multistage or continuous dynamical systems is of interest.

References 1. Ananyev, B.I.: Some aspects of statistically uncertain minimax estimation. IFAC-PapersOnLine 51(32), 195–200 (2018) 2. Bertsecas, D.P., Shreve, S.E.: Stochastic Optimal Control: The Discrete-Time Case. Academic Press, Inc. (1978) 3. Himmelberg, C.: Measurable relations. Fund. Math. 87, 53–72 (1974) 4. Kurzhanski, A.B., Varaiya, P.: Dynamics and Control of Trajectory Tubes, Theory and Computation. In: Systems & Control: Foundations & Applications, vol. 85. Birkhäuser, Basel (2014)

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5. Miranda, E., Couso, I., Gil, P.: Random intervals as a model for imprecise information. Fuzzy Sets Syst. 154, 386–412 (2005) 6. Miranda, E., Couso, I., Gil, P.: Upper probabilities and selectors of random sets. In: Grzegorzewski, P., Hryniewicz, O., Gil, M.A. (eds.) Soft Methods in Probability. Statistics and Data Analysis, pp. 126–133. Physica-Verlag, Heidelberg (2002)

Chapter 16

The First Boundary Value Problem for Multidimensional Pseudoparabolic Equation of the Third Order in the Domain with an Arbitrary Boundary Murat Beshtokov

Abstract The work is devoted to the study of the first initial boundary value problem for multidimensional pseudoparabolic equation of the third order with variable coefficients. Locally one-dimensional difference schemes are constructed. A priori estimates are obtained using the maximum principle for solutions of locally onedimensional difference schemes in a uniform metric in the norm of C. Stability and convergence of locally one-dimensional difference scheme are proved.

16.1 Introduction It is well known that pseudoparabolic equations arise while describing liquid filtration processes in porous media [1, 2], heat transfer in heterogeneous medium [3, 4], moisture transfer in soils [5], [6, p. 137]. Among such tasks the most complex from the point of view of numerical implementation are considered to be multidimensional (with respect to spatial variables) problems consisting of a significant increase in the amount of computation that occurs during the transition from one-dimensional to multidimensional problems. In this regard, great importance acquires the construction of economical difference schemes for numerical solution of multidimensional problems. A large number of works are devoted to the study of various initial and initial boundary value problems for one-dimensional pseudoparabolic equations with variable coefficients [7–17]. The works [18–20] are devoted to construction of vector operator-difference additive schemes for solving boundary value problems for multidimensional pseudoparabolic equations.

M. Beshtokov (B) Institute of Applied Mathematics and Automation, Kabardino-Balkariya, Nalchik 360000, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_16

169

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16.2 Statement of the Problem In the cylinder Q T = G × [0 ≤ t ≤ T ], we consider the following problem: ∂u ∂ = Lu + α Lu + f (x, t), (x, t) ∈ Q T , ∂t ∂t   u  = μ(x, t), 0 ≤ t ≤ T, G = G + Γ,

(16.2)

u(x, 0) = u 0 (x), x ∈ G,

(16.3)

Γ

(16.1)

where Lu =

p  k=1

L k u,

Lku =

∂ ∂xk

 Θk (x, t)

∂u ∂xk

 − qk (x, t)u, k = 1, 2, . . . , p,

(16.4) α > 0, 0 < c0 ≤ Θk (x, t) ≤ c1 , qk > 0, c0 , c1 , c2 —some positive constants. Γ —the domain boundary G, x = (x1 , x2 , . . . , x p )—a point of p-dimensional Euclidean space R p . Two assumptions are used regarding the G domain ([21], p. 486): (a) the intersection of the area G with the line Ck , parallel to the coordinate axis Oxk , consists of a single interval Δk ; (b) in a closed domain G = G ∪ Γ it is possible to construct a connected grid ω¯ h with steps h k , k = 1, 2, . . . , p. The set ωh of internal grid nodes consists of points x = (x1 , x2 , . . . , x p ) ∈ G of the intersection of the hyperplanes xk = i k h k , i k = 0, ±1, ±2, . . . , k = 1, 2, . . . , p, whereas the set γh of boundary nodes consists of the points of intersection of lines Ck , k = 1, 2, . . . , p passing through internal nodes x ∈ ωh , with the boundary Γ . We denote by γh,k —the set of boundary nodes in the direction of xk , γh —the set ∗ —the set of near-boundary nodes in the direction of all boundary nodes x ∈ Γ , ωh,k ∗ ∗∗ —the set of irregular nodes in the of xk , ωh - the set of all near-boundary nodes, ωh,k ∗∗ direction of xk , ωh —the set of all irregular nodes, ωh,k —the set of regular nodes in the direction of xk , and ωh —the set of all regular nodes. As noted in [6], the second term on the right-hand side of Eq. (16.1) is very small when absorption takes place and large when evaporation takes place. In the future, we will assume that the coefficients of the equation and boundary conditions (16.1)–(16.3) satisfy the necessary conditions of smoothness providing the necessary smoothness of the solution u(x, t) in the cylinder Q T .

16 The First Boundary Value Problem for Multidimensional Pseudoparabolic …

171

16.2.1 Locally One-Dimensional Difference Scheme (LOS) 1

Convert Eq. (16.1) by multiplying both sides of (16.1) by α1 e α t , and integrating the obtained expression with respect to ξ from 0 to t, we get Bu = Lu +  f (x, t), where Bu =

1 α

t

1

e− α (t−ξ) u ξ dξ,

0

1  f (x, t) = α

t

1

1

e− α (t−ξ) f (x, ξ)dξ − e− α Lu 0 (x), α > 0.

0

Instead of the problem (16.1)–(16.3), we consider the following problem with a small parameter: f (x, t), εu εt + Bu ε = Lu ε + 

L=

p 

L k , (x, t) ∈ Q T ,

(16.5)

k=1

  u ε  = μ(x, t), 0 ≤ t ≤ T, G = G + Γ,

(16.6)

u ε (x, 0) = u 0 (x), x ∈ G,

(16.7)

Γ

where ε— positive constant → 0. Since at t = 0 the initial conditions for Eqs. (16.1) and (16.5) coincide, then in neighborhood t = 0 the derivative u εt has no singularity of the boundary layer type. Choose a spatial grid being uniform in each direction Oxk with increments h k = lk , k = 1, 2, . . . , p : Nk  lk = i k h k : i k = 0, 1, . . . , Nk , h k = , k = 1, 2, . . . , p}, ω¯ = ω¯ h k . Nk k=1 p

ω¯ h k =

{xk(ik )

On the segment [0, T ], we introduce the grid ω¯ τ = {0, t j+ kp =

 j+

 k τ, p

j = 0, 1, . . . , j0 − 1, k = 1, 2, . . . , p},

containing dummy nodes t j+ kp , k = 1, 2, . . . , p − 1 along with the nodes t j = jτ . We will denote by ωτ , the set of grid nodes ω¯ τ , for which t > 0. By analogy with [21], to Eq. (16.5), we put into correspondence a chain of “onedimensional” equations; for this purpose, Eq. (16.5) is rewritten in

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£u ε = εu εt + Bu ε − Lu ε − f = 0, or

p 

£k u ε = 0, £k u ε =

k=1

ε ε 1 ε u + Bu − L k u ε − f k , p t p

where f k (x, t), k = 1, 2, . . . , p—arbitrary functions having the same smoothness p as f (x, t) and satisfying the condition f k = f. k=1 

k , t On every half-interval Δk = t j+ k−1 j+ p , k = 1, 2, . . . , p, we will succesp sively solve the problems £k ϑ(k) = 0, x ∈ G, t ∈ Δk , k = 1, 2, . . . , p,

(16.8)

ϑ(k) = μ(x, t) i f x ∈ Γk , each time assuming ϑ(1) (x, 0) = u 0 (x), ϑ(1) (x, t j ) = ϑ( p) (x, t j ), ) = ϑ(k−1) (x, t j+ k−1 ), k = 2, 3, . . . , p; ϑ(k) (x, t j+ k−1 p p

(16.9)

j = 0, 1, .., j0 − 1,

Γk —the set of boundary points in the direction of xk . Let us call the solution of this problem for t = t j+1 the function ϑ(t j+1 ) = ϑ( p) (t j+1 ). Find the discrete analogue of Bu: t j+ k

1 α



p

e

  − α1 t j+ k −ξ p

pj+k  1  t sp − α1 u ξ dξ = e α s=1 t s−1

  t j+ k −ξ p

u(x, ˙ ξ)dξ =

p

0

pj+k  1  t sp − α1 = e α s=1 t s−1

  t j+ k −ξ p



¯ − t¯) dξ = u(x, ˙ t¯) + u(x, ¨ ξ)(ξ

p









1 1 pj+k  pj+k  1  t sp − α t j+ kp −ξ 1  t sp − α t j+ kp −ξ ¯ = e u(x, ˙ t¯)dξ + e u(x, ¨ ξ)(ξ − t¯)dξ ≤ α α s=1 t s−1 s=1 t s−1 p p

     pj+k pj+k  t s  − α1 t j+ k −ξ p 1  sp t sp − α1 t j+ kp −ξ 1τ p M ≤ u e dξ + e dξ = α s=1 t¯ t s−1 αp s=1 t s−1 p

p

16 The First Boundary Value Problem for Multidimensional Pseudoparabolic …



pj+k

=

s p

u t¯ e

   − α1 t j+ k −ξ t sp



p

s=1

t s−1 p

173

     pj+k − α1 t j+ k −t s−1 Mτ  − α1 t j+ kp −t sp p p + −e e = p s=1

 s  pj+k    − 1 t k−s Mτ  − α1 t j+ k−s − α1 t j+ k−s+1 − α1 t j+ k−s+1 p α j+ p p p p e u t¯ + e , −e −e = p s=1 s=1 pj+k

where t = t sp − 21p , u˙ =

∂2u ∂u , u¨ = 2 , t s−1 < ξ < ξ, u(x, ¨ ξ) ≤ M. p ∂t ∂t

Consider the second expression      pj+k  Mτ  − α1 t j+ k−s Mτ Mτ τ − α1 t j+ k−s+1 − α1 t j+ k p p p e = ≤ 1−e =O . −e p s=1 p p p Therefore, we have 1 α



t j+ k

p

e

  − α1 t j+ k −ξ p

0 s

s p

 s     − 1 t k−s τ − 1 t k−s+1 e α j+ p − e α j+ p u t¯p + O , p s=1 (16.10)

pj+k

u ξ dξ =

s−1 p

. where u tp = u −uτ p Each of the Eq. (16.8) is replaced by a difference scheme taking into account (16.10)  s pj+k  1  − α1 t j+ k−s ε j+ kp − α1 t j+ k−s+1 p p e yt p = y + −e p t¯ p s=1

 k k−1 j+ k = Λk σk y j+ p + (1 − σk )y j+ p + ϕk p , x ∈ ωh , k = 1, 2, . . . , p, (16.11) k k y j+ p |γh,k = μ j+ p , j = 0, 1, . . . , j0 − 1, y(x, 0) = u 0 (x), k = 1, 2, . . . , p; where σk —arbitrary parameters, γh,k —set of boundary nodes in the direction of xk ,  lk  , x ∈ ω¯ h = xi = (i 1 h 1 , . . . , i p h p ) ∈ G, i k = 0, 1, . . . Nk , h k = Nk j+ kp

dk



 j+ k = qk x, t j+ kp , ϕk p = f k x, t j+ kp , t = t j+ 21 , k = 1, 2, . . . , p,

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k μ j+ p = μ x, t j+ kp ,

s

yt¯p =

s

yp −y τ p

s−1 p

.

The difference operator Λk ∼ L k has the following form: (1) In regular nodes: 

k k j+ k Λk y j+ p = ak yx¯k p − dk y j+ p , a (+1) = ai+1 , ai = Θi−1/2 (t). xk

(2) In irregular nodes: Λk y(k) =

Λk y(k)

1 hk

1 = hk

  y (+1k ) − y y − y (−1k ) ak,ik +1 − dk y(k) , x (−1k ) ∈ γh,k − ak,ik hk h ∗k   y (+1k ) − y y − y (−1k ) ak,ik +1 − dk y(k) , x (+1k ) ∈ γh,k , − ak,ik h ∗k hk (16.12)

where h ∗k —distance from irregular node x to boundary node x (+1k ) or x (−1k ) . If both ∗ nodes x (+1k ) and x (−1k ) adjacent to x ∈ ωh,k are boundary, that is, x (±1k ) ∈ γh,k , then Λk y

j+ kp

1 = hk

  k y (+1k ) − y y − y (−1k ) ak,ik +1 − dk y j+ p , − ak,ik h ∗k h ∗k

where h ∗k ± —the distance between x and x (+1k ) , h ∗k ± ≤ h k . In regular nodes, Λk has a second order of approximation, Λk u − kk u = O(h 2k ), whereas in irregular nodes Λk u − L k u = O(1). (see [21], p. 223).

16.2.2 Approximation Error of LOS We proceed to the study of approximation error (residuals) of the locally onedimensional scheme and make sure that each separate Eq. (16.11) with the number k does not approximate Eq. (16.5), but the sum of the approximation errors ψ = ψ 1 + ψ2 + · · · + ψ p tends to zero at τ and |h| both tending to zero, where |h|2 = h 21 + h 22 + · · · + h 2p . We assume that σk = 1, k = 1, 2, . . . , p. Let u = u(x, t) be the solution of probk lem (16.5) and y j+ p is the solution of difference problem (16.11). The characteristic of the accuracy of a locally one-dimensional scheme is the difference y j+1 − u j+1 = k k z j+1 . Intermediate values y j+ p we shall compare with u j+ p = u(x, t j+ kp ), assum-

16 The First Boundary Value Problem for Multidimensional Pseudoparabolic … k

k

k

k

k

175

k

ing that z j+ p = y j+ p − u j+ p . Substituting y j+ p = z j+ p + u j+ p into difference Eq. (16.11), we arrive at k k j+ k ε j+ kp 1 zt + Bτ z j+ p = Λk z j+ p + ψk p , p p

(16.13)

k

z j+ p |γh,k = 0, z(x, 0) = 0, j+ k ψk p

= Λk u

j+ kp

+

j+ k ϕk p

 s pj+k  1  − α1 t j+ k−s ε j+ k − α1 t j+ k−s+1 p p e u tp − u t¯ p , − −e p s=1 p

 s   − 1 t k−s − α1 t j+ k−s+1 α j+ p p e z tp . = −e pj+k

Bτ z

j+ kp

s=1

 j+ 21 ◦ Introducing the notation ψk = L k u + f k − εp u t − 1p Bu and noting that p p k ◦ ∗ ◦ j+ ψk = 0, if f k = f, we represent ψk = ψk p in the form of ψk = ψk +ψ k ,

k=1

k=1

   j+ k

j+ 21 p j+ kp j+ 21 − + ϕk − f k ψ k = Λk u − Lku

where

∗

 −

k 1 1 1 Bτ u j+ p − (Bu) j+ 2 p p



 −

ε j+ kp ε j+ 1 u t¯ − u t 2 p p

∗

 .

∗

It is clear that ψ k = O(h 2k + τ ) in the regular nodes, ψ k = O(1) in the irregular nodes, since each of the schemes (16.11) with the number approximates the corresponding Eq. (16.5) in the ordinary sense. Thus, ψ=

p  k=1

  p  p  ◦ ∗ ∗ ψk = ψk + ψk = ψ k = O(|h|2 + τ ), k=1 ◦

ψk = O(1),

k=1 p 

◦

ψk = 0

k=1

in regular grid nodes ωh , i.e., LOS (16.11) has total approximation of O(|h|2 + τ ) in regular grid nodes ωh . In irregular nodes ψ = O(1).

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16.2.3 Stability of LOS We obtain a priori estimate in a uniform metric, in the norm C for the solution of the difference problem (16.11), which expresses the stability of the locally onedimensional scheme on the initial data and the right-hand side. The study of the stability of the difference scheme (16.11) will be carried out using the maximum principle [21, p. 226]. To this end we represent the solution of problem (16.11) as a sum, at σ = 1 y = y + v + w, where y is the solution of homogeneous Eq. (16.11) with inhomogeneous boundary and initial conditions k

k

y j+ p |γh,k = μ j+ p ,

y(x, 0) = u 0 (x),

where v, w—the solution of inhomogeneous Eq. (16.11) with homogeneous boundary and initial conditions. Thus, we get three problems:  s pj+k  k 1  − α1 t j+ k−s ε j+ kp − 1 t k−s+1 p − e α j+ p e y¯t¯ + y tp = Λk y j+ p , x ∈ ωh , p p s=1 k

k

y j+ p |γh,k = μ j+ p ,

(16.14)

y(x, 0) = u 0 (x),

 s pj+k  k ◦ j+ p k ε j+ kp 1  − α1 t j+ k−s − 1 t k−s+1 p − e α j+ p e vtp = Λk v j+ p + ϕk , x ∈ ωh , vt¯ + p p s=1 (16.15) k v j+ p |γh,k = 0, v(x, 0) = 0,  s pj+k  k ∗ j+ p k ε j+ kp 1  − α1 t j+ k−s − α1 t j+ k−s+1 p p wt¯ e wtp = Λk w j+ p + ϕk , x ∈ ωh , + −e p p s=1 (16.16) k w j+ p |γh,k = 0, w(x, 0) = 0, k ◦ j+ p

where ϕk

k ∗ j+ p

, ϕk

are defined by conditions k ◦ j+ p

ϕk

k ∗ j+ p

ϕk

◦

k ◦ j+ p

∗

k ∗ j+ p

= ϕk , x ∈ ω h , ϕ k = ϕk , x ∈ ω h , ϕ k

∗

= 0, x ∈ ω h , ◦

= 0, x ∈ ω h ,

16 The First Boundary Value Problem for Multidimensional Pseudoparabolic … ◦

∗

177

∗

so that ϕk + ϕk = ϕk at x ∈ ωh , i.e., ϕk is different from zero only in the border nodes. We get the estimate for y¯ . To do this, we rewrite the Eq. (16.14) in the canonical form. At the point P = P(xik , t j+ kp ), we have  

 a 1 τ j+ k 1 ak,ik ak,ik +1 j+ kp k,i k +1 y¯ik p = ε + 1 − e− α p + + + d y¯ + k ∗ ∗ τ h k h k+ h k h k− h k h ∗k+ ik +1 +

  j+ k−1 ak,ik j+ kp 1 − α1 t 1 −1t2 p + e α p ε + 1 − 2e y¯ik p + y ¯ + i k −1 ∗ h k h k− τ   j+ k−2 1 − α1 t 1p − α1 t 2 − α1 t 3 p p y¯ik p + · · · + e + − 2e +e τ

+

(16.17)

 1    1 − α1 t j+ k−2 1 − α1 t j+ k−1 − 1 t k−1 −1t k −1t k p − 2e α j+ p + e α j+ p p − e α j+ p y¯ikp + y¯i0k . e e τ τ

In [21], the maximum principle is proved and a priori estimates are obtained for the solution of the grid equations of the general form A(P)y(P) =



B(P, Q)y(Q) + F(P),

P ∈ Ω,

(16.17)

Q∈W  (P)

y(P) = μ(P) i f

P ∈ S,

where P, Q are the grid nodes of Ω+S, W (P)—the neighborhood of node P, not containing the node P itself. The coefficients A(P), B(P, Q) satisfy the conditions A(P) > 0,

B(P, Q) > 0,

D(P) = A(P) −



B(P, Q) ≥ 0.

(16.18)

Q∈W  (P)

We denote by P(x, t  ), where x ∈ ωh , t  ∈ ωτ a node of ( p + 1)-dimensional grid Ω = ωh × ωτ , by S—the boundary Ω, consisting of nodes P(x, 0) at x ∈ ω h as well as of nodes P(x, t j+ kp ) at t j+ kp ∈ ωτ and x ∈ γh,k for all k = 1, 2, . . . , p; j = ∗

∗

0, 1, . . . , j0 , Ω k —the set of nodes P(x, t j+ kp ), where x ∈ ω h,k —the node of the grid ω¯ h , a near-boundary one in the direction of xk . It is obvious that the coefficients of Eq. (16.17) at the point P = P(xik , t j+ kp ) satisfy the conditions (16.18) and D(P)=0. It follows from Theorem 4 [22, p. 347], that for the solution of problem (16.14), the following estimate is valid: μ(x, t  )Cγ , y j C ≤ u 0 C + max  0 0, τ

P(k) = P(x, t j+ kp ),  , Q ∈ Wk , for all Q ∈ Wk−1

 1 τ 1

ε + 1 − e− α p + τ

A(P(k) ) > 0,

B(P(k) , Q) > 0,

(16.21)

16 The First Boundary Value Problem for Multidimensional Pseudoparabolic …



B(P(k) , Q) =

 Q∈Wk−1

179



2  1 τ 1 > 0, ε + 1 − e− α p τ

2 − α1 τp ε + 1 − e 1  < 1,

B(P(k) , Q) = − α1 τp D  (P(k) )  ε + 1 − e Q∈Wk−1 

where W

  P x,t j+ k

  = Wk + Wk−1 , Wk —the set of nodes Q = Q(ξ, tk ) ∈ W(P(x,t , k ))

p

  Wk−1 —the set of nodes Q = Q(ξ, tk−1 ) ∈ W(P(x,t . k−1 )) On the basis of Theorem 4 [22, p. 347], by virtue of (16.21), we obtain the estimate

τ

k

v j+ p C ≤

j+

ε+1−e j+ kp

Let us estimate ϕk

j+ k ϕk p

− α1

τ p

ϕk

k p

C +

2 1 τ ε + 1 − e− α p ε+1−e

− α1

τ p

v j+

k−1 p

C .

(16.22)

C , where k ◦ j+ p

= ϕk

  j+ k−2 1 − α1 t 1p − α1 t 2 p vik p − e + −e τ

 s  pj+k−2  s−1 1  − α1 t j+ k−s − α1 t j+ k−s+1 p p p p e vik − vik = −e − τ s=1 k ◦ j+ p

= ϕk +

+

  1 − α1 t j+ k−1 −1t k p − e α j+ p vi0k + e τ

 1  1 − α1 t j+ k−2 − 1 t k−1 −1t k p − 2e α j+ p + e α j+ p vikp + · · · + e τ   j+ k−2 1 − α1 t 1p − α1 t 2 − α1 t 3 p p e vik p . + − 2e +e τ

(16.23)

Since the expressions in parentheses are positive, then from (16.23), we obtain the estimate   k ◦ j+ p s j+ kp 1 − α1 t 1p − α1 t 2 p max v j+ p C . e ϕk C ≤ ϕk C + −e (16.24) 0≤s≤k−2 τ With the help of (16.24), from (16.22), we find

180

M. Beshtokov s

τ

s

max v j+ p C ≤ max v j+ p C +

0≤s≤k

0≤s≤k−1

s ◦ j+ p

1 τ

ε + 1 − e− α p

max ϕ

0≤s≤k

C .

(16.25)

Summing (16.25) first by k = 1, 2, . . . , p, then by j  = 0, 1, . . . , j, we get the estimate j p s   ◦ j +p τ v j+1 C ≤ max ϕ C . (16.26) ε k=1 0≤s≤k j  =0 Now consider the problem (16.16) for w. We rewrite the problem (16.16) in canonical form 

  a 1 τ j+ kp 1 ak,ik ak,ik +1 j+ kp k,i k +1 ε + 1 − e− α p + + + d = w + k wi k ∗ ∗ τ h k h k+ h k h k− h k h ∗k+ ik +1   j+ kp j+ k−1 ak,ik 1 − α1 t 1 − α1 t 2 p p p w ε + 1 − 2e + w + + e + i −1 i k h k h ∗k− k τ +

  j+ k−2 1 − α1 t 1p −1t2 −1t3 e − 2e α p + e α p wik p + · · · + τ

+

  1 1 − α1 t j+ k−2 − 1 t k−1 −1t k p − 2e α j+ p + e α j+ p e wipk + τ   k ∗ j+ p 1 − α1 t j+ k−1 − α1 t j+ k p p + wi0k + ϕk , e −e τ k

w j+ p = 0, at x ∈ γh,k , w(x, 0) = 0, i.e., w = 0 on the boundary S of the grid Ω, i.e., w(P) = 0 at P ∈ S. ∗ ∗ The right-hand side ϕ is not equal to zero only in the nodes (x, t  ), x ∈ ω h . In these nodes, due to the homogeneous boundary condition w = 0, we have D(P) ≥ min k

c0 c0 = 2 , h = max h k . h h k h ∗k± h

(16.27)

On the basis of Theorem 3 [22, p. 344], we get ∗

 max |w(P)| ≤ max  Ω+S

t ∈wτ

ϕ(x, t  ) D

∗ ≤ C

∗ 1 max h 2 ϕ ∗ C c0 0 0 uniformly to the solution of differential problem (16.1)–(16.3) with 

2 converges 1 the rate O h 1 + τ 3 . τ3

References 1. Barenblatt, G.I., Zheltov, Y.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous fluids in fissurized rocks. J. Appl. Math. Mech. 24(5), 1286–1303 (1960) 2. Dzektser, E.S.: Equations of motion of ground water with free surface in multilayered media. Dokl. Akad. Nauk SSSR 220(3), 540–543 (1975) 3. Rubinshtein, L.I.: On the process of heat transfer in heterogenous media. Izv. Akad. Nauk SSSR. Ser. Geogr. 12(1), 27–45 (1948) 4. Ting, T.W.: A cooling process according to two temperature theory of heat conduction. J. Math. Anal. Appl. 45(9), 23–31 (1974) 5. Hallaire, M.: Le potentiel efficace de leau dans le sol en regime de dessechement. LEau et la Production Vegetale. Paris, Institut National de la Recherche Agronomique 9, 27–62 (1964) 6. Chudnovskii, A.F.: Thermal Physics of Soils, p. 353. Nauka, Moscow (1976) 7. Colton, D.L.: Pseudoparabolic equations in one space variable. J. Differ. Equ. 12, 559–565 (1972) 8. Coleman, B.D., Duffin, R.J., Mizel, V.J.: Instability, uniqueness, and nonexistence theorems for the equation u t = u x x − u x xt on a strip. Arch. Rat. Mech. Anal. 19, 100–116 (1965) 9. Rundell, W.: The construction of solutions to pseudoparabolic equations noncylindrical domains. J. Differ. Equ. 28, 394–404 (1978) 10. Shkhanukov, M.K.: On some boundary value problems for third-order equations arising in the modeling of flows in porous media. Differ. Equ. 18(4), 689–699 (1982) 11. Showalter, R.E., Ting, T.W.: Pseudoparabolic Partial Differential Equations. SIAM J. Math. Anal. 1(1), 1–26 (1970)

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12. Ting, T.W.: Certain non-steady flows of second-order fluids. Arch. Ration Mech. Anal. 14(1), 1–26 (1963) 13. Sveshnikov, A.A., Alshin, A.B., Korpusov, M.O., Pletner, Y.D.: Linear and Nonlinear SobolevType Equations, (Moscow: Fizmatlit, 2007), p. 736 14. Kozhanov, A.I.: On a nonlocal boundary value problem with variable coefficients for the heat equation and the Aller equation. Differ. Uravn. 40, 763 (2004) 15. Beshtokov, M.KH.: On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation. Differ. Uravn. 52(10), 1–14 (2016) 16. Beshtokov, M.KH.: difference method for solving a nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients. Comp. Math. Math. Phys. 56(10), 1763–1777 (2016) 17. Beshtokov, M.KH.: Local and nonlocal boundary value problems for degenerating and nondegenerating pseudoparabolic equations with a riemannliouville fractional derivative. Differ. Uravn. 54(6), 758–774 (2018) 18. Vabishchevich, P.N.: On a new class of additive (splitting) operator-difference schemes. Math. Comput. 81(277), 267–276 (2012) 19. Vabishchevich, P.N., Grigoriev, A.V.: Splitting schemes for pseudoparabolic equations. Differ. Uravn. 49(7), 837–843 (2013) 20. Shkhanukov-Lafishev, M.K., Arhestova, S.M., Tkhamokov, M.B.: Vector additive schemes for certain classes of hyperbolic equations. Vladikavkazskii matematicheskii zhurnal, 15(1), 71–84 (2013) 21. Samarskii, A.A.: Theory of Difference Schemes, p. 616. Nauka, Moscow (1983) 22. Samarskii, A.A., Gulin A.V.: Stability of Difference Schemes (Nauka, Moscow, 1973) p. 415

Chapter 17

Difference Methods of the Solution of Local and Non-local Boundary Value Problems for Loaded Equation of Thermal Conductivity of Fractional Order M. H. Beshtokov and M. Z. Khudalov Abstract We study local and non-local boundary value problems for a onedimensional space-loaded differential equation of thermal conductivity with variable coefficients with a fractional Caputo derivative, as well as difference schemes approximating these problems on uniform grids. For the solution of local and nonlocal boundary value problems by the method of energy inequalities, a priori estimates in differential and difference interpretations are obtained, which implies the uniqueness and stability of the solution from the initial data and the right side, as well as the convergence of the solution of the difference problem to the solution of the corresponding differential problem at the rate of O(h 2 + τ 2 ).

17.1 Introduction An important section in the theory of differential equations is loaded equations. The first works were devoted to the loaded integral equations. Works [1–4] are devoted to this class of the loaded equations. The importance of studying such equations was emphasized by Krylov, Smirnov, Tikhonov, and Samarsky, who gave examples of applied problems from engineering and physics reduced to loaded integral equations. Scientists [5–8] made a great contribution to the development of the theory of loaded differential equations. In the review works of Nakhushev, the practical and theoretical importance of studies of the loaded differential equations is shown by M. H. Beshtokov (B) Institute of Applied Mathematics and Automation, Kabardino-Balkariya, 360000 Nalchik, Russia e-mail: [email protected] M. Z. Khudalov North Ossetian State University after K.L. Khetagurov, Northern Ossetia, 362000 Vladikavkaz, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_17

187

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M. H. Beshtokov and M. Z. Khudalov

numerous examples. One of the methods of approximate solution of boundary value problems for differential equations is the method of reduction of integro-differential equations to the loaded differential equations proposed by Nakhushev. For the first time, this work [6] presented the relationship of non-local problems with loaded equations. Non-local problems of the Bitsadze–Samarsky type for the Laplace equations and thermal conductivity are equivalent to reduced local problems for loaded differential equations. Boundary value problems for differential equations of fractional order arise now in the description of many physical processes of stochastic transfer [9–11], in the study of fluid filtration in a highly porous (fractal) environment [12]. Sometimes these equations are also called the equations of slow diffusion (subdiffusion). It should be noted that the order of the fractional derivative is associated with the dimension of the fractal [10, 11], and the principles of calculating the fractal dimension are well known [13]. The works of the authors [14–16] are devoted to various boundary value problems for loaded partial differential equations.

17.2 Statement of the Problem In the rectangle Q T = {(x, t) : 0 ≤ x ≤ l, 0 ≤ t ≤ T } consider the following problem: ∂0tα u =

∂ ∂x

  ∂u ∂u k(x, t) + r (x, t) (x0 , t) − q(x, t)u + f (x, t), ∂x ∂x 0 < x < l, 0 < t ≤ T,

(17.1)

u(0, t) = u(l, t) = 0, 0 ≤ t ≤ T,

(17.2)

u(x, 0) = u 0 (x), 0 ≤ x ≤ l,

(17.3)

0 < c0 ≤ k(x, t) ≤ c1 , |k x (x, t), r (x, t), q(x, t)| ≤ c2 ,

(17.4)

where  t u τ (x,τ ) 1 ∂0tα u = Γ (1−α) 0 (t−τ )α dτ —a fractional derivative in the sense of Caputo of the order α, 0 < α < 1 [17], ci , i = 0, 1, 2—positive constant numbers, x0 —an arbitrary point of the interval [0, l]. In the future statement, we will assume that the problem (17.1)–(17.3) has the only solution that has the necessary derivatives in the course of presentation. We also assume that the coefficients of the equation and the boundary conditions satisfy the necessary conditions of smoothness, which provides the necessary order of approximation of the difference scheme. In the course of the presentation, we will also use positive constant numbers Mi , i = 1, 2, . . ., depending only on the input data of the problem.

17 Difference Methods of the Solution of Local and Non-local Boundary …

189

1. A priori estimation in differential form. To obtain an a priori estimate of the solution of the problem (17.1)–(17.3) in differential form, multiply Eq. (17.1) scalar by U = ∂0tα u − u x x :            ∂0tα u, U = ku x x , U + r u x (x0 , t), U − qu, U + f, U ,

(17.5)

     l where a, b = 0 abdx, a, a = a20 , where a, b—given by [0, l] functions. Transforming the integrals entering in the identity (17.5), using (17.2), Cauchy inequality with ε [18, p. 100] and by Lemma 1 [19], after simple transformations from (17.5), we find ∂0tα u x 20 + ∂0tα u20 + u x x 20

≤ ε1 M1 ∂0tα u20 + εM2 u x x 20 + M3ε,ε1 u x 20 + M4ε,ε1  f 20 .

Choosing ε1 =

1 ,ε 2M1

=

1 , 2M2

(17.6)

from (17.6), we find

∂0tα u x 20 + ∂0tα u20 + u x x 20 ≤ M5 u x 20 + M6  f 20 .

(17.7)

Applying to both parts of the inequality (17.7) the operator of fractional integration −α , we obtain D0t   −α u x 20 + D0t ∂0tα u20 + u x x 20   −α −α ≤ M7 D0t u x 20 + M8 D0t  f 20 + u 0 (x)20 .

(17.8)

On the basis of Lemma 2 [19] from (17.8) we find a priori evaluation of     −α −α ∂0tα u20 + u x x 20 ≤ M D0t  f 20 + u 0 (x)20 , u x 20 + D0t

(17.9)

−α where M − const > 0 is depending only on the input data (17.1)–(17.3). D0t u=  t udτ 1 —fractional Riemann–Liouville integral of order α, 0 < α < 1. Γ (α) 0 (t−τ )1−α

Theorem C 1,0 (Q T ), r (x, t), q(x, t), f (x, t) ∈ C(Q T ), u(x, t) ∈   17.11,0If k(x, t) ∈ α 2,0 QT ∩ C Q T , ∂0t u(x, t) ∈ C(Q T ) and conditions (17.4) are fulfilled, then C for the solution to problem (17.1)–(17.3) a priori estimate (17.9) holds true. From (17.9) follow the uniqueness and stability of the solution on the right side and the initial data. 2. Stability and convergence of the difference scheme for problem (17.1)–(17.3). To solve the problem (17.1)–(17.3), we apply the method of finite difference. Then in a closed cylinder Q T we introduce a uniform grid ω hτ = ω h × ω τ , where ω h = {xi = i h, i = 0, N , h = l/N }, ω τ = {t j = jτ , j = 0, 1, . . . , j0 , τ = T /j0 }. On

190

M. H. Beshtokov and M. Z. Khudalov

the uniform grid ω hτ differential problem (17.1)–(17.3) we put in accordance the difference scheme of the order of approximation O(h 2 + τ 2 ):   Δα0t j+σ yi = a j yx(σ) ¯

x,i

  j j (σ) j (σ) − (σ) + + ri yx,i x + y x + ϕi , ˚ 0 i0 ˚ 0 +1 i 0 − di yi x,i

(17.10)

y0(σ) = y N(σ) = 0,

(17.11)

y(x, 0) = u 0 (x),

(17.12)

j 1−α s where Δα0t j+σ y = Γ τ(2−α) s=0 c(α,σ) j−s yt is a discrete analogue of the fractional derivative of Caputo of order α, 0 < α < 1 with the order of approximation O(h 2 + τ 3−α ) [20].     a0(α,σ) = σ 1−α , al(α,σ) = l + σ bl(α,σ) =

− l −1+σ

1−α

, l ≥ 1,

 1

(l + σ)2−α − (l − 1 + σ)2−α 2−α  1

− (l + σ)1−α + (l − 1 + σ)1−α , l ≥ 1, 2

(17.13)

j = 0, c0(α,σ) = a0(α,σ) ;

if if

1−α

j > 0, cs(α,σ) = a0(α,σ) + b1(α,σ) , s = 0,

(α,σ) cs(α,σ) = as(α,σ) + bs+1 − bs(α,σ) , 1 ≤ s ≤ j − 1,

cs(α,σ) = a (α,σ) − b(α,σ) , s = j, j j   α j j (α,σ) ai = k xi−0.5 , t j+σ , ϕi = f (xi , t j+σ ), σ = 1 − , cs−1 > cs(α,σ) , 2 cs(α,σ) >

1−α j (s + σ)−α > 0, y (σ) = σ y j+1 + (1 − σ)y j , di = d(xi , t j+σ ), 2 ( j+σ)

r0 = r (0, t) = r0

( j+σ)

≤ 0, r N = r (l, t) = r N

xi0 ≤ x0 ≤ xi0 +1 , xi−0 =

≥ 0,

xi0 +1 − x0 + x0 − xi0 , xi0 = . h h

We find a priori estimate by the method of energy inequalities by introducing scalar products and the norm: N −1       u, v = u i vi h, u, u = 1, u 2 = u20 , i=1

17 Difference Methods of the Solution of Local and Non-local Boundary …



191

N    u, v = u i vi h, 1, u 2 = u]|20 . i=1

Multiply (17.10) scalar by y¯ = Δα0t j+σ y − yx(σ) ¯x :      Δα0t j+σ y, y¯ = ayx(σ) , y¯ ¯ x         j (σ) − (σ) + (σ) , y ¯ − dy + ri yx,i x + y x , y ¯ + ϕ, y ¯ . ˚ 0 i0 ˚ 0 +1 i 0 x,i

(17.14)

Estimate the amounts included in (17.14):     Δα0t j+σ y, y¯ = Δα0t j+σ y, Δα0t j+σ y − yx(σ) ¯x       2 = 1, Δα0t j+σ y − Δα0t j+σ y, yx(σ) = Δα0t j+σ y20 ¯x   (σ) α N α −yx(σ) ¯ Δ0t j+σ y|0 + yx¯ , Δ0t j+σ yx¯ 1 ≥ Δα0t j+σ y20 + Δα0t j+σ yx¯ ]|20 . 2

(17.15)

Converted amounts included in the identity (17.15), taking into account (17.11) and Lemma 1 [20]:        (σ)  (σ) α σ , y ¯ = ay , Δ y − (ay ) , y ) ayx(σ) 0t j+σ x¯ x ¯ x¯ x¯ x x x       (σ) α (σ) (σ) 2 = ayx¯ Δ0t j+σ y|0N − ayx¯ , Δα0t j+σ yx¯ − ax yx¯ , yx(σ) − a (+1) , (yx(σ) ¯x ¯x ) c0 α (σ) 2 (σ) 2 2 ε Δ yx¯ ]|20 − c0 yx(σ) (17.16) ¯ x 0 + εyx¯ x 0 + M1 yx¯ ]|0 . 2 0t j+σ        (σ) − j (σ) − (σ) (σ) + + r, Δα0t j+σ y − yx¯ x ri yx,i ˚ 0 x i 0 + yx,i ˚ 0 +1 x i 0 , y = yx,i ˚ 0 x i 0 + yx,i ˚ 0 +1 x i 0 ≤−

(σ) 2 ε,ε1 2 ≤ ε1 c22 Δα0t j+σ y20 + εc22 yx(σ) ¯ x 0 + M2 yx¯ ]|0 .

(17.17)

    2 − dy (σ) , y = − dy (σ) , Δα0t j+σ y − yx(σ) = ε1 Δα0t j+σ y20 + εyx(σ) ¯x ¯ x 0 (σ) 2 ε,ε1 2 +M3ε,ε1 y (σ) ]|20 ≤ ε1 Δα0t j+σ y20 + εyx(σ) ¯ x 0 + M4 yx¯ ]|0 ,

(17.18)



   (σ) (σ) 2 ε,ε1 α 2 2 ϕ, y¯ = ϕ, Δα 0t j+σ y − yx¯ x ≤ ε1 Δ0t j+σ y0 + εyx¯ x 0 + M5 ϕ0 .

(17.19) Taking into account the transformations (17.15)–(17.19), from (17.14), we find

192

M. H. Beshtokov and M. Z. Khudalov 2 α 2 Δα0t j+σ yx¯ ]|20 + Δα0t j+σ y20 + yx(σ) ¯ x 0 ≤ ε1 M6 Δ0t j+σ y0 (σ) 2 ε,ε1 ε,ε1 2 2 +εM7 yx(σ) ¯ x 0 + M8 yx¯ ]|0 + M9 ϕ0 .

Choosing ε1 =

1 ,ε 2M6

=

1 , 2M7

(17.20)

from (17.20) find

(σ) 2 2 2 Δα0t j+σ yx¯ ]|20 + Δα0t j+σ y20 + yx(σ) ¯ x 0 ≤ M10 yx¯ ]|0 + M11 ϕ0 .

(17.21)

Rewrite (17.21) in another form σ yx¯ Δα0t j+σ yx¯ ]|20 ≤ M12

j+1 2 ]|0

σ + M13 yx¯ ]|20 + M11 ϕ20 . j

(17.22)

Based on Lemma 7 [21] of (17.22), we obtain j+1 2 ]|0

yx¯

  j 2 ≤ M yx0¯ ]|20 + max ϕ  0 ,  0≤ j ≤ j

(17.23)

where M − const > 0 is independent of h and τ . Theorem 17.2 Let the conditions (17.4) be satisfied, then there is such a small τ0 , that is, if τ ≤ τ0 , then for the solution of the difference problem (17.10)–(17.12) the a priori estimate (17.23) is valid. From (17.23) follow the uniqueness and stability of the solution of the problem (17.10)–(17.12) on the initial data and the right side. j Let u(x, t) be the solution of the problem (17.1)–(17.3) and y(xi , t j ) = yi be the solution of the difference problem (17.10)–(17.12). To estimate the accuracy j of the difference scheme (17.10)–(17.12) consider the difference between the z i = j j j yi − u i , where u i = u(xi , t j ). Then, substituting y = z + u in the ratio (17.10)– (17.12), we obtain the problem for the function z   Δα0t j+σ z i = a j z (σ) x¯

x,i

  j j (σ) j (σ) − + + ri z (σ) x + z x ˚ 0 i0 ˚ 0 +1 i 0 − di z i + Ψi , x,i x,i

(17.24)

z 0(σ) = z (σ) N = 0,

(17.25)

z(x, 0) = 0,

(17.26)

  where Ψ = O h 2 + τ 2 —errors of approximation of the differential problem (17.1)– (17.3) ) by the difference scheme (17.10)–(17.12) in class of the solution u = u(x, t) of the problem (17.1)–(17.3).

17 Difference Methods of the Solution of Local and Non-local Boundary …

193

Applying the a priori estimate (17.23) to the solution of the problem (17.24)– (17.26), we obtain the inequality 

j+1

Ψ j 20 , z x¯ ]|20 ≤ M max  0≤ j ≤ j

(17.27)

where M − const > 0 is independent of h and τ . From the a priori estimate (17.27) follow the convergence of the solution of the difference problem (17.10)–(17.12) to the solution of the differential problem (17.1)– j+1 (17.3) in the sense of the norm z x¯ ]|20 0 on each layer so that there is τ0 , such that at τ ≤ τ0 the estimate j+1

yx¯

  j+1 − u x¯ ]|20 ≤ M h 2 + τ 2

is valid. 3. Setting a non-local boundary value problem and a priori estimation in differential form Consider now the non-local boundary value problem for Eq. (17.1) k(0, t)u x (0, t) = β11 (t)u(0, t) + β12 (t)∂0tα u(0, t) − μ1 (t),

(17.28)

−k(l, t)u x (l, t) = β21 (t)u(l, t) + β22 (t)∂0tα u(l, t) − μ2 (t), where 0 < c0 ≤ k, β12 , β22 ≤ c1 , |β11 , β21 , r, q, k x | ≤ c2 .

(17.29)

Multiply Eq. (17.1) scalar by U = u + ∂0tα u − u x x :            ∂0tα u, U = ku x x , U + r u x (x0 , t), U − qu, U + f, U .

(17.30)

Transform the components included in the identity (17.30)       ∂0tα u, U = (1, u∂0tα u) + 1, (∂0tα u)2 − u x x , ∂0tα u      1 1, ∂0tα u 2 + 1, (∂0tα u)2 − u x ∂0tα u|l0 + 1, u x ∂0tα u x ≥ 2 1 1 (17.31) = ∂0tα u20 + ∂0tα u x 20 + ∂0tα u20 − u x ∂0tα u|l0 , 2 2

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            ku x x , U = ku x x , u + ku x x , ∂0tα u − ku x x , u x x     = kuu x |l0 − (k, u 2x ) − k, u x ∂0tα u x + ku x ∂0tα u|l0 − k x u x , u x x   c0 − k, u 2x x ≤ kuu x |l0 − c0 u x 20 − ∂0tα u x 20 2 2 α l 2 −c0 u x x 0 + ku x ∂0t u|0 + εu x x 0 + M1ε u x 20 , (17.32)       r u x (x0 , t), U = u x (x0 , t)(r, u) + u x (x0 , t) r, ∂0tα u − u x (x0 , t) r, u x x   ≤ ε1 M2 ∂0tα u20 + εM3 u x x 20 + M4ε1 ,ε2 u 2x (x0 , t) + u20   ≤ ε1 M2 ∂0tα u20 + εM5 u x x 20 + M6ε,ε1 u20 + u x 20 , (17.33)         − qu, U = − q, u 2 − q, u∂0tα u + q, uu x x ≤ ε1 ∂0tα u20 + εu x x 20 + M7ε,ε1 u20 , 

 f, U

=



(17.34)

       f, u + ∂0tα u − u x x = f, u + f, ∂0tα u − f, u x x

≤ ε1 ∂0tα u20 + εu x x 20 + M8 u20 + M9ε,ε1  f 20 .

(17.35)

Taking into account the transformations (17.31)–(17.35), from (17.30), we find   1 c  1 α 0 ∂0t u20 + + ∂0tα u x 20 + ∂0tα u20 + c0 u x 20 + u x x 20 2 2 2 l ≤ uku x |0 + u x ∂0tα u|l0 + ∂0tα uku x |l0 + ε1 M10 ∂0tα u20   ε,ε1 ε,ε1 u20 + u x 20 + M13 +εM11 u x x 20 + M12  f 20 . (17.36) Transform the first, second, and third terms in the right part (17.36) uku x |l0 = u(l, t)k(l, t)u x (l, t) − u(0, t)k(0, t)u x (0, t) = μ2 (t)u(l, t) − β21 u 2 (l, t) − β22 (t)u(l, t)∂0tα u(l, t) +μ1 (t)u(0, t) − β11 (t)u 2 (0, t) β22 (t) α 2 β12 α 2 −β12 (t)u(0, t)∂0tα u(0, t) ≤ − ∂0t u (l, t) − ∂ u (0, t) 2 0t   2  +M13 u20 + u x 20 + M14 μ21 (t) + μ22 (t) .

(17.37)

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   α u|l = u (l, t) + μ (t) − β (t)u(l, t) − β (t)∂ α u(l, t) ∂ α u(l, t) u x + ku x ∂0t x 2 21 22 0t 0t 0   α α + μ1 (t) − u x (0, t) − β11 (t)u(0, t) − β12 (t)∂0t u(0, t) ∂0t u(0, t) 2 β   β22  α 12 α u(0, t) 2 ∂0t u(l, t) − ∂0t ≤− 2 2     2 ε 2 ε μ2 (t) + μ2 (t) . εu x x 0 + M15 u0 + u x 20 + M16 (17.38) 1 2

Accounting the transformations (17.37), (17.38) of (17.36), we find 1 c  1 α 0 ∂0t u20 + + ∂ α u x 20 + ∂0tα u20 2 2 2 0t   β (t) β12 α 2 22 ∂0tα u 2 (l, t) + ∂ u (0, t) +c0 u x 20 + u x x 20 + 2 2 0t   ε,ε1 ≤ ε1 M10 ∂0tα u20 + εM14 u x x 20 + M15 u20 + u x 20 ε,ε1 +M16  f 20 .

Choosing ε =

c0 , 2M14

ε1 =

1 , 2M10

(17.39) from (17.39), we find

∂0tα u2W 1 (0,l) + u x 20 + u x x 20 + ∂0tα u20 2   ≤ M19 u2W 1 (0,l) + M20  f 20 + μ21 + μ22 , 2

(17.40)

where u2W 1 (0,l) = u20 + u x 20 . 2

−α to both parts of the inequality Applying the operator of fractional integration D0t (17.40), we obtain an a priori estimate based on Lemma 2 [19] after simple transformations   −α u x 20 + u x x 20 + ∂0tα u20 u2W 1 (0,l) + D0t 2     −α ≤ M D0t (17.41)  f 20 + μ21 + μ22 + u 0 2W 1 (0,l) , 2

where M—positive constant that depends only on the input data of the problem (17.1), (17.28), (17.3). Theorem C 1,0 (Q T ), r (x, t), q(x, t), f (x, t) ∈ C(Q T ), u(x, t) ∈   17.31,0If k(x, t) ∈ α 2,0 QT ∩ C Q T , ∂0t u(x, t) ∈ C(Q T ) and conditions (17.4), (17.29) are fulC filled, then for the solution to problem (17.1), (17.28), (17.3) a priori estimate (17.41) holds true. From (17.41) follow the uniqueness and stability of the solution on the right side and the initial data.

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4. Stability and convergence of the difference scheme for problem (17.1), (17.28), (17.3). On a uniform grid ω hτ differential problem (17.1), (17.3), (17.28) we put in correspondence the difference scheme of the order of approximation O(h 2 + τ 2 ):   Δα0t j+σ yi = a j yx(σ) ¯

x,i

  j j (σ) j (σ) − (σ) + + ri yx,i x + y x + ϕi , ˚ 0 i0 ˚ 0 +1 i 0 − di yi x,i

(17.42)

  (σ) (σ) − (σ) + ˜ (σ) ˜ 1 + β˜12 Δα y0 , + 0.5hr0 yx,i x + y x a1 yx,0 0t j+σ ˚ 0 i0 ˚ 0 +1 i 0 = β11 y0 − μ x,i

(17.43)

  (σ) (σ) − (σ) + ˜ (σ) ˜ 2 + β˜22 Δα y N , + 0.5hr N yx,i − a N yx,N 0t j+σ ¯ ˚ 0 x i 0 + yx,i ˚ 0 +1 x i 0 = β21 y N − μ (17.44) (17.45) y(x, 0) = u 0 (x), x ∈ ω h , where j β˜11 (t j+σ ) = β11 (t j+σ ) + 0.5hd0 , β˜12 (t j+σ ) = β12 (t j+σ ) + 0.5h, j β˜21 (t j+σ ) = β21 (t j+σ ) + 0.5hd N , β˜22 (t j+σ ) = β22 (t j+σ ) + 0.5h, j

j

μ˜ 1 (t j+σ ) = μ1 (t j+σ ) + 0.5hϕ0 , μ˜ 2 (t j+σ ) = μ2 (t j+σ ) + 0.5hϕ N . Find a priori estimate by the method of energy inequalities, for this we multiply Eq. (17.42) by scalar y¯ = y (σ) + Δα0t j+σ y − yx(σ) ¯x : 

       j (σ) − (σ) + Δα0t j+σ y, y¯ = (ayx(σ) y ¯ ) , y ¯ + r x + y x x ¯ i ˚ 0 i0 ˚ 0 +1 i 0 , y x,i x,i     − dy (σ) , y¯ + ϕ, y¯ ,

where [u, v] =

N

(17.46)

u i vi ,  = 0.5h, i = 0, N  = h, i = 0, N ,

i=0

[u, u] = [1, u 2 ] = |[u]|20 , (u, v] =

N −1

u i vi h.

i=1

Transform the first and second terms in the right part (17.46)       (σ) (σ) (σ) (σ) α (ayx(σ) ) , y ¯ = (ay ) , y + Δ y − (ay ) , y x x x 0t j+σ ¯ x¯ x¯ x¯ x     N  

(σ) (σ) (σ) (σ) α (σ) α + ay y , y + Δ y + Δ y − a y , y − ayx(σ)

x ¯ x ¯ 0t j+σ 0t j+σ ¯ x¯ x¯ x¯ x¯ x 0

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  c0 α 2 Δ ≤ −c0 yx(σ) − a (+1) , yx(σ) yx¯ ]|20 ¯x ¯ ]|0 − 2 0t j+σ   N

(σ) 2 (σ) 2 (σ) 2 ε y (σ) + Δα0t j+σ y . (17.47) −c0 yx(σ) ¯ x 0 + εyx¯ x 0 + M1 yx¯ ]|0 + ayx¯ 0

Accounting (17.47), from (17.46), we get   c0 α 2 2 Δ yx¯ ]|20 + c0 yx(σ) Δα0t j+σ y, y¯ + c0 yx(σ) ¯ ]|0 + ¯ x 0 2 0t j+σ   N

(σ) (σ) (σ) α y ≤ ayx(σ) + Δ y

+ εyx¯ x 20 + +M2ε yx¯ ]|20 0t j+σ ¯ 0    x0 − xi0    (σ)   (σ) x i 0 +1 − x 0 (σ) , y ¯ − dy + r yx,i , y ¯ + ϕ, y ¯ . + y ˚ 0 ˚ 0 +1 x,i h h (17.48) Transform the first term in the right part (17.48), then we get     N

(σ) y (σ) + Δα0t j+σ y = a N yx,N y N(σ) + Δα0t j+σ y N ayx(σ) ¯ ¯ 0      (σ) (σ) α −a1 yx,0 y0 + Δ0t j+σ y0 = μ2 − β21 y N(σ) − β22 Δα0t j+σ y N y N(σ) + Δα0t j+σ y N   x0 − xi0  (σ) x i 0 +1 − x 0 (σ) + y +0.5h ϕ N − d N y N(σ) − Δα0t j+σ y N + r N yx,i ˚ 0 ˚ 0 +1 x,i h  h    (σ) (σ) (σ) α α α ∗ y N + Δ0t j+σ y N + μ1 − β11 y0 − β12 Δ0t j+σ y0 y0 + Δ0t j+σ y0

 x0 − xi0  (σ) x i 0 +1 − x 0 (σ) +0.5h ϕ0 − d0 y0(σ) − Δα0t j+σ y0 + r0 yx,i + yx,i ˚ 0 ˚ 0 +1 h h   (σ) α ∗ y0 + Δ0t j+σ y0 . (17.49) Transform the first and third terms in the right part (17.49), then we get    μ2 − β21 y N(σ) − β22 Δα0t j+σ y N y N(σ) + Δα0t j+σ y N    + μ1 − β11 y0(σ) − β12 Δα0t j+σ y0 y0(σ) + Δα0t j+σ y0  2  2 β β12 α 22 α Δ0t j+σ y N2 − Δ y2 ≤ −β22 Δα0t j+σ y N − β12 Δα0t j+σ y0 − 2 2 0t j+σ 0  2  2   2 +ε2 Δα0t j+σ y0 + ε3 Δα0t j+σ y N + M3ε2 ,ε3 |[y (σ) ]|20 + yx(σ) ]| 0 ¯    2 β  2 β β 22 12 22 α Δα0t j+σ y N − Δα0t j+σ y0 − Δ y2 +M4ε2 ,ε3 μ21 + μ22 ≤ − 2 2 2 0t j+σ N     β12 α ε2 ,ε3 2 2 2 Δ0t j+σ y02 + M3ε2 ,ε3 |[y (σ) ]|20 + yx(σ) + M μ (17.50) − ]| + μ 0 1 2 . 4 ¯ 2

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Considering (17.49) and (17.50) from (17.48) we find

 c0 α 2 2 Δ Δα0t j+σ y, y (σ) + Δα0t j+σ y + c0 yx(σ) yx¯ ]|20 + c0 yx(σ) ¯ ]|0 + ¯ x 0 2 0t j+σ 2 β  2 β β22  α β12 α 12 22 α Δ0t j+σ y N + Δα0t j+σ y0 + Δ0t j+σ y N2 + Δ0t j+σ y02 + 2 2 2 2       (σ) 2 2 ε (σ) 2 2 2 + εyx(σ) ≤ Δα0t j+σ y, yx(σ) ¯x ¯ x 0 + M6 |[y ]|0 + yx¯ ]|0 + M7 μ1 + μ2

  x0 − xi0  (σ) (σ) x i 0 +1 − x 0 (σ) + r yx,i + yx,i , y + Δα0t j+σ y ˚ 0 ˚ 0 +1 h h 



  (σ) (σ) α (σ) − dy , y + Δ0t j+σ y + ϕ, y + Δα0t j+σ y + dy (σ) , yx(σ) ¯x       x − x x − x 0 0 i0 (σ) i 0 +1 (σ) − r yx,i + yx,i , yxσ¯ x + ϕ, yx(σ) (17.51) ¯x . ˚ 0 ˚ 0 +1 h h

Transform the terms included in (17.51):



  2  Δα0t j+σ y, y (σ) + Δα0t j+σ y = 1, y (σ) Δα0t j+σ y + 1, Δα0t j+σ y ≥



1 α Δ |[y]|20 + |[Δα0t j+σ y]|20 . 2 0t j+σ

(17.52)

N    1 α

(σ) α α 2 Δα0t j+σ y, yx(σ) = yx(σ) ¯x ¯ Δ0t j+σ y − 1, yx¯ Δ0t j+σ yx¯ ≤ − Δ0t j+σ yx¯ ]|0 0 2     2  2  (σ) 2 (σ) 2 + ε2 Δα0t j+σ y0 + M5ε1 ε2 (yx,0 . +ε1 Δα0t j+σ y N ) + (y ) ¯ x,N ¯ (17.53)

  x0 − xi0  (σ) (σ) x i 0 +1 − x 0 (σ) α r yx,i + y , y + Δ y 0t ˚ 0 ˚ 0 +1 x,i j+σ h h 

  x − x x − x i +1 0 0 i (σ) 0 (σ) 0 (σ) α + y r, y = yx,i + Δ y 0t ˚ 0 ˚ 0 +1 x,i j+σ h h 2 2



2 x − x 1  (σ) xi0 +1 − x0 0 i0 (σ) (σ) α yx,i + y ≤ + ε r, y + ε r, Δ y 0t j+σ ˚ 0 +1 x,i 4ε ˚ 0 h h   ε,ε1 2 2 |[y (σ) ]|20 + yx(σ) ≤ ε1 M6 |[Δα0t j+σ y]|20 + εyx(σ) ¯ x 0 + M7 ¯ ]|0 .

(17.54)





 − dy (σ) , y (σ) + Δα0t j+σ y = − d, (y (σ) )2 − dy (σ) , Δα0t j+σ y ≤ ε1 |[Δα0t j+σ y]|20 + M8ε1 |[y (σ) ]|20 .

(17.55)

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 ϕ, y (σ) + Δα0t j+σ y = ϕ, y (σ) + ϕ, Δα0t j+σ y ≤ ε|[Δα0t j+σ y]|20 + M9ε |[y (σ) ]|20 + |[ϕ]|20 .

 x0 − xi0  (σ)  (σ) x i 0 +1 − x 0 (σ) + y , yx¯ x − r yx,i ˚ 0 ˚ 0 +1 x,i h h  x0 − xi0  (σ)  (σ) x i 0 +1 − x 0 (σ) r, yx¯ x = − yx,i + y ˚ 0 ˚ 0 +1 x,i h h (σ) 2 ε,ε1 ε 2 yx(σ) ≤ εM10 ¯ x 0 + M12 yx¯ ]|0 .     (σ) 2 ε (σ) 2 dy (σ) , yx(σ) − ϕ, y ≤ |[ϕ]|20 + εyx(σ) ¯x x¯ x ¯ x 0 + M13 |[y ]|0 .

(17.56)

(17.57) (17.58)

Accounting the transformations (17.52)–(17.58), from (17.51), we get c 1 α 1 α 0 (σ) 2 2 Δ0t j+σ |[y]|2 + |[Δα0t j+σ y]|20 + + Δ yx¯ ]|20 + c0 yx(σ) ¯ ]|0 + c0 yx¯ x 0 2 2 2 0t j+σ 2 β  2 β β22  α β12 α 12 22 α Δ0t j+σ y N + Δα0t j+σ y0 + Δ0t j+σ y N2 + Δ + y2 2 2 2 2 0t j+σ 0  2  2 2 α 2 ≤ ε2 Δα0t j+σ y N + ε3 Δα0t j+σ y0 + εM14 yx(σ) ¯ x 0 + ε1 M15 |[Δ0t j+σ y]|0     ε,ε1 ,ε2 ,ε3 ε,ε1 ,ε2 ,ε3 2 |[y (σ) ]|20 + yx(σ) |[ϕ]| + μ21 + μ22 . (17.59) +M16 ¯ ]|0 + M17 Choosing ε =

c0 , ε1 2M14

=

1 , 2M15

ε2 =

β22 , 2

ε3 =

β12 , 2

from (17.59), we find

(σ) 2 2 α 2 Δα0t j+σ |[y]|2W 1 (0,l) + yx(σ) ¯ ]|0 + yx¯ x 0 + |[Δ0t j+σ y]|0 2   ≤ M18 |[y (σ) ]|2W 1 (0,l) + M19 |[ϕ]|20 + μ21 + μ22 , 2

(17.60)

where |[y]|2W 1 (0,l) = |[y]|20 + yx¯ ]|20 . 2 Rewrite (17.60) in another form σ |[y j+1 ]|2W 1 (0,l) Δα0t j+σ |[y j ]|2W 1 (0,l) ≤ M20 2 2   σ j 2 2 +M21 |[y ]|W 1 (0,l) + M22 |[ϕ]|0 + μ21 + μ22 . 2

(17.61)

Based on Lemma 7 [21] from (17.61) we find a priori estimate    j 2 2 2 |[ϕ , |[y j+1 ]|2W 1 (0,l) ≤ M |[y 0 ]|2W 1 (0,l) + max ]| + μ + μ 0 1 2  2

2

0≤ j ≤ j

where M − const > 0 is independent of h and τ .

(17.62)

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Theorem 17.4 Let the conditions (17.4), (17.29) be satisfied, then there is such a small τ0 , that is, if τ ≤ τ0 , then for the solution of the difference problem (17.42)– (17.45) the a priori estimate (17.62) is valid. From (17.62) follow the uniqueness and stability of the solution of the problem (17.42)–(17.45) on the initial data and the right side. j Let u(x, t) be a solution of the problem (17.1), (17.3), (17.28) and y(xi , t j ) = yi , a solution of the difference problem (17.42)–(17.45). For assessing the accuracy of the j j j difference scheme (17.42)–(17.45) consider the difference between z i = yi − u i j and u i = u(xi , t j ). Then, substituting y = z + u in the ratio (17.42)–(17.45), we obtain the problem for the function z     j j (σ) j (σ) − + + ri z (σ) Δα0t j+σ z = a j z (σ) x¯ ˚ 0 x i 0 + z x,i ˚ 0 +1 x i 0 − di z i + Ψi , x,i

(17.63)

  (σ) − (σ) + α ˜ (σ) ˜ z + 0.5hr x + z x ˜1 , a1 z (σ) 0 x,0 ˚ 0 i0 ˚ 0 +1 i 0 = β11 z 0 + β12 Δ0t j+σ z 0 − ν x,i x,i

(17.64)

x

  (σ) − (σ) + −a N z (σ) z + 0.5hr x + z x N x,N ¯ ˚ 0 i0 ˚ 0 +1 i 0 x,i x,i α ˜ ˜2 , = β˜21 z (σ) N + β22 Δ0t j+σ z N − ν

(17.65)

z(x, 0) = 0, x ∈ ω h ,

(17.66)

      where Ψ = O h 2 + τ 2 , ν˜1 = O h 2 + τ 2 , ν˜2 = O h 2 + τ 2 —the errors of approximation of the differential problem (17.1), (17.4), (17.28) of difference scheme (17.42)–(17.45) in the class of the solution u = u(x, t) of the problem (17.1), (17.4), (17.28). Applying a priori estimate (17.62) to the solution of the problem (17.63)–(17.66), we obtain the inequality |[z j+1 ]|2W 1 (0,l) ≤ M max  2

0≤ j ≤ j





j 2

j 2

|[Ψ j ]|20 + ν1 + ν2

 ,

(17.67)

where M − const > 0 is independent of h and τ . From the a priori assessment (17.67) follow convergence of the solution of the difference problem (17.42)–(17.45) to the solution of the differential problem (17.1), (17.28), (17.4) ) in the sense of the norm |[z j+1 ]|2W 1 (0,l) for each layer so that there is 2 a τ0 , such that if τ ≤ τ0 fair assessment   |[y j+1 − u j+1 ]|W21 (0,l) ≤ M h 2 + τ 2 .

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References 1. Knezer, A.: Belastete Integralgleichungen. Rend. Sircolo Mat. Palermo 37, 169–197 (1914) 2. Lichtenstein, L.: Vorlesungen uber einege Klassen nichtlinear Integralgleichungen und Integraldifferentialgleihungen nebst Anwendungen. Springer, Berlin (1931) 3. Nazarov, N.N.: On a new class of linear integral equations. Tr. Inst. Mat. Mekh. Akad. Nauk, Uz. SSR, 4, 77–106 (1948) 4. Gabib-Zade, A.S.: Investigation of solutions to a class of linear loaded integral equations. Tr. Inst. Fiz. Mat. Akad. Nauk Az. SSR Ser. Mat. 8, 177-182 (1959) 5. Budak, V.M., Iskenderov, A.D.: On a class of boundary value problems with unknown coefficients. Sov. Math. Dokl. 8, 786–789 (1967) 6. Nakhushev, A.M.: Loaded equations and applications. Diff. Urav. 19(1), 86–94 (1983) 7. Kaziev, V.M.: The tricomi problem for a loaded Lavrentev-Bitsadze equation. Diff. Urav. 15(1), 173–175 (1979) 8. Krall, A.M.: The development of general differential and general differential boundary systems. Rock Mount. J. Math. 5(4), 493–542 (1975) 9. Chukbar, K.V.: Stochastic transport and fractional derivatives. Zh. Eksp. Teor. Fiz. 108, 1875– 1884 (1995) 10. Kobelev, V.L., Kobelev, Y.L., Romanov, E.P.: Non-debye relaxation and diffusion in fractal space. Dokl. Akad. Nauk 361(6), 755–758 (1998) 11. Kobelev, V.L., Kobelev, Ya L., Romanov, E.P.: Self-maintained processes in the case of nonlinear fractal diffusion. Dokl. Akad. Nauk 369(3), 332–333 (1999) 12. Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with the fractal geometry. Phys. Stat. Sol. (b) 133, 425-430 (1986) 13. Dinariev, O.Y.: Filtering in a cracked medium with fractal crack geometry. Izv. Ross. Akad. Nauk, Ser. Mekh. Zhidkosti Gaza 5, 66-70 (1990) 14. Beshtokov, M.KH.: The third boundary value problem for loaded differential Sobolev type equation and grid methods of their numerical implementation. IOP Conf. Ser.: Mater. Sci. Eng. 158(1), (2016) 15. Beshtokov, M.KH.: Differential and difference boundary value problem for loaded third-order pseudo-parabolic differential equations and difference methods for their numerical solution. Comput. Math. Math. Phys. 57(12), 1973–1993 (2017) 16. KHudalov, M.Z.: A nonlocal boundary value problem for a loaded parabolic type equation. Vladikavkazskii matematicheskii zhurnal 4(4), 59–64 (2002) 17. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Integrals and Derivatives of Fractional Order and Their Certain Applications Nauka i Tekhnika, p. 688. Minsk (1987) 18. Samarskii, A.A.: Theory of Difference Schemes, p. 616. Nauka, Moscow (1983) 19. Alikhanov, A.A.: A priori estimates for solutions of boundary value problems for equations of fractional order. Differ. Equ. 46(5), 660–666 (2010) 20. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015) 21. Beshtokov, M.KH.: To boundary-value problems for degenerating pseudoparabolic equations with gerasimov caputo fractional derivative. Russ. Math. 62(10), 1–14 (2018)

Chapter 18

A Class of Semilinear Degenerate Equations with Fractional Lower Order Derivatives Marina V. Plekhanova and Guzel D. Baybulatova

Abstract Unique solution existence is proved for the generalized Showalter– Sidorov problem to semilinear evolution equations with a degenerate linear operator at the highest fractional Gerasimov–Caputo derivative and with some constraints on the nonlinear operator. The nonlinear operator in the equation depends on lower order fractional derivatives. Abstract result is used for the study of an initial boundary value problem for the system of the dynamics of Kelvin–Voigt viscoelastic fluid, in which rheological relation is defined by a fractional order equation.

18.1 Introduction Let X, Y be Banach spaces. Consider the problem (P x)(k) (t0 ) = xk , k = 0, . . . , m − 1,

(18.1)

for the semilinear equation of fractional order Dtα L x(t) = M x(t) + N (t, Dtα1 x(t), Dtα2 x(t), . . . , Dtαn x(t)),

(18.2)

where L ∈ L(X; Y) (linear and continuous operator from X into Y), M ∈ Cl(X; Y), i. e., it is linear closed operator having a dense domain D M in the space X and acting into Y, N : R × Xn → Y is a nonlinear operator, n ∈ N, Dtα , Dtα1 , Dtα2 , . . . , Dtαn are the fractional Gerasimov–Caputo derivatives, 0 ≤ α1 < α2 < · · · < αn ≤ m − 1 < α ≤ m ∈ N. Suppose that ker L = {0}, in this case, the equation is called degenerate. The projection P on the complement X1 of the degeneracy subspace will be defined M. V. Plekhanova (B) South Ural State University, Chelyabinsk, Russia e-mail: [email protected] M. V. Plekhanova · G. D. Baybulatova Chelyabinsk State University, Chelyabinsk, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_18

203

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further. A class of semilinear degenerate evolution equations with fractional lower order derivatives is investigated here. Firstly, we prove the theorem on unique solution existence of the Cauchy problem for the corresponding nondegenerate equation. The next reasoning on unique solvability of problem (18.1), (18.2) is based on this theorem. Abstract result is used for the study of an initial boundary value problem for the system of the dynamics of Kelvin–Voigt viscoelastic fluid, in which rheological relation [1] is described by a fractional order equation [2, 3]. Initial boundary value problems for fractional order with respect to time evolution equations and systems of equations, arising in mathematical physics, are studied by many authors [4–6]. Degenerate linear fractional differential equations were investigated at many works [7–10]. The closest results on unique solvability of fractional order degenerate semilinear equations, but with integer order lower derivatives are obtained in [11–13].

18.2 Nondegenerate Equations t Denote gδ (t) := t δ−1 / (δ), Jtδ h(t) := t0 gδ (t − s)h(s)ds for δ > 0, t > 0, g˜ δ (t) := (t − t0 )δ−1 / (δ). Let m − 1 < α ≤ m ∈ N, Dtm is the derivative of the integer order m ∈ N, and Jt0 is the identical operator. The Gerasimov–Caputo derivative of h has the form (see [14, p. 11])  Dtα h(t) := Dtm Jtm−α h(t) −

m−1 

 h (k) (t0 )g˜ k+1 (t) , t ≥ t0 .

k=0

Define Dt0 h(t) := h(t) also. Let Z be a Banach space. Consider the Cauchy problem z (k) (t0 ) = z k , k = 0, 1, . . . , m − 1,

(18.3)

for the inhomogeneous differential equation Dtα z(t) = Az(t) + f (t), t ∈ [t0 , T ],

(18.4)

where A ∈ L(Z) := L(Z; Z), T > t0 , the mapping f : [t0 , T ] → Z is known. A solution of problem (18.3), (18.4) is such z ∈ C m−1 ([t0 , T ]; Z), that  Jtm−α z −

m−1 

 z (k) (t0 )g˜ k+1

k=0

and equalities (18.3), (18.4) are valid.

∈ C m ([t0 , T ]; Z)

18 A Class of Semilinear Degenerate Equations with Fractional …

205

Theorem 18.1 ([11]) Let α > 0, A ∈ L(Z), f ∈ C([t0 , T ]; Z), z k ∈ Z, k = 0, . . . , m − 1. Then there exists a unique solution of problem (18.3), (18.4). It has the form z(t) =

m−1 

α

t

(t − t0 ) E α,k+1 (A(t − t0 ) )z k + k

k=0

(t − s)α−1 E α,α (A(t − s)α ) f (s)ds.

t0

(18.5) Here E α,β (z) :=

∞  n=0

zn (αn+β)

is the Mittag–Leffler function.

Let Z be an open set in R × Zn , n ∈ N, B : Z → Z, 0 ≤ α1 < α2 < · · · < αn ≤ m − 1, z k ∈ Z, k = 0, 1, . . . , m − 1. Consider the problem z (k) (t0 ) = z k , k = 0, 1, . . . , m − 1,

(18.6)

Dtα z(t) = Az(t) + B(t, Dtα1 z(t), Dtα2 z(t), . . . , Dtαn z(t)).

(18.7)

A solution of (18.6), (18.7) on a segment [t0 , t1 ] is such z ∈ C m−1 ([t0 , t1 ]; Z), that  Jtm−α z −

m−1 

 z (k) (t0 )g˜ k+1

∈ C m ([t0 , t1 ]; Z),

k=0

for any t ∈ [t0 , t1 ] (t, Dtα1 z(t), Dtα2 z(t), . . . , Dtαn z(t)) ∈ Z , equalities (18.6) and (18.7) for all t ∈ [t0 , t1 ] are valid. We shall use the inequality β

Dt hC([t0 ,t1 ];Z) ≤ ChC l ([t0 ,t1 ];Z) ,

(18.8)

which is valid for all h ∈ C l ([t0 , t1 ]; Z), where l − 1 < β ≤ l ∈ N (see [15, Lemma 1]). Using (18.8) and Theorem 18.1, we can easily obtain the next assertion. Lemma 18.1 Let α > 0, A ∈ L(Z), B ∈ C(Z ; Z), 0 ≤ α1 < α2 < · · · < αn ≤ m − 1. Then a function z ∈ C m−1 ([t0 , t1 ]; Z) is a solution of problem (18.6), (18.7), if and only if for all t ∈ [t0 , t1 ] (t, Dtα1 z(t), . . . , Dtαn z(t)) ∈ Z and z(t) =

m−1 

(t − t0 )k E α,k+1 (A(t − t0 )α )z k +

k=0

t + t0

(t − s)α−1 E α,α (A(t − s)α ) B(s, Dtα1 z(s), Dtα2 z(s), . . . , Dtαn z(s))ds. (18.9)

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Denote x¯ = (x1 , x2 , . . . , xn ). Here, Sδ (x) ¯ = { y¯ ∈ Zn : yk − xk Z ≤ δ, k = 1, 2, . . . , n}. A mapping B : Z → Z is called locally Lipschitz continuous in z¯ , ¯ ⊂Z if for all (t, x) ¯ ∈ Z we have such δ > 0 and l > 0, that [t − δ, t + δ] × Sδ (x) and for any ¯ y¯ ) − B(s, v¯ )Z ≤ l (s, y¯ ), (s, v¯ ) ∈ [t − δ, t + δ] × Sδ (x)B(s,

n k=1

yk − vk Z .

Using the initial data z 0 , z 1 , . . . , z m−1 from (18.6) denote z˜ (t) := z 0 + z 1 (t − t0 ) + · · · +

z m−1 (t − t0 )m−1 , (m − 1)!

z˜ 1 := Dtα1 |t=t0 z˜ (t), z˜ 2 := Dtα2 |t=t0 z˜ (t), . . . , z˜ n := Dtαn |t=t0 z˜ (t). Theorem 18.2 Let A ∈ L(Z), a set Z be open in R × Zn , B ∈ C(Z ; Z) be locally Lipschitz continuous in z¯ , z k ∈ Z, k = 0, 1, . . . , m − 1, such that (t0 , z˜ 1 , z˜ 2 , . . . , z˜ n ) ∈ Z . Then there exists a unique solution of problem (18.6), (18.7) on a segment [t0 , t1 ] with some t1 > t0 . Proof Due to Lemma 18.1, it is sufficient to prove that (18.9) has a unique solution z ∈ C m−1 ([t0 , t1 ]; Z) for some t1 > t0 . Choose τ > 0 and δ > 0 such that V := [t0 , t0 + τ ] × Sδ (˜z ) ⊂ Z . Denote by S the set of all y ∈ C m−1 ([t0 , t0 + τ ]; Z) such that for all t ∈ [t0 , t0 + τ ] we (k) Z ≤ δ, k = 0, 1, . . . , m − 1. Define on S the metric d(y, v) := have m−1y (t) − z k(k) sup y (t) − v(k) (t)Z . Then S is a complete metric space and z˜ ∈ S at k=0 t∈[t0 ,t0 +τ ]

sufficiently small τ > 0. Due to (18.8) at small τ > 0 for y ∈ S and for all t ∈ [t0 , t0 + τ ], we have (t, Dtα1 y(t), , Dtαn y(t)) ∈ Z , define G(y)(t) =

m−1 

(t − t0 )k E α,k+1 (A(t − t0 )α )z k +

k=0

t +

(t − s)α−1 E α,α (A(t − s)α )B(s, Dtα1 y(s), Dtα2 y(s), . . . , Dtαn y(s))ds

t0

and show that operator G is contraction on S, if τ > 0 is sufficiently small. Indeed, for r = 0, 1, . . . , m − 1 G (r ) (y)(t) =

r −1  k=0

(t − t0 )α+k−r AE α,α+k+1−r (A(t − t0 )α )z k +

18 A Class of Semilinear Degenerate Equations with Fractional …

+

m−1 

207

(t − t0 )k−r E α,k+1−r (A(t − t0 )α )z k +

k=r

t +

(t − s)α−r −1 E α,α−r (A(t − s)α )B(s, Dtα1 y(s), Dtα2 y(s), . . . , Dtαn y(s))ds.

t0

Put K = y∈S

max B(t, Dtα1 z˜ (t), Dtα2 z˜ (t), . . . , Dtαn z˜ (t))Z . According to (18.8), for

t∈[t0 ,t0 +τ ]

B(t, Dtα1 y(t), Dtα2 y(t), . . . , Dtαn y(t))Z ≤

≤ B(t, Dtα1 y(t), . . . , Dtαn y(t)) − B(t, Dtα1 z˜ (t), . . . , Dtαn z˜ (t))Z + K ≤ n 

≤l

α

α

Dt k y(t) − Dt k z˜ (t)Z + K ≤ Cln

m−1 

k=1

≤ Cln

m−1 

sup

k=0 t∈[t0 ,t0 +τ ]

(k)

y (t) − z k Z +

sup

k=0 t∈[t0 ,t0 +τ ] m−1 

y (k) (t) − z˜ (k) (t)Z + K ≤

 sup

k=0 t∈[t0 ,t0 +τ ]

(k)

˜z (t) − z k Z + K ≤

≤ 2Clmnδ + K . Then for all t ∈ [t0 , t0 + τ ] G (r ) (y)(t) − zr Z ≤

r −1 

τ α+k−r AL(Z) E α,α+k+1−r (AL(Z) τ α )z k Z +

k=0 m−1    +  E α,1 (A(t − t0 )α )zr − zr Z + τ k−r E α,k+1−r (AL(Z) τ α )z k Z + k=r +1

+



τ α−r E α,α−r AL(Z) τ α (2Clmnδ + K ) ≤ δ α−r

for sufficiently small τ . Therefore, G : S → S. Moreover, for sufficiently small τ at all t ∈ [t0 , t0 + τ ], r = 0, 1, . . . , m − 1, y, v ∈ S G (r ) (y)(t) − G (r ) (v)(t)Z ≤  d(y, v) τ α−r E α,α−r (AL(Z) τ α )Cln . sup y (k) (t) − v(k) (t)Z ≤ α −r 2m k=0 t∈[t0 ,t0 +τ ] m−1

≤

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Thus, d(G(y), G(v)) ≤ d(y, v)/2 and the mapping G has a unique fixed point in S. It is a solution of problem (18.6), (18.7) on [t0 , t0 + τ ].

18.3 Degenerate Case Suppose L ∈ L(X; Y), ker L = {0}, M ∈ Cl(X; Y), D M is a domain of M with the norm  ·  D M :=  · X + M · Y . Denote ρ L (M) := {μ ∈ C : (μL − M)−1 ∈ L(Y; X)}, σ L (M) := C\ρ L (M), RμL (M) := (μL − M)−1 L, L μL := L(μL − M)−1 . An operator M is called (L , σ )-bounded, if there exists a > 0 such that for all μ ∈ C, if |μ| > a, then μ ∈ ρ L (M). Take γ := {μ ∈ C : |μ| = r > a}. Then P =:

1 2πi

 RμL (M) dμ ∈ L(X),

Q :=

γ

1 2πi

 L μL (M) dμ ∈ L(Y) γ

are projections [16]. Define X0 := ker P, X1 := im P, Y0 := ker Q, Y1 := imQ. k Denote by L k (Mk ) the restriction of the operator

on X (D0Mk :=

DM ∩ 1 L (M) k 1 X ), k = 0, 1. In [16], it is proved that M1 ∈ L X ; Y , M0 ∈ Cl X ; Y0 , L k ∈ k k

; Y , k = 0, 1, there exist inverse operators M0−1 ∈ L Y0 ; X0 , L −1 1 ∈L L X Y 1 ; X1 . Denote G := M0−1 L 0 . For p ∈ N0 := N ∪ {0} operator M is called (L , p)bounded, if it is (L , σ )-bounded, G p = 0, G p+1 = 0. For the degenerate equation Dtα L x(t) = M x(t) + N (t, Dtα1 x(t), Dtα2 x(t), . . . , Dtαn x(t)),

(18.10)

consider the generalized Showalter–Sidorov problem (P x)(k) (t0 ) = xk , k = 0, 1, . . . , m − 1,

(18.11)

where 0 ≤ α1 < α2 < · · · < αn ≤ m − 1 < α ≤ m ∈ N. Let r − 1 < αn ≤ r ∈ N, N : X → Y be a mapping, where X is an open subset of R × Xn . A solution of problem (18.11), (18.10) is a function x ∈ C([t0 , t1 ]; D M ) ∩ C r ([t0 , t1 ]; X), for which L x ∈ C m−1 ([t0 , t1 ]; Y),  Jtm−α

Lx −

m−1 

 (k)

(L x) (t0 )g˜ k+1

∈ C m ([t0 , t1 ]; Y),

k=0

for each t ∈ [t0 , t1 ] (t, Dtα1 x(t), Dtα2 x(t), . . . , Dtαn x(t)) ∈ X and equalities (18.11), (18.10) for all t ∈ [t0 , t1 ] hold. Denote

18 A Class of Semilinear Degenerate Equations with Fractional …

x˜ = x0 +

209

x1 x2 xm−1 (t − t0 ) + (t − t0 )2 + · · · + (t − t0 )m−1 , 1! 2! (m − 1)!

for xk from (18.11), k = 0, 1, . . . , m − 1, ˜ x˜2 = Dtα2 |t=t0 x(t), ˜ . . . , x˜n = Dtαn |t=t0 x(t). ˜ x˜1 = Dtα1 |t=t0 x(t), ], M is (L , 0)-bounded, X is an open set in Theorem 18.3 Assume that r ≤ [ m−1 2 R × Xn , N : X → Y; V := X ∩ (R × (X1 )n ), for each (t, z˜ 1 , z˜ 2 , . . . , z˜ n ) ∈ X , such that (t, P z˜ 1 , P z˜ 2 , . . . , P z˜ n ) ∈ V , the equality N (t, z˜ 1 , . . . , z˜ n ) = N1 (t, P z˜ 1 , . . . , P z˜ n ) holds with some N1 : V → Y, the mapping Q N1 : V → Y is locally Lipschitz continuous in x, ¯ (I − Q)N1 ∈ C r (V ; Y). Then for all xk ∈ X1 , k = 0, 1, . . . , m − 1, such that (t0 , x˜1 , x˜2 , . . . , x˜n ) ∈ V , there exists a unique solution of problem (18.10), (18.11) on [t0 , t1 ] for some t1 > t0 . Proof Multiply (18.10) from the left by M0−1 (I − Q) and L −1 1 Q alternately. Then we obtain the problem αn α1 α2 Dtα v(t) = S1 v(t) + L −1 1 Q N1 (t, Dt v(t), Dt v(t), . . . , Dt v(t)), (k) v (t0 ) = xk , k = 0, 1, . . . , m − 1,

(18.12)

0 = w(t) + M0−1 (I − Q)N1 (t, Dtα1 v(t), Dtα2 v(t), . . . , Dtαn v(t))

(18.13)

−1 for v(t) := P x(t), w(t) := (I − P)x(t). Here S1 = L −1 1 M1 . Since L 1 Q N1 is locally Lipschitzian with respect to v¯ , there exists a unique solution v of problem (18.12) on [t0 , t1 ] by Theorem 18.2. Then we have the equality

w(t) = −M0−1 (I − Q)N1 (t, Dtα1 v(t), Dtα2 v(t), . . . , Dtαn v(t)). The inequality αn + r ≤ m − 1 and the inclusion v ∈ C m−1 ([t0 , t1 ]; X1 ) imply that w ∈ C r ([t0 , t1 ]; X0 ), besides, Lw ≡ 0. Consequently, problem (18.11), (18.10) is uniquely solvable with the solution x = v + w. Remark 18.1 In [15] problem (18.11), (18.10) is investigated for (L , p)-bounded operator M, p ∈ {0} ∪ N, but with other restrictions on the nonlinear operator N .

18.4 Example Consider initial boundary value problem ∂kv (x, 0) = ψk (x), x ∈ , k = 0, 1, . . . , m − 1, ∂t k

(18.14)

v(x, t) = 0, (x, t) ∈ ∂ × [t0 , T ),

(18.15)

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− χ Dtα v = νv − (v · ∇)v − Dtα0 vt − r + f, (x, t) ∈ × [t0 , T ), (18.16) ∇ · v = 0, (x, t) ∈ × [t0 , T ).

(18.17)

Here ⊂ Rd is a region, which is having a smooth boundary ∂ , χ , ν ∈ R. Functions v = (v1 , v2 , . . . , vd ) of the velocity and r = (r1 , r2 , . . . , rd ) of the pressure gradient are unknown, f : × [t0 , T ) → Rd is given, m − 1 < α ≤ m ∈ N, α0 ∈ [0, α). To reduce initial boundary value problem (18.14)–(18.17) to abstract problem (18.11), (18.10), denote the spaces L2 := (L 2 ( ))d , H1 := (W21 ( ))d , H2 := (W22 ( ))d . A closure of L := {v ∈ (C0∞ ( ))d : ∇ · v = 0} in L2 is denoted by Hσ ; H1σ is closure of L in H1 . Also, we use the denotation H2σ := H1σ ∩ H2 . An orthogonal complement for Hσ in L2 is denoted by Hπ . The corresponding orthoprojectors are  : L2 → Hσ ,  = I − . Consider the operator  =  in L. It is known that this operator with a domain H2σ , extended to a closed operator in Hσ , has a real, negative discrete spectrum of finite multiplicity, condensing at −∞ only. Its eigenvalues {λk } are numbered in nonincreasing order counting their multiplicities. The orthonormal system of the corresponding eigenfunctions {ϕk } forms a basis in Hσ . Define (18.18) X = H2σ × Hπ , Y = L2 = Hσ × Hπ  L=

−χ A O −χ  O



 ∈ L(X; Y),

M=

νA O ν −I

∈ L(X; Y).

(18.19)

Lemma 18.2 Let X and Y have form (18.18), L and M have form (18.19), χ = 0. Then M is (L , 0)-bounded operator and  P=

 I O I O . , Q= OO A−1 O

(18.20)

Proof We have −(μχ + ν)−1 A−1 ∈ L(Hσ ; H2σ ) at μ = −νχ −1 , A−1 ∈ L (Hσ ; Hπ ). For such μ, we have  μL − M =

 −(μχ + ν)A O −(μχ + ν)−1 A−1 O . , (μL − M)−1 = I −A−1 −(μχ + ν) I

Therefore, operator (μL − M)−1 : Y → X is continuous for μ = −νχ −1 ,  RμL (M) =

 O χ (μχ + ν)−1 I O χ (μχ + ν)−1 I . , L μL (M) = O O χ (μχ + ν)−1 A−1 O

Consequently, projectors P, Q have form (18.20). The form of P implies that ker L = {0} × Hπ = ker P = X0 , that is why operator M is (L , 0)-bounded.

18 A Class of Semilinear Degenerate Equations with Fractional …

211

Theorem 18.4 Let χ = 0, α0 ∈ [0, 1], α > 2, d ∈ {1, 2, 3, 4, 5, 6}. Then problem (18.14)–(18.17) has a unique solution on [t0 , t1 ] at some t1 ∈ (t0 , T ). Proof The form of the projector P implies that (18.14) are the Showalter–Sidorov conditions. Define N (v, Dtα0 v, r, Dtα0 r ) = −(v · ∇)v − Dtα0 v, therefore by the Sobolev’s embedding theorem (v · ∇)v2L2 ≤ Cv2L2q  v2W1 ≤ Cv4H2 , 2q

N (v, Dtα0 v, r, Dtα0 r )2Y = (v · ∇)v + Dtα0 v2L2 ≤ ≤ 2(v · ∇)v2L2 + 2Dtα0 v2L2 ≤ 2Cv4H2 + 2Dtα0 v2H2 . q 1 ( ))d , q  = q−1 , q > 1 is chosen so close to 1, Here L2q  = (L 2q  ( ))d , W12q = (W2q d 1 that is, q ≤ d−2 for d ∈ {3, 4, 5, 6} in order to have H 2 ( ) ⊂ W2q ( ). Embedding d 2 H ( ) ⊂ L 2q  ( ) holds for d ≤ 4 or for q ≥ 4 while d = 5 or d = 6. Besides, (h · ∇)v + (v · ∇)h2L2 ≤ Cv2H2 h2H2 ,

therefore the Freshet derivative of operator N is continuous, so N is locally Lipschitz continuous. By Theorem 18.3 with parameter r = 1 this theorem is proved. Acknowledgements The reported study was funded by Act 211 of Government of the Russian Federation, contract 02.A03.21.0011, and by Ministry of Science and Higher Education of the Russian Federation, task number 1.6462.2017/BCh.

References 1. Zvyagin, V.G., Turbin, M.V.: The study of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids. J. Math. Sci. 168(2), 157–308 (2010) 2. Mainardi, F., Spada, G.: Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur. Phys. J., Spec. Top. 193, 133–160 (2011) 3. Jaishankar, A., McKinley, G.H.: Power-law rheology in the bulk and at the interface: quasiproperties and fractional constitutive equations. Proc. R. Soc. A 469, 20120284 (2013) 4. Bajlekova, E.G.: The abstract Cauchy problem for the fractional evolution equation. Fract. Calc. Appl. Anal. 1(3), 255–270 (1998) 5. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Publishing, Amsterdam, Boston, Heidelberg (2006) 6. Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, New York (2011) 7. Debbouche, A., Torres, D.F.M.: Sobolev type fractional dynamic equations and optimal multiintegral controls with fractional nonlocal conditions. Fract. Calc. Appl. Anal. 18, 95–121 (2015) 8. Fedorov, V.E., Gordievskikh, D.M.: Resolving operators of degenerate evolution equations with fractional derivative with respect to time. Russ. Math. 59, 60–70 (2015)

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9. Fedorov, V.E., Gordievskikh, D.M., Plekhanova, M.V.: Equations in Banach spaces with a degenerate operator under a fractional derivative. Differ. Equ. 51, 1360–1368 (2015) 10. Fedorov, V.E., Plekhanova, M.V., Nazhimov, R.R.: Degenerate linear evolution equations with the Riemann–Liouville fractional derivative. Siberian Math. J. 59(1), 136–146 (2018) 11. Plekhanova, M.V.: Nonlinear equations with degenerate operator at fractional Caputo derivative. Math. Methods Appl. Sci. 40(17), 6138–6146 (2016) 12. Plekhanova, M.V.: Distributed control problems for a class of degenerate semilinear evolution equations. J. Comput. Appled Math. 312, 39–46 (2017) 13. Plekhanova, M.V.: Strong solutions to nonlinear degenerate fractional order evolution equations. J. Math. Sci. 230(1), 146–158 (2018) 14. Bajlekova, E.G.: Fractional Evolution Equations in Banach Spaces. PhD thesis. University Press Facilities, Eindhoven University of Technology, Eindhoven (2001) 15. Plekhanova, M.V., Baybulatova, G.D.: Semilinear equations in Banach spaces with lower fractional derivatives. In: Area, I., Cabada, A., Cid., J.A. (eds.) Nonlinear Analysis and Boundary Value Problems. NABVP 2018, Santiago de Compostela, Spain, September 4–7. Springer Proceedings in Mathematics and Statistics, vol. 292. Springer Nature Switzerland AG, Cham, 81–93 (2019) 16. Sviridyuk, G.A., Fedorov, V.E.: Linear Sobolev Type Equations and Degenerate Semigroups of Operators. VSP, Utrecht, Boston (2003)

Chapter 19

Program Packages Method for Solution of a Linear Terminal Control Problem with Incomplete Information Nikita Strelkovskii and Sergey Orlov

Abstract Closed-loop terminal control problem for the controlled linear dynamical system is considered in case of incomplete information about the actual initial state of this system. It is assumed, though, that the finite set containing the actual initial state is known. The program packages method is applied for obtaining solution of the problem. The package terminal problem is formulated and is proven to be equivalent to the closed-loop terminal control problem. The package terminal problem is further reduced to a finite-dimensional open-loop terminal control problem which can be solved using a numerical algorithm. Such an algorithm is provided in detail and applied to an illustrative example.

19.1 Introduction Terminal control problems comprise an important class of applied optimal control problems. One of the classical applications is motion control processes, for example, guiding a moving object to the desired position or launching a rocket to a maximum altitude [1]. Being precisely modeled with nonlinear systems of differential equations, these processes are often approximated on the design stage or short time intervals with simpler linear models. Motion control processes in the real world are often subject to disturbances of various nature and incomplete information. One of such problem settings was suggested in [2] for a nonlinear positional guidance problem with incomplete information about initial states of the considered dynamical system. A novel program packages method was suggested to solve this problem [2, 3] by Osipov and Kryazhimskii. This N. Strelkovskii (B) · S. Orlov International Institute for Applied Systems Analysis, Schlossplatz, 1, Laxenburg, Austria e-mail: [email protected] S. Orlov e-mail: [email protected] S. Orlov Lomonosov Moscow State University, GSP-1, Leninskiye Gory, 1-52, Moscow, Russia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_19

213

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method originates from the theory of closed-loop control developed by the school of Krasovskii [4–9]. The method was further developed in more detail for the linear case in [10]. A solvability criterion for linear guidance problems [11] as well as numerical algorithms for guiding controls was developed [12]. However, in all these papers, the goal of the controlling side was to guide motions of the considered system on a preset closed compact set. In [13], the authors extended the problem setting to guidance on a preset system of sets. In this sense, the authors of this paper go further and suggest a logical generalization of the problem, i.e., consider a terminal control problem for a given linear system and the set of its possible initial states. This problem requires minimizing a given convex and continuous functional. It is clear that a guidance problem is a special case of a terminal control problem.

19.2 Problem Statement The following controlled linear dynamical system is considered: x(t) ˙ = A(t)x(t) + B(t)u(t) + c(t),

(19.1)

where t is time variable, t ∈ [t0 , ϑ], t0 < ϑ < ∞; x(t) ∈ Rn is the state of the system at time t; and u(t) ∈ Rm is the value of the control function at time t. Functions A(·), B(·), c(·) are defined and continuous on [t0 , ϑ] and take values in sets Rn×n , Rn×m , and Rn respectively. The instantaneous control resource P ⊂ Rm is assumed to be a convex compact set. An arbitrary function u(·) : [t0 , ϑ] → P, which is piecewise continuous from the left and continuous at the endpoints of [t0 , ϑ] is called an openloop control (a program). The set of all open-loop controls is denoted further by U . A motion of the system from initial state x0 ∈ Rn under the action of an open-loop control u(·) ∈ U is defined as a solution of ODE (19.1) with the initial condition x(t0 ) = x0 on the segment [t0 , ϑ] and is denoted by x(·|x0 , u(·)). We assume that the initial state of the system (19.1) belongs to a priori known finite set X 0 = {x01 , . . . , x0N }, which is called the set of admissible initial states. However, the initial state itself is not known. We assume that the motion of system (19.1) can be observed using a signal function y(·): y(t) = Q(t)x(t), t ∈ [t0 , ϑ], where Q(·) is a matrix observation function, which is defined and piecewise continuous from the left on [t0 , ϑ] and takes its values in Rq×n . A closed-loop terminal control problem has to be solved, i.e., the given continuous convex scalar function (terminal functional) ϕ(x(·|x01 , u(·)), . . . , x(·|x0N , u(·))) should be minimized at the terminal time ϑ for all initial states x0 ∈ X 0 with a given accuracy ε > 0. Formally, for any given ε > 0 such a number J∗ should be found and such an admissible closed-loop strategy with memory S ε (in the sense defined in [2]) should

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be built that ϕ(x(ϑ|x01 , S ε ), . . . , x(ϑ|x0N , S ε )) ≤ J∗ + ε and for any J < J∗ no such strategy exists. We call such a strategy Sε ε-optimal. The closed-loop terminal control problem can be reduced to the package terminal control problem (Sect. 19.3) and further to a finite-dimensional open-loop terminal control problem (Sect. 19.4), which allows the application of the numerical solution algorithm (Sect. 19.5).

19.3 Package Terminal Control Problem Consider the homogeneous system x(t) ˙ = A(t)x(t)

(19.2)

corresponding to equation (19.1), with initial condition x(t0 ) = x0 . The Cauchy formula yields its solution x(t) = F(t, t0 )x0 , where F(t, s), t, s ∈ [t0 , ϑ] is the Cauchy matrix. For any admissible initial state x0 ∈ X 0 define corresponding homogeneous signal as follows gx0 (·): gx0 (t) = Q(t)F(t, t0 )x0 , t ∈ [t0 , ϑ].

(19.3)

Note that the observation of the signal from the original system (19.1) can be reduced to observation of homogeneous signal (19.3) using the Cauchy formula (more details are available in [12, Sect. 6]). A homogeneous signal corresponding to an arbitrary (not specified) admissible initial state is simply called a homogeneous signal and is denoted g(·). Set of all admissible initial states corresponding to homogeneous signal g(·) up to time τ ∈ [t0 , ϑ] is denoted as X 0 (τ |g(·)). Thus, X 0 (τ |g(·)) = {x0 ∈ X 0 : g(·)|[t0 ,τ ] = gx0 (·)|[t0 ,τ ] }. Here and further g(·)|[t0 ,τ ] , where τ ∈ [t0 , ϑ], is the restriction of homogeneous signal g(·) on segment [t0 , τ ]. For arbitrary homogeneous signal g(·) let us define set ˜ ∈ G : g(t) ˜ ≡ g(t), t ∈ [t0 , t0 + σ ], σ = σ (g) ˜ > 0)} G 0 (g(·)) = {g(·) of homogeneous signals initially coinciding with g(·) and time  τ1 (g(·)) = max τ ∈ [t0 , ϑ] :

max

 max |g(t) ˜ − g(t)| = 0 ,

g(·)∈G ˜ 0 (g(·)) t∈[t0 ,τ ]

which is called the first splitting moment of homogeneous signal g(·). Similarly, for i = 1, 2, . . . let us define sets   G i (g(·)) = g(·) ˜ ∈ G i−1 (g(·)) : g(t) ˜ ≡ g(t), t ∈ [τi (g(·)), τi (g(·)) + σ ], σ = σ (g) ˜ >0

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of all homogeneous signals from set G i−1 (g(·)), coinciding with g(·) in a right-sided neighborhood of time τi (g(·)), and times  τi+1 (g(·)) = max τ ∈ (τi (g(·)), ϑ] :

 max

max

g(·)∈G ˜ i (g(·)) t∈[τi (g(·)),τ ]

|g(t) ˜ − g(t)| = 0 ,

which are called the (i + 1)-th splitting moments of the homogeneous signal g(·). Let us also define set T (g(·)) = {τ j (g(·)) : j = 1, . . . , k g(·) } of all splitting moments of homogeneous signal g(·) and set T = ∪g(·)∈G T (g(·)) of all splitting moments of all homogeneous signals. Since it is proven that set T is finite [11, Sect. 2], consider it in the form T = {τ1 , . . . , τ K }, where τ1 < . . . < τ K . For each k = 1, . . . , K define set X0 (τk ) = {X 0 (τk |g(·)) : g(·) ∈ G} which is called cluster position at time τk while all its elements X 0 j (τk ), j = 1, . . . , J (τk ) are called clusters of initial states at this time. Here J (τk ) is the number of clusters in cluster position X0 (τk ), k = 1, . . . , K . Family of open-loop controls (u x0 (·))x0 ∈X 0 is called program package if it satisfies the following non-anticipatory condition: for any homogeneous signal g(·), time τ ∈ (t0 , ϑ] and admissible initial states x0 , x0 ∈ X 0 (τ |g(·)) equality u x0 (t) = u x0 (t) holds for all t ∈ [t0 , τ ]. Program package (u x0 (·))x0 ∈X 0 is called optimal, if it minimizes function ϕ(x(ϑ|x01 , u x01 (·)), . . . , x(ϑ|x0N , u x0N (·))). The package terminal control problem comprises finding an optimal program package. Definition 19.1 Let us call admissible closed-loop strategy S ε , ε > 0, εcorresponding to program package (u x0 (·))x0 ∈X 0 , if for all x0 ∈ X 0 holds |x(ϑ|x0 , S ε ) − x(ϑ|x0 , u x0 (·))| ≤ ε. Definition 19.2 Let us call program package (u x0 (·))x0 ∈X 0 corresponding to admissible closed-loop strategy S if for all x0 ∈ X 0 and t ∈ [t0 , ϑ] holds u x0 (t) = u(x(t|x0 , S)). Let J¯∗ be the optimal value of the terminal functional in the closed-loop terminal control problem and J∗ be the optimal value of the terminal functional in the package terminal control problem. Theorem 19.1 (Equivalence of the closed-loop and package terminal control problems) The following statements hold true: (1) J¯∗ = J∗ , (2) If optimal program package (u ∗x0 (·))x0 ∈X 0 solves the package terminal control problem, then for any given ε > 0 there exists ε-optimal admissible positional strategy with memory S ε for the closed-loop terminal control problem. Proof Let us first prove that J¯∗ = J∗ . Assume that it does not hold. Then there are two alternatives, either J¯∗ < J∗ or J¯∗ > J∗ .

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If J¯∗ < J∗ , then there exists an ε-optimal closed-loop strategy S∗ε . Let us take J∗ − J¯∗ ε = ε0 = , and denote by (u εx00 (·))x0 ∈X 0 the program package corresponding 2 to closed-loop strategy S∗ε0 . Then, obviously, ϕ(x(ϑ|x01 , u ε01 (·)), . . . , x(ϑ|x0N , u ε0N (·))) = ϕ(x(ϑ|x01 , S∗ε0 ), . . . , x(ϑ|x0N , S∗ε0 )) ≤ x0

x0

J∗ − J¯∗ J∗ + J¯∗ ≤ J¯∗ + ε0 = J¯∗ + = < J∗ . 2 2

Thus, we have found a program package that delivers the terminal functional a lower value than J∗ , which is a contradiction to assumed optimality of value J∗ . Next, consider the case where J¯∗ > J∗ . Let (u ∗x0 (·))x0 ∈X 0 be optimal program package. According to the procedure introduced in [12], for any δ > 0, it is possible to construct admissible closed-loop strategy S∗δ which is δ-corresponding to optimal program package (u ∗x0 (·))x0 ∈X 0 , i.e., for all x0 ∈ X 0 holds |x(ϑ|x0 , S∗δ ) − x(ϑ|x0 , u ∗x0 (·))| ≤ δ. Due to continuity of ϕ(·) with respect to all N variables, for any ε > 0 there exists δ = δ(ε) such that |ϕ(x(ϑ|x01 , S∗δ ), . . . , x(ϑ|x0N , S∗δ )) − ϕ(x(ϑ|x01 , u ∗x 1 (·)), . . . , x(ϑ|x0N , u ∗x N (·)))| ≤ ε. 0

0

J¯∗ − J∗ and δ < δ(ε0 ). Then, remembering that 2 N 1 ∗ ∗ ϕ(x(ϑ|x0 , u x 1 (·)), . . . , x(ϑ|x0 , u x N (·))) = J∗ , we rewrite the above inequality in 0 0 the form |ϕ(x(ϑ|x01 , S∗δ ), . . . , x(ϑ|x0N , S∗δ )) − J∗ | ≤ ε0 . Let us take ε = ε0 =

J∗ + J¯∗ < J¯∗ . Thus, we have 2 found admissible closed-loop strategy S∗δ that delivers the terminal functional a lower value than J¯∗ , which is a contradiction to assumed optimality of the value J¯∗ . It is obvious that constructed above admissible closed-loop strategy S∗δ , δ < δ(ε), is ε-optimal for the closed-loop terminal control problem, which proves the second statement of the theorem.  Then ϕ(x(ϑ|x01 , S∗δ ), . . . , x(ϑ|x0N , S∗δ )) ≤ J∗ + ε0 =

19.4 Extended Open-Loop Terminal Control Problem For arbitrary h = 1, 2, . . . define extended space Rh as a finite-dimensional Euclidean space of all vector families (r x0 )x0 ∈X 0 from Rh with scalar product ·, · Rh of the form (r x 0 )x0 ∈X 0 , (r x0 )x0 ∈X 0 Rh =

 x0 ∈X 0

r x 0 , r x0 Rh , (r x 0 )x0 ∈X 0 , (r x0 )x0 ∈X 0 ∈ Rh .

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Let us introduce P ⊂ Rm — a set of all families (u x0 )x0 ∈X 0 of vectors from P. Let us call an arbitrary function (u x0 (·))x0 ∈X 0 : [t0 , ϑ] → P which is piecewise continuous from the left on the segment [t0 , ϑ] an extended open-loop control. For each k = 1, . . . , K let us denote by Pk the set of all families of vectors (u x0 )x0 ∈X 0 ∈ P, such that for any cluster X 0 j (τk ) ∈ X0 (τk ), j = 1, . . . , J (τk ), and any admissible initial states x0 , x0 ∈ X 0 j (τk ) equality u x0 = u x0 holds. Let us call extended open-loop control (u x0 (·))x0 ∈X 0 admissible, if for each k = 1, . . . , K holds (u x0 (t))x0 ∈X 0 ∈ Pk for all t ∈ (τk−1 , τk ] in the case of k > 1 and for all t ∈ [t0 , τ1 ] in the case of k = 1. Let us denote the set of all admissible extended open-loop controls by U, and let us call the set of their values (a subset of the set P, constrained by the geometric conditions outlined above) the extended instantaneous control resource and denote ˆ it P(t), t ∈ [t0 , ϑ]. Let us consider a dynamical system in space Rn consisting of copies of the original system (19.1); each copy is parametrized by a different admissible initial state x0 ∈ X 0 . A motion of the copy parameterized by an admissible initial state x0 starts from this initial state at time t0 and is controlled by open-loop control u x0 (·) on the segment [t0 , ϑ]. Such system is called the extended system and has the form 

x˙ x0 (t) = A(t)x x0 (t) + B(t)u x0 (t) + c(t), x x0 (t0 ) = x0 ∈ X 0 .

(19.4)

Let us constrain open-loop control families (u x0 (·))x0 ∈X 0 used to control the extended system with set U. For any admissible extended open-loop control (u x0 (·))x0 ∈X 0 ∈ U, a corresponding motion of the extended system is the function (x(·|x0 , u x0 (·)))x0 ∈X 0 : [t0 , ϑ] → Rn . Let us call admissible extended open-loop control (u x0 (·))x0 ∈X 0 optimal if it minimizes the function ϕ(x(ϑ|x0 , u x0 (·))x0 ∈X 0 ). The extended open-loop terminal control problem comprises finding an optimal admissible extended open-loop control. Theorem 19.2 An admissible extended open-loop control is an optimal program package if and only if it is optimal. Proof From [11, Lemma 3], it follows that extended open-loop control (u x0 (·))x0 ∈X 0 is a program package if and only if it is admissible. By definition, an admissible extended open-loop control (u x0 (·))x0 ∈X 0 is an optimal program package if and only if it minimizes function ϕ(x(ϑ|x01 , u x01 (·)), . . . , x(ϑ|x0N , u x0N (·))) or, which is the same, it minimizes the function ϕ(x(ϑ|x0 , u x0 (·))x0 ∈X 0 ). The latter statement is precisely the definition of the fact that extended open-loop control (u x0 (·))x0 ∈X 0 is optimal.  Thus, the package terminal problem and the extended open-loop terminal control problem are essentially the same problems. Corollary 19.1 It is obvious that a program package (u x0 (·))x0 ∈X 0 corresponding to an admissible closed-loop strategy S is an admissible extended open-loop control

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due to coincidence of the control set P in the closed-loop and open-loop terminal control problems and due to definition of the program package. The extended terminal control problem comprises a well-known terminal control problem for which a solving algorithm exists [14]. However, in our case, the problem ˆ has to be solved in the extended space Rn and the control set P(t), t ∈ [t0 , ϑ], depends on time. In [15, 16], an algorithm for solving a guidance problem on a fixed set at the terminal time under the above conditions was suggested. We provide details on modification of this algorithm based on theory developed in [14] in the next section.

19.5 Solving Algorithm On the first stage of the algorithm, the procedure suggested in [16] is used to solve the extended guidance problem on set Mϕ0 = {x ∈ Rn : ϕ(x) ≤ ϕ0 }, where ϕ0 = min x∈Rn ϕ(x). If the solution (u ∗x0 (·))x0 ∈X 0 is found then it is the optimal program package. If the solution does not exist, then we proceed to the second stage of the algorithm which is explained below. For an arbitrary δ > 0 let us denote Mδ = {x ∈ Rn : ϕ(x) ≤ δ}. We modify the algorithm from [15] as follows: 1. The algorithm is initialized with δ ∗(0) = ϕ0 and (u (0) x0 (·))x0 ∈X 0 ≡ 0, t ∈ [t0 , ϑ]. 2. We use a smoothing technique from the paper [16] for approximating extended ˆ control set P(t), t ∈ [t0 , ϑ], with a strictly convex set. 3. No compression of the attainability set is assumed, i.e., on all steps i = 0, 1, . . . of the algorithm we assume that parameter a ∗(i) corresponding to compression of the attainability set [15] equals 1. 4. On each step of the algorithm we consider guidance on Mδ∗(i) , and thus function γˆ (·) takes the form 

 ∗(i) )x0 ∈X 0 , δ ∗(i) = p l x0 , x 0 + γˆ (l x∗(i) 0 x0 ∈X 0  ⎞ ⎛  K τk      P ⎠ dt− + ρ− ⎝ D(t)l x∗(i) 0   k=1 τ X 0 j (τk )∈X0 (τk ) x0 ∈X 0 j (τk )  k−1∗(i) + −ρ (l x0 )x0 ∈X 0 |Mδ∗(i) . and depends on δ ∗(i) (parameter of compression of the set Mδ on step (i) instead of a ∗(i) (parameter of compression of the attainability set). )x0 ∈X 0 , δ ∗(i) = 0 with 5. On each step of the algorithm, we solve equation γˆ (l x∗(i) 0  respect to δ ∗(i) (instead of solving equation γˆ (l x∗(i) )x0 ∈X 0 , a ∗(i) = 0 with respect 0 ∗(i) to a ). 6. The other calculations on each step of the algorithm, i.e., estimation of u ∗(i) X 0 j (τk ) (t) (t ∈ [τk−1 , τk ]), χ ∗(i) , and z ∗(i) remains identical to the original algorithm.

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7. The algorithm stops when     ∗(i) ∗(i) ϕ(x(ϑ|x01 , u x 1 (·)), . . . , x(ϑ|x0N , u x N (·))) − δ ∗(i)  ≤ , 0

0

where  > 0 is desired accuracy of the algorithm. The convergence of this algorithm follows from convergence of the algorithm [14] and remark 4 of the paper [11] which states a bijection between the elements of the extended space Rn and elements of a Euclidean space of a larger (i.e., n N ) dimension.

19.6 Example Let us consider a linear dynamical controlled system x˙1 (t) = x2 (t), x˙2 (t) = u(t)

(19.5)

on time segment [t0 , ϑ] = [0, 2]. The control function values are constrained with set P = [−1, 1]. The initial admissible states set consists of two different points, i.e.,   X 0 = x0 , x0 , where x0 = (2, 1/2)T , x0 = (−1, −1/2)T .

(19.6)

The states of system (19.5) are not observable on time segment [0, 1/2] and are fully observable on half-segment (1/2, 2]. Thus, matrix observation function Q(·) is as follows:    (0, 0), t ∈ 0, 21  , Q(t) = (1, 0), t ∈ 21 , 2 . The terminal functional ϕ(ϑ) = ϕ(x x0 (ϑ), x x0 (ϑ)) = x x0 (ϑ)2 + x x0 (ϑ)2 has to be minimized, which is a typical terminal control problem setting; however, here it is complicated by lack of information about the exact initial state of the considered dynamical system. Let us transit to the extended system ⎧ x˙ x0 1 (t) = x x0 2 (t), ⎪ ⎪ ⎪ ⎨ x˙  (t) = u  (t), x0 2 x0  ⎪ x˙ x 1 (t) = x x0 2 (t), ⎪ ⎪ ⎩ 0 x˙ x0 2 (t) = u x0 (t),

 x x0 1 (0) = x01 ,

 x x0 2 (0) = x02 ,   x x0 1 (0) = x01 ,

 x x0 2 (0) = x02 .

The extended instantaneous control set has the form  P1 , t ∈ [0, 1/2), ˆ P(t) = P2 , t ∈ [1/2, 2],

(19.7)

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where   P1 = (u x0 )x0 ∈X 0 ∈ [−1, 1] × [−1, 1] : u x0 = u x0 ,   P2 = (u x0 )x0 ∈X 0 ∈ [−1, 1] × [−1, 1] . Let us substitute the variables ⎞ ⎛ vx0 1 (t)  ⎜ vx  2 (t) ⎟ V1 , t ∈ [0, 1/2), 0 ⎟ ⎜ ˆ ˆ ˆ ∈ V(t) = B P(t) = v(t) = B u(t) ˆ = ⎝  vx0 1 (t) ⎠ V2 , t ∈ [1/2, 2], vx0 2 (t) where

⎛

⎞ ⎛ ⎞ 0 00 ⎜ u x  (t) ⎟ ⎜ ⎟ 0 ⎟ ˆ ⎜1 0⎟ u(t) ˆ =⎜ ⎝ 0 ⎠, B = ⎝0 0⎠ u x0 (t) 01

and   V1 = v ∈ R4 : vx0 1 = vx0 1 = 0, vx0 2 = vx0 2 ∈ [−1, 1] ,   V2 = v ∈ R4 : vx0 1 = vx0 1 = 0, vx0 2 , vx0 2 ∈ [−1, 1] . The system (19.7) takes the form ⎧ x˙ x0 1 (t) = x x0 2 (t) + vx0 1 (t), ⎪ ⎪ ⎪ ⎨ x˙  (t) = v  (t), x0 2 x0 2  ⎪ x˙ x 1 (t) = x x0 2 (t) + vx0 1 (t), ⎪ ⎪ ⎩ 0 x˙ x0 2 (t) = vx0 2 (t),

 x x0 1 (0) = x01 ,

 x x0 2 (0) = x02 ,   x x0 1 (0) = x01 ,

(19.8)

 x x0 2 (0) = x02 .

The set V(t) can be smoothed using the procedure described in [16] and then solving algorithm described in Sect. 19.5 can be applied. It is obvious that ϕ0 = min ϕ(x  , x  ) = x  2 + x  2 = 0. The extended x  ,x  ∈R2

guidance problem on the set Mϕ0 is not solvable since the solvability criterion [11, Theorem 2] does not hold. Thus, we initialize the solving algorithm with δ ∗(0) = 0 and (u (0) x0 (·))x0 ∈X 0 ≡ 0, t ∈ [t0 , ϑ]. After 12 iterations the solving algorithm has reached the required accuracy of ε = 0.005. The optimal value of the optimized functional, i.e., δ ∗ equals 1.529. The projections of the system’s (19.8) motions onto R2 are depicted in Fig. 19.1. The motions at time ϑ are guided on the boundary of set Mδ∗ . An ε-optimal closed-loop strategy S ε can be easily constructed based on the retrieved optimal extended open-loop control using the procedure described in [12].

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Fig. 19.1 Motions of the system (19.8) guided on the boundary of the set Mδ ∗ at time ϑ x2

x0 x1

δ

x0

The algorithm was coded and executed in the Maple 18 software. The operating time of the program on a laptop with the Intel Core [email protected] processor was 90.3 s.

References 1. Tyatyushkin, A.I.: Numerical methods for control optimization in linear systems. Comput. Math. Math. Phys. (2015). https://doi.org/10.1134/S0965542515050152 2. Osipov, Yu.S.: Control packages: an approach to solution of positional control problems with incomplete information. Russ. Math. Surv. (2006). https://doi.org/10.4213/rm1760 3. Kryazhimskiy, A.V., Osipov, Y.S.: Idealized program packages and problems of positional control with incomplete information. Proc. Steklov Inst. Math. (Suppl.) (2010). https://doi. org/10.1134/S0081543810050123 4. Krasovskii, N.N.: Game Problems on the Encounter of Motions. Nauka, Moscow (1970). [in Russian] 5. Krasovskii, N.N., Subbotin, A.I.: Positional Differential Games. Nauka, Moscow (1974). [in Russian] 6. Krasovskii, N.N.: Control of a Dynamical System: Problem on the Minimum of Guaranteed Result. Nauka, Moscow (1985). [in Russian] 7. Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York (1988) 8. Krasovskii, A.N., Krasovskii, N.N.: Control Under Lack of Information. Birkhäuser, Boston (1995) 9. Subbotin, A.I., Chentsov, A.G.: Guarantee Optimization in Control Problems. Nauka, Moscow (1981). [in Russian] 10. Kryazhimskiy, A.V., Osipov, Yu.S.: On the solvability of problems of guaranteeing control for partially observable linear dynamical systems. Proc. Steklov Inst. Math. (2012). https://doi. org/10.1134/S0081543812040104 11. Kryazhimskii, A.V., Strelkovskii, N.V.: An open-loop criterion for the solvability of a closedloop guidance problem with incomplete information: linear control systems. Proc. Steklov Inst. Math. (2015). https://doi.org/10.1134/S0081543815090084

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12. Strelkovskii, N.V.: Constructing a strategy for the guaranteed positioning guidance of a linear controlled system with incomplete data. Moscow Univ. Comput. Math. Cybern. (2015). https:// doi.org/10.3103/S0278641915030085 13. Maksimov V.I., Surkov P.G.: On the solvability of the problem of guaranteed package guidance to a system of target sets. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki. (2017). https:// doi.org/10.20537/vm170305 [in Russian] 14. Gindes, V.B.: On the problem of minimizing a convex functional on a set of finite states of a linear control system. USSR Comput. Math. Math. Phys. (1966). https://doi.org/10.1016/ 0041-5553(66)90159-5 15. Strelkovskii, N.V., Orlov, S.M.: Algorithm for constructing a guaranteeing program package in a control problem with incomplete information. Mosc. Univ. Comput. Math. Cybern. (2018). https://doi.org/10.3103/S0278641918020061 16. Orlov, S.M., Strelkovskii, N.V.: Calculation of elements of a guiding program package for singular clusters of the set of initial states in the package guidance problem. In: Proceedings of the Institute of Mathematics and Mechanics (2019). https://doi.org/10.21538/0134-48892019-25-1-150-165 [in Russian]

Chapter 20

Application of Correcting Control in the Problem with Unknown Parameter Vladimir Ushakov and Aleksandr Ershov

Abstract The approach problem for a control system containing an unknown constant parameter is considered. The actual value of the parameter in this control system (described by a vector differential equation) is not known to the person who controls the system; only some set containing the value of the parameter is known. The recovery of the unknown parameter is carried out by applying to the control system for a short period of time test control and observing the corresponding change in the motion of the system. After the approximate determination of the unknown parameter, we can construct the resolving program control by the standard method; however, we need to take into account the additional error associated with the approximate determination of the parameter. In this paper, we reduce the error in the solution of the approach problem by the method of separation of control into basic and correcting controls, which was previously successfully used for simpler approach problems that do not contain an unknown parameter.

20.1 Introduction The work is devoted to the study of the approach problem of a nonlinear control system with a compact target set in a finite-dimensional phase space (see, for example, [1, 2]). A feature of the problem considered in this paper is the presence in the system of an unknown constant parameter. Earlier, the work [3] presented a scheme for constructing a program control that solves, with a certain degree of accuracy, V. Ushakov (B) · A. Ershov N.N. Krasovskii Institute of Mathematics and Mechanics, 16 S.Kovalevskaya Str., Yekaterinburg 620108, Russia e-mail: [email protected] A. Ershov e-mail: [email protected] A. Ershov Ural Federal University named after the first President of Russia B.N. Yeltsin, 19 Mira Str., Yekaterinburg 620002, Russia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_20

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such an approach problem. It used the assumption that we can exactly measure the motion of a control system at any given time. In [4], this assumption was replaced by the condition that phase-parameter measurements have a limited error. However, the estimates obtained in it can be improved using the method of separation of control into basic and correcting controls introduced in [5, 6] for problems without parameters. The aim of this work is to obtain new estimates of the error of constructing a solution to the approach problem using this method in the presence of an unknown constant parameter.

20.2 The Problem Statement Let us consider a control system dx = f (t, x(t), u(t), α), dt

(20.1)

where the time t ∈ [t0 , ϑ] (t0 < ϑ < ∞), the phase vector x(t) ∈ Rn , the control vector u(t) ∈ P ∈ comp(R p ), the parameter α ∈ L ∈ comp(Rq ); here, Rk is the Euclidean space of dimension k, comp(Rk ) is the space of the compacts in Rk with the Hausdorff metric d(·, ·). Assume that the following conditions are satisfied. A. The vector-valued function f (t, x, u, α) is defined and continuous on the time interval [t0 , θ ] × Rn × P × L , and there exists such a constant L() ∈ (0, ∞) for any bounded and closed domain  ⊂ [t0 , θ ] × Rn , that || f (t, x (1) , u, α) − f (t, x (2) , u, α)||  L()||x (1) − x (2) ||,

(20.2)

(t, x (i) , u, α) ∈  × P × L , i = 1, 2. B. There exists such a constant γ ∈ (0, ∞), that || f (t, x, u, α)||  γ (1 + ||x||), (t, x, u, α) ∈ [t0 , θ ] × Rn × P × L . C. Fα (t, x) = f (t, x, P, α) = { f (t, x, u, α) : u ∈ P}, (t, x, α) ∈ [t0 , θ ] × Rn × L is the convex set in Rn . Here || f || is the norm of vector f in the Euclidean space. D. Denote F (u ∗ ) (t0 , x (0) ) = { f (t0 , x (0) , u ∗ , α) : α ∈ L }. Then there exists such an unique mapping α(·) : F (u ∗ ) (t0 , x (0) ) −→ L and a function e(λ) ↓ 0, α ↓ 0, that e

f (t0 , x (0) , u ∗ , α( f )) = f,

(t0 , x (0) , u ∗ ) ∈  × P, f ∈ F (u ∗ ) (t0 , x (0) );

20 Application of Correcting Control in the Problem …

||α( f ∗ ) − α( f ∗ )||  e(|| f ∗ − f ∗ ||),

227

f ∗ ∈ F (u ∗ ) (t0 , x (0) ), f ∗ ∈ F (u ∗ ) (t0 , x (0) ).

e

Remark 20.1 By the condition B, there exists a sufficiently large cylinder D = [t0 , ϑ] × B n (0, R), B n (0, R) = {x ∈ Rn : ||x||  R}, which contains all possible motion of the system (20.1). In this regard, we will use the Lipschitz constant L = L(D) in what follows. In addition to it, we will use the constant K = max{|| f (t, x, u, α)|| : (t, x, u, α) ∈ D × P × L }. Remark 20.2 From condition A, we obtain that the functions, ω(1) (δ) = max{|| f (t∗ , x, u, α) − f (t ∗ , x, u, α)|| : (t∗ , x, u, α) and (t ∗ , x, u, α) of D × P × L , |t∗ − t ∗ |  δ}, δ ∈ (0, ∞), ω(3) (δ) = max{|| f (t, x, u ∗ , α) − f (t, x, u ∗ , α)|| : (t, x, u ∗ , α) and (t, x, u ∗ , α) of D × P × L , ||u ∗ − u ∗ ||  δ}, δ ∈ (0, ∞), ω(4) (δ) = max{|| f (t, x, u, α∗ ) − f (t, x, u, α ∗ )|| : (t, x, u, α∗ ) and (t, x, u, α ∗ ) of D × P × L , ||α∗ − α ∗ ||  δ}, δ ∈ (0, ∞), satisfy the limit relations ω(k) (δ) ↓ 0 at δ ↓ 0, k = 1, 3, 4. Remark 20.3 Condition C is not critical for describing an approximate solution scheme for the approach problem; it was introduced in order to avoid unnecessarily complicated calculations. Introduce some mathematical concepts that are well known and which will be used in the following discussion. By the admissible control u(t), t ∈ [t0 , ϑ], we mean a Lebesgue-measurable vector-valued function from [t0 , ϑ] into P. Let X α (t ∗ , t∗ , x∗ ) be the reachable set in Rn of the system (20.1), corresponding to position (t∗ , x∗ ), t0  t∗  t ∗  θ, x∗ ∈ Rn , α ∈ L ; the moment t ∗ and  the initial ∗ X α (t∗ , x∗ ) = t ∗ ∈[t∗ ,θ] (t , X α (t ∗ , t∗ , x∗ )) ⊂ [t∗ , θ ] × Rn be the integral funnel of n . the system (20.1) with the initial position (t∗ , x∗ ) ∈ [t0 , θ ] × R ∗ ∗ ∗ (t , t , X ) = Under the conditions A, B, C, the set X α ∗ x∗ ∈X ∗ X α (t , t∗ , x ∗ )  and the set X α (t∗ , X ∗ ) = x∗ ∈X ∗ X α (t∗ , x∗ ) are compacts, respectively, in Rn and [t∗ , θ ] × Rn at any t∗ and t ∗ (t0  t∗  t ∗  θ ) and X ∗ ∈ comp(Rn ). Let M be some compact in Rn representing the target set for the system (20.1). Before proceeding with the formulation and discussion of the problems related to the convergence of the system (20.1) with M, we specify the informational conditions under which the system (20.1) is controlled. Namely, we assume that some value α∗ ∈ L of the parameter α ∈ L appears in (20.1) at the initial moment of time t0 of the interval [t0 , ϑ], and this value is actual in the system (20.1) throughout [t0 , ϑ]; on the other hand, at time t0 the person who controls system (20.1), that is, chooses the strategy u(t), does not know α∗ . We assume that this person is only aware of the constraint L .

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This informational condition was considered in the work [3] subject to exact measurement of the phase variable x(t). As in [4], here we suppose that we can measure the phase variable x(t) only with an error not exceeding δx , i.e., ||x(t) ˆ − x(t)||  δx ,

(20.3)

where x(t) ˆ is the result of measurement x(t) at time t. Our approach problem can be divided into two problems. Problem 1. It is required to determinate the value α∗ ∈ L present in the system (20.1). Problem 2. It is required to determine the existence of an admissible program control that translates the motion x(t) of the system (20.1) to M at the time ϑ and to construct this control in the case of the positive answer. Note that Problem 1 and Problem 2 were solved in [3] subject to the availability of accurate information about the phase variable of the control system, and these problems were solved in [4] and in such case the phase variable was measured with an error. The aim of this paper is to improve the solution of Problem 2 by applying the method of separation of control into basic and correcting, which was introduced in [5]. For convenience, we assume that the parameter α has already been recovered at the initial time t0 with some accuracy δα , i.e., the person who controls the system knows such its approximate value α ∗ that ||α ∗ − α∗ ||  δα .

20.3 The Solving Algorithm For the possibility of constructing admissible control by separating the control into the basic and correcting controls, we impose on the control system (20.1) the following additional conditions, which are analogues of the conditions E, F, C’ of [3]. E. There exists a strictly monotone function ρ(), defined on [0, ∞] such that ρ() ↓ 0 as  ↓ 0, ρ(0) = 0, and it satisfies inclusions ˇ ×L, Xˇ α (t + , t, x) ⊂ x +  f (t, x, B p (u, ρ())), (t, x, u, α) ∈ D × P(δ) (20.4) where Xˇ α (t + , t, x) is the reachable set of the system (20.1) at the time moment t +  with initial position (t, x) and with admissible controls consisted of a class ˇ of Lebesgue-measurable functions with values from Pˇ = P(ρ) = P −˙ B p (0, ρ) = p p {u ∈ R : u + B (0, ρ) ⊂ P}. Note that the function δ(ρ) makes sense to choose as small as possible. F. The set Pˇ is not empty. Like condition C, only to reduce the calculations [5], we will consider the following condition to be fulfilled. ˇ = { f (t, x, u, α) : u ∈ P} ˇ is convex for any G. The set Fα (t, x, ) = f α (t, x, P) n (t, x, α) ∈ [t0 , ϑ] × R × L .

20 Application of Correcting Control in the Problem …

229

Problem 2 can be divided into two subproblems. Problem 2.1. It is required to find the set Wα∗ ⊂ [t0 , ϑ] × Rn of all initial positions (t∗ , x∗ ) for which there are admissible controls on [t0 , ϑ] transferring the phase vector x(t) of the system (20.1) to the target set M at the time ϑ. Problem 2.2. For the point (t0 , x (0) ) ∈ Wα∗ , it is necessary to find an admissible control u ∗ (t), t ∈ [t0 , ϑ], generating such a motion x ∗ (t), x ∗ (t0 ) = x (0) , of the system (20.1) that x ∗ (ϑ) ∈ M. Adhering to the terminology accepted in the scientific literature (for example, [7]), we call Wα∗ the resolvability set of the approach problem. In order to calculate Wα∗ , we introduce the “reverse” time τ = t0 + ϑ − t ∈ [t0 , ϑ] (along with the “direct” time t ∈ [t0 , ϑ]) and associate the system (20.1) with the control system dz = g(τ, z(τ ), v(τ ), α), τ ∈ [t0 , ϑ], dτ

(20.5)

where g(τ, z, v, α) = − f (t0 + ϑ − τ, z, v, α), (τ, z, v, α) ∈ [t0 .ϑ] × Rn × Pˇ ×L. Introduce the following notation: Z α (τ ∗ , τ∗ , z ∗ ) is the reachable set of the system (20.5) at time τ ∗ from the initial position (τ∗ , z ∗ ), t0 τ∗  τ ∗  ϑ, z ∗ ∈ Rn , and with some α ∈ L ; Z α (τ ∗ , τ∗ , Z ∗ ) = z∗ ∈Z ∗ Z α (τ ∗ , τ∗ , z ∗ ) is the reachable set of the system (20.5) ∗ n at time τ ∗ from the  initial set∗(τ∗ , Z ∗∗), t0  τ∗  τ  ϑ, Z ∗ ∈ comp(R ); Z α (τ∗ , z ∗ ) = τ ∗ ∈[τ∗ ,ϑ] (τ , Z α (τ , τ∗ , z ∗ )) is the integral funnel of the system n (20.5) from the initial  position∗ (τ∗ , z∗∗ ), t0  τ∗  ϑ, z ∗ ∈ R ; Z α (τ∗ , Z ∗ ) = τ ∗ ∈[τ∗ ,ϑ] (τ , Z (τ , τ∗ , Z ∗ )) is the integral funnel of the system (20.5) from the initial set (τ∗ , Z ∗ ), t0  τ∗  ϑ, Z ∗ ∈ comp(Rn ). Formulate the algorithm approximately solving Problem 2.1. Denote by (δ) (·) the mapping that “thin out” the set, i.e., to any bounded set  = (δ) (A), consisting, if possible, of a smaller A ⊂ Rn it associates a finite set A number of its points and having the property:   δ. d(A, A)  are given in [5, p. 549]. Methods to build such a “thinned out” set A ˇ where u is an arbitrary selected positive constant.  = (u ) ( P), Denote P Introduce the mapping Z ∗ , α) = Z () (τ ∗ , τ∗ , 



 α)} {z + (τ ∗ − τ∗ )g(τ∗ , z, P,

z∈  Z∗

=

 

{z + (τ ∗ − τ∗ )g(τ∗ , z,  u , α)}.

 z∈  Z∗  u∈ P

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On the axis of the “reverse” time τ , we introduce a finite partition = {τ0 = t0 , τ1 , τ2 , . . . , τi , . . . , τ N = ϑ} of the time interval [t0 , ϑ], where diameter  = ( ) satisfies the relation  = τi+1 − τi = N −1 · (ϑ − t0 ), i = 0, N − 1. Choose in some way the constant x > 0 and define the sets  Z j = (x ) (Z (ρ) (τ j , τ j−1 ,  Z j−1 , α ∗ ),

 Z 0 = (x ) (M),

j = 1, N ,

i =  Z j , i = 0, N , j = N − i, W  Z=

N  j=0

(τ j ,  Z j ),

= W

N 

i ). (ti , W

i=0

During the construction of finite sets  Z j , j = 1, N , for each point z ( j) ∈  Z j we ( j−1) ∈ Z j−1 and the control v ( j−1) = const which will remember the “parent” point z satisfy the following equations: z ( j) = z ( j−1) +  · g(τ j−1 , z ( j−1) , v ( j−1) , α ∗ ),

j = 1, N .

This information will be used in solving the Problem 2.2.  is an approximate solution of the Problem 2.1. The finite set W Formulate an algorithm for solving Problem 2.2. 0 ) < x . Then So, suppose that the initial position (t0 , x (0) ) is such that h(x (0) , W (0) (0) (0)  we can choose the point x ∈ W0 such that ||x − x ||  x . Next, suppose that the point x (0) = z (N ) is obtained from some point x (1) = z (N −1) by applying the constant control v (N −1) on the interval τ = [τ N −1 , τ N ]. Define the desired piecewise-constant control u ∗ (t) = v(N −1) + w(N −1) on the interval [t0 , t1 ], where the constant w(N −1) is the solution of the problem 

z˙ (τ ) = −g(τ, z(t), v(N −1) + w(N −1) , α ∗ ), τ ∈ (τ N −1 , τ N ), z(τ N −1 ) = x (1) , z(τ N ) = x (0) .

By the choice of the function ρ(), there exists a solution w N −1 of the problem that the norm ||w N −1 ||  ρ(). We will also assume that we can solve this problem precisely. Next, let the point x (1) = z (N −1) be obtained from the point x (2) = z (N −2) by applying the constant control v (N −2) at τ ∈ [τ N −2 , τ N −1 ]. Define the desired control u ∗ (t) = v(N −2) + w N −2 on the interval [t1 , t2 ], where constant vector w(N −2) is the solution of the problem 

z˙ (τ ) = g(τ, z(t), v(N −2) + w(N −2) , α ∗ ), τ ∈ (τ N −2 , τ N −1 ), z(τ N −2 ) = x (2) , z(τ N −1 ) = x (1) .

Applying this scheme, we will construct program piecewise-constant control u ∗ (t) = v(N −i) + w(N −i) at t ∈ [ti−1 , ti ] for i = 1, N , where constant correcting vec-

20 Application of Correcting Control in the Problem …

231

tor w(N −i) ∈ B p (0, ρ()) is the solution of the problem 

z˙ (τ ) = −g(τ, z(t), v (N −i) + w(N −i) , α ∗ ), τ ∈ (τ N −i , τ N −i+1 ), z(τ N −i ) = x (i) , z(τ N −i+1 ) = x (i−1) .

Here the point x (i−1) is “obtained” from the point x (i) by applying the control v N −i . when we define the set W

20.4 Estimations Denote by Wˇ α∗ the resolvability set of the approach set of the system (20.1) when ˇ replacing P with Pˇ = P(ρ), where ρ = ρ(). In addition, to reduce the notations, we denote Z α∗ = Z α∗ (τ0 , M) and Zˇ α∗ =   Zˇ α∗ (τ0 , M) = τ ∗ ∈[t0 ,ϑ] z∗ ∈M Zˇ α∗ (τ ∗ , t0 , z ∗ ), where Zˇ α (τ ∗ , τ∗ , z ∗ ) differs from ˇ Z α (τ ∗ , τ∗ , z ∗ ) replacing the restriction P by P. For any set A ⊂ [t0 , ϑ] × R, we will denote its section at time t∗ by A(t∗ ) = {x ∈ Rn : (t∗ , x) ∈ A} ⊂ Rn , t∗ ∈ [t0 , ϑ].  ω(3) (ρ)  L(ϑ−t0 ) e Theorem 20.1 There is an estimation h(Wα∗ , Wˇ α∗ )  −1 . L Proof Note that from the definitions of the function g(τ, z, v, α) = − f (t0 + ϑ − τ, z, v, α) and the function ω(3) (·) follows the estimate ||g(τ, z, v∗ , α) − g(τ, z, v∗ , α)||  ω(3) (||v − vˇ ||), (τ, z, α) ∈ D × L , z ∗ ∈ P, z ∗ ∈ P.

Denote by z(τ ) the motion of the system (20.5) under the application of an arbitrarily chosen control v(t) ∈ P and with the initial point z(t0 ) ∈ M; denote by zˇ (τ ) the motion of the system (20.5) from same initial point zˇ (t0 ) = z(t0 ) under the appliˇ By the definition of the set P, ˇ we can choose such a control cation of control vˇ (t) ∈ P. vˇ (τ ) that ||v(τ ) − vˇ (τ )||  ρ for all τ ∈ [t0 , ϑ]. Define the control vˇ (τ ), τ ∈ [t0 , ϑ] just like that. The equality holds z˙ (τ ) − z˙ˇ (τ ) = g(τ, z(τ ), v(τ ), α∗ ) − g(τ, vˇ (τ ), vˇ (t), α∗ ), τ ∈ (t0 , ϑ). From it we obtain the following estimation: d ||z(τ ) − zˇ (τ )||  ||˙z (τ ) − z˙ˇ (τ )||  ||g(τ, z(τ ), v(τ ), α∗ ) − g(τ, zˇ (τ ), vˇ (τ ), α∗ )|| dτ  ||g(τ, z(τ ), v(τ ), α∗ ) − g(τ, zˇ (τ ), v(τ ), α∗ )|| + ||g(τ, zˇ (τ ), v(τ ), α∗ ) − g(τ, zˇ (τ ), vˇ (τ ), α∗ )||

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 L||z(τ ) − zˇ (τ )|| + ω(3) (ρ). Last estimation implies that ||z(τ ) − zˇ (τ )|| 

 ω(3) (ρ)  L(τ −t0 ) e − 1 , τ ∈ [t0 , ϑ]. L

By virtue of the arbitrary specification of the motion z(τ ) we obtain that h(Z α∗ , Zˇ α∗ ) 

 ω(3) (ρ)  L(ϑ−t0 ) e −1 . L

Hence, by virtue of the well-known (e.g., [3]) equalities Wα∗ = Z α∗ and Wˇ α∗ = ˇ Z α∗ follows the statement of the theorem.  Theorem 20.2 The inequality holds (1) (3) (4)    )  ω () + ω (u ) + ω (δα ) e L(ϑ−t0 ) − 1 + x e L(ϑ−t0 ) . h(Wˇ α∗ , W L

Proof Let zˇ (τ ) be a motion of the system (20.5) from initial position (t0 , x (0) ) under application of some admissible control vˇ (τ ) ∈ Pˇ for almost everyone τ ∈ [t0 , ϑ]. To  ) = h( Zˇ α∗ ,  Z ) we construct the Euler broken line z(τ ) close to estimate h(Wˇ α∗ , W the motion zˇ (τ ) in the Hausdorff metric and one of those that actually form the set  Z. 0 = (x ) (M), we can choose such Define the Euler broken line z(τ ). Since W (0) (0)  its initial position z ∈ Z 0 that ||z − z(τ0 )||  x . Define the piecewise-constant control, under which z(τ ) is formed, as follows: v(t) = v ( j) , t ∈ (t j , t j+1 ],  will be determined later. where the values of the constant vectors v ( j) ∈ P  Note that the set Z actually consists of broken line values defined this way at time moments τ j , j = 0, N . Calculate the absolute difference zˇ (τ ) and z(τ ). Consider the first time interval τ ∈ [τ0 , τ1 ]. The following inequality holds on it: τ     ||ˇz (τ ) − z(τ )||  ˇz (0) + g(ξ, zˇ (ξ ), vˇ (ξ ), α∗ )dξ − z (0) − (τ − τ0 )g(τ0 , z (0) , v (0) , α ∗ ) τ0

 ||ˇz

(0)

τ  τ    − z || +  g(ξ, zˇ (ξ ), vˇ (ξ ), α∗ )dξ − g(τ0 , zˇ (ξ ), vˇ (ξ ), α∗ )dξ  (0)

τ0

τ0

20 Application of Correcting Control in the Problem …

233

τ  τ    + g(τ0 , zˇ (ξ ), vˇ (ξ ), α∗ )dξ − g(τ0 , z (0) , vˇ (ξ ), α∗ )dξ  τ0

τ0

τ  τ    (0) + g(τ0 , z , vˇ (ξ ), α∗ )dξ − g(τ0 , z (0) , v (0) , α∗ )dξ  τ0

τ0

τ  τ    (0) (0) + g(τ0 , z , v , α∗ )dξ − g(τ0 , z (0) , v (0) , α ∗ )dξ  τ0

τ0

 ||ˇz (0) − z (0) || + ω(1) (τ − τ0 ) + (τ − τ0 )L · max ||ˇz (ξ ) − z (0) || ξ ∈[τ0 ,τ ]

τ  τ    (0) + g(τ0 , z , vˇ (ξ ), α∗ )dξ − g(τ0 , z (0) , v (0) , α∗ )dξ  + ω(4) (||α∗ − α ∗ ||). τ0

τ0



We will separately evaluate 

τ

τ0

g(τ0 , z (0) , vˇ (ξ ), α∗ )dξ −



τ τ0

  g(τ0 , z (0) , v(0) , α∗ )dξ .

ˇ α ∗ ) is convex, then, Since set g(τ0 , z (0) , P, the τ ˇ α ∗ ), g(τ0 , z (0) , vˇ (ξ ), α∗ )dξ ∈ (τ − τ0 )g(τ0 , z (0) , P, firstly, τ0

secondly, =

τ

τ0

there

exists g(τ0 , z (0) , vˇ (ξ ), α∗ )dξ .

such

a

vector

v ∈ Pˇ

that

(τ − τ0 )g(τ0 , z (0) , v, α∗ )

 that Then there exists such v(0) ∈ P τ   τ   (0)  g(τ0 , z , vˇ (ξ ), α∗ )dξ − g(τ0 , z (0) , v (0) , α∗ )dξ  τ0

τ0

τ  τ    (0) =  g(τ0 , z , v, α∗ )dξ − g(τ0 , z (0) , v (0) , α∗ )dξ  τ0

τ 

τ0

||g(τ0 , z (0) , v, α∗ ) − g(τ0 , z (0) , v (0) , α∗ )||dξ

τ0

 (τ − τ0 )ω(3) (||v − v (0) ||)  (τ − τ0 )ω(3) (u ). Since this inequality, we obtain the estimation

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||ˇz (τ ) − z(τ )||  ||ˇz (0) − z (0) || + (τ − τ0 ) ω(1) (τ − τ0 ) + L max ||ˇz (ξ ) − z (0) || ξ ∈[τ0 ,τ ]

 +ω(3) (u ) + ω(4) (||α∗ − α ∗ ||) . Note that the estimation on the right monotonically increases in τ . This means that the same estimate is also valid for the maximum ||ˇz (τ ) − z(τ )|| with respect to τ , i.e.,

max ||ˇz (τ ) − z(τ )||  ||ˇz (0) − z (0) || + (τ − τ0 ) ω(1) (τ − τ0 ) + L max ||ˇz (ξ ) − z (0) ||

ξ ∈[τ0 ,τ ]

ξ ∈[τ0 ,τ ]

 +ω(3) (u ) + ω(4) (||α∗ − α ∗ ||) ,

 (1 − L(τ − τ0 )) max ||ˇz (ξ ) − z(ξ )||  ||ˇz (0) − z (0) || + (τ0 ) ω(1) () + ω(3) (u ) + ω(4) (δα ) , ξ ∈[τ0 ,τ ]



||ˇz (τ ) − z(τ )||  max ||ˇz (ξ ) − z(ξ )|| τ ∈[τ0 ,τ ]

 (0) (0) ||ˇz − z || + (τ − τ0 ) ω(1) () + ω(3) (u ) + ω(4) (δα ) 1 − L(τ − τ0 )

(20.6) .

Now introduce the partition γ = {τ (0) = τ0 , ..., τ (k) = τ0 + kn −1 , ..., τ (n) = τ1 } on the interval [τ0 , τ1 ]. The estimate (20.6) implies the inequality ||ˇz (τ (1) ) − z(τ (1) )|| 

 ||ˇz (0) − z (0) || + (τ (1) − τ (0) ) ω(1) () + ω(3) (u ) + ω(4) (δα ) 1 − L(τ (1) − τ (0) )

Similar arguments implies that ||ˇz (τ ( j+1) ) − z(τ ( j+1) )|| 

 ||ˇz (τ ( j) ) − z(τ ( j) )|| + (τ (1) − τ (0) ) ω(1) () + ω(3) (u ) + ω(4) (δα ) 1 − L(τ (1) − τ (0) )

,

j = 0, N − 1.

Using the geometric progression sum formula, we can calculate that ||ˇz (1) − z (1) ||

.

20 Application of Correcting Control in the Problem … 

ω(1) () + ω(3) (u ) + ω(4) (δα ) L



1 1 − Ln −1

n

−1 +

235 1 1 − Ln −1

n

||ˇz (0) − z (0) ||.

Passing to the limit as n → ∞, we obtain the inequality ||ˇz (1) − z (1) || 

 ω(1) () + ω(3) (u ) + ω(4) (δα )  L e − 1 + e L ||ˇz (0) − z (0) ||. L

By similar reasoning, we can obtain that ||ˇz ( j+1) − z ( j+1) || 

 ω(1) () + ω(3) (u ) + ω(4) (δα )  L e − 1 + e L ||ˇz ( j) − z ( j) ||, L

j = 0, N − 1.

In the same way, from this we get that ||ˇz ( j) − z ( j) || 

 ω(1) () + ω(3) (u ) + ω(4) (δα ) Lj e − 1 + e L ||ˇz (0) − z (0) ||, j = 1, N . L

By the arbitrary choice of the control vˇ (τ ) and the estimate ||ˇz (0) − z (0) ||  x it follows that i ) = h( Zˇ (τ j ),  Z j) h(Wˇ α∗ (ti ), W 

 ω(1) () + ω(3) (u ) + ω(4) (δα )  Lj e − 1 + e Lj x , i = N − j, j = 0, N . L However, we can estimate  

N 1 + K 2, Wˇ (ti )  h Wˇ α∗ , i=0 2

so

 (1) (3) (4)   )  ω () + ω (u ) + ω (δα ) e L(ϑ−t0 ) − 1 + e L(ϑ−t0 ) x +  1 + K 2 . h(Wˇ α∗ , W L 2

Remark 20.4 If the initial position x (0) is known with a measurement error δx , then we will have the estimation ||ˇz (0) − z (0) ||  x + δx and, accordingly, the estimation ) h(Wˇ , W 

 ω(1) () + ω(3) (u ) + ω(4) (δα ) L(ϑ−t0 )  − 1 + e L(ϑ−t0 ) (x + δx ) + 1 + K 2. e L 2

Corollary 20.1 We have the following estimation:

236

V. Ushakov and A. Ershov (1) (3) (3) (4)    )  ω () + ω (u ) + ω (ρ()) + ω (δα ) e L(ϑ−t0 ) − 1 h(Wα∗ , W L

+e L(ϑ−t0 ) (x + δx ) +

 1 + K 2. 2

Theorem 20.3 Let the system (20.1) satisfies the conditions A, B, C, D, E, G, H; the control u ∗ (t) is determined according to the algorithm described in Sect. 20.2. Denote by x ∗ (t) the motion of the system (20.1) generated by u ∗ (t). Then the distance ρ(x ∗ (ϑ), M) = max ||x ∗ (ϑ) − y||  ω(4) (δα ) y∈M

e L(ϑ−t0 ) − 1 + δx e L(ϑ−t0 ) , L

where δx is the maximum possible measurement error of the initial position. Proof The correcting control at each step eliminates the distance between Euler’s broken line and the real motion of the system (20.1), which can be generated by an ˇ Therefore, the reasons for the discrepancy admissible control with values from P. can only be an inaccurately determined parameter α and an inaccurately measured initial position (t0 , x (0) ). Let x (0) be an initial point measurement with the error not exceeding δx , i.e., (0) ||x − x (0) ||  δx . Denote by x∗ (t) the system motion generated by the control u ∗ (t), but with the real value α ∗ of the parameter α and the initial position (t0 , x (0) ). Notice that x∗ (ϑ) ∈ M. Then ||x ∗ (t0 ) − x∗ (t0 )|| = ||x (0) − x (0) ||, d||x ∗ (t) − x∗ (t)||  ||x˙ ∗ (t) − x˙∗ (t)||  || f (t, x ∗ (t), u ∗ (t), α∗ ) − f (t, x∗ (t), u ∗ (t), α ∗ )|| dt

 || f (t, x ∗ (t), u ∗ (t), α∗ ) − f (t, x∗ (t), u ∗ (t), α∗ )|| +|| f (t, x∗ (t), u ∗ (t), α∗ ) − f (t, x∗ (t), u ∗ (t), α ∗ )||  L||x ∗ (t) − x∗ (t)|| + ω(4) (δα ). Hence, e Lt

 d ∗ ||x (t) − x∗ (t)||e−Lt  ω(4) (α ). dt

Integrating the last inequality in t from t0 to ϑ, we obtain that ||x ∗ (t) − x∗ (t)||  ω(4) (δα )

e L(t−t0 ) − 1 + ||x (0) − x (0) ||e L(t−t0 ) . L

Defining t = ϑ, we obtain the statement of the theorem.



Applying the same arguments to the “reverse” system (20.5), we can prove the following estimation:

20 Application of Correcting Control in the Problem …

 , Wα∗ )  ω(4) (δα ) Theorem 20.4 h(W

237

e L(ϑ−t0 ) − 1 + δx e L(ϑ−t0 ) . L

 ) = h(Wˇ α∗ , W  ). Corollary 20.2 d(Wˇ α∗ , W

20.5 Conclusions The application of the method of separation of control into basic and correcting controls led to a significant increase in the accuracy of the arrival of the system motion to the target set using program control compared to [4]. However, a drawback of this method is the narrowing of the resolvability set relative to the case without using the separation of control. Acknowledgements This research was supported by the Russian Science Foundation (project no. 19-11-00105).

References 1. Krasovskii, N.N.: Game problems on the encounter of motions [Igrovye zadachi o vstreche dvizhenii]. Nauka, Moscow (1970) (in Russian) 2. Krasovskii, N.N., Subbotin, A.I.: Positional differential games [Pozitsionnye differetsial’nye igry]. Nauka, Moscow (1974) (in Russian) 3. Ershov, A.A., Ushakov, V.N.: An approach problem for a control system with an unknown parameter. Sbornik Math. 208(9), 1312–1352 (2017). https://doi.org/10.1070/SM8761 4. Ushakov, V.N., Ershov, A.A., Ushakov, A.V.: An approach problem with an unknown parameter and inaccurately measured motion of the system. IFAC-PapersOnLine 51(32), 234–238 (2018). https://doi.org/10.1016/j.ifacol.2018.11.387 5. Ushakov, V.N., Ershov, A.A.: On the solution of control problems with fixed terminal time. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki. 26(4), 543–564 ( 2016). https://doi.org/10.20537/vm160409 (in Russian) 6. Ershov, A.A., Ushakov, A.V., Ushakov, V.N.: An approach problem for a control system with a compactum in the phase space in the presence of phase constraints. Sbornik Math. 210(8), 1092–1128 (2019). https://doi.org/10.1070/SM9141 7. Kurjanskii, A.B.: Selected Works [Izbrannye trudy]. Moscow University Press, Moscow (2009) (in Russian)

Chapter 21

On Solving Dynamic Reconstruction Problems with Large Number of Controls Evgenii Aleksandrovitch Krupennikov

Abstract The problem of dynamic reconstruction of the normal control generating some trajectory of a dynamic control system is considered in this paper. Known inaccurate measurements of the realized trajectory, which arrive in real time, are used. A class of dynamic control systems with dynamics linear in controls and nonlinear in state variables is considered. The dimension of the control parameter is supposed to be greater than or equal to the dimension of the state variables vector. A new algorithm for solving dynamic control reconstruction problem for this class of dynamics is suggested. The material presented in this paper continues the study of a new approach for solving inverse problems of optimal control theory. The feature of this approach is using constructions based on the necessary conditions of optimality in auxiliary variational problems on extremum of a convex-concave integral functional. An example with a model from the area of medicine is exposed.

21.1 Introduction The problem of dynamic reconstruction of the normal control is considered in this paper. It is the problem of real-time reconstruction of the control that generates a trajectory of a dynamic control system and has the least possible norm. The reconstruction is done based on inaccurate measurements of the realized trajectory, which arrive in real time. A well-known approach to the dynamic control reconstruction problems has been developed by Kryazhimskii and Osipov [1, 2]. This approach uses a regularized (Tikhonov regularization [3]) procedure of so-called extremal aiming on an auxiliary model. It has roots in the works of N. N. Krasovskii’s school on the theory of optimal feedback control [4, 5]. This approach has been afterward developed in particular by E. A. Krupennikov (B) N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences (IMM UB RAS), 16 S.Kovalevskaya Str., Yekaterinburg 620108, Russia e-mail: [email protected] Ural Federal University, 620002 19 Mira street, Ekaterinburg, Russia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_21

239

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V. I. Maksimov, M. S. Blizorukova, et al. Paper [2] offers a survey of methods based on this approach. Another approach to dynamic reconstruction problems was suggested by Subbotinaet al. [6–12]. It relies on auxiliary variational problems on extremum of a regularized integral functional. The innovation of the suggested method consists of using a convex-concave functional in the auxiliary problems instead of a convex one. The novel results presented in this paper include a modification of this method intended for solving dynamic reconstruction problems with discrete input data. An illustrative example is exposed. A medical model, suggested in [16], with unknown variable parameters is considered. The unknown parameters are reconstructed in real time by known inaccurate discrete measurements of the state variables.

21.2 Dynamics We consider dynamics of the form x(t) ˙ =G(x(t), t)u(t) + f (x(t), t), x(·) : [0, T ] → Rn , u(·) : [0, T ] → Rm , t ∈ [0, T ], m ≥ n, G(·) : Rn × [0, T ] → Rn×m , f (·) : Rn × [0, T ] → Rn , T < ∞. (21.1)

Here x(t) is the vector of the state coordinates and u(t) is the vector of the controls. The admissible controls are continuous functions satisfying the restriction u(t) ∈ U = {u ∈ Rm : u ≤ U¯ > 0}, t ∈ [0, T ].

(21.2)

21.3 Input Data A trajectory x ∗ (·) : [0, T ] → Rn of system (21.1) is generated in real time by some admissible control. The discrete inaccurate measurements {ykδ : ykδ − x ∗ (tk ) ≤ δ, tk = kh δ , k = 0, . . . , K , of the trajectory x ∗ (·) are obtained with step h δ in real time.

K = T / h δ ∈ N} (21.3)

Assumption 21.1 There exist a constant δ0 > 0 and a compact  ⊂ Rn such that (1) For any δ ∈ (0, δ0 ] and any measurement step h δ ∈ (0, T ] 

B2δ0 [ykδ ] ⊂ ,

k=0,...,K , K = T / h δ

where Br [x] is the closed ball of the radius r with the center in x.

21 On Solving Dynamic Reconstruction Problems …

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(2) The elements of the matrix function G(x, t) and the vector-function f (x, t) are continuously differentiable with respect to all variables for each (x, t) ∈  × [0, T ]. (3) The rows of the matrix G(x, t) are linearly independent for each (x, t) ∈  × [0, T ].

21.4 Reconstruction Problem Let U ⊂ C([0, T ]) be the set of admissible controls generating x ∗ (·). It is nonempty and may consist from more than one element. Let us prove that the normal control u ∗ (·) ∈ U that has the least possible L 2 [0, T ] norm exists and is unique. Lemma 21.1 For the trajectory x ∗ (·) of system (21.1)

∗

T

∗

∃!u (·) ∈ U : u (·) = argmin u∈U

u(τ )2 dτ. 0

We consider the function u¯ ∗ (·)  G + (x ∗ (·), ·)(x˙ ∗ (·) − f (x ∗ (·), ·)),

(21.4)

de f

where G + = G T (GG T )−1 is the generalized matrix inverse. For any u(·) ∈ U we have that x˙ ∗ (·) = G(x ∗ (·), ·)u(·) + f (x ∗ (·), ·). Therefore, u¯ ∗ (·) = G + (x ∗ (·), ·)G(x ∗ (·), ·)u(·), ∀u(·) ∈ U. The matrix G + G is idempotent. Indeed, (G + G)(G + G) = G T (GG T )−1 GG T (GG T )−1 G = G T (GG T )−1 G = G + G. An idempotent matrix has only eigenvalues 0 or 1 [14]. Therefore, the spectral norm G + G2 ≤ 1. The spectral norm is consistent with the Euclidean vector norm. So, u¯ ∗ (t) ≤ G + (x ∗ (t), t)G(x ∗ (t), t)2 u(t) ≤ u(t), u(·) ∈ U, t ∈ [0, T ]. (21.5) In particular, (21.5) means that u¯ ∗ (t) ∈ U, t ∈ [0, T ] by Definition (21.2). Moreover, after substituting (21.4) into (21.1), one can check that u¯ ∗ (·) generates x ∗ (·). So, u¯ ∗ (·) ∈ U. Since (21.5), u¯ ∗ (·) L 2 [0,T ] ≤ u(·) L 2 [0,T ] for any u(·) ∈ U. Thus, we obtained that u¯ ∗ (·) has the minimal L 2 [0, T ] norm on U. Now, let us prove that U is convex. We consider two arbitrary elements {u 1 (·), u 2 (·)} ⊂ U. Let us check that u 1 (·) + (u 2 (·) − u 1 (·))θ = u 3 (·) ∈ U, θ ∈ [0, 1].

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G(x ∗ (t), t)u 3 (t) + f (x ∗ (t), t) = G(x ∗ (t), t)(u 1 (t) + (u 2 (t) − u 1 (t))θ) + f (x ∗ (t), t) ± f (x ∗ (t), t)θ = x˙ ∗ (t) + (x˙ ∗ (t) − x˙ ∗ (t))θ = x˙ ∗ (t), t ∈ [0, T ].

(21.6) So, u 3 (·) generates x ∗ (·). Moreover, the set U is convex. Then, u 3 (t) = u 1 (t) + (u 2 (t) − u 1 (t))θ ∈ U, t ∈ [0, T ]. Therefore, u 3 (·) ∈ U and the set U is convex. T The functional 0 u(τ )2 dτ is strongly convex. Therefore, the set of it’s global minimums on U can consist from not more than one point [15]. Thus, u¯ ∗ (·) is the unique element with the minimal L 2 [0, T ] norm on U. Lemma is proven.  Let us consider the following dynamic reconstruction problem: for measurements {ykδ } (21.3) arriving in real time, to find a function u(·, δ) = u δ (·) : [0, T ] → Rm , which is extended step-by-step each time new measurements ykδ arrive, such that C1. The function u δ (·) is an admissible control; C2. The control u δ (·) generates the trajectory x(·, δ) = x δ (·) : [0, T ] → Rn of system (21.1) with the boundary condition x δ (0) = y0δ such that lim x δ (·) − x ∗ (·)C[0,T ] = 0; C3. lim u δ (·) − u ∗ (·) L 2 [0,T ] = 0.

δ→0

δ→0

Remark 21.1 Lemma 21.1 means that formula (21.4) can be used to construct a solution of reconstruction problem C1–C3. Yet, this approach involves matrix inversion. The suggested below approach reduces the inverse problem to integration of a system of linear differential equations. Numerical integration may be more efficient than matrix inversion in some problems, as it was shown in [12].

21.5 Algorithm for Constructing Solution of the Reconstruction Problem The construction of the solution is carried out stepwise with step h δ (with the arrival of new measurements). The algorithm relies on the procedures described in [12, 13].

21.5.1 Input Data Interpolation This method suggested in [12, 13] implies that the measurements are a continuous function y β (·) : [0, T ] → Rn , β > 0 that satisfies the condition y β (·) − x ∗ (·)C[0,T ] ≤ β.

(21.7)

So, let us consider a cubic spline interpolation y β (·) of the discrete input data (21.3): y β (t)  ak t 3 + bk t 2 + ck t + dk , ak , bk , ck , dk ∈ Rn , t ∈ [tk−1 , tk ], k = 1, . . . , K .

(21.8)

21 On Solving Dynamic Reconstruction Problems …

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The coefficients ak , bk , ck , dk are defined by the conditions δ y β (tk−1 ) = yk−1 ,

y˙ β (tk−1 ) =

δ ykδ −yk−2 2h δ

 y  k−1 ,

y β (tk ) = ykδ , y˙ β (tk ) =

except for the first interval [0, t1 ], where y˙ β (0) =

δ δ yk+1 −yk−1 2h δ

(21.9)

 yk ,

y1δ − y0δ . The interpolation is hδ

extended in real time on each algorithm step. The following lemma about the properties of the function y β (·) holds. Lemma 21.2 There exist parameters β δ = β(δ, h δ ) > 0, Y 2,δ 2,δ 0, Y = Y (δ, h δ ) > 0 such that 1,δ L1. max  y˙ β (t) ≤ Y ;

1,δ

=Y

1,δ

(δ, h δ ) >

t∈[0,T ]

L2. max  y¨ β (t) ≤ Y t∈[0,T ]

2,δ

;

L3. y β (·) − x ∗ (·)C[0,T ] ≤ β δ

δ→0,h δ →0

−→

0.

Proof One can derive from (21.8) the algebraic equations that define the coefficients ak , bk , ck , dk of the cubic splines y β (·). They have the form of linear equations. After solving them, and provided that x ∗ (tk ) − ykδ  ≤ δ, we obtain that for t ∈ [tk−1 , tk ]  (t−t )(t−t )6(y δ −y δ )−2h (y  −y  ) h  y  (t−t )2 +y  (t−t )2   k δ δ k−1 k   k−1 k k−1 k k−1 k k−1  y˙ (t) =  −  3 hδ 3  δ hδδ   (yk −yk−1 )    ≤ 6 h δ  + 3y k  + 3y k−1   ∗  ∗    ∗     (x (tk )−x ∗ (tk−2 ))   (x (tk+1 )−x ∗ (tk−1 ))   2δ   x (tk )−x ∗ (tk−1 )  + 1.5 + 1.5 + 12 ≤ 6    h δ .    hδ hδ hδ (21.10) It follows from Assumption 21.1, that x ∗ (t) ∈ , t ∈ [0, T ]. Thus, β

G(x ∗ (t), t)2 ≤ G 

max

x∈,t∈[0,T ]

G(x, t)2 < ∞, G(x ∗ (t), t)u ∗ (t) ≤ GU , t ∈ [0, T ].

(21.11) Analogically,  f (x ∗ (t), t) ≤ F < ∞, t ∈ [0, T ]. These estimates provide that x ∗ (τ2 ) − x ∗ (τ1 ) ≤ |τ2 − τ1 |(GU + F), ∀τ1 , τ2 ∈ [0, T ].

(21.12)

We get by applying (21.12) to (21.10) that  y˙ β (t) ≤ 12(GU + F) + 24 hδδ  Y

1,δ

, t ∈ [0, T ].

Thus, statement L1 is proven. Analogically, we can obtain that for t ∈ [tk−1 , tk ]

(21.13)

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   y¨ β (t) = 2 −





h δ y  k (t−tk−1 )−y  k−1 (t−tk ) hδ 3

≤

12 GUh+F δ



 +

+

δ 3(ykδ −yk−1 )−h δ (y  k +y  k−1 )

12 hδ 2 δ

Y

2,δ

  

(t−tk−1 )+(t−tk )

hδ 3

. (21.14)

Statement L2 is proven. It follows from (21.12) and (21.13) that x ∗ (t) − y β (t) ≤ x ∗ (t) − x ∗ (tk ) + x ∗ (tk ) − ykδ  + ykδ − y β (t) ≤ 13h δ (GU + F) + 25δ  β δ

δ→0,h δ →0

−→

0, t ∈ [0, T ], k = t/ h δ .

So, statement L3 and Lemma 1 are proven.

(21.15) 

21.5.2 Auxiliary Problem To construct the solution of problem C1–C3 on the first step, we introduce the following auxiliary variational problem (AVP) for a function y β (·) : [0, h δ ] → Rn . We consider the set of pairs of continuously differentiable functions Fxu = {{x(·), u(·)} : x(·) : [0, T ] → Rn , u(·) : [0, T ] → Rm } that satisfy differential equations (21.1) and the following boundary conditions x(0) = y β (0), u(0) = G + (y(0), 0)( y˙ β (0) − f (y(0), 0)).

(21.16)

The AVP consists of finding a pair {x(·), u(·)} ∈ Fxu such that it provides an extremum (minimum) for the functional h δ  x(t) − y β (t)2 α 2 u(t)2 − + dt, I (x(·), u(·)) = 2 2

(21.17)

0

where α is a small regularizing parameter [3]. Remark 21.2 The suggested algorithm uses constructions based on the necessary optimality conditions in the AVP. However, it is not verified if the extremum is actually reached in the AVP. In the same way it was shown in [11], the necessary optimality conditions in the AVP can be obtained in the form of the Hamiltonian system, where s(t) is the adjoint variables vector x(t) ˙ = −α −2 G(x(t), t)G T (x(t), t)s(t) + f (x(t), t), β (x(t), t)G T (x(t), t)s(t) + s(t), ∂∂xfi (x(t), t), s˙i (t) = xi (t) − yi (t) + α12 s(t), ∂∂G xi i = 1, . . . , n. (21.18) The boundary conditions are

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  x(0) = y β (0), s(0) = −α 2 G + (y β (0), 0) y˙ β (0) − f (y β (0), 0) .

(21.19)

21.5.3 Solution of the Reconstruction Problem On the first step let us first consider a linearized version of system (21.18) x(t) ˙ = −α −2 G(y β (t), t)G T (y β (t), t)(t)s(t) + f (y β (t), t), s˙ (t) = x(t) − y β (t), t ∈ [0, h δ ]

(21.20)

with boundary conditions (21.19). It is a heterogeneous linear system of ODEs with coefficients that are continuous functions on time t. Therefore, its solution {x(·), s(·)} : [0, h δ ] → R2n exists and is unique and extendable on [0, h δ ]. This solution will be used as a base for construction of the solutions of inverse problem C1–C3. We introduce the so-called cutoff function

β,α β,α u 0 (t) , u 0 (t) ∈ U, β,α β,α uˆ 0 (t) = argminu (t) − w , u β,α (t) ∈ (21.21) / U, 0 0 w∈U

β,α

where U is the set from (21.2) and u 0 (·)  −α −2 G T (x0 (·), ·)s0 (·). Here x0 (·), s0 (·) is the solution of system (21.20) with boundary conditions (21.19). Basing on the results exposed in [13], the following lemma and proposition can be proven. Lemma 21.3 Let Assumptions 21.1 hold. Let a function y β (·) satisfy (21.7) for 0 < β ≤ 0.5δ0 and lim β(Y

β→0

1,β

+Y

2,β

) = 0, Y

1,β

= max  y˙ β (t), Y

2,β

t∈[0,T ]

= max  y¨ β (t). t∈[0,T ]

(21.22) Then there exists a function R(β) that is defined by δ0 from Assumption 21.1, U¯ and 1,β 2,β the parameters Y , Y such that if β→0

β → 0, α = α(β) : α(β)R(β) −→ 0, β,α

the function uˆ 0 (t) (21.21) meets the conditions β,α C1b. The function uˆ 0 (·) belongs to the set of admissible controls; C2b. lim x β (·) − x ∗ (·)C[0,T ] = 0, where x β (·) is the solution of δ→0

β,α

x(t) ˙ = G(x(t), t)uˆ 0 (t) + f (x(t), t), x(0) = y0δ ; β,α

C3b. lim uˆ 0 (·) − u ∗ (·) L 2 [0,T ] = 0. δ→0

(21.23)

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Lemmas 21.2 and 21.3 provide the validity of the proposition Proposition 21.1 If Assumptions 21.1 holds and if δ→0

α = α(δ) : α(δ)R(δ) −→ 0,

δ δ→0 −→ 0, hδ

(21.24)

β then the function uˆ δ,α 0 (t) (21.21) constructed for the functions y (·) (21.9) meets conditions C1–C3.

To construct the solution on a k th (k > 1) step let us consider system (21.20) with boundary conditions x(tk−1 ) = xk−1 (tk−1 ), s(tk−1 ) = sk−1 (tk−1 ),

(21.25)

where {xk−1 (·), sk−1 (·)} is the solution of system (21.20) constructed on the previous step. β,α We consider the functions uˆ k (·) (21.21) constructed for solutions of system (21.20) with boundary conditions (21.25). Based on Proposition 21.1, Lemmas 21.2 and 21.3, and the results exposed in [13], the following proposition can be proven. Proposition 21.2 If Assumptions 21.1 and condition (21.24) hold, then the function β,α uˆ k (t) (21.21), constructed on a finite kth step for the function y β (·) (21.9) and solutions of system (21.20) with boundary conditions (21.25), meets conditions C1– C3. Propositions 21.1 and 21.2 mean that for a finite time interval [0, T ] the functions β,α uˆ k (t) are the solution to problem C1–C3.

21.6 Example As an example let us consider an inverse problem for a model describing the process of penicillin fermentation in organism. It was suggested in [16]: ˙  X (t) μ(t) = G(X (t), S(t), t) + f (X (t), S(t), t), ˙ S(t) ρ(t)  X X 2 /P ∗ , G(X, S, t) = −X/Yx/s −X/Y p/s + X (S f − S)/P ∗   XC/P ∗ X K deg + , f (X, S, t) = −m s S X/(K m + S) −K deg (S f − S) − C(S f − S)/P ∗ ) X (·) : [0, T ] → R, S(·) : [0, T ] → R, μ(·) : [0, T ] → R, ρ(·) : [0, T ] → R, |μ(t)| ≤ 1, |ρ(t)| ≤ 1, T = 15,

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247

Fig. 21.1 Reconstructed approximation of μ(t) and ρ(t)

β

Fig. 21.2 Reconstructed trajectories X β (t) and S β (t) and interpolated measurements y X (t) and β yS (t)

P ∗ = 9, Yx/s = 0.47, Y p/s = 1.2, S f = 500, K m = 0.0001, m s = 0.029,

K deg = 0.01, C = 0.095.

The state variables are X (·), S(·) (concentration of biomass and substrate). The unknown parameters μ(t), ρ(t) (specific biomass growth rate and penicillin production rate) play the role of the controls that are to reconstruct. The data from [16] has been randomly perturbed to simulate the inaccurate measurements {X kδ , Skδ , k = 0, . . . , K }. It is known that S(t) ∈ [1, 20], X (t) ∈ [0, 30], so Assumption 21.1 is fulfilled for  = [0, 30] × [2, 20] and δ0 = 0.5. The graphs of the reconstructed μδ,α(δ) (t), ρ δ,α(δ) (t) are shown on Fig. 21.1. The reconstructed trajectories are shown on Fig. 21.2.

21.7 Comparison with Another Approach The novel feature of the method described in this paper is using functional convex in the control parameters and concave in the discrepancy of the state variables. This method is close to another approach. A survey of algorithms based on this approach is presented in [2]. The approach discussed in [2] implies minimizing constructions convex in all variables. The main difference of these two approaches is that the

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new method allows to obtain solutions that have oscillating character instead of exponential. In the future papers, a more accurate comparison of the approaches will be presented.

21.8 Conclusion This paper continues the developing ([6, 7, 9–13]) of a new method for solving inverse reconstruction problems. In particular, a modification of this approach for solving dynamic reconstruction problems with discrete input data is presented. The perspective directions for the future research on the suggested method include expanding the class of the considered control systems. In particular, a regularization procedure that allows to omit the condition m ≥ n in (21.1) is under development.

References 1. Kryazhimskij, A.V., Osipov, Yu.S.: Modelling of a control in a dynamic system. Eng. Cybern. 21(2), 38–47 (1983) 2. Osipov, Yu.S., Kryazhimskii, A.V., Maksimov, V.I.: Some algorithms for the dynamic reconstruction of inputs. P. Steklov Inst. Math. (2011). https://doi.org/10.1134/S0081543811090082 3. Tikhonov, A.N.: Ob ustoichivosti obratnih zadach [on the stability of inverse problems]. Doklady Academii Nauk SSSR. Moscow 39, 195–198 (1943) 4. Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York (1988) 5. Krasovskii, N.N., Subbotin, A.I.: Positcionnie differentcialnie igri [Positional differential games]. Nauka, Moscow (1974) 6. Subbotina, N.N., Tokmantsev, T.B., Krupennikov, E.A.: On the solution of inverse problems of dynamics of linearly controlled systems by the negative discrepancy method. P. Steklov Inst. Math. (2015). https://doi.org/10.1134/S0081543815080209 7. Subbotina, N.N., Krupennikov, E.A.: Dynamic programming to identification problems. World J. Eng. Technol. (2016). https://doi.org/10.4236/wjet.2016.43D028 8. Subbotina, N.N., Tokmantsev, T.B., Krupennikov, E.A.: Dynamic programming to reconstruction problems for a macroeconomic model. IFIP Adv. Inf. Comm. Te. (2017). https://doi.org/ 10.1007/978-3-319-55795-3_45 9. Subbotina, N.N., Krupennikov, E.A.: The method of characteristics in an identification problem. P. Steklov Inst. Math. (2017). https://doi.org/10.1134/S008154381709022X 10. Krupennikov, E.A.: On control reconstruction problems for dynamic systems linear in controls. In: Static & Dynamic Game Theory: Foundations & Applications, pp. 89–120 (2018). https:// doi.org/10.1007/978-3-319-92988-0_7 11. Krupennikov, E.A.: Solution of inverse problems for control systems with large control parameter dimension. IFAC-PapersOnLine (2018). https://doi.org/10.1016/j.ifacol.2018.11.423 12. Krupennikov, E.A.: A new approximate method for construction of the normal control. IFACPapersOnLine (2018). https://doi.org/10.1016/j.ifacol.2018.11.407 13. Krupennikov, E.A.: On estimates of the solutions of inverse problems of optimal control. Mathematical Optimization Theory and Operations Research (Chap. 39) (MOTOR 2019, CCIS 1090) (2019). https://doi.org/10.1007/978-3-030-33394-2_39 14. Neudecker, H., Magnus J.R.: Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd edn. Wiley (2019). https://doi.org/10.1002/9781119541219

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15. Vasilev, F.P.: Methods for solving problems [Metodi resheniya ekstremalnih zadach.] Moscow (1981) 16. Zhai, C., Qiu, T., Palazoglu, A., Sun, W.: The emergence of feedforward periodicity for the fedbatch vpenicillin fermentation process. IFAC-PapersOnLine (2018). https://doi.org/10.1016/j. ifacol.2018.11.367

Chapter 22

A Class of Initial Value Problems for Distributed Order Equations with a Bounded Operator Vladimir E. Fedorov and Aliya A. Abdrakhmanova

Abstract We study an initial value problem for a class of inhomogeneous distributed order equations in a Banach space with a linear bounded operator in the right-hand side of the equation, solved with respect to the distributed Riemann–Liouville derivative. The unique solvability theorem for the problem is proved; the form of the solution is found. The deduced general result is applied to the unique solvability study of the initial value problem for a system of integrodifferential equations and of an initial boundary value problem for a class of distributed order in time equations with polynomials of a self-adjoint elliptic differential operator with respect to the spatial variables.

22.1 Introduction In the last three decades distributed order differential equations arise in applied problems in describing certain physical or technical processes: in the theory of viscoelasticity [1], in the kinetic theory [2], and so on (see, e.g., [3–7]). At the same time, equations with distributed fractional derivatives increasingly are studied in pure mathematical works (see the works of Nakhushev [8, 9], Pskhu [10, 11], Umarov and Gorenflo [12], Atanackovi´c et al. [13], Kochubei [14] and others). In [15, 16] the unique solvability of the Cauchy problem z (k) (0) = z k , k = 0, 1, . . . , m − 1, is studied for distributed order equation

V. E. Fedorov (B) Chelyabinsk State University, Chelyabinsk, Russia e-mail: [email protected] South Ural State University (National Research University), Chelyabinsk, Russia A. A. Abdrakhmanova Ufa State Aviation Technical University, Ufa, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_22

251

252

V. E. Fedorov and A. A. Abdrakhmanova

b

ω(α)CDtα z(t)dα = Az(t) + g(t), t ∈ [0, T ),

a

with the Gerasimov–Caputo derivative CDtα and with a linear bounded operator A in a Banach space Z, where m − 1 < b ≤ m ∈ N, 0 ≤ a < b, T > 0, g ∈ C([0, T ); Z). The form of the solution is found also. The deduced general results are applied in the analysis of initial boundary value problems to systems of distributed order partial differential equations. The present work is the continuation of the papers [15, 16]. We study the analogous distributed order equation b

ω(α)Dtα z(t)dα = Az(t) + g(t), t ∈ (0, T ),

(22.1)

a

with the the Riemann–Liouville derivative Dtα . Here A is a linear bounded operator in a Banach space Z, a < b ≤ 1, it is possible a < 0, T > 0, g ∈ C([0, T ); Z). The natural initial value problem for this equation has the unusual form b

ω(α)Dtα−1 z(0)dα = z 0 .

(22.2)

max{a,0}

Theorem on unique solvability of problem (22.1), (22.2) and the form of the problem solution are obtained. The results of the present paper develop the theory of resolving operators families for the distributed order equations using the Laplace transform in the spirit of the operator semigroup theory [17] and its generalizations for the integral evolution equations [18, 19], fractional order evolution equations [20, 21]. Abstract results for problem (22.1), (22.2) are used for the unique solvability research of the initial value problem for a system of integrodifferential equations and of an initial boundary value problem for a class of distributed order in time equations with polynomials of a self-adjoint elliptic differential operator with respect to the spatial variables.

22.2 The Initial Value Problem for the Inhomogeneous Equation For β > 0, t > 0 denote gβ (t) := t β−1 / (β), where  is the Euler gamma function, β Jt h(t)

t := 0

1 gβ (t − s)h(s)ds = (β)

t 0

(t − s)β−1 h(s)ds.

22 A Class of Initial Value Problems for Distributed Order …

253

Let 0 < α ≤ 1, Dt1 is the usual first-order derivative, Dtα is the Riemann–Liouville fractional derivative, i.e., Dtα h(t) := Dt1 Jt1−α h(t). We shall use the equality Dt−α h(t) := Jtα h(t) for α > 0. Let Z be a Banach space. The Laplace transform of the function h : R+ → Z is denoted by L[h]. We have for β > 0, α ∈ (0, 1] β

L[Jt h](λ) = λ−β L[h](λ), L[Dtα h](λ) = λα L[h](λ) − Dtα−1 h(0).

(22.3)

Denote by L(Z) the Banach space of all linear continuous operators from Z to Z. For A ∈ L(Z), b ∈ (0, 1] consider the initial problem b

ω(α)Dtα−1 z(0)dα = z 0

(22.4)

max{a,0}

to the distributed order equation b

ω(α)Dtα z(t)dα = Az(t) + g(t), t ∈ (0, T ),

(22.5)

a

where Dtα−1 , Dtα are the Riemann–Liouville fractional derivatives, −∞ < a < b ≤ 1, ω : (a, b) → C. By a solution of problem (22.4), (22.5) we mean a function z ∈ b C((0, T ); Z), such that there exists ω(α)Dtα z(t)dα ∈ C((0, T ); Z) and equalities a

(22.4) and (22.5) are fulfilled. We denote γ :=

3 

γk , γ1 := {λ ∈ C : |λ| = r0 , arg λ ∈ (−π, π )},

k=1

γ2 := {λ ∈ C : arg λ = π, λ ∈ [−r0 , −∞)}, b γ3 := {λ ∈ C : arg λ = −π, λ ∈ (−∞, −r0 ]}, W (λ) :=

ω(α)λα dα,

a

Z (t) :=

1 2πi



(W (λ)I − A)−1 eλt dλ.

γ

Denote by Lap(Z) the set of functions z : R+ → Z, such that there exists L[z].

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V. E. Fedorov and A. A. Abdrakhmanova

Theorem 22.1 Let A ∈ L(Z), ω ∈ L 1 (a, b), for some β > 1 W (λ) be a holomorphic function on the set Sβ := {λ ∈ C : |λ| ≥ β, arg λ ∈ (−π, π )}, satisfying the condition (22.6) ∃C1 > 0 ∃δ ∈ (0, 1) ∀λ ∈ Sβ |W (λ)| ≥ C1 |λ|δ ,   r0 = max β, (2 A L(Z) /C1 )1/δ , z 0 ∈ Z, g ∈ C([0, T ); Z). Then the function t z(t) = Z (t)z 0 +

Z (t − s)g(s)ds 0

is a unique solution to problem (22.4), (22.5) in the space Lap(Z). Proof For λ ∈ γ with the given r0 the inequality |W (λ)| ≥ 2 A L(Z) holds, there exists (W (λ)I − A)−1 ∈ L(Z), and (W (λ)I − A)−1 L(Z) ≤

1  |W (λ)| 1 −

A L(Z) |W (λ)|

≤

2 2 ≤ . C1 |λ|δ C1r0δ

(22.7)

Thus, at t > 0 the integral Z (t) converges. Let R > r0 , R =

4 

k,R , 1,R = γ1 , 2,R = {λ ∈ C : |λ| = R, arg λ ∈ (−π, π )},

k=1

3,R = {λ ∈ C : arg λ = π, λ ∈ [−r0 , −R]}, 4,R = {λ ∈ C : arg λ = −π, λ ∈ [−R, −r0 ]}. So  R is the closed loop, oriented counter-clockwise. Consider also the contours 5,R = {λ ∈ C : arg λ = π, λ ∈ (−R, −∞)}, 6,R = {λ ∈ C : arg λ = −π, λ ∈ (−∞, −R)}, then γ = 5,R ∪ 6,R ∪  R \ 2,R . Due to (22.7) for some ε > 0 Z (t) L(Z)

1 ≤ C1 π

 γ

etReλ ds ≤ K e(r0 +ε)t t δ−1 , t > 0, |λ|δ

(22.8)

22 A Class of Initial Value Problems for Distributed Order …

because 1 π

1 π

 γk



etReλ er 0 t ds ≤ |λ|δ πr0δ−1

γ1

etReλ t δ−1 ds ≤ |λ|δ π

−tr0 −∞

2π

er0 t (cos ϕ−1) dϕ ≤

0

255

2er0 t r0δ−1

, t ≥ 0,

ex t δ−1 (1 − δ) , k = 2, 3, t > 0. dx ≤ |x|δ π

Therefore, Z ∈ Lap(Z). Under the condition Reμ > r0 we have the equality L[Z ](μ) =



1 2πi

γ

1 (W (λ)I − A)−1 dλ. μ−λ

Due to (22.6) these integrals converge and 1 R→∞ 2πi



lim

s,R

1 (W (λ)I − A)−1 dλ = 0, s = 2, 5, 6. μ−λ

Therefore, by the Cauchy integral formula 1 L[Z ](μ) = lim R→∞ 2πi

 R

1 (W (λ)I − A)−1 dλ = (W (μ)I − A)−1 . μ−λ

So L[Dtα−1 Z ] = μα−1 (W (μ)I − A)−1 . Hence, L[Z ] and L[Dtα−1 Z ] have holomorphic extensions on {μ ∈ C : |μ| > r0 , arg μ ∈ (−π, π )}, Dtα−1 Z (t)



1 = 2πi

λα−1 (W (λ)I − A)−1 eλt dλ.

γ

For α < δ this integral converges uniformly with respect to t ∈ [0, 1], Dtα−1 Z (0)

 =

λα−1 (W (λ)I − A)−1 dλ =

γ

⎛ ⎜ =⎝





R

+ 5,R

 + 6,R

 −

⎞ ⎟ α−1 (W (λ)I − A)−1 dλ = 0, ⎠λ

2,R

since the integral on  R is equal to zero by the Cauchy theorem, and other integrals tend to zero as R → ∞, because λα−1 (W (λ)I − A)−1 L(Z) ≤ 2C1−1 |λ|−1−δ+α .

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Thus, we have b

ω(α)Dtα−1 Z (0)dα

b = lim

t→0+

ω(α)Dtα−1 Z (t)dα =

a

max{a,0}

1 t→0+ 2πi



= lim

γ

W (λ) (W (λ)I − A)−1 eλt dλ = λ

⎞ ⎛    ∞ 1 ⎝ eλt Ak eλt = lim dλ + dλ⎠ I = I, k t→0+ 2πi λ λ(W (λ)) k=1 γ

γ

since for λ ∈ γ due to (22.7) ∞    Ak      λ(W (λ))k 

≤

L(Z)

k=1

2 A L(Z) 2 A L(Z) ≤ . |λW (λ)| C1 |λ|1+δ

Further, using formula (22.3) for the Laplace transform, we can write at b ≤ 1 ⎡ L⎣

b

⎤ ω(α)Dtα Z (t)z 0 dα ⎦ (μ) = W (μ) (W (μ)I − A)−1 z 0 −

a

b −

ω(α)Dtα−1 Z (0)z 0 dα = A(W (μ)I − A)−1 z 0 = L[AZ (t)z 0 ](μ).

max{a,0}

We can apply the inverse Laplace transform on the both parts of the equality and obtain equality Dtα Z (t)z 0 = AZ (t)z 0 in all continuity points of function Z (·)z 0 , i.e., for all t > 0. t Denote z g (t) := 0 Z (t − s)g(s)ds, this integral converges due to inequality (22.8). Define g(t) = 0 for t ≥ T , then we have the convolution z g = Z ∗ g, and L[z g ] = L[Z ]L[g]. Hence,   L Dtα−1 z g (μ) = μα−1 (W (μ)I − A)−1 L[g](μ), Dtα−1 z g (t)

t = 0

Dtα−1 Z (t − s)g(s)ds,

Dtα−1 z g ≡ 0, α < δ,

22 A Class of Initial Value Problems for Distributed Order …

⎡

⎤

⎡ b ⎤  ⎥ ω(α)Dtα−1 z g dα ⎦ (μ) = L ⎣ ω(α)Dtα−1 z g dα ⎦ (μ) =

b

⎢ L⎣

257

a

max{a,0}

=

W (μ) (W (μ)I − A)−1 L[g](μ), μ b

ω(α)Dtα−1 z g (0)dα =

max{a,0}

1 t→0+ 2πi

t 

= lim

0

γ

W (λ) (W (λ)I − A)−1 eλ(t−s) dλg(s)ds = λ

⎞ ⎛ t   λ(t−s) t   ∞ 1 ⎝ e eλ(t−s) Ak dλg(s)ds · I + = lim dλg(s)ds ⎠ = t→0+ 2πi λ λ(W (λ))k k=1 0

γ

0

γ

t = lim

g(s)ds = 0

t→0+ 0

due to the continuity of g. Hence, ⎡ L⎣

b

⎤ ω(α)Dtα z g dα ⎦ (μ) = W (μ) (W (μ)I − A)−1 L[g](μ) =

a

= L[g](μ) + A (W (μ)I − A)−1 L[g](μ). Acting by the inverse Laplace transform on the both sides of this equality, obtain b

ω(α)Dtα z g (t)dα = g(t) + A(Z ∗ g)(t) = g(t) + Az g (t)

a

due to the continuity of the linear operator A. If there exist two solutions z 1 , z 2 of problem (22.4), (22.5) from the class Lap(Z), then their difference y = z 1 − z 2 ∈ Lap(Z) is a solution of equation Dtα y(t) = Ay(t) b and satisfies the initial condition max{a,0} ω(α)Dtα−1 y(0)dα = 0. Application of the Laplace transform to the both sides of equation (22.5) gives the equality

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V. E. Fedorov and A. A. Abdrakhmanova

W (λ)L[y](λ) = AL[y](λ). Therefore, for λ ∈ Sβ we have L[y](λ) ≡ 0. It means that y ≡ 0.  Remark 22.1 It can be shown, that, for example, for ω ∈ C([a, b]; R), such that ω(b) = 0, the condition (22.6) holds (see [16]).

22.3 Applications 22.3.1 A System of Integrodifferential Equations Consider the problem 1

Dtα−1 v(s, 0)dα = v0 (s), s ∈ ,

(22.9)

0

1 −1

Dtα v(s, t)dα

 =

K (s, ξ )Bv(ξ, t)dξ + f (s, t), (s, t) ∈ × (0, T ).

(22.10) Here ⊂ Rd is a bounded region, B is a (n × n)-matrix, f ∈ C([0, T ); L 2 ( )), K : × → Rn are given, K ∈ L 2 ( × ), v(s, t) = (v1 (s, t), v2 (s, t), . . . , vn (s, t)) is an unknown vector-function. We take Z = L 2 ( )n , z 0 = v0 = (v01 , v02 , . . . , v0n ) ∈ L 2 ( )n , f (·, t) = g(t), t ∈ [0, T ), (Aw)(s) = K (s, ξ )Bw(ξ )dξ for a vector-function w = (w1 , w2 , . . . ,

wn ) ∈ L 2 ( )n . Then A ∈ L(L 2 ( )n ), ω ≡ 1, W (λ) =

λ − λ−1 ln λ

satisfies the condition (22.6) with any δ ∈ (0, 1). By Theorem 22.1 problem (22.9), (22.10) has a unique solution in Lap(L 2 ( )n ).

22.3.2 A Class of Initial Boundary Value Problems n n Let Pn (λ) = i=0 ci λi , Q n (λ) = i=0 di λi , ci , di ∈ C, i = 0, 1, . . . , n, cn = 0, d ⊂ R be a bounded region with a smooth boundary ∂ , operators pencil , B1 , B2 , . . . , Br be regularly elliptic [22], where

22 A Class of Initial Value Problems for Distributed Order …

(u)(s) =



aq (s)

|q|≤2r

(Bl u)(s) =



blq (s)

|q|≤rl

∂ q1 s

259

∂ |q| u(s) , aq ∈ C ∞ ( ), ∂ q 1 s1 ∂ q 2 s2 . . . ∂ q d sd

∂ |q| u(s) , blq ∈ C ∞ (∂ ), l = 1, 2, . . . , r, q q 1 ∂ 2 s2 . . . ∂ d sd

q = (q1 , q2 , . . . , qd ) ∈ Nd0 . Define the operator 1 ∈ Cl(L 2 ( )) with domain D1 = 2r ( ) [22] by the equality 1 u = u. Let 1 be the self-adjoint operator and it H{B l} has a bounded from the right spectrum. Then the spectrum σ (1 ) of the operator / σ (1 ), {ϕk : k ∈ N} is 1 is real, discrete, and condensed at −∞ only. Let 0 ∈ the orthonormal in L 2 ( ) system of the operator 1 eigenfunctions, numbered in according to nonincreasing of the corresponding eigenvalues {λk : k ∈ N}, taking into account their multiplicity. Consider the initial boundary value problem b

ω(α)Dtα−1 v(s, 0) = v0 (s), s ∈ ,

(22.11)

0

Bl k v(s, t) = 0, k = 0, 1, . . . , n − 1, l = 1, 2, . . . , r, (s, t) ∈ ∂ × (0, T ), (22.12) b ω(α)Dtα Pn ()v(s, t)dα = Q n ()v(s, t) + f (s, t), (s, t) ∈ × (0, T ), a

(22.13) where a ≤ 0 < b ≤ 1, ω : (a, b) → R, f ∈ × [0, T ) → R. Under the condition Pn (λk ) = 0 for all k ∈ N set Z = {u ∈ H 2r n ( ) : Bl k u(s) = 0, k = 0, 1, . . . , n − 1, l = 1, 2, . . . , r, s ∈ ∂ },

L = Pn (),

M = Q n (),

A = L −1 M, g(t) = L −1 f (·, t), z 0 = v0 (·).

Then A ∈ L(Z) and problem (22.11)–(22.13) is presented in form (22.4), (22.5) and by Theorem 22.1 we obtain the next result. Theorem 22.2 Let the spectrum σ (1 ) do not contain zero point and roots of the polynomial Pn (λ), for some β > 1 W (λ) is a holomorphic function on the set Sβ := {λ ∈ C : |λ| ≥ β, arg λ ∈ (−π, π )}, satisfying condition (22.6),  r0 = max β,



2C1−1

Q n (λk ) sup k∈N Pn (λk )

1/δ  ,

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v0 ∈ Z, f ∈ C([0, T ); L 2 ( )). Then there exists a unique solution of problem (22.11)–(22.13) in Lap(Z). Let n = 1, P1 (λ) = 1 − λ, Q 1 (λ) = d1 λ + d0 , u = u, r = 1, B1 = I , f ≡ 0. Then problem (22.11)–(22.13) has the form b

ω(α)Dtα−1 v(s, 0) = v0 (s), s ∈ .

a

v(s, t) = 0, (s, t) ∈ ∂ × R+ , b

ω(α)Dtα (1 − )v(s, t)dα = d1 v(s, t) + d0 v(s, t), (s, t) ∈ × R+ .

a

Acknowledgements The reported study was funded by Act 211 of Government of the Russian Federation, contract 02.A03.21.0011; by Ministry of Science and Higher Education of the Russian Federation, task number 1.6462.2017/BCh; and by Russian Foundation for Basic Research, project number 19-41-450001.

References 1. Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002) 2. Sokolov, I.M., Chechkin, A.V., Klafter, J.: Distributed-order fractional kinetics. Acta Phys. Pol. B 35, 1323–1341 (2004) 3. Caputo, M.: Mean fractional order derivatives. Differential equations and filters. Annali dell’Universita di Ferrara. Sezione VII. Scienze Matematiche XLI, 73–84 (1995) 4. Caputo, M.: Distributed order differential equations modeling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, 421–442 (2001) 5. Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations. Part 2. Int. J. Appl. Math. 2(8), 965–987 (2000) 6. Diethelm, K., Ford, N.J.: Numerical solution methods for distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 531–542 (2001) 7. Jiao, Z., Chen, Y., Podlubny, I.: Distributed-Order Dynamic System: Stability, Simulations, Applications and Perspectives. Springer, London (2012) 8. Nakhushev, A.M.: On continual differential equations and their difference analogues. Doklady Akademii nauk 300(4), 796–799 (1988). (In Russian) 9. Nakhushev, A.M.: Positiveness of the operators of continual and discrete differentiation and integration, which are quite important in the fractional calculus and in the theory of mixed-type equations. Differ. Equ. 34(1), 103–112 (1998) 10. Pskhu, A.V.: On the theory of the continual and integro-differentiation operator. Differ. Equ. 40(1), 128–136 (2004) 11. Pskhu, A.V.: Partial Differential Equations of Fractional Order. Nauka Publ, Moscow (2005). (In Russian) 12. Umarov, S., Gorenflo, R.: Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations. Zeitschrift für Analysis und ihre Anwendungen 24, 449–466 (2005)

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13. Atanackovi´c, T.M., Oparnica, L., Pilipovi´c, S.: On a nonlinear distributed order fractional differential equation. J. Math. Anal. Appl. 328, 590–608 (2007) 14. Kochubei, A.N.: Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl. 340, 252–280 (2008) 15. Streletskaya, E.M., Fedorov, V.E., Debbouche, A.: The Cauchy problem for distributed order equations in Banach spaces. Math. Notes NEFU 25(1), 63–72 (2018). (In Russian) 16. Fedorov, V.E., Streletskaya, E.M.: Initial-value problems for linear distributed-order differential equations in Banach spaces. Electron. J. Differ. Equ. 2018(176), 1–17 (2018) 17. Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. American Mathematical Society, Providence (1957) 18. Prüss, J.: Evolutionary Integral Equations and Applications. Springer, Basel (1993) 19. Kosti´c, M.: Abstract Volterra Integro-Differential Equations. CRC Press, Boca Raton (2015) 20. Bajlekova, E.G.: Fractional Evolution Equations in Banach Spaces. University Press Facilities, Eindhoven University of Technology, Eindhoven (2001). PhD Thesis 21. Li, C.-G., Kosti´c, M., Li, M.: Abstract multi-term fractional differential equations. Kragujev. J. Math. 38(1), 51–71 (2014) 22. Triebel, H.: Interpolation Theory. Function Spaces. Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin (1977)

Chapter 23

A Solution Algorithm for Minimax Closed-Loop Propellant Consumption Control Problem of Launch Vehicle A. F. Shorikov and V. I. Kalev

Abstract This paper considers the minimax closed-loop propellant consumption terminal control problem of liquid-propellant launch vehicle. Given initial model of plant is approximated by linear discrete-time dynamical controllable system, which concludes state vector, control vector, and disturbance vector. Control vector is constrained by finite set and disturbance vector is limited by convex compact set (polytope). Main problem of minimax closed-loop terminal control and auxiliary problem of minimax open-loop control are formulated for approximated system. To solve these two problems an approach and numerical algorithms are provided. The efficiency of proposed approach and algorithms is illustrated on the numerical example of launch vehicle’s third stage propellant consumption terminal control problem.

23.1 Introduction The problem of propellant consumption control is one of the most important tasks (such as guidance, navigation, or attitude control) that launch vehicle’s on-board control system must be able to solve. The history of design of propellant consumption control systems goes back to 1950s, generally to the works of Petrov and his followers [1]. Every liquid-propellant launch vehicle has oxidizer and fuel tanks, which in sum with combustion chamber make together a propulsion engine. The problem at hand consists in total and synchronous consumption of oxidizer and fuel tanks at prescribed time (engines cutoff). It can be formalized as the minimization of final state vector value and desired (reference) final state vector value mismatch, i.e., as the classical terminal control problem. In existing literature [1], this problem was solved and the solution is “statistically optimal”. Actually, statistical (distribution) laws of plant parameters are unknown. A. F. Shorikov (B) Ural Federal University, Yekaterinburg, Russia e-mail: [email protected] V. I. Kalev Scientific and Production Association of Automatics, Yekaterinburg, Russia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_23

263

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Therefore, it is correct to consider the unknown parameters to be “within” some given set of values. In this paper, we take initial discrete-continuous (hybrid) nonlinear mathematical model of typical propellant consumption process and approximate it by linear discrete-time dynamical system that includes state vector, control vector, and disturbance vector, which represents the error of approximation. We assume that the constraint of control vector values is the finite set and that disturbance vector lies within convex compact set, namely, convex, closed, and limited polyhedron with finite number of vertices (later we will use simple name of “polytope” to describe all the properties). Under these constraints in approximated system, the main minimax closed-loop propellant consumption terminal control problem is formulated and it decomposes to number of auxiliary minimax open-loop terminal control problems. To solve considered problem we implement the ideas and approaches described by works [2, 3]. Based on general recurrent algebraic method of reachable set computation [3] with some modifications we provide numerical algorithms for auxiliary and main problems. Effectiveness of proposed approach and designed numerical algorithms toward launch vehicle’s propellant consumption task is demonstrated on simple modeling example of launch vehicle’s third stage engine propellant consumption control.

23.2 Mathematical Model of Propellant Consumption Process Let us consider nonlinear discrete-continuous (hybrid) mathematical model of launch vehicle’s propellant consumption process on the time interval [θ0 , θ f ], where θ0 is turn-on time of propulsion system, θ f is engine cutoff time. Control u(t) ∈ R1 is applied in discrete-time instances {θ0 , θ1 , . . . , θT −1 } ⊂ [θ0 , θ f ], (t ∈ 0, T − 1, T ∈ N), where θT = θ f . Control u(t) changes the angle αth (t) of throttle in fuel pipeline in accordance with following recurrence equation αth (t + 1) = αth (t) + c0 u(t),

αth (0) = 0,

(23.1)

which accordingly changes the mass rates of oxidizer and fuel flow: (P + c1 αth (θ )2 + c2 αth (θ ))(K + c5 αth (θ )) , (I + c3 αth (θ )2 + c4 αth (θ ))(1 + K + c5 αth (θ )) P + c1 αth (θ )2 + c2 αth (θ ) , m f (θ ) = (I + c3 αth (θ )2 + c4 αth (θ ))(1 + K + c5 αth (θ )) m o (θ ) =

(23.2)

where P, I, K are reference values of propulsive force, specific propulsion burn, and oxidizer-fuel ratio, respectively; c0 , . . . , c5 are propulsion engine dynamic coefficients.

23 A Solution Algorithm for Minimax Closed-Loop …

265

Masses of oxidizer Mo (θ ) and fuel M f (θ ) in corresponding tanks can be calculated by θ f Mo (θ ) = M0.o −

m o (θ )dθ , θ0

θ f M f (θ ) = M0. f −

m f (θ )dθ ,

(23.3)

θ0

where M0.o , M0. f are initial masses of oxidizer and fuel in corresponding tanks. Initial nonlinear model (23.1)–(23.3) is linearized along reference trajectory m roe f =

PK , (I + I K )

Mor e f (θ ) = M0.o − θ m roe f ,

ref

mf

=

P , (I + I K )

ref

(23.4) ref

M f (θ ) = M0. f − θ m f ,

and then is discretized with partitioning {θ0 , θ1 , . . . , θT −1 } ⊂ [θ0 , θ f ]. Then, as a result of the implementation of such a procedure, we have approximating linear discrete-time dynamical system on integer-valued time interval 0, T x(t + 1) = A(t)x(t) + B(t)u(t) + D(t)v(t),

x(0) = x0 ,

(23.5)

where x(t) ∈ R4 is a state vector (x(0) = x0 is a given initial state); u(t) ∈ R1 is a control; v(t) ∈ R2 is a disturbance vector, which represents the error of approximation of initial model by linear discrete-time system; A(t), B(t), D(t) are known real-valued matrices with corresponding orders. We assume that control u(t) and disturbance v(t) are constrained as u(t) ∈ U1 (t),

v(t) ∈ V1 (t),

t ∈ 0, T − 1,

(23.6)

besides set U1 (t) ⊂ R1 is a finite set and set V1 (t) ⊂ R2 is a polytope. Let at the time interval τ, T ⊆ 0, T the set of all feasible realizations of open-loop controls is denoted by U(τ, T ) and the set of all feasible realizations of disturbance vector is denoted by V(τ, T ). Let us introduce also the set of all feasible τ -positions w(τ ) = {τ, x(τ )} as W(τ ) = {τ } × R4 . For given feasible data triple (w(τ ), u τ (·), vτ (·)) ∈ W(τ ) × U(τ, T ) × V(τ, T ), the cost function at time interval τ, T is a convex terminal functional γ : W(τ ) × U(τ, T ) × V(τ, T ) → R1 with the following values γτ,T (w(τ ), u τ (·), vτ (·)) = xτ (T ) − xd 4 = (xτ (T )),

(23.7)

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where w(τ ) = {τ, x(τ )}; xτ (T ) = x(T ¯ ; τ, T , x(τ ), u τ (·), vτ (·)) is a final state of approximating system (23.5) trajectory; xd ∈ R4 is a desired value of final state vector;  : R4 → R1 is a convex functional;  · 4 is a Euclidean norm in R4 .

23.3 Problem Statement To solve the main problem let us firstly introduce an auxiliary problem of minimax open-loop terminal control [3] for approximating system (23.5)–(23.7). Problem 23.1 Given integer-valued time interval τ, T ⊆ 0, T and τ -position w(τ ) = {τ, x(τ )} ∈ W(τ ) in system (23.5)–(23.7) it is required to find a set U(e) (τ, T , w(τ )) ⊆ U(τ, T ) of open-loop controls u (e) (·) = {u (e) (t)}t∈τ,T −1 ∈ U(τ, T ) that satisfies to minimax condition [2, 3]  U(e) (τ, T , w(τ )) = u (e) (·) |u (e) (·) = {u (e) (t)}t∈τ,T −1 ∈ U(τ, T ), cγ(e) (τ, T , w(τ )) = γτ,T (w(τ ), u (e) (·), v(e) (·)) = = max γτ,T (w(τ ), u (e) (·), v(·)) = v(·)∈V(τ,T )  = min max γτ,T (w(τ ), u(·), v(·) .

(23.8)

u(·)∈U(τ,T ) v(·)∈V(τ,T )

Let us call the set U(e) (τ, T , w(τ )) a minimax open-loop control set and number an optimal guaranteed (minimax) result of this problem. It has been shown in [3] that since the set U(τ, T ) is finite the solution of Problem 23.1 exists and can be reduced to finite sequence of only one-step operations using general recurrent algebraic method of reachable set computation [3]. Now let us describe the problem of minimax closed-loop terminal control for system (23.5)–(23.7). On the time interval 0, T it is required to compute control u(·) = {u(t) ∈ U1 (t)}t∈0,T −1 using closed-loop control policy with whole information about t-position w(t) = {t, x(t)} ∈ W(t) at every time step t ∈ 0, T − 1 so, that functional (23.7) at final step will take a minimum possible value in the presence of possible the worst disturbance v(·) = {v(t)}t∈0,T −1 ∈ V(0, T ), that is, trying to maximize the functional. To formalize this description of problem we need a few definitions. A feasible propellant consumption closed-loop control policy Ua in discrete-time dynamical system (23.5)–(23.7) at time interval 0, T is a mapping Ua : W(τ ) → U1 (τ ), which for every τ ∈ 0, T − 1 and τ -position w(τ ) = {τ, x(τ )} ∈ W(τ ) assigns a set Ua (w(τ )) ⊆ U1 (τ ) of controls u(τ ) ∈ U1 (τ ). Let us denote by Ua∗ the set of all feasible propellant consumption closed-loop policies. A minimax closed-loop control policy of system (23.5)–(23.7) at time interval 0, T is a specific policy Ua(e) = Ua(e) (w(τ )) ∈ Ua∗ described by the following expressions: cγ(e) (τ, T , w(τ ))

(1) for all τ ∈ 0, T − 1 and τ -positions w(e) (τ ) = {τ, x (e) (τ )} ∈ W(τ ) let

23 A Solution Algorithm for Minimax Closed-Loop …

267

 Ua(e) (w(e) (τ )) = u (e) (τ )|u (e) (τ ) ∈ U1 (τ ),

 u (e) (·) = {u (e) (t)}t∈τ,T −1 ∈ U(e) (τ, T , w(e) (τ )) ,

(23.9)

where x (e) (τ ) = x(τ ¯ ; 0, τ , x0 , u a(e) (·), va(e) (·)); u a(e) (·) ∈ U(0, τ ) and va(e) (·) ∈ V(0, τ ) are the realizations generated by the specific policy Ua(e) ; the set U(e) (τ, T , w(e) (τ )) is the solution of Problem 23.1; (2) for all τ ∈ 0, T − 1 and τ -positions w(τ ) ∈ {W(τ )\w(e) (τ )} let Ua(e) (w(τ )) = U1 (τ ).

(23.10)

Note that for all τ ∈ 0, T − 1 and τ -positions w(τ ) ∈ W(τ ) the sets Ua(e) (w(τ )) are nonempty. Let the controls u a(e) (·) = {u a(e) (t)}t∈0,T −1 ∈ U(0, T ) and the disturbances va (·) = {va (t)}t∈0,T −1 ∈ V(0, T ) are generated using minimax closed-loop control policy Ua(e) = Ua(e) (w(τ )) ∈ Ua∗ at time interval 0, T and let disturbance vector satisfies the expression

=

γT −1,T (wa(e) (T − 1), u a(e) (T − 1), va(e) (T − 1)) = γT −1,T (wa(e) (T − 1), u a(e) (T − 1), v(T − 1)), max

(23.11)

v(T −1)∈V1 (T −1)

where wa(e) (T −1) = {T −1; xa(e) (T −1)} is a (T − 1)-position of system (23.5)–(23.7) ¯ − 1; 0, T , x0 , u a(e) (·), va (·)). such, that xa(e) (T − 1) = x(T (e) Let ∀t ∈ 0, T − 2 : va (t) = va (t), va(e) (·) = {va(e) (t)}t∈0,T −1 ∈ V(0, T ), where (e) va (T − 1) ∈ V1 (T − 1) satisfies a condition (23.8). Then, let us call a number (e) (0, T , w0 ) = γ0,T (w0 , u a(e) (·), va(e) (·)) ca,γ

(23.12)

the minimax (optimal guaranteed) result of applying minimax closed-loop control policy Ua(e) = Ua(e) (w(τ )) ∈ Ua∗ to discrete-time dynamical system (23.5)–(23.7). Thus, we ready to formalize main propellant consumption minimax closed-loop terminal control problem. Problem 23.2 Given time interval 0, T and initial position w0 = {0, x0 } ∈ W(0) in system (23.5)–(23.7) it is required to find a minimax closed-loop control policy Ua(e) = Ua(e) (w(τ )) ∈ Ua∗ , w(t) ∈ W(t), t ∈ 0, T − 1, which satisfies (23.9), (23.10), using an algorithm with finite sequence of only one-step operations. Taking into account system (23.5)–(23.7) properties and results of [2, 3] we can hold that solution of Problem 23.2 exists. Moreover, the solution result of Problem 23.2 can only improve the solution result of Problem 23.1 in case the disturbances do not realized purposefully in the worst way.

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23.4 The Solution Algorithms for Problems 23.1 and 23.2 A main contribution in our solution algorithms has general recurrent algebraic method of reachable sets computation. Let us introduce the following definition. Definition 23.1 A generalized reachable set (GRS) of system (23.5)–(23.7) at time T corresponding to the tuple (X (τ ), u ∗ (·)), X (0) = x0 in presence of feasible open-loop control u ∗ (·) = {u ∗ (t)}t∈τ,T −1 ∈ U(τ, T ) is a set G(τ, X (τ ), u ∗ (·); T ) = {x(T )|x(T ) ∈ R4 , x(t + 1) = A(t)x(t) + B(t)u ∗ (t) + D(t)v(t), t ∈ τ, T − 1, x(τ ) ∈ X (τ ), v(t) ∈ V1 (t)}.

(23.13)

Let us now provide a basic algorithm of general recurrent algebraic method of reachable sets computation. Algorithm 23.1 (GRS computation) 1. Input: X (τ ) = {x(τ )}, u ∗ (·) = {u ∗ (t)}t∈τ,T −1 ∈ U(τ, T ). 2. For all t ∈ τ, T − 1 do: X (t + 1) = RemoveRedundancy((A(t)X (t) + B(t)u ∗ (t)) ⊕ ⊕D(t)V1 (t)); 3. Output: G(τ, X (τ ), u ∗ (·); T ) = X (T ). As the sets V1 (t), t ∈ 0, T − 1 are polytopes, GRS is also polytopes. An operation RemoveRedundancy in Algorithm 23.1 means the finding of all extreme points among the set of points inside brackets (reduces to the solution of multiple linear programs (LP) [3]). An operation ⊕ is a Minkowski sum of two sets [3, 4]. Note that all sets in Algorithm 23.1 are polytopes with vertex representation [4]. Therefore, let us present numerical algorithms for solving Problem 23.1. Algorithm 23.2 (Minimax open-loop terminal control) 1. Generating of set U(τ, T ) = {u ( j) (·)} j∈1,N (ascend ordering by index j); 2. For all j ∈ 1, N : 2.1. Computation of GRS G(τ, X (τ ), u ( j) (·); T ) = X (T ) by Algorithm 23.1 [2]; 2.2. Optimization of convex functional (23.7) under constraint set G(τ, X (τ ), u ( j) (·); T ) = X (T ), that is, finding following value: ( j)

=

γ˜τ,T = γτ,T (x0 , u ( j) (·), v(e) (·)) = = max γτ,T (x0 , u ( j) (·), v(·)) = v(·)∈V(τ,T )   x(T ) − xd 4 = x (e) (T ) − xd 4 . max

x(T )∈G(τ,X (τ ),u ( j) (·);T )

(23.14)

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269

3. Finding the minimax open-loop control set U(e) (τ, T , w(τ )) and optimal guaranteed result cγ(e) (τ, T , w(τ )) by solving the following optimization problem: U(e) (τ, T , w(τ )) = {u (e) (·)|u (e) (·) ∈ U(τ, T ), ( j) (e) min max γτ,T (x0 , u ( j) (·), v(·)) = min γ˜τ,T = γτ,T }.

u(·)∈U(τ,T ) v(·)∈V(τ,T )

(23.15)

j∈1,N

Let in system (23.5)–(23.7) τ -position w(e) (τ ) = {τ, x (e) (τ )} ∈ W(τ ), w(e) (0) = w0 is already generated. Thus, for τ ∈ 0, T − 1 a minimax closed-loop control policy Ua(e) ∈ Ua∗ can be implemented using the following finite sequence of operations. Algorithm 23.3 (Minimax closed-loop terminal control) 1. Compute at time interval τ, T a minimax open-loop control set U(e) (τ, T , w(τ )) (e) using Algorithm 23.2. and minimax result cγ(e) (τ, T , w(τ )) = γτ,T (e) (e) 2. Compute Ua (w (τ ) ⊆ U1 (τ ) using expressions (23.9), (23.10). 3. Choose any control u (e) (τ ) ∈ Ua(e) (w(e) (τ ). 4. Using initial nonlinear discrete-continuous model (23.1)–(23.3) compute (τ +1)¯ +1; τ, τ + 1, x (e) (τ ), u (e) (τ ))}. position w(e) (τ +1) = {τ +1, x (e) (τ +1) = x(τ 5. If (τ + 1) ≤ T − 1 go to 1, otherwise go to 6. 6. Using expressions (23.11), (23.12) compute optimal guaranteed (minimax) result (e) (0, T , w0 ) = γ0,T (w0 , u a(e) (·), va(e) (·)) corresponding to realizations of control ca,γ (e) u a (·) = {u a(e) (t)}t∈0,T −1 ∈ U(0, T ) and disturbance va (·) = {va (t)}t∈0,T −1 ∈ V(0, T ), which are generated by policy Ua(e) = Ua(e) (w(τ )) ∈ Ua∗ at time interval 0, T . Note that proposed Algorithm 23.3 consists in solving of finite number of linear and convex programs, one-step operations under polytopes and conversions between vertex and halfspace representations of polytopes.

23.5 Numerical Example Let us demonstrate effectiveness of proposed approach in model example of propellant consumption closed-loop terminal control for launch vehicle’s third stage propulsion system. Propellant consumption process is described by Eqs. (23.1)–(23.3) at time interval [0, θ f ] with parameters from Table 23.1. The approximating linear discrete-time dynamical system corresponding to initial model (23.1)–(23.3) is described by following vector-matrix equation: x(t + 1) = Ax(t) + Bu(t) + Dv(t), t ∈ 0, T − 1, T = 4, where x(t) = {x1 (t), x2 (t), x3 (t), x4 (t)} ∈ R4 ; x1 (t) is mass rate of oxidizer; x2 (t) is mass of oxidizer in tank; x3 (t) is mass rate of fuel; x4 (t) is mass of fuel in tank;

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Table 23.1 Parameters of initial nonlinear discrete-time model I (s)

P (tf)

K

M0.o (kg)

M0. f (kg)

c0

320

528

2.57

119,610

45,220

13

c1

c2

c3

c4

c5

θ f (s)

1/125

−6700

9000

−15

−12

100

u(t) is scalar control constrained by set U1 (t) = {0.6; 0.3; 0; −0.3; −0.6}, ∀t ∈ 0, 3; v(t) is disturbance (model approximation error), constrained by set V1 (t) = {v(t) | |v1 (t)| ≤ 0.4, |v2 (t)| ≤ 0.4} ⊂ R2 , ∀t ∈ 0, 3; initial state vector x(0) = x0 = ( 1182.074 119610 459.832 45220 ) ; matrices A, B, D equal to ⎛

1 ⎜ −25 A=⎜ ⎝ 0 0

0 1 0 0

⎞ ⎛ ⎞ ⎛ ⎞ 0 0 20.073 10 ⎜ ⎟ ⎜ ⎟ 0 0⎟ 0 ⎟, B = ⎜ ⎟, D = ⎜ 0 0 ⎟. ⎠ ⎝ ⎠ ⎝ 1 0 −10.473 0 −1 ⎠ −25 1 0 00

A cost function is a convex terminal functional at time T = 4 (x(T )) = (x1 (T ) − 1200)2 + x2 (T )2 + (x3 (T ) − 450)2 + x4 (T )2 . Implemented Algorithm 23.2 obtained the solution of auxiliary Problem 23.1, that is, the minimax open-loop control set U(e) (0, T , w(0)) containing a single feasible open-loop control u (e) (·) = {u (e) (t)}t∈0,3 = {0.6, 0.6, −0.3, 0}, which guarantees a result no worse than cγ(e) (0, T , w0 ) = 158.498. Implemented Algorithm 23.3 obtained the solution of main Problem 23.2, i.e., minimax closed-loop control policy had generated the set Ua(e) (w(e) (t)) containing a single feasible control u a(e) (·) = {u a(e) (t)}t∈0,3 = {0.6, 0.3, 0.6, −0.6}, which guarantees a result no worse (e) (0, T , w0 ) = 63.375. than ca,γ Obtained minimax open-loop u (e) (·) and closed-loop u a(e) (·) controls are applied to initial model (23.1)–(23.3) and the results are presented in Table 23.2. The trajectories of initial (23.1)–(23.3) and approximating system (23.5)–(23.7) using obtained minimax controls are presented in Fig. 23.1. The program for simulation is written in MATLAB language. Table 23.2 The results of numerical simulation System

Control

x1 (t)

x2 (t)

x3 (t)

x4 (t)

(x(T ))

Initial

u ref (t) = 0

1200

0

450

0

0

Approx.

u (e) (·)

1198.54

107.7

452

−116.3

158.5

Initial

u (e) (·)

1199.55

91.2

450.24

−45.6

101.97

Approx.

u a(e) (·)

1198.91

−56.8

450.95

28.1

63.375

Initial

(e) u a (·)

1199.55

−56.8

450.24

28.1

63.36

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Fig. 23.1 The projections of state trajectories according to the results of solving Problems 23.1 and 23.2

23.6 Conclusion This paper provides the algorithm of minimax closed-loop control policy for solving the propellant consumption control problem of launch vehicle. This is done by approximation of initial nonlinear discrete-continuous model by discrete-time dynamical system and using the general recurrent algebraic method of reachable set computation [3] for solving the auxiliary minimax open-loop control problem. The results of the numerical simulation demonstrate a good performance of represented algorithms. Thus, we can conclude that proposed approach is applicable for propellant consumption control problems of launch vehicles. Acknowledgements This work was supported by RFBR (project # 18-01-00544).

References 1. Petrov, B.N., et al.: On-board terminal control systems. In: Principles of Construction and Elements of Theory, 200 pp. Mashinostroenie, Moscow (1983) (in Russian) 2. Krasovskii, N.N.: Theory of the Control of Motion, 476 pp. Nauka, Moscow (1968) (in Russian) 3. Shorikov, A.F.: Minimax Estimation and Control in Discrete-Time Dynamical Systems, 242 pp. Ural University Publication, Ekaterinburg (1997) (in Russian) 4. Ziegler, G.M.: Lectures on Polytopes, 384 pp. Springer, New York, (1998)

Chapter 24

A Problem of Dynamic Optimization in the Presence of Dangerous Factors Sergey M. Aseev

Abstract We consider an optimal control problem with a mixed functional and free stopping time. Dynamics of the system is given by means of a differential inclusion. The integral term of the functional contains the characteristic function of a given open set M ⊂ Rn which can be interpreted as a “risk” or “dangerous” zone. The statement of the problem can be treated as a weakening of the statement of the classical optimal control problem with state constraints. We study relationships between these two problems. An illustrative example is presented as well.

24.1 Problem Formulation Let M be a given open set in Rn . Denote by δ M (·) its characteristic function:  δ M (x) =

1, x ∈ M, 0, x ∈ / M.

The problem (P) with discontinuous integrand is stated as follows: T J (T, x(·)) = ϕ(T, x(0), x(T )) +

λ(x(t))δ M (x(t)) dt → min,

(24.1)

0

x(t) ˙ ∈ F(x(t)), x(0) ∈ M0 ,

x(T ) ∈ M1 .

(24.2) (24.3)

S. M. Aseev (B) Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_24

273

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Here the state vector x belongs Rn , the sets M0 and M1 in Rn are nonempty and closed, the multivalued mapping F : Rn ⇒ Rn is locally Lipschitz with nonempty convex compact values, ϕ : [0, ∞) × Rn × Rn → R1 is a locally Lipschitz function, and λ : Rn → (0, ∞) is a continuously differentiable function. We assume that both sets M and G = Rn \ M are nonempty, and the interior of the Clarke tangent cone TG (x) is also nonempty for any x ∈ G (see [1]). Further, let stopping time T > 0 be free. Assign by H (F(x), ψ) = max f ∈F(x)  f, ψ the value of the Hamiltonian of the differential inclusion (24.2) and by ∂ H (F(x), ψ) the Clarke subdifferential of H (F(·), ·) at a point (x, ψ) ∈ Rn × Rn . Also, denote by ˆ ∂φ(T, x1 , x2 ) the generalized gradient of function φ(·, ·, ·) at (T, x1 , x2 ) ∈ [0, ∞) × Rn × Rn . Denote by N A (a) and Nˆ A (a) the Clarke normal cone and the cone of generalized normals to a closed set A ⊂ Rn at a point a ∈ A, respectively. All Caratheodory solutions x(·) of differential inclusion (24.2) that are defined on various intervals [0, T ], T > 0, and satisfy endpoint constraints (24.3) are considered as admissible trajectories in (P). We will call all such pairs (T, x(·)) admissible in (P). An admissible pair (T∗ , x∗ (·)), T∗ > 0, is optimal if the functional (24.1) takes the minimal possible value at (T∗ , x∗ (·)). The functional (24.1) contains the discontinuous function δ M (·) in the integrand. This provides the main peculiarity to problem (P). This integral term penalizes the presence of the states of the system in the set M. Such dangerous sets could appear in considerations of various applied problems (for example, in economics, ecology, and engineering) when there is an admissible but undesirable set of states M ⊂ Rn . In the literature the presence of an undesirable set M in Rn is considered usually via additional state constraint (see [2, Chap. 6]) x(t) ∈ G = Rn \ M,

t ∈ [0, T ].

This constraint prohibits the presence of the system in the set M. It is assumed usually that the set G is closed. Earlier an optimal control problem similar to (P) was considered in [3] in the case of a closed convex set M, linear control system and on some additional regularity assumptions on behavior of an optimal trajectory x∗ (·). Then the case of time-dependent closed convex set M = M(t), t ∈ [0, T ], was considered in [4] on the same linearity and regularity assumptions. In [5, 6] the problem of optimal crossing a given closed set M was studied and necessary conditions for optimality were developed without any regularity assumptions for affine in control systems. This result (again in the case when the set M is closed) was extended to a more general functional in [7]. Notice that the approach developed in [5–7] is not applicable if the set M is open. Nevertheless, the situation when M is open is the most important. In this case the problem (P) is closely related to the classical problem with state constraints. In fact its statement can be treated as a weakening of the statement of problem with state constraints in this case. Notice, that nondegenerate necessary optimality conditions for problem (P) have been developed recently in [8, 9]. These conditions are similar to ones obtained earlier in [10] for the problem with state constraints.

24 A Problem of Dynamic Optimization in the Presence of Dangerous Factors

275

The goal of the present paper is to study relationships between problem (P) and the optimal control problem for differential inclusion with state constraints. Our main result (Theorem 24.2 in Sect. 24.2) states that under suitable controllability assumptions problem (P) can provide an exact penalization of the corresponding problem with state constraints. In Sect. 24.3 we present an illustrative example.

24.2 Problem ( P) and Problem with State Constraints Consider the following optimal control problem with state constraint (Q): J˜(T, x(·)) = ϕ(T, x(0), x(T )) → min, x(t) ˙ ∈ F(x(t)), x(t) ∈ G, x(0) ∈ M0 ,

(24.4)

t ∈ [0, T ],

(24.5)

x(T ) ∈ M1 .

(24.6)

Here G = Rn \ M defines states constraints. All Caratheodory solutions x(·) of (24.4) that are defined on various intervals [0, T ], T > 0, satisfy the state constraint (24.5) and the endpoint constraints (24.6) are considered as admissible trajectories in problem (Q). All such pairs (T, x(·)) will be called admissible in (Q). For an arbitrary real λ > 0 consider also the following problem (Pλ ) which is a special case of (P) with λ(x) ≡ λ, x ∈ Rn : T Jλ (T, x(·)) = ϕ(T, x(0), x(T )) + λ

δ M (x(t)) dt → min,

(24.7)

0

x(t) ˙ ∈ F(x(t)), x(0) ∈ M0 ,

x(T ) ∈ M1 .

Notice that the sets of admissible pairs (T, x(·)) in problems (P) and (Pλ ) coincide. Everywhere in this section we assume that either the set M0 or the M1 is compact ˜ and M0 ∩ M1 = ∅. Assume also that there is at least one admissible trajectory x(·) ], T  > 0, in (Q) (and hence in (P)). In particular this defined on an interval [0, T means that x0 ∈ G and x1 ∈ G. We assume also that all admissible trajectories x(·) in problems (Q) and (P) are extendable to [0, ∞). Therefore, we will always assume such trajectories to be defined on [0, ∞), unless otherwise stated. Since the multivalued mapping F(·) is locally Lipschitz and its values F(x) are nonempty convex compacts for all x ∈ Rn the Theorem 3 in [11, Sect. 7] implies that

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the sets of all admissible trajectories {x(·)} in problems (Q) and (P) considered on [0, T ], are compacts in the space C([0, T ], Rn ) of all continuous Rn -valued functions on [0, T ] with the uniform metric. Assumption ( A1). The following condition takes place: lim

inf

T →∞ (x0 ,x1 )∈M0 ×M1

ϕ(T, x0 , x1 ) = ∞.

If (A1) holds then due to the standard existence result (see [12, Theorem 9.2.i]) an optimal admissible trajectory x∗ (·) in (Q) exists. Similarly, if Assumption (A1) is valid then due to the lower semi-continuity of the integral term in the functional (24.7) with a fixed T > 0 (see [8, Theorem 1]) an optimal admissible trajectory x∗ (·) in (Pλ ) also exists for any λ > 0. Theorem 24.1 Assume (A1) is valid, and let {(TN , x N (·))}∞ N =1 be a sequence of optimal pairs (TN , x N (·)) in (PN ), N = 1, 2, . . . . Then passing if necessary to a subsequence without loss of generality we assume lim TN = T∗ > 0,

N →∞

lim x N (·) = x∗ (·) uniformly on [0, T∗ ],

N →∞

where (T∗ , x∗ (·)) is a solution in (Q). Proof Due to (A1) the positive sequence {TN }∞ N =1 is bounded. Hence, passing if necessary to a subsequence we can assume that lim N →∞ TN = T∗ ≥ 0. Since M0 ∩ M1 = ∅ and at least one of the sets M0 or M1 is compact we have T∗ > 0. Consider the sequence {x N (·)}∞ N =1 on [0, T∗ ]. Since the set {x(·)} of admissible trajectories of (P) considered on [0, T∗ ] is compact in C([0, T∗ ], Rn ), and all trajectories x N (·) are admissible in (P), we can assume that there is a trajectory x∗ (·) of differential inclusion (24.2) such that x N (·) → x∗ (·) in C([0, T∗ ], Rn ) as N → ∞. Since x N (0) ∈ M0 , x N (TN ) ∈ M1 , N = 1, 2, . . . , and lim N →∞ TN = T∗ we get x∗ (0) ∈ M0 and x∗ (T∗ ) ∈ M1 . Thus (T∗ , x∗ (·)) is admissible in (P) and hence in (PN ) for any N = 1, 2, . . . For any N = 1, 2, . . . we have TN φ(TN , x N (0), x N (TN )) + N

, x(0), )), δ M (x N (t)) dt ≤ φ(T ˜ x( ˜ T

0

T , x(·)) where (T ˜ is an admissible pair in (Q). Hence, lim N →∞ 0 ∗ δ M (x N (t)) dt = 0. This implies x∗ (t) ∈ G for all t ∈ [0, T∗ ]. Thus, (T∗ , x∗ (·)) is an admissible pair in (Q). This pair is optimal in (Q). Indeed, let (Tˆ , x(·)) ˆ is an arbitrary admissible pair ˆ is admissible in (Q). Then since the pair (TN , x N (·)) is optimal and the pair (Tˆ , x(·)) in (PN ), N = 1, 2, . . . , for any N = 1, 2, …we have

24 A Problem of Dynamic Optimization in the Presence of Dangerous Factors

TN φ(TN , x N (0), x N (TN )) + N

277

δ M (x N (t)) dt ≤ φ(Tˆ , x(0), ˆ x( ˆ Tˆ )).

0

From the other hand we have lim N →∞ φ(TN , x N (0), x N (TN )) = φ(T∗ , x∗ (0), ˆ x( ˆ Tˆ )). x∗ (T∗ )). These imply φ(T∗ , x∗ (0), x∗ (T∗ )) ≤ φ(Tˆ , x(0), Let us introduce the following controllability type assumption. Assumption (A2). For any point ξ ∈ ∂G there are an ε > 0 and a L > 0 such that for any trajectory x(·) of the differential inclusion (24.2) which is defined on a time interval [a, b], 0 ≤ a < b, satisfies inclusions x(a) ∈ ∂G and x(b) ∈ ∂G, and lies in M ∩ int Bε (ξ ) for all t ∈ (a, b), where Bε (ξ ) = {x ∈ Rn : x − ξ  ≤ ε}, there is an other trajectory x(·) ˆ of the differential inclusion (24.2) which is defined on a time interval [a, c], a < c, satisfies boundary conditions x(a) ˆ = x(a) and x(c) ˆ = x(b), lies in G for al t ∈ [a, c], and we have c − a ≤ Lx(b) − x(a). The following result demonstrates that if Assumptions (A1) and (A2) are satisfied then for all large enough values λ > 0 the problems (Pλ ) provide exact penalizations of the problem (Q) with state constraints. Theorem 24.2 Let (A1) and (A2) hold. Then there is a real λˆ ≥ 0 such that for any λ > λˆ the problem (Pλ ) is equivalent to the problem (Q), i.e., the sets of their solutions and the corresponding values of the functionals coincide. Proof Assume that assertion of the theorem fails. Then, since (A1) holds, for any λ N = N , N = 1, 2, . . . there is an optimal admissible pair (TN , x N (·)), TN > 0, in (PN ) such that x N (·) does not satisfy the state constraint (24.5), i.e., at some point ξ N ∈ (0, TN ) we have x N (ξ N ) ∈ M. Due to Theorem 24.1 without loss of generality we assume lim N →∞ TN → T∗ > 0, lim N →∞ x N (·) → x∗ (·) in C([0, T∗ ], Rn ), there trajectory x∗ (·) is optimal in (Q). Hence, we can assume that ξ N → ξ ∈ [0, T∗ ] as N → ∞. Due to (A2) there are an εξ > 0 and an L ξ > 0 such that for any trajectory x(·) of differential inclusion (24.2) which is defined on a time interval [a, b], 0 ≤ a < b, satisfies inclusions x(a) ∈ ∂G and x(b) ∈ ∂G, and lies in M ∩ int Bεξ (ξ ) for all t ∈ (a, b), there is an other trajectory x(·) ˆ of the differential inclusion (24.2) which is defined on a time interval [a, c], a < c, satisfies boundary conditions x(a) ˆ = x(a) and x(c) ˆ = x(b), lies in G for al t ∈ [a, c], and we have c − a ≤ L ξ x(b) − x(a). Due to optimality of x N (·) and admissibility of x∗ (·) in (PN ), N = 1, 2, . . . , we have lim N →∞ meas{t ∈ [0, TN ] : x N (t) ∈ M} = 0. Hence, lim N →∞ (b N − a N ) = 0 and there is a natural Nξ such that for any N ≥ Nξ we have [a N , b N ] ⊂ Bεξ (ξ ). Hence, for any N ≥ Nξ there is an other trajectory xˆ N (·) of differential inclusion (24.2) which is defined on a time interval [a N , c N ], a N < c N , satisfies boundary conditions xˆ N (a N ) = x N (a N ) and xˆ N (c N ) = x(b N ), lies in G for all t ∈ [a N , c N ], and we have c N − a N ≤ L ξ x(b N ) − x(a N ). Without loss of generality we can assume that |c N − b N | < TN . For N ≥ Nξ define an admissible trajectory x˜ N (·) of differential inclusion (24.2) N = TN + c N − b N > 0, as follows: N ], T on time interval [0, T

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⎧ ⎪ ⎨x N (t), t ∈ [0, a N ], x˜ N (t) = xˆ N (t), t ∈ [a N , c N ], ⎪ ⎩ x N (t + b N − c N ), t ∈ [c N , TN + c N − b N ]. Due to the local Lipschitz continuity of the function φ(·, ·, ·) there is a constant C ≥ 0 such that for the corresponding value of the functional we get N , x˜ N (·)) = ϕ(T N , x N (0), x N (TN )) + N J N (T

TN δ M (x˜ N (t)) dt 0



≤ ϕ(TN , x N (0), x N (TN )) + C  TN − TN + N

TN

b N δ M (x N (t)) dt − N

δ M (x N (t)) dt aN

0

≤ J N (TN , x N (·)) + C |c N − b N | − N (b N − a N )  N − a N ), (24.8) ≤ J N (TN , x N (·)) − (N − C)(b

 is a constant. Inequality (24.8) implies that for all large enough natural N we where C N , x˜ N (·)) < JN (TN , x N (·)). The last inequality contradicts the optimality have JN (T of the pair (TN , x N (·)) in (PN ). Notice that if the minimal λˆ satisfying conditions of Theorem 24.2 is positive then due to continuity of the functional Jλ (T, x(·)) in λ ≥ 0 (see (24.7)) the optimal values of functionals in problems (Pλˆ ) and (Q) are the same. However, as it can be seen in the example below the set of optimal trajectories in (Pλˆ ) can be large than in (Q).

24.3 Example λ ): Consider the following problem ( P J˜λ (T, x(·)) = T + λ

T δ M (x(t)) dt → min,

(24.9)

0

x˙ 1 (t) = x 2 (t), x˙ 2 (t) = u(t), x0 = (x 1 (0), x 2 (0)) = (0, 0),

|u| ≤ 1, x1 = (x 1 (T ), x 2 (T )) = (2, 0).

Here T > 0 is a free terminal time, λ > 0 is a constant, and the open set M =  x = (x 1 , x 2 ) ∈ R2 : |x 2 | > 1 is a risk zone.

24 A Problem of Dynamic Optimization in the Presence of Dangerous Factors

279

λ ) is a particular case of problem (P). Its statement weakens the Obviously, ( P  statement of the following time optimal problem with state constraints ( Q): J˜(T ) = T → min,

(24.10)

x˙ 1 (t) = x 2 (t),

(24.11)

x˙ 2 (t) = u(t),

|u| ≤ 1,

(24.12)

 x(t) ∈ G = x = (x 1 , x 2 ) ∈ R2 : |x 2 | ≤ 1 = R2 \ M, x0 = (x 1 (0), x 2 (0)) = (0, 0),

x1 = (x 1 (T ), x 2 (T )) = (2, 0).

(24.13)

It is easy to show directly that T∗ = 3 is the optimal time, and u ∗ (·) defined as ⎧ ⎪ t ∈ [0, 1], ⎨1, u ∗ (t) = 0, t ∈ [0, 2], ⎪ ⎩ −1, t ∈ [2, 3],

(24.14)

 is the corresponding (unique) optimal control in ( Q). Since all assumptions of Theorem 24.2 are satisfied, there is a minimal λˆ ≥ 0   ˆ such that for all λ > √λ the problems ( Pλ ) are equivalent to ( Q). As far as T˜∗ = 2 2 < T∗ is the optimal time in the time optimal problem (24.10)– (24.12), (24.13) without any state constraints, it can be seen that for all small enough  This imply that λ ) are smaller than in ( Q). λ > 0 the optimal values in problems ( P λˆ > 0. λ ). It is easy to see that due to the For arbitrary 0 < λ ≤ λˆ consider the problem ( P classical Pontryagin’s maximum principle for time optimal problems without state λ ) either moves along parabolas x 1 = constraints any optimal trajectory xλ (·) in ( P



 2 2 1 x 2 + C with u(t) ≡ 1 or x 1 = − 21 x 2 + C with u ∗ (t) ≡ −1 (see [2]) (on the 2 corresponding open time intervals), or belongs the boundary ∂G = {(x 1 , x 2 ) : |x 2 | = 1} of G. Here C ∈ R1 is a constant. Since δ M (x) = 0 in G, any optimal trajectory

2 xλ (·) moves along parabola x 1 = 21 x 2 on the initial time interval [0, 21 ] and along

2 ∗ > parabola x 1 = − 21 x 2 + 2 on the final time interval [Tλ − 1, Tλ ], there Tλ ≥ T 1  1 is the optimal time in ( Pλ ). On the intermediate time interval [ 2 , Tλ − 1] the

2

2 trajectory xλ (·) could move along parabolas x 1 = 21 x 2 + C or x 1 = − 21 x 2 + C in M or it belongs the line {(x 1 , x 2 ) : x 2 = 1} ⊂ ∂G. Let I ⊂ [ 21 , Tλ − 1] be an open set where the trajectory xλ (·) lies in M. (1) If I = ∅ then I = ∪i (αi , βi ) is a union of a finite or a countable number of open intervals (αi , βi ), αi < βi , xλ (αi ) ∈ ∂G, xλ (βi ) ∈ ∂G. We can move all these intervals to the left side of the interval [ 21 , Tλ − 1] putting α1 = 21 , β1 = α2 , . . . By this way we get a new admissible trajectory x˜λ (·) defined on the same time interval [0, Tλ ]. It is easy to see that this operation does not change the value of the

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λ ). Due to functional Jλ (·, ·) (see (24.9)). Hence (Tλ , x˜λ (·)) is an optimal pair in ( P its optimality the constructed trajectory x˜λ (·) has a simple structure. It moves along

2  ], 0 < γ = i (βi − αi ) ≤ Tλ − 23 , parabola x 1 = 21 x 2 on a time interval [0, 1+γ 2

 2 then along parabola x 1 = − 21 x 2 + 1 + γ on the time interval [ 1+γ , 21 + γ ], then 2 along the line {(x 1 , x 2 ) : x 2 = 1} ⊂ ∂G on the time interval [ 21 + γ , Tλ − 1], and

2 finally along parabola x 1 = − 21 x 2 + 2 on the time interval [Tλ − 1, Tλ ]. The direct calculations show that in the case 0 < λ < λˆ the optimality of x˜λ (·) implies γ√= 1. Hence, x˜λ (·) ≡ xλ (·) is a unique optimal trajectory in this case, and Tλ = 2 2 < T∗ while the corresponding optimal control u˜ λ (·) is the following: 

√ 1, t ∈ [0, 2], √ √ u˜ λ (t) = −1, t ∈ [ 2, 2 2].

(24.15)

The direct calculations give us the corresponding optimal value of the functional: √ √ J˜λ (Tλ , xλ (·)) = 2 2 + 2λ( 2 − 1). This implies λˆ =

√ 3−2 2 . √ 2( 2 − 1)

(2) If I = ∅ then xλ (·) lies in G. Due to Theorem 24.2 this implies λ = λˆ . √ Thus, for any 0 < λ < λˆ the√optimal time √ is Tλ = 2 2, the optimal value of the functional is J˜λ (Tλ , xλ (·)) = 2 2 + 2λ( 2 − 1), and the √ corresponding (unique) optimal control u˜ λ (·) is defined on the time interval [0, 2 2] by (24.15). For λ = λˆ ˜ the optimal value of the functional in ( P˜λˆ ) equals 3. In this √case ( Pλˆ ) has two solutions. The first solution is defined on the time interval [0, 2 2] by (24.15). The second solution is defined on the time interval [0, 3] by (24.14). This solution is exactly the  For λ > λˆ the optimal value of the functional in ( P˜ˆ ) equals same as in problem ( Q). λ 3. In this case the problem ( P˜λˆ ) has a unique solution which coincide with solution  (see (24.14)). of problem ( Q)

References 1. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) 2. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962) 3. Pshenichnyi, B.N., Ochilov, S.: On the problem of optimal passage through a given domain. Kibern. Vychisl. Tekhn. 99, 3–8 (1993) 4. Pshenichnyi, B.N., Ochilov, S.: A special problem of time-optimal control. Kibern. Vychisl. Tekhn. 101, 11–15 (1994)

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5. Aseev, S.M., Smirnov, A.I.: The Pontryagin maximum principle for the problem of optimal crossing of a given domain. Dokl. Math. 69(2), 243–245 (2004) 6. Aseev, S.M., Smirnov, A.I.: Necessary first-order conditions for optimal crossing of a given region. Comput. Math. Model. 18(4), 397–419 (2007) 7. Smirnov, A.I.: Necessary optimality conditions for a class of optimal control problems with discontinuous integrand. Proc. Steklov Inst. Math. 262, 213–230 (2008) 8. Aseev, S.M.: An optimal control problem with a risk zone. In: 11th International Conference on Large-Scale Scientific Computing, LSSC 2017, Sozopol, Bulgaria, 5–9 June, 2017. Lecture Notes in Computer Science, vol. 10665. Springer, Cham, pp. 185–192 (2018) 9. Aseev, S.M.: On an optimal control problem with discontinuous integrand. Proc. Steklov Inst. Math. 24(Suppl. 1), S3–S13 (2019) 10. Arutyunov, A.V., Aseev, S.M.: Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints. SIAM J. Control Optim. 35(3), 930–952 (1997) 11. Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer, Dordrecht (1988) 12. Cesari, L.: Optimization is Theory and Applications. Problems with Ordinary Differential Equations. Springer, New York (1983)

Chapter 25

On Piecewise Linear Minimax Solution of Hamilton–Jacobi Equation with Nonhomogeneous Hamiltonian L. G. Shagalova

Abstract Terminal Cauchy problem for nonhomogeneous Hamilton–Jacobi equation is considered in the case when state space is Euclidean plane. The Hamiltonian and the terminal function are piecewise linear. This problem reduces to a problem with a homogeneous Hamiltonian in three-dimensional state space. A finite algorithm for the exact construction of the minimax and/or viscosity solution is developed. The algorithm consists of a finite number of consecutive stages, at each of which elementary problems of several types are solved and the continuous gluing of these solutions are carried out. The solution built by the algorithm is a piecewise linear function. Cases are also indicated when the original problem with a nonhomogeneous Hamiltonian can be solved on a plane without moving to the problem in three-dimensional space.

25.1 Introduction Hamilton–Jacobi equations arise in many areas of mechanics, mathematical physics, and optimal control. As a rule, such equations do not have classical solutions. Solutions of such equations are understood in a generalized sense. This paper is devoted to the construction of a minimax solution [1, 2] of the terminal Cauchy problem for Hamilton–Jacobi equation. For considered problem minimax solution is equivalent to viscosity solution introduced in [3]. Exact formulas for such solutions can be obtained only in very few cases. In most practical problems, solutions have to be found using numerical methods. In [4, 5] the Hamilton–Jacobi equation with positively homogeneous piecewise linear input data (Hamiltonian and terminal function) was considered in twoL. G. Shagalova (B) N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russia e-mail: [email protected] Ural Federal University named after the first President of Russia B.N.Yeltsin, Yekaterinburg, Russia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_25

283

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dimensional state space, and a finite algorithm for the exact construction of the minimax solution was developed. The homogeneity of the Hamiltonian was significant in [4, 5]. Here the case of the nonhomogeneous Hamiltonian is considered, and an algorithm for the exact construction of the minimax solution is proposed and justified. In this case, the original problem is reduced to a problem with a homogeneous Hamiltonian in three-dimensional state space. The solution built by the algorithm is a piecewise linear function. This algorithm was used in [6] for exact constructing value function of a differential game with simple motions. Also, here some cases are indicated where the problem with the nonhomogeneous Hamiltonian can be solved without increasing the dimension of state space. Presented results are of independent interest, and they can also be useful for piecewise linear approximation of minimax solutions for Hamilton–Jacobi equations with Hamiltonians of general type.

25.2 Problem Statement Consider the following Cauchy problem: ∂ω(t, x) +H ∂t



∂ω(t, x) ∂x

 = 0, t ≤ ϑ, x ∈ R n

ω(ϑ, x) = σ (x), x ∈ R n .

(25.1) (25.2)

Here ϑ is a given positive number, the Hamiltonian H (·) and the function σ (·) are assumed to be globally Lipschitz continuous. It is known [1, 2] that minimax (and/or viscosity [3]) solution ω(·) of the problem (25.1), (25.2) exists and is unique. But finding this solution in the general case is a difficult task. If at least one of the functions H (·) or σ (·) is convex or concave, explicit expressions for the minimax solution can be obtained using Hopf-Lax and Pshenichyi-Sagaidak formulas (see [7–9]). Also, if the state space is two-dimensional, and the Hamiltonian H (·) and the terminal function σ (·) are piecewise linear, to construct the minimax solution ω(·) exactly, one can use the finite algorithm described in [4, 5]. This algorithm has been developed under the assumption that functions H (·) and σ (·) are positively homogeneous, i.e., satisfy the following conditions: σ (λx) = λσ (x), x ∈ R n , λ ∈ R, λ > 0,

(25.3)

H (λs) = λH (s), s ∈ R n , λ ∈ R, λ > 0.

(25.4)

Both conditions (25.3), (25.4) are very significant. The aim of this paper is to generalize algorithm proposed in [4, 5] to the case of a nonhomogeneous Hamiltonian H (·), that is, when condition (25.4) is not fulfilled.

25 On Piecewise Linear Minimax Solution of Hamilton–Jacobi …

285

25.3 Known Facts. Reduced Problem This section contains the information from [1, 2] and facts that can be easily obtained from this information. The facts presented here will be used below to develop the algorithm for constructing the piecewise linear solution to the problem described above. Assume the limit exists s  = H0 . (25.5) lim r H r ↓0 r Let us introduce functions H  (s, r ) =



|r |H ( |rs | ), if r = 0, if r = 0, H0 ,

(s, r ) ∈ R n × R,

σ  (x, y) = σ (x) + y, x ∈ R n , y ∈ R.

(25.6) (25.7)

It is easy to see that the Hamiltonian H  (·) is positively homogeneous with respect to the variable s = (s, r ). H (λs) = λH (s), s ∈ R n+1 , λ ∈ R, λ > 0. Consider the following Cauchy problem: ∂u(t, x, y) + H ∂t



∂u(t, x, y) ∂u(t, x, y) , ∂x ∂y

 = 0, t ≤ ϑ, (x, y) ∈ R n × R

u(ϑ, x, y) = σ  (x, y), x ∈ R n , y ∈ R.

(25.8) (25.9)

The following assertion was proved in [1]. Theorem 25.1 The function ω(t, x) is a minimax solution of problem (25.1), (25.2) if and only if the function u(t, x, y) = ω(t, x) + y is the minimax solution of problem (25.8), (25.9). Minimax solution u(t, x, y) for problem (25.8), (25.9) with Lipschitz continuous Hamiltonian H  (·) satisfies the relation  u(t, x, y) = (ϑ − t)u 0,

y x , ϑ −t ϑ −t

 x ∈ R n , y ∈ R.

(25.10)

Based on (25.10), one can move from problem (25.8), (25.9) to a reduced problem. It is required in the reduced problem to find function ϕ(x, y) = u(0, x, y) x ∈ R n , y ∈ R.

(25.11)

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Function ϕ(·) is a minimax solution of the first-order PDE  ∂ϕ(x, y) ∂ϕ(x, y) + x, +y· H − ϕ(x, y) = 0, ∂x ∂y (25.12) where x ∈ R n , y ∈ R, x, z denotes the inner product of vectors x and z. PDE (25.12) is considered together with the limit relation 



∂ϕ(x, y) ∂ϕ(x, y) , ∂x ∂y

lim αϕ α↓0





x y = σ  (x, y), x ∈ R n , y ∈ R. , α α

(25.13)

The minimax solution of (25.12) can be defined as a continuous function satisfying the pair of differential inequalities. One can write these inequalities in various equivalent forms. It is convenient for us to present these inequalities as follows: H  (l, m) + l, x + m · y ≤ ϕ(x, y), x ∈ R n , y ∈ R, (l, m) ∈ D − ϕ(x, y), (25.14) H  (l, m) + l, x + m · y ≥ ϕ(x, y), x ∈ R n , y ∈ R, (l, m) ∈ D + ϕ(x, y). (25.15) Here symbols D − ϕ(x, y) and D + ϕ(x, y) denote the subdifferential and the superdifferential of function ϕ(·) at the point (x, y), respectively.

25.4 Exact Constructing the Solution for Reduced Problem For the situation when the state space is Euclidean plane (i.e., n = 2), and functions σ (·) and H (·) are piecewise linear, the minimax solution ω(·) of problem (25.1), (25.2) is piecewise linear too. It is possible to receive ω(·) exactly. We propose the algorithm here with the help of which one can construct function ϕ(·). Knowing ϕ(·), one can obtain function ω(·) by using Theorem 25.1 and relation (25.10).

25.4.1 Splitting of Terminal and Limit Functions Note y+ = max{0; y},

y− = min{0; y};

σ+ (x) = max{0; σ (x)}, σ− (x) = min{0; σ (x)}; 



σ+ (x, y) = σ+ (x) + y+ , σ− (x, y) = σ− (x) + y− , where y ∈ R, x ∈ R n .

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Function σ  (·) (25.7) can be presented as the sum of positive and negative components   (25.16) σ  (x, y) = σ+ (x, y) + σ− (x, y), x ∈ R n , y ∈ R. Let ϕ+ (·) and ϕ− (·) be solutions of problem (25.12), (25.13) corresponding to   limit functions σ+ (·) and σ− (·), respectively. Then, for the solution ϕ(·) of problem (25.12), (25.13) the representation holds ϕ(x, y) = ϕ+ (x, y) + ϕ− (x, y), x ∈ R n , y ∈ R.

(25.17)

Under conditions described in the following subsection, constructing functions ϕ+ (·) and ϕ− (·) does not differ in essence.

25.4.2 Assumptions on Input Data Algorithm for constructing minimax solution ϕ(·) of the reduced problem is developed under conditions specified in this subsection. A1. Hamiltonian H (·) is piecewise linear. It is formed by “sewing together” a finite number of linear functions of the form H i (s) =< h i , s > + pi ,

i ∈ 1, n H , h i ∈ R 2 , pi ∈ R, s ∈ R 2 .

(25.18)

Remark If A1 is satisfied, then limit (25.5) exists. A2. Terminal function σ (·) is piecewise linear and satisfies condition (25.3). It is formed by “sewing together” a finite set of linear functions σ i (x) =< s i , x >, i ∈ 1, n σ , s i ∈ R 2 , x ∈ R 2 . Denote by Z the set of vectors defining linear functions that form σ (·) Z = {s i |i ∈ 1, n σ }.

(25.19)

Taking into account representations (25.16), (25.17), one can assume also, without loss of generality, that A3. Function σ (·) is non-negative σ (x) ≥ 0, x ∈ R 2 .

(25.20)

So, one can consider the algorithm for constructing the function ϕ(·), corresponding to the limit function σ  (x, y) = σ (x) + y+ , x ∈ R 2 , y ∈ R.

(25.21)

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25.4.3 Instruments for Formalization of Constructions Here the concept of simple piecewise linear function (SPLF) introduced in [4, 5] will also be useful. We omit the full definition of this object and point out only the main property of SPLF. If ψ(·) is an SPLF, then for an arbitrary point x∗ ∈ R 2 in its domain, there exists a neighborhood Oε (x∗ ) where function ψ(·) has one of three possible representations: ψ(x) =< si , x > +h i , ψ(x) = max{< si , x > +h i , < s j , x > +h j }, ψ(x) = min{< si , x > +h i , < s j , x > +h j }. Here si and s j are vectors in R 2 , and h i and h j are numbers. So, there is no point in the domain of SPLF such that in small neighborhoods of the point three or more linear functions are glued together. Formal definition of SPLFs based on use structural matrices. The structural matrix (SM) contains an information about all linear functions, that are forming the corresponding SPLF. Knowing the corresponding SM, one can easily calculate the value of the SPLF at every point from its domain. Remark If condition A3 is satisfied, function σ : R 2 −→ R + is SPLF in region R 2 \ 0. Here 0 is zero vector, R + = {x ∈ R|x ≥ 0}. An example of the possible behavior of function σ level lines is shown on Fig. 25.1. Using the apparatus of simple piecewise linear functions allows us to formalize the construction of the minimax solution.

Fig. 25.1 Level lines of terminal function σ

20 15 10

x2 5 0 −5 −10 −10

−5

x1

0

5

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25.4.4 Elementary Problems As a matter of fact, constructing function ϕ(·) is a process which consists in sequential solving some elementary problems of several types. These problems arise in a certain order. This order is specified using structural matrices. Then solutions of elementary problems are gluing together to create continuous function which satisfies differential inequalities (25.14), (25.15). Let us describe elementary problems here. Denote ς + (x, y) = max{< a, x > +y, < b, x > +y}, ς − (x, y) = min{< a, x > +y, < b, x > +y}, where a, b, x are vectors from R 2 , y ∈ R. Problems 1 and 2. Given linearly independent vectors a ∈ R 2 and b ∈ R 2 . It is required in elementary problem 1 [problem 2] to construct minimax solution of (25.12), (25.13), (25.18) with σ  = ς + [with σ  = ς − ]. Function ς + is convex, and the function ς − is concave. So, using Hopf formulas, one can obtain explicit expressions for solutions of problems 1 and 2. The solutions of these problems are functions φ + (x, y) = max φl (x, y), φ − (x, y) = min φl (x, y), l∈[a,b]

l∈[a,b]

where [a, b] = {λa + (1 − λ)b | λ ∈ [0, 1]}, φl (x, y) =< l, x > +y + H (l). The first stage of the algorithm for constructing the solution ϕ(·) of reduced problem (25.12), (25.13), (25.21) consists in solving problems 1 and 2. Actual problems needed to solve are determined by function σ (·) and its structural matrix. It may be that by gluing together the solutions of elementary problems at the first stage, we construct a function defined on the whole space R 3 . Then the algorithm ends. But such a situation is rare. As a rule, construction should be continued in the second and subsequent stages. In these further stages we have to solve elementary problems of another type. Let s¯ = (s1 , s2 , s3 ) ∈ R 3 . Denote ϕs¯ (x, y) =< s, x > +s3 · y + H  (¯s ), x ∈ R 2 , y ∈ R, where the vector s ∈ R 2 is formed by the first two components of the vector s¯ , s = (s1 , s2 ). Note that if s3 = 1, then H  (¯s ) = H (s) and ϕs¯ (x, y) = φs (x, y). For given set F we will denote its closure by the symbol cl F and its boundary by the symbol ∂ F.

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Problems 3 and 4. Given linearly independent vectors a¯ = (a1 , a2 , a3 ) ∈ R 3 and b¯ = (b1 , b2 , b3 ) ∈ R 3 , and a number r > 0. Let ϕ ∗ (x, y) = max{ϕa¯ (x, y), ϕb¯ (x, y)}, ϕ∗ (x, y) = min{ϕa¯ (x, y), ϕb¯ (x, y)}, G ∗ = {(x, y) ∈ R 3 |ϕ ∗ (x, y) < r }, G ∗ = {(x, y) ∈ R 3 |ϕ∗ (x, y) < r }. In problem 3 it is required to find a continuous function ϕ 0 : clG ∗ → R which is a minimax solution of the PDE (25.12) in G ∗ and satisfies relations ϕ 0 (x, y) < r, ∀(x, y) ∈ G ∗ ; ϕ 0 (x, y) = r, ∀(x, y) ∈ ∂G ∗ . In problem 4 it is required to find a continuous function ϕ0 : clG ∗ → R which is a minimax solution of the PDE (25.12) in G ∗ and satisfies relations ϕ0 (x, y) < r, ∀(x, y) ∈ G ∗ ; ϕ0 (x, y) = r, ∀(x, y) ∈ ∂G ∗ . The solution of Problem 3 is the function ¯ ¯ b), ϕ 0 (x, y) = max ϕs¯ (x, y) for s¯ ∈ Sr (a, s¯

where

¯ = {¯s ∈ con(a, ¯ < s, w0 > +H  (¯s ) = r }, ¯ b) ¯ b)| Sr (a, ¯ = {λa¯ + μb)|λ ¯ con(a, ¯ b) ≥ 0, μ ≥ 0},

and point w0 ∈ R 2 is the solution of linear system of two equations ¯ = r. ¯ = r, < b, w0 > +H  (b) < a, w0 > +H  (a) Components of vectors a ∈ R 2 and b ∈ R 2 coincide with the first two components ¯ respectively. of a¯ and b, The solution of Problem 4 in general case, for arbitrary vectors a¯ and b¯ may not exist. But in the cases which arise in the construction of function ϕ(·) to problem (25.12), (25.13), (25.21) this solution exists. It is the function ¯ ¯ b). ϕ0 (x, y) = min ϕs¯ (x, y) for s¯ ∈ Sr (a, s¯

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25.4.5 The Structure of Piecewise Linear Solution Denote by  the set of points in the space R 3 where the Hamiltonian H  : R 3 → R is nondifferentiable, and by 0 the zero vector in R 3 . On the base of the set Z ⊂ R 2 (25.19) of vectors forming function σ , we define the set Z  ⊂ R 3 Z  = {¯s = (s1 , s2 , s3 ) ∈ R 3 |s = (s1 , s2 ) ∈ Z , s3 = 1}. Theorem 25.2 Suppose conditions A1–A3 are satisfied. Then (A) Function ϕ(·) constructed as a result of algorithm described above is a nonnegative function. It formed by sewing together linear functions ϕs¯ (x, y) =< s, x > +s3 · y + H  (¯s ), s¯ ∈ L , where the set L is finite, and Z  ⊂ L , (L \ Z  ) ⊂ ( ∪ 0). (B) For every y∗ ∈ R function ϕ(x, y∗ ) in the region {x ∈ R 2 |ϕ(x, y∗ ) > 0} is formed by sewing together a finite collection of SPLFs. (C) Function ϕ(·) is the minimax solution of problem (25.12), (25.13), (25.21). Proof Parts (A) and (B) of the theorem follow directly from the description of proposed algorithm given in previous subsections. So, it is necessary to prove part (C) and show that the function ϕ(·) is really a minimax solution. To do this, one should verify the inequalities (25.14), (25.15) and the limit relation (25.13). Omitting the detailed description, one can say that function ϕ(·) is glued together from solutions of elementary problems, so inequalities (25.14), (25.15) are satisfied. Now prove the limit relation (25.13). Consider an arbitrary point (x∗ , y∗ ), x∗ ∈ R 2 , y∗ ∈ R, y∗ ≥ 0. If x∗ is the zero vector, then for any α > 0 the vector xα∗ is also zero. For y∗ = 0 we have   0 0 , = lim αϕ(0, 0) = 0 = σ  (0, 0). lim αϕ α↓0 α↓0 α α If y∗ = 0, it follows from the algorithm that there exist a number α∗ > 0 and a vector s ∈ R 2 such that for all 0 < α < α∗ the following relation is valid.  y  y∗ ∗ = + H (s), ϕ 0, α α and we get

 lim αϕ α↓0

0 y∗ , α α



= y∗ = σ  (0, y∗ ).

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Now let x∗ = 0. If there exists a neighborhood O(x∗ ) in which the function σ (·) is linear σ (x) = a, x , x ∈ O(x∗ ) ⊂ R 2 , then we have from the algorithm that there exists a number α∗ > 0 such that for all 0 < α < α∗  y  x y ∗ ∗ = a, + ϕ x, + H (a). α α α Thus, lim αϕ α↓0

y∗  = a, x∗  + y∗ = σ  (x∗ , y∗ ). α α

x

∗

,

It remains to consider the case when in the neighborhood O(x∗ ) function σ (·) is glued together from two linear functions . For definiteness, let us assume that linear functions are glued by the maximum operation σ (x) = max {a, x , b, x} x ∈ O(x∗ ).  Then a number α∗ > 0 exists such that for all 0 < α < α∗ the point xα∗ , yα∗ belongs to the region in which function ϕ coincides with the solution of the first elementary problem defined by the vectors a and b, and ϕ



y∗  x∗ y∗ = max λa + (1 − λ)b, + + H (λa + (1 − λ)b) . λ∈[0,1] α α α α

x

∗

,

We get lim αϕ α↓0

y∗  = max {a, x , b, x} + y = σ  (x∗ , y∗ ). α α

x

∗

,

So, the verification of the limit relation (25.13) is complete.

25.5 Problem with Prehomogeneous Hamiltonian Definition 25.1 A nonhomogeneous Hamiltonian H : R n → R will be called prehomogeneous if there exists a vector d ∈ R n such that the Hamiltonian Hd (x) = H (x + d) is positively homogeneous. Consider the Cauchy problem ∂ωd (t, x) + Hd ∂t



∂ωd (t, x) ∂x

 = 0, t ≤ ϑ, x ∈ R n

ωd (ϑ, x) = σ (x) − d, x, x ∈ R n .

(25.22) (25.23)

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Theorem 25.3 Function ωd (t, x) is the minimax solution of problem (25.22), (25.23) if and only if function ω(t, x) = ωd (t, x) + d, x is the minimax solution of problem (25.1), (25.2). Proof It is obviously that (25.2) is valid iff (25.23) is satisfied. It is not difficult to see that for all (t, x) ∈ (0, ϑ) × R n the following equalities are valid for sub- and superdifferentials of functions ω and ωd : D − ω(t, x) = {(a, s + d)|a ∈ R, s ∈ R n ; (a, s) ∈ D − ωd (t, x)},

(25.24)

D + ω(t, x) = {(a, s + d)|a ∈ R, s ∈ R n ; (a, s) ∈ D + ωd (t, x)}.

(25.25)

Thus, the assertion of theorem follows from (25.24), (25.25) and the definition of the minimax solution [1, 2] in the form of inequalities for sub- and superdifferentials. So, it follows from Theorem 25.3 that in the case n = 2 the problem (25.1), (25.2) with piecewise linear input data and prehomogeneous Hamiltonian can be solved with the help of described in [4, 5] algorithm for problem with homogeneous Hamiltonian. In this case, there is no need to increase the dimension of the state space.

25.6 Conclusion The algorithm for exact constructing minimax solution of Hamilton–Jacobi equation with nonhomogeneous Hamiltonian is presented in the paper. The state space is the plane R 2 . Input data (Hamiltonian and terminal function) are piecewise linear. The original equation reduces to an equation with a homogeneous Hamiltonian in threedimensional state space. Based on the additive structure of new terminal function, one can reduce the construction of minimax solution to solving elementary problems of several types. The minimax solution is obtained as a result of continuous gluing of the solutions for these elementary problems. The solution built by the algorithm is a piecewise linear function. Also, the case of prehomogeneous Hamiltonian is picked out. In this case, there is no need to increase the dimension of the state space. The minimax solution can be founded with the help of solution for problem with homogeneous Hamiltonian in two-dimensional space. The presented results can be used for piecewise linear approximation of minimax solutions of the Hamilton–Jacobi equations with general Hamiltonians. Acknowledgements This work was supported by the Russian Fund for Basic Research (project 20-01-00362) and Act 211 Government of the Russian Federation (contract 02.A03.21.0006).

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References 1. Subbotin, A.I.: Minimax Inequalities and Hamilton-Jacobi Equations. Nauka, Moscow (1991). (in Russian) 2. Subbotin, A.I.: Generalized Solutions of First Order PDEs. The Dynamical Optimization Perspective. Birkhäuser, Boston (1995) 3. Crandall, M., Lions, P.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983) 4. Subbotin, A.I., Shagalova, L.G.: A piecewise linear solution of the Cauchy problem for the Hamilton-Jacobi equation. Ross. Akad. Nauk Doklady 325(5), 932–936 (1992) (in Russian; English transl., Russian Acad. Sci. Dokl. Math., 46(1), 144–148 (1993)) 5. Shagalova, L.G.: A piecewise linear minimax solution of the Hamilton-Jacobi equation. In: Kirillova, F.M., Batukhtin V.D. (eds.) Nonsmooth and Discontinuous Problems of Control and Optimization. Chelyabinsk, 1998, Proceedings of IFAC Conference on Nonsmooth and Discontinuous Problems of Control and Optimization, pp. 193–197. Pergamon, Elsevier Science, Oxford (1999) 6. Shagalova, L.G.: The value function of a differential game with simple motions and an integroterminal payoff functional. IFAC-PapersOnLine 51(32), 861–865 (2018) 7. Hopf, E.: Generalized solutions of nonlinear equations of first order. J. Math. Mech. 14, 951– 973 (1965) 8. Pshenichyi, B.N., Sagaidak, M.I.: Differential games of prescribed duration. Kibernetika, 2, 54–63 (1970). (in Russian; English transl., Cybernetics 6, 72–83 (1970)) 9. Bardi, M., Evans, L.: On Hopf’s formulas for solutions of Hamilton-Jacobi equations. Nonlinear Anal. Theory Methods Appl. 8(11), 1373–1381 (1984)

Chapter 26

Investigation of Stability of Elastic Element of Vibration Device Petr A. Velmisov and Andrey V. Ankilov

Abstract The stability of solutions of the boundary value problem for a coupled nonlinear system of integrodifferential partial differential equations, describing the dynamics of a deformable element of a vibrating device, is investigated. The definitions of the stability of a deformable body adopted in the work correspond to the concept of the stability of dynamic systems according to Lyapunov. The deformable element is surrounded by a subsonic flow of an ideal fluid. The effect of fluid (in an ideal medium model) is determined from the asymptotic equations of aerohydromechanics. To solve the aerohydrodynamic part of the problem, the methods of the theory of functions of a complex variable are used. To describe the dynamics of an elastic element, the nonlinear theory of a solid deformable body is used, taking into account its transverse and longitudinal deformation. The stability study is based on the construction of positive definite Lyapunov type functional. The sufficient stability conditions for the solutions of the proposed system of equations are obtained. Based on the Galerkin method, the numerical experiments for specific examples of mechanical systems were carried out, confirming the reliability of the investigations.

26.1 Introduction At the design and exploitation of structures, interacting with fluid, it is necessary to study the stability of their deformable elements, since the influence of the flow can lead to its loss. At the same time, for the operation of some technical devices, the phenomenon of excitation of oscillations during aerohydrodynamic action, indicated above as negative, is necessary. Examples of such devices related to vibration technology and used to intensify technological processes are devices for preparing homogeneous mixtures and emulsions. The main part of a wide class of such devices P. A. Velmisov (B) · A. V. Ankilov Ulyanovsk State Technical University, Severny Venets Str., 32, 432027 Ulyanovsk, Russia e-mail: [email protected] A. V. Ankilov e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_26

295

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is a flow channel, on the walls of which (or inside) the elastic elements are located. The operation of such devices is based on the vibration of elastic elements during the flow of fluid inside the channels. The stability of elastic bodies interacting with the gas and fluid is devoted to many theoretical and experimental studies. Among the studies we should be noted studies [1–7] and many others. Among the works of the authors of this article about fluid–structure interaction, note the articles [8–13]. The paper investigates the problem of the dynamics and stability of an elastic element of a vibrating device. The definitions of stability of an elastic body accepted in the work correspond to the concept of stability of dynamical systems according to Lyapunov. The device is a flow channel with a deformable element modeled by an elastic plate and located on the channel wall. A subsonic flow of an ideal incompressible medium flows inside the channel. To study the dynamics of elastic elements, the nonlinear equations are used. These equations describe the longitudinal–transverse vibrations of elastic plates. Aerohydrodynamic load is determined from the asymptotic equations of aerohydromechanics. At the inlet and outlet of the channel, the law of variation of the longitudinal component of the fluid velocity is given. At the inlet of the channel, the fluid flow velocity is considered constant and directed along the axis of the channel.

26.2 Statement of Problem and Solution of Aerohydrodynamic Part  Consider the plane motion of a fluid in a rectilinear channel J = (x, y) ∈ R 2 : 0 < x < x0 , 0 < y < y0 }. The velocity of the unperturbed fluid flow is equal V and directed along the axis O x. The part of the wall y = y0 at x ∈ [b, c] is deformable (Fig. 26.1). We introduce the following notation: u(x, t) and w(x, t) are the functions describing the longitudinal and transverse components of the deformation of the elastic element, respectively; ϕ (x, y, t) is potential of the velocity of the disturbed flow. The mathematical formulation of the problem has the form:

Fig. 26.1 The channel whose wall contains a deformable element

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ϕx x + ϕ yy = 0, (x, y) ∈ J, t ≥ 0;  ϕ y (x, y0 , t) =

(26.1)

wt (x, t) + V wx (x, t), x ∈ (b, c) , t ≥ 0, 0, x ∈ (0, b] ∪ [c, x0 ) , t ≥ 0;

(26.2)

ϕ y (x, 0, t) = 0, x ∈ (0, x0 ) , t ≥ 0; ϕx (0, y, t) = 0, ϕx (x0 , y, t) = 0,

(26.3)

y ∈ [0, y0 ], t ≥ 0;

(26.4)

⎧ EF  ⎪ ⎪ − 2u x (x, t) + wx2 (x, t) x + Mu tt (x, t) = 0, ⎪ ⎪ ⎪ ⎨ E2F

  wx (x, t) 2u x (x, t) + wx2 (x, t) x + Mwtt (x, t) + Dwx 4 (x, t)+ (26.5) − 2 ⎪ ⎪ ⎪ ⎪ +N wx x (x, t) + β0 w(x, t) + β1 wt (x, t) + β2 wx 4 t (x, t) = ⎪ ⎩ = −ρ (ϕt (x, y0 , t) + V ϕx (x, y0 , t)) , x ∈ (b, c), t ≥ 0, w(b, t) = wx x (b, t) = u(b, t) = w(c, t) = wx x (c, t) = u(c, t) = 0, t ≥ 0. (26.6) Indices x, y, and t below denote the partial derivatives with respect to x, y, and t, respectively; ρ is the density of the fluid in a homogeneous undisturbed flow; D, M are the flexural rigidity and linear mass of the elastic element; N is compressive (N > 0) or tensile (N < 0) elastic element force; β1 , β2 are the coefficients of external and internal damping; β0 is the stiffness coefficient of the base; E is the modulus of elasticity of the material of the element; F is the cross-sectional area of the element. The equation (26.1) describes the dynamics of an ideal incompressible fluid flow; (26.2), (26.3) are impermeability conditions; (26.4) are the law of variation of the longitudinal component of the fluid velocity in the boundary sections of the channel; the equations (26.5) describe the dynamics of the elastic element of the channel wall; (26.6) are the conditions of hinged fastening of the ends of the elastic element. Thus, a nonlinear boundary value problem (26.1)–(26.6) was obtained for determining of the three unknown functions—the deformation of the elastic element u(x, t), w(x, t) and the velocity potential of the fluid ϕ(x, y, t). We introduce the complex potential W = f (z, t) = ϕ + iψ, z = x + i y in the region J and consider the analytic function f z (z, t) = ϕx − iϕ y . Using the function 1 0 −i y0 ) √ 2dt 2 2 is ζ (z) = sn K (k)i(2z−2x , where snx is the elliptic sine, K (k) = y0 0

(1−t )(1−k t )

the full elliptic  integral of the first kind, the module k is determined from the relation √ 2 K 1 − k y0 = 2K (k)x0 , we map the rectangle J conformally onto the upper half-plane H = {ζ : I mζ > 0} of the complex variable ζ = ξ + iη. The points real axis ξ with abscissas –1/k, –1, 1, 1/k in a ζ -plane will be responded to the boundary points A(0, y0 ), B(x0 , y0 ), C(x0 , 0), D(0, 0) of the rectangle J . At this mapping a segment [−β, −α] on the real axis will correspond to the elastic element, moreover β = ζ (c), α = ζ (b).

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According to the boundary conditions (26.2)–(26.4), for the analytic function i f z (z(ζ ), t) we have a mixed boundary value problem inthe upper half-plane. The solution of this problem, limited in the points ξ = ±1, ±1 k, is given by the formula: f z (z(ζ ), t) = −

Q(ζ ) iπ

−α −β

(wt (x(τ ), t) + V wx (x(τ ), t)) dτ  , (τ 2 − 1)(1 − k 2 τ 2 )(τ − ζ )

(26.7)

 where the function Q(ζ ) = (1 − ζ 2 )(1 − k 2 ζ 2 ) and it is considered the branch of the root, which is positive in the interval (−1, 1). In this case the condition c (wt + V wx ) dx = 0

(26.8)

b

must be satisfied. The condition (26.8) reflects the incompressibility of the medium. Further, since Wζ = f z · z ζ , then, integrating over ζ , we obtain y0 f (z(ζ ), t) = − 2π K (k)

−α −β

(wt (x(τ ), t) + V wx (x(τ ), t)) ln(τ − ζ )dτ  . (τ 2 − 1)(1 − k 2 τ 2 )

(26.9)

We proceed to the limit in (26.7), (26.9) at ζ → ξ ∈ (−β, −α) (where in z → x + i y0 , x ∈ (b, c)). Equating the real parts, according to the Sokhotsky formula, we obtain −ρ (ϕt (x, y0 , t) + V ϕx (x, y0 , t)) = −α ρy0 (wtt (x(τ ), t) + V wxt (x(τ ), t)) ln |τ − ξ |  dτ + = 2π K (k) (τ 2 − 1)(1 − k 2 τ 2 ) −β  −α 2 ρV (ξ − 1)(1 − k 2 ξ 2 ) (wt (x(τ ), t) + V wx (x(τ ), t)) dτ  , + 2π (τ 2 − 1)(1 − k 2 τ 2 )(τ − ξ )

(26.10)

−β

ξ ∈ (−β, −α). In view of condition (26.8) and substituting |ξ | = −ζ (x), |τ | = −ζ (τ1 ) and then replacing τ1 on τ , we obtain a system (26.5) as

26 Investigation of Stability of Elastic Element of Vibration Device

299

⎧ EF  ⎪ ⎪ − 2u x (x, t) + wx2 (x, t) x + Mu tt (x, t) = 0, ⎪ ⎪ 2 ⎪ ⎪   EF

⎪ ⎪ wx (x, t) 2u x (x, t) + wx2 (x, t) x + Mwtt (x, t) + Dwx 4 (x, t)+ − ⎪ ⎪ ⎪ ⎪ ⎨ +N2(t)wx x (x, t) + β0 w(x, t) + β1 wt (x, t) + β2 w 4 (x, t) = x t c c ⎪ ρ ρV ⎪ ⎪ ⎪ = (wtt (τ, t) + V wτ t (τ, t)) G(τ, x)dτ − (wt (τ, t)+ ⎪ ⎪ π π ⎪ ⎪ ⎪ b b ⎪ ⎪ ⎪ ⎩ +V wτ (τ, t)) ∂G(τ, x) dτ, x ∈ (b, c), t ≥ 0, ∂x (26.11)     cd 2K (k)i(x − b)y −1 − cd 2K (k)i(x − c)y −1  0 0   0 0   , (26.12) G(τ, x) = ln    cd 2K (k)i(x0 − τ )y0−1 − cd 2K (k)i(x0 − x)y0−1  where cd x = cnx/dnx = −sn (x − K (k)), dnx is the delta amplitude, cnx is the elliptic cosine. It is easy to see that G(τ, x) = G(x, τ ) ≥ 0, i.e., the kernel is symmetric and non-negative. Let’s set the initial conditions u(x, 0) = f 1 (x), w(x, 0) = f 2 (x), u t (x, 0) = f 3 (x), wt (x, 0) = f 4 (x), (26.13) which must be consistent with the boundary conditions (26.6). Thus, a nonlinear initial boundary value problem (26.6), (26.11), (26.13) was obtained for determining of the two unknown functions—the deformations of the elastic element u(x, t), w(x, t).

26.3 Investigation of Stability We obtain the sufficient conditions of the stability of the zero solution u(x, t) ≡ 0, w(x, t) ≡ 0 of the system of integrodifferential equations (26.11) with respect to perturbations of the initial conditions (26.13), if the functions u(x, t), w(x, t) satisfy the boundary conditions (26.6). To do this, we introduce the functional c 

(t) =

M(u 2t

+

wt2 )

b

ρ + π

+

Dwx2 x

c

c wt (x, t)wt (τ, t)G(τ, x)dτ −

dx b

ρV 2 − π

c

b

c wx (x, t)wτ (τ, t)G(τ, x)dτ.

dx b

   1 2 2 2 2 + E F u x + wx − N wx + β0 w dx+ 2

b

(26.14)

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For functions u(x, t), w(x, t), that are solutions of the system of equations (26.11) and satisfy the boundary conditions (26.6), the expression for the derivative t (t) takes the form: c   β1 w2 + β2 wx2 x dx.

t (t) = −2 b

Let the conditions β1 ≥ 0, β2 ≥ 0

(26.15)

be satisfied, then t (t) ≤ 0. Integrating from 0 to t, we obtain

(t) ≤ (0).

(26.16)

Consider the boundary value problem for the equation ψx 4 (x) = −λψx x (x), x ∈ [b, c] with boundary conditions (26.6) for the function w(x, t). This problem is selfadjoint and fully defined. From the Rayleigh inequality [14] we obtain c

c wx2 x (x, t)dx

≥ λ1

b

wx2 (x, t)dx,

(26.17)

b

where λ1 is the smallest eigenvalue of the considered boundary value problem. Using the obvious inequality ±2ad ≤ a 2 + d 2 , symmetry of the kernel G(τ, x), conditions (26.6), (26.8), we obtain c ±

c wt (x, t)wt (τ, t)G(τ, x)dτ = ±

dx b

c

b

b

|wt (x, t)| · |wt (τ, t)| ×

dx b

wt (x, t)wt (τ, t)×

dx

b c

c

× (G(τ, x) + g(x) + g(τ )) dτ ≤

c

b

(26.18)

c

× |G(τ, x) + g(x) + g(τ )| dτ ≤ K 1

wt2 (x, t)dτ, b

c ±

c wx (x, t)wτ (τ, t)G(τ, x)dτ = ±

dx b

c

b

b

c

|wx (x, t)| · |wτ (τ, t)| ×

dx b

wx (x, t)wτ (τ, t)×

dx b

c

× (G(τ, x) + g(x) + g(τ )) dτ ≤

c

b

c

× |G(τ, x) + g(x) + g(τ )| dτ ≤ K 1

wx2 (x, t)dτ, b

(26.19)

26 Investigation of Stability of Elastic Element of Vibration Device

where K 1 = supx∈(b,c)

c

301

|G(τ, x) + g(x) + g(τ )| dτ ; g(x) is an arbitrary integrable

b

on (b, c) function chosen for reasons of attaining the smallest possible value K 1 . Taking into account the expression (26.14) and inequalities (26.17), (26.18), (26.19), we estimate the right and left sides of inequality (26.16) c 

(t) ≥ b

   2 ρ K1 1 wt2 (x, t) + E F u x (x, t) + wx2 (x, t) + Mu 2t (x, t) + M − π 2



ρV 2 K 1 + Dλ1 − N − π c 

(0) ≤ b



 wx2 (x, t)

+ β0 w (x, t) dx, 2

(26.20)

   2 ρ K1 1 wt2 (x, 0) + E F u x (x, 0) + wx2 (x, 0) + Mu 2t (x, 0) + M + π 2 

+Dwx2 x (x, 0) +

  ρV 2 K 1 2 2 − N wx (x, 0) + β0 w (x, 0) dx. π

(26.21)

Using the conditions (26.6) and Bunyakovsky’s inequality, we have c w (x, t) ≤ (c − b)

c wx2 (x, t)dx,

2

u (x, t) ≤ (c − b)

u 2x (x, t)dx.

2

b

(26.22)

b

According to inequalities (26.16), (26.20), (26.21), the partial derivatives u t (x, t), u x (x, t), wt (x, t), wx (x, t) are stable with respect to perturbations of the initial data u t (x, 0), u x (x, 0), w(x, 0), wt (x, 0), wx (x, 0), wx x (x, 0), and therefore, under the conditions β0 ≥ 0,

M−

ρ K1 > 0, π

Dλ1 − N −

ρV 2 K 1 > 0, π

(26.23)

according to (26.22), the functions u(x, t), w(x, t) are stable with respect to perturbations of these initial data. Thus, the next theorem is proved. Theorem 26.1 Let conditions (26.15), (26.23) be satisfied. Then the solution u(x, t), w(x, t) of the system of equations (26.11) and partial derivatives u t (x, t), u x (x, t), wt (x, t), wx (x, t) are stable with respect to perturbations of the initial values u t (x, 0), u x (x, 0), w(x, 0), wt (x, 0), wx (x, 0), wx x (x, 0), if the functions u(x, t), w(x, t) satisfy the boundary conditions (26.6). Remark 1 In a similar way, we can prove that Theorem 26.1 is also true, if the functions u(x, t), w(x, t) satisfy in any combination (other 3 combinations) by the boundary conditions:

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(1) rigid fastening: w(x1 , t) = wx (x1 , t) = u(x1 , t) = 0; (2) hinged fastening: w(x1 , t) = wx x (x1 , t) = u(x1 , t) = 0, where x1 = b or x1 = c.

26.4 Investigation of Dynamics The solution of the system of equations (26.11) will be sought by the Galerkin method, subordinating the desired functions w(x, t), u(x, t) to the boundary conditions (26.6). According to the Galerkin method, the solution of the system of equations (26.11) is sought in the form w(x, t) =

m  n=1

wn (t) sin γn (x − b), u(x, t) =

m  n=1

u n (t) sin γn (x − b), γn =

πn , c−b

where the functions wn (t), u n (t) are determined from the condition of the orthogonality of the residuals of the equations of system (26.11) to the functions sin γn (x − b), n = 1, ·, m, and therefore, are the solution to the system of 2m nonlinear ordinary differential equations. To find the initial conditions wn (0), wnt (0), u n (0), u nt (0), we use the orthogonality of the residuals of equations (26.13) to the functions sin γn (x − b), n = 1, ·, m. Thus, we obtain the Cauchy problem for a system of 2m ordinary differential equations, the solution of which was found using the Wolfram Mathematica 9 mathematical software package. We take the following parameters of the studied mechanical system: E F = 3.8 · 108 ; M = 100; D = 810; N = 2 · 104 ; ρ = 840; x0 = 2; y0 = 0.5; b = 0.8; c = 1.2;β0 = 4; β1 = 0.1; β2 = 0.2; m = 3. All values are given in the SI system. √ 1 − k 2 = 8K (k) we find the module k = 1.4 · 10−5 . From the equation K Choosing a function g(x) = 0.27 · sin (2.5π(x − b)) + 0.49 · cos (2.5π(x − b)) − 0.79, we find K 1 = 0.37. Therefore, the conditions (26.15) and the first two conditions (26.23) are satisfied. Given that under conditions (26.6) the smallest eigenvalue λ1 = 6.25π 2 , the third condition (26.23) takes the form N < 49964.87 − 98.93 · V 2 . This inequality defines the stability region on the plane (V ; N ) shown in Fig. 26.2 (gray region). Choose the points P1 (15; 2 · 104 ), P2 (24; 2 · 104 ). Figure 26.2 shows these points. The point P1 belongs to the stability region, and the point P2 does not belong to this region. We take the initial conditions in the form f 1 (x) = f 2 (x) = f 3 (x) = 0, f 4 (x) = 0.02 · sin (2.5π(x − b)) . Figs. 26.3 and 26.4 show the nature of the vibrations of the elastic element at x = 1 in the points P1 , P2 . The numerical calculations in Fig. 26.3 confirm the stable nature of the vibrations of the elastic element, and in Fig. 26.4 we have the instability of the vibrations of the elastic element.

26 Investigation of Stability of Elastic Element of Vibration Device

303

Fig. 26.2 Stability region

Fig. 26.3 Oscillations of a point x = 1 at V = 15

Fig. 26.4 Oscillations of a point x = 1 at V = 24

26.5 Conclusion In the article an example of the investigation of a device with one element is given. Similar studies were also carried out in the case of an arbitrary number of deformable elements located both on one or two walls of the channel, and inside it (some results are presented in [15]). The obtained results are intended for use at the design stage of vibration devices used for the intensification of technological processes. Acknowledgements The work was supported by the Russian foundation for basic research, grants No. 18-41-730015.

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References 1. Moditis, K., Paidoussis, M., Ratigan, J.: Dynamics of a partially confined, discharging, cantilever pipe with reverse external flow. J. Fluids Struct. 63, 120–139 (2016). https://doi.org/ 10.1016/j.jfluidstructs.2016.03.002 2. Kontzialis, K., Moditis, K., Paidoussis, M.P.: Transient simulations of the fluid-structure interaction response of a partially confined pipe under axial flows in opposite directions. J. Press. Vessel. Technol. Trans. ASME 139(3) (2017) https://doi.org/10.1115/1.4034405 3. Mogilevich, L.I., Popov, V.S., Popova, A.A., Christoforova, A.V., Popova, E.V.: Mathematical modeling of three-layer beam hydroelastic oscillations. Vibro Eng. Procedia 12, 12–18 (2017). https://doi.org/10.21595/vp.2017.18462 4. Abdelbaki, A.R., Paidoussis, M.P., Misra, A.K.: A nonlinear model for a free-clamped cylinder subjected to confined axial flow. J. Fluids Struct. 80, 390–404 (2018). https://doi.org/10.1016/ j.jfluidstructs.2018.03.006 5. Blinkov, Y.A., Blinkova, A.Y., Evdokimova, E.V., Mogilevich, L.I.: Mathematical modeling of nonlinear waves in an elastic cylindrical shell surrounded by an elastic medium and containing a viscous incompressible liquid. Acoust. Phys. 64(3), 274–279 (2018). https://doi.org/10.1134/ S106377101803003X 6. Mogilevich, L.I., Popov, V.S., Popova, A.A.: Longitudinal and transverse oscillations of an elastically fixed wall of a wedge-shaped channel installed on a vibrating foundation. J. Mach. Manuf. Reliab. 47(3), 227–234 (2018). https://doi.org/10.3103/S1052618818030093 7. Abdelbaki, A.R., Paidoussis, M.P., Misra, A.K.: A nonlinear model for a hanging tubular cantilever simultaneously subjected to internal and confined external axial flows. J. Sound Vib. 449, 349–367 (2019). https://doi.org/10.1016/j.jsv.2019.02.031 8. Ankilov, A.V., Velmisov, P.A.: Stability of solutions to an aerohydroelasticity problem. J. Math. Sci. (U. S.) 219(1), 14–26 (2016). https://doi.org/10.1007/s10958-016-3079-4 9. Velmisov, P.A., Ankilov, A.V.: Dynamic stability of plate interacting with viscous fluid. Cybern. Phys. 6(4), 262–270 (2017) 10. Velmisov, P.A., Ankilov, A.V., Semenova, E.P.: Mathematical modeling of aeroelastic systems. AIP Conf. Proc. 1910, 1–8 (2017). https://doi.org/10.1063/1.5013977 11. Badokina, T.E., Begmatov, A.B., Velmisov, P.A., Loginov, B.V.: Methods of bifurcation theory in multiparameter problems of hydroaeroelasticity. Differ. Equ. 54(2), 143–151 (2018). https:// doi.org/10.1134/S0012266118020015 12. Velmisov, P.A., Ankilov, A.V.: Stability of solutions of initial boundary-value problems of aerohydroelasticity. J. Math. Sci. (U. S.) 233(6), 958–974 (2018). https://doi.org/10.1007/ s10958-018-3975-x 13. Velmisov, P.A., Ankilov, A.V.: Dynamic stability of deformable elements of designs at supersonic mode of flow. J. Samara State Tech. Univ. Ser. Phys. Math. Sci. 22(1), 96–115 (2018) 14. Kollatz, L.: Problems on eigenvalues, p. 503. Science, Moscow (1968) 15. Ankilov, A.V., Velmisov, P.A.: Mathematical modeling in problems of dynamic stability of deformable elements of constructions at aerohydrodynamic influence, p. 322. Ulyanovsk State Technical University, Ulyanovsk (2013)

Chapter 27

A Geometric Approach to a Class of Optimal Control Problems Dmitrii R. Kuvshinov

Abstract A global optimum search algorithm is presented for a class of optimal control problems which have ODE dynamics, a terminal functional, and a state constraint in the following form: along any admissible motion a given function must reach maximum at the right end. The algorithm presented in the paper is based upon binary search in the course of which auxiliary control problems are being solved.

27.1 Problem Setting Let the dynamics of the system in consideration be governed by the following ODE: ˙ = f (t, x(t), u(t)). t ∈ [0, T ], x(0) = x0 , x(t)

(27.1)

Final time moment T ∈ (0, +∞) is fixed. State x(t) belongs Rd . Control u(t) is a measurable map [0, T ] → P, where P ⊂ R p is a compact set. An optimal control problem to be solved has the following form: γ (t, x(t)) ≤ γ (T, x(T )),

(27.2)

ϕ(x(T )) → max.

(27.3)

u(·)

Let for some compact set D ⊆ [0, T ] × Rd the following conditions hold: 1. (0, x0 ) ∈ D and { x ∈ Rd | (t, x) ∈ D } = ∅ for all t ∈ [0, T ]. 2. Function f (t, x, u) is continuous over D × P. 3. f (t, x, u) ≤ m(t) for some integrable on [0, T ] function m : [0, T ] → R and all (t, x, u) ∈ D × P. D. R. Kuvshinov (B) Yeltsin Federal University and Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_27

305

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4. f (t, x  , u) − f (t, x  , u) ≤ L x  − x  for some L > 0 and all (t, x  ) ∈ D, (t, x  ) ∈ D, u ∈ P. 5. The set { f (t, x, u) | u ∈ P } is convex for all (t, x) ∈ D. 6. Function γ (t, x) is continuous over D. 7. Function ϕ(x) is continuous over { x ∈ Rd | (T, x) ∈ D }. When some control u(·) is chosen conditions 1–4 provide us with a unique Carathéodory solution of (27.1) x : [0, T ] → D (“motion” x(·)) [1]. Conditions 1–5 are sufficient for the attainable set of (27.1) to be closed in [0, T ] × D. Conditions 6–7 are used in proofs of lemmas presented below. “Admissible motions” are motions of (27.1) satisfying state constraint (27.2). The setting where some function must reach its maximum in the final point along the motion comes from [2] where this function is actually a value function in a zerosum game against a player whom we want to motivate to cooperate. In that setting it is rational for the player to follow an admissible motion because otherwise the payoff will be less than in the end of the admissible motion. This approach was used to derive a numerical algorithm for constructing solutions in a class of Stackelberg games [3]. Attempts to improve this algorithm [4, 5] led to results described here.

27.2 Solution Method Methods for finding a solution of the problem (27.2)–(27.3) that are described in this section are based upon searching across a parameterized family of simpler (optimal) control problems. We consider two methods. The first method is based upon global maximum search of a scalar function. The second method checks if a given functional value is attainable and uses binary search to approximate the maximal attainable value.

27.2.1 Scalar Optimization Consider the following optimal control problem family: γ (T, x(T )) = g, γ (t, x(t)) ≤ g, ϕ(x(T )) → max. u(·)

(27.4) (27.5)

The dynamics is governed by (27.1). The only change is the state constraint. Solutions of (27.1) which satisfy (27.4) are called “g-admissible motions”. There are no gadmissible motions for g < g∗ and g > g ∗ where g∗ = γ (0, x0 ), g ∗ = max{ γ (T, x) | (T, x) ∈ D }.

27 A Geometric Approach to a Class of Optimal Control Problems

307

Remark 27.1 It must be pointed out that in general case assumptions listed in Sect. 27.1 are not sufficient to solve such optimal control problems in practice. A survey of numerical methods for solving optimal control problems of (27.4)–(27.5) kind or more general Bolza problems may be found, e.g., in [6, 7]. Consider the following function: cmax (g) =

max

x(·) is g-admissible

ϕ(x(T )).

(27.6)

Computation of cmax (g) is equivalent to solving optimal control problem (27.4)– (27.5). A solution of the original problem (27.2)–(27.3) corresponds to a global maximum of cmax (g), g ∈ [g∗ , g ∗ ]. Thus we may try any scalar derivative-free global maximum search algorithm to find it. However, cmax (g) may be neither unimodal nor Lipschitz continuous and so in general case we have no guarantee of converging to a global maximum in finite time.

27.2.2 Recurrent Relation Consider the following optimal control problem family: γ (t, x(t)) ≤ g, ϕ(x(T )) ≥ c,

(27.7)

γ (T, x(T )) → max.

(27.8)

u(·)

Solutions of (27.1) which satisfy (27.7) are called “g, c-admissible motions”. If x(·) is g, c-admissible and γ (T, x(T )) = g then x(·) is admissible. Thus for some c a value g exists such that g, c-admissible motions exist iff ϕ(x ∗ (T )) ≥ c for a motion x ∗ (·) corresponding to a solution of the original problem (27.2)–(27.3). Consider the following function: g max (g, c) =

max

x(·) is g,c− admissible

γ (T, x(T )).

(27.9)

Computation of g max (g, c) is equivalent to solving optimal control problem (27.7)– (27.8). By (27.7) we have that for any g ∈ [g∗ , g ∗ ] if g max (g, c) is defined then g max (g, c) ≤ g and g max (g  , c ) is defined for any g  ≥ g and c ≤ c as well. Lemma 27.1 For any g  ∈ [g∗ , g ∗ ), g  ∈ (g  , g ∗ ] such that g max (g  , c) is defined the inequality g max (g  , c) ≤ g max (g  , c) is true. Proof By definition any g  , c-admissible motion is also g  , c-admissible, so maximum on a subset can’t be greater than maximum on its superset. 

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Consider the following recurrent relation: gc0 = g ∗ , gci = g max (gci−1 , c).

(27.10)

Due to Lemma 27.1 we have that gci ≤ gci−1 if gci is defined. Lemma 27.2 If the set Gc = { g ∈ [g∗ , g ∗ ] | g max (g, c) = g }

(27.11)

is not empty, then it contains its supremum: sup Gc ∈ Gc . Proof Suppose the opposite: sup Gc ∈ / G c . As Gc = ∅ the value g˜ = g max (sup Gc , c) is defined. By our assumption g˜ < sup Gc . Here Gc can not be a finite set, so ∀ε > 0 ∃η ∈ (0, ε) g max (sup Gc − η, c) = sup Gc − η. Now for ε = sup Gc − g˜ we get some η ∈ (0, ε) and g  = sup Gc − η such that g  < sup Gc ∧ g max (g  , c) = g  > g˜ = g max (sup Gc , c). According to Lemma 27.1 we have a contradiction.



So it is correct to write “max Gc ” instead of “sup Gc ”. Lemma 27.3 Let c ∈ R be such that Gc = ∅. Then max Gc ≤ gci for any i ∈ N. Proof By definitions (27.10) and (27.11) we have that max Gc ≤ gc0 . We need to prove that gci cannot “jump over” max Gc . Suppose there is g  ∈ Gc such that gci+1 < g  < g i for some i ∈ N. But now we have g max (g i , c) = gci+1 < g  = g max (g  , c). According to Lemma 27.1 we have a contradiction.  Lemma 27.4 Let limi→∞ gci = gc ∈ [g∗ , g ∗ ] be defined. Then gc = max Gc . Proof Denote X ci = { x(T ) | x(·) is gci , c -admissible }. These sets are compact (e.g., ∞ X ci is also a nonempty compact set. see [8, p. 150]) and X ci+1 ⊆ X ci . So X c = ∩i=0 Due to definition (27.10) we have a sequence { xci } of endpoints of gci , c-admissible motions such that γ (T, xci ) = gci+1 and so limi→∞ γ (T, xci ) = gc . As { xci } ⊂ X c0 there is a converging subsequence { xcik }: x  = limk→∞ xcik . As γ (t, x) is continuous we have γ (t, x  ) = gc . Obviously, for all i x  ∈ X ci and so x  ∈ X c . Thus we may define a sequence of motions { xci (·) } such that for any t ∈ [0, T ] γ (t, xci (t)) ≤ gci and xci (T ) = x  . According to Theorem 6 [1, p. 84] and Arzelà– Ascoli theorem there is a subsequence { xcik (·) } uniformly converging to some solution of (27.1) x ∗ (·), such that x ∗ (T ) = x  and for any t ∈ [0, T ] for any i ∈ N γ (t, x ∗ (t)) ≤ gci . That means that x ∗ (·) is gc , c-admissible and so g max (gc , c) = gc .  Due to Lemma 3 max Gc ≤ gc so max Gc = gc .

27 A Geometric Approach to a Class of Optimal Control Problems

309

The algorithm for solving problem (27.2)–(27.3) presented below in pseudocode is based upon computing gci series and is an improved version of “Leader” algorithm from [5]. It is called “RR-algorithm” here due to being based on recurrent relation (27.10). The algorithm was implemented in C++. RR-algorithm consists of two nested loops. The outer loop does binary search over possible values ϕ(x(T )) = c ∈ [c∗ , c∗ ]. The inner loop computes gci series. There is no need to start gci from g ∗ every time because in the process of the search we always move to lower g-values. The last “good” value is stored in variable g0 , which is used as gc0 for the next c. In the case of slow convergence the algorithm uses cmax (g) to find the actual value of c corresponding to the current g. If it is not enough, we slightly decrease g. RR-algorithm Input: • bracket g∗ and g ∗ ; • bracket c∗ and c∗ such that solution ϕ(x(T )) ∈ [c∗ , c∗ ]; • solution accuracy ε > 0; • gci equality tolerance tolg ≥ 0; • gci limit tolerance tollim > 0; • gci limit maximum iterations maxiters ∈ N. Assign: l ← c∗ , u ← c∗ , g0 ← g ∗ . While u − l > ε repeat: • assign: c ← (l + u)/2, g ← g0 , i ← 0; • repeat: – // At first, we need to check if there are any g, c-admissible motions. – if g max (g, c) is undefined ∨ g max (g, c) < g∗ then: // no g, c-admissible motions · assign u ← c; · break repeat; – // Now we check if we reach the limit of gci . – if g − g max (g, c) < tolg then: // close enough to be considered equal · assign l ← c, g0 ← g; · break repeat; – assign: i ← i + 1; – if i < maxiters ∧ g − g max (g, c) ≥ tollim then: // proceed to the limit · assign: g ← g max (g, c); // next gci · continue repeat; – // We are iterating too long or too slowly. – // We may use cmax (g) to find c for which g ∈ Gc . – assign c ← cmax (g), g0 ← min(g, g0 − tolg ), g ← g0 , i ← 0; – if c − l > min(2ε, (u − l)/4) then: // new c is not too close to l

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· assign: l ← c; · break repeat; – // If new c is too close to l or to old c we may try to improve it. – assign: c ← (3l + u)/4. It is assumed that computation of g max (g, c) (when this value is defined) produces a corresponding approximate g, c-admissible motion and control. It is also assumed that computation of cmax (g) approximates it from below and produces a corresponding approximate g-admissible motion and control. The result of the algorithm is the motion among those computed which delivers maximal ϕ(x(T )).

27.2.3 Auxiliary Control Problems Consider the following control problem: given c, y and g find any motion x(·) of (27.1) satisfying the following state constraint or determine there are no such motions: ϕ(x(T )) ≥ c, γ (T, x(T )) ≥ y, γ (t, x(t)) ≤ g. (27.12) Given g and bracket cmax (g) ∈ [c∗ , c∗ ] we can approximate cmax (g) using binary search along parameter c, because this problem has no solutions for c > cmax (g) (choosing y = g). Given g and c we can approximate g max (g, c) using binary search along parameter y ∈ [g∗ , g], because this problem has no solutions for y > g max (g, c).

27.2.4 Geometric Interpretation Consider the following level sets: c = { x ∈ Rd | ϕ(x) ≥ c }, g (t) = { x ∈ Rd | γ (t, x) ≥ g }. A motion x(·) of (27.1) such that x(t) ∈ / int g (t) for all t ∈ [0, T ] is “g-possible”. These motions form an attainable set of (27.1) with state constraint γ (t, x) ≤ g: A g (t) = { y ∈ Rd | ∃x (x(·) is g-possible motion ∧ y = x(t)) }. A g-possible motion x(·) is g-admissible if x(T ) ∈ g (T ) and so cmax (g) = max{ ϕ(x) | x ∈ A g (T ) ∩ g (T ) }.

27 A Geometric Approach to a Class of Optimal Control Problems

311

A g-possible motion x(·) is g, c-admissible if x(T ) ∈ c and so g max (g, c) = max{ γ (T, x) | x ∈ A g (T ) ∩ c }. On the one hand, a g-possible motion x(·) is a solution of control problem (27.12) if x(T ) ∈ c ∩  y (T ). On the other hand, any motion within the attainable set of (27.1) in reverse time respecting condition x(t) ∈ / int g (t) starting from c ∩  y (T ) and ending in x0 is a solution of control problem (27.12). In the case of linear dynamics, it is possible to build pipes of convex polyhedra lying within the corresponding attainable set time sections instead of building approximations of full attainable sets [4]. RR-algorithm may be adapted for an implementation based upon approximating level sets g and attainable sets A g because if we compute g max (g, c) then it may be relatively easy to compute cmax (g) as well by reusing g (T ) and A g (T ).

27.3 Model Problem The model problem from [3–5] has the following dynamics: x(t) ˙ = (T − t)u(t), x(0) = (−0.5, 0) , where T = 1, vectors x(t) ∈ R2 , u(t) ∈ { u ∈ R2 | u ≤ 2 }, · is Euclidean norm. Functions γ (t, x) and ϕ(x) are the following:  γ (t, x) = max 0, x −

√ 1− 2 (T 4

 − t)2 ,

ϕ(x) = − x − a . Three choices of a ∈ R2 are considered: • a1 = (0.3, 0.25) (max cmax (g) ≈ −0.0508316); • a2 = (0.3, 0.27) (max cmax (g) ≈ −0.05620998); • a3 = (0.25, 0) (max cmax (g) ≈ −0.02124683). We compare two methods: global cmax (g) maximum search and RR-algorithm. Three different solution target accuracy parameter ε values are considered (thus making 18 numerical results in total, see the tables). The former is done using AGS solver [9] with the following parameters: eps = ε, r = 3, itersLimit = 1000, epsR = 10ε, evolventDensity = 12, refineSolution = true, mixedFastMode = false. For both methods we use g∗ = −0.35355, g ∗ = −ε. Functions cmax (g) and max g (g, c) are approximated within ε absolute error by solving auxiliary control problems (27.12).

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Table 27.1 Computation results for ε = 10−3 max Method-a max cnumerical abs. error AGS-a1 RR-a1 AGS-a2 RR-a2 AGS-a3 RR-a3

−0.0511022 −0.0509413 −0.0567575 −0.0564973 −0.0214844 −0.0217209

0.00027 0.00011 0.00055 0.00029 0.00024 0.00047

Table 27.2 Computation results for ε = 10−4 max Method-a max cnumerical abs. error AGS-a1 RR-a1 AGS-a2 RR-a2 AGS-a3 RR-a3

−0.0509115 −0.0508370 −0.0562648 −0.0562192 −0.0213013 −0.0212914

· 10−5

8.0 5.5 · 10−6 5.5 · 10−5 9.2 · 10−6 5.4 · 10−5 4.5 · 10−5

Table 27.3 Computation results for ε = 10−5 max Method-a max cnumerical abs. error AGS-a1 RR-a1 AGS-a2 RR-a2 AGS-a3 RR-a3

−0.0508341 −0.0508357 −0.0562155 −0.0562132 −0.0212479 −0.0212548

2.5 · 10−6

4.1 · 10−6 5.5 · 10−6 3.2 · 10−6 1.0 · 10−6 7.9 · 10−6

g max evals

cmax evals

aux evals

0 17 0 12 0 15

38 30 32 27 47 29

342 93 288 52 376 113

g max evals

cmax evals

aux evals

0 37 0 27 0 26

122 98 54 93 95 76

1464 340 648 209 1140 307

g max evals

cmax evals

aux evals

0 162 0 83 0 96

252 331 137 248 228 206

4032 1941 2192 816 3420 1452

RR-algorithm was used with the following parameters: c∗ = − a , c∗ = 0, tolg = 10 , tollim = min(10−3 , 10ε), maxiters = 10000. The results are presented in Tables 27.1, 27.2 and 27.3. The columns of the tables are as follows: −8

• Method-a: method (AGS or RR-algorithm) and a choice (a1 , a2 or a3 ); max • max cnumerical : value of the found maximum of ϕ(x(T )) on admissible motions; • abs. error: absolute error of the numerical solution functional value max cmax (g) − max ; max cnumerical max • g evals: how many times g max (g, c) was evaluated; • cmax evals: how many times cmax (g) was evaluated; • aux evals: how many instances of auxiliary control problems have been solved in total.

27 A Geometric Approach to a Class of Optimal Control Problems Table 27.4 Computation results for ε = 10−5 max Method-x(0) max cnumerical abs. error AGS-1 RR-1 AGS-2 RR-2

−9.0283 −8.0007 −8.1858 −7.2001

1.0283 6.7 · 10−4 0.986 1.1 · 10−4

313

g max evals

cmax evals

0 32 0 24

182 10 116 4

Now consider another system governed by the same ODE, but with T = 2, control u(t) ∈ { (u, v) ∈ R2 | |u| + |v| ≤ 1 }. and another γ (t, x) and ϕ(x):      tπ tπ 2−t tπ tπ  − y sin  − x sin + y cos , γ (t, (x, y) ) = x cos 4 4 4 4  3   1 1 ϕ((x, y) ) = − 3 |x| + 3 |y| + 3 |x + y| + 3 |x − y| . 2 2 

Two choices of x(0) are considered (for both cases g ∗ = 2 and c∗ = 0): • x 1 (0) = (1, 2) (max cmax (g) = −8, g∗ = −1, c∗ = −42); • x 2 (0) = (1, 1.9) (max cmax (g) = −7.2, g∗ = −0.9, c∗ ≈ −40.5). Auxiliary control problems are solved by building discrete-time approximations of attainable sets in the same way as it was done in [3]. Time step is 1/64. c are approximated using 4096 radial rays. g (t) are approximated with step 1/32 along x-axis, then applying rotation by tπ/4 angle. The implementation uses the Computational Geometry Algorithms Library [10]. RR-algorithm accuracy parameter values: ε = 10−3 , tollim = 10−4 , tolg = 10−5 , maxiters = 20. AGS solver was used with the following parameters: eps = ε, itersLimit = 200, epsR = 10−2 , all other parameters being the same as in the previous example. The results are presented in Table 27.4. In this example cmax (·) is not Lipschitz continuous and AGS converges to a local maximum, which is not the global one. RR-algorithm converges to the global maximum.

27.4 Conclusion The “Leader” algorithm in [5] is formulated in game-theoretic background by geometrical means. There the function γ (t, x) is the value function in an auxiliary positional differential game. RR-algorithm introduced in this paper is an improved version of “Leader” algorithm that is described in a more general optimal control framework by means of auxiliary (optimal) control problems. The first improvement compared to “Leader” algorithm is using cmax (g) as the first aid to cope with slow

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convergence. The second improvement is tracking the last good g value instead of beginning series with g ∗ each time. Here the choice of γ (t, x) is limited by our capability of solving auxiliary control problems.

References 1. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht (1988) 2. Klejmenov, A.F.: Stackelberg strategies for hierarchical differential two-person games. Probl. Control Inf. Theory. 15, 399–412 (1986) 3. Osipov, S.I.: Realization of the algorithm for constructing solutions for a class of hierarchical Stackelberg games. Autom. Remote Control. (2007). https://doi.org/10.1134/ S0005117907110148 4. Kuvshinov, D.R., Osipov, S.I.: Numerical construction of Stackelberg solutions in a linear positional differential game based on the method of polyhedra. Autom. Remote Control. (2018). https://doi.org/10.1134/S0005117918030074 5. Kuvshinov, D.R., Osipov, S.I.: Numerical Stackelberg solutions in a class of positional differential games. IFAC-PapersOnLine (2018). https://doi.org/10.1016/j.ifacol.2018.11.404 6. Pytlak, R.: Numerical Methods for Optimal Control Problems with State Constraints. SpringerVerlag, New York (1999) 7. Conway, B.A.: A survey of methods available for the numerical optimization of continuous dynamic systems. J. Optim. Theory Appl. (2012). https://doi.org/10.1007/s10957-011-9918-z 8. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990) 9. Sovrasov, V.V.: AGS NLP solver. https://github.com/sovrasov/ags_nlp_solver (2019). Accessed 4 Aug 2019 10. The Computational Geometry Algorithms Library. https://www.cgal.org (2019). Accessed 10 Sep 2019

Chapter 28

Approximating Systems in the Exponential Stability Problem for the System of Delayed Differential Equations R. I. Shevchenko

Abstract The problem of exponential stability for a delay differential system is considered. In our research we use the special approximating systems of differential equations with piecewise-constant arguments. We obtain conditions on the approximation parameter under which exponential stability of the approximating systems implies exponential stability of the initial system.

28.1 Introduction Stability and control problems for delay differential equations were investigated by method of approximating ordinary differential equations in [1–4]. In the paper [1] Krasovskii proved the convergence of such approximations for the systems with constant delay. Repin in [2] obtained results for nonlinear systems and Kurjanskii in [3] extended the convergence results on the case of variable delay. Later, in the paper [4] these approximations were applied to the exponential stability problem for linear systems with constant delay. In this paper we suggest the special approximating systems with piecewiseconstant arguments for the exponential stability analysis. We show that, under suitable conditions on the approximation parameter, exponential stability of the approximating systems implies exponential stability of the initial system.

R. I. Shevchenko (B) Ural Federal University, Mira str. 19, Yekaterinburg, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_28

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28.2 Problem Statement We consider the following system of differential equations with constant delay dx(t) = A1 (t)x(t) + A2 (t)x(t − τ ), t ≥ t0 ≥ 0, dt

(28.1)

where x : [−τ, +∞) → Rn ; A1 (·), A2 (·) are bounded measurable matrix-valued functions; τ is constant delay. The initial value problem for (28.1) is determined by the condition x(t0 + ϑ) = ϕ(ϑ), ϕ ∈ C = C [−τ, 0]. Let us introduce the following notations ϕC = max ϕ(t) , t∈[−τ,0]

x = max |xi | . 1≤i≤n

Definition 28.1 System (28.1) is called exponentially stable if, for every t0 ≥ 0 and every solution x (t, t0 , ϕ) with an initial function ϕ ∈ C, the relation ˜ 0) x (t, t0 , ϕ) ≤ Pe−α(t−t ϕC , t ≥ t0 ,

is true for some constant P, α˜ > 0 which do not depend upon ϕ and the initial time t0 . The exponential stability problem for (28.1) is replaced by the analogous problem for the system of differential equations dy0 (t) = A1 (t)y0 (t) + A2 (t)y N (t), dt        tN τ tN τ N dyi (t) = yi−1 − yi , i = 1, ..., N , dt τ τ N τ N

(28.2)

(28.3)

where [a] is the integer part of a, yi ∈ Rn , i = 0, ..., N , N ≥ 1, t0 is the initial time. Further we use the simplifying notation [t] N = t τN Nτ . Definition 28.2 System (28.2), (28.3) is exponentially stable uniformly on N ≥ N0 ≥ 1 if, for every t0 ≥ 0 and arbitrary y 0 , y −1 ∈ Rn×(N +1) the solution of the initial value problem with y(t0 ) = y 0 , y ([t0 ] N ) = y −1 satisfies the inequality  



    y t, t0 , y 0 , y −1  ≤ K˜ e−α(t−t0 )  y 0  +  y −1  , t ≥ t0 . N N N Here K˜ , α > 0 do not depend upon t0 , y 0 , y −1 , N ≥ N0 ; y = {yk }0N , y N = max yk .

0≤k≤N

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In this paper we prove that the system (28.1) is exponentially stable if and only if for N sufficiently large the system (28.2), (28.3) is exponentially stable. In addition, we obtain the condition on N under which exponential stability of the system (28.2), (28.3) implies exponential stability of the system (28.1).

28.3 General Solution of the Approximating System Let us determine the form of the general solution of the system (28.3) with a given bounded locally measurable

y0 (t), t ≥ [t0 ] N ≥ 0. The right part of (28.3) is constant  on the intervals t ∈ t j , t j+1 , j ≥ 0, where t j = [t0 ] N + j Nτ for j ≥ 1. Hence the

 solution {yi (t)}1N of (28.3) is a continuous linear function on t j , t j+1 , j ≥ 0. Taking the limit t → t j+1 we obtain the solution value {yi (t j+1 )}1N . The above remarks imply that the values yi (t j ), j ≥ 0, 1 ≤ i ≤ N , determine the unique solution of the system (28.3) for t ≥ t0 and any initial points y 0 , y −1 . Lemma 28.1 Suppose y0 (t), t ≥ [t0 ] N ≥ 0, is a given bounded locally measurable function. Then for any initial conditions yi (t0 ) = yi0 , yi ([t0 ] N ) = yi−1 , 1 ≤ i ≤ N , y0−1 = y0 ([t0 ] N ) , the sequence yi (t j , t0 , y 0 , y −1 ), j ≥ 1, 1 ≤ i ≤ N , is given by the formulas  N  −1 −1 0 yi t j , t0 , y 0 , y −1 = yi− j+1 + (t1 − t0 ) τ yi− j − yi− j+1 , 1 ≤ j ≤ i, 1 ≤ i ≤ N ,

(28.4)





yi t j , t0 , y 0 , y −1 = y0 t j−i , j ≥ i + 1, 1 ≤ i ≤ N .

(28.5)

Proof Using the fact that [t] N = [t0 ] N , t ∈ [t0 , t1 ), [t] N = t j , t ∈ [t j , t j+1 ), j ≥ 1, and integrating both sides of (28.3) from t j to t, where t ∈ [t j , t j+1 ), j ≥ 0, we obtain yi (t) − yi (t0 ) =

N (yi−1 ([t0 ] N ) − yi ([t0 ] N )) (t − t0 ), t ∈ [t0 , t1 ), 1 ≤ i ≤ N , τ







N yi−1 t j − yi t j (t − t j ), t ∈ [t j , t j+1 ), j ≥ 1, 1 ≤ i ≤ N . yi (t) − yi t j = τ

We now take in the above formulas the limit t → t j+1 , j ≥ 0, and apply the initial conditions yi (t0 ) = yi0 , yi ([t0 ] N ) = yi−1 , i ≥ 1. As a result, we derive the formula (28.4) for j = 1, 1 ≤ i ≤ N , and the recurrent relation



yi t j+1 , t0 , y 0 , y −1 = yi−1 t j , t0 , y 0 , y −1 , j ≥ 1, 1 ≤ i ≤ N .

(28.6)

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The formula (28.5) for i = 1 follows immediately from (28.6). Now assume that the formulas (28.4), (28.5) hold for some arbitrary 2 ≤ i ≤ N − 1, and prove them for i + 1. From (28.4), (28.5), (28.6) we derive



N 0 yi+1 t j , t0 , y 0 , y −1 = yi t j−1 , t0 , y 0 , y −1 = yi+1− j+1 + (t1 − t0 ) τ  −1 −1 × yi+1− j − yi+1− j+1 , 1 ≤ j ≤ i + 1,



yi+1 t j , t0 , y 0 , y −1 = yi t j−1 , t0 , y 0 , y −1 = y0 (t j−i−1 ), j ≥ i + 2. This proves the formulas (28.4), (28.5) for i + 1.



The next result follows immediately from the above discussion. Lemma 28.2 For any given bounded locally measurable y0 (t), t ≥ [t0 ] N ≥ 0, and any given initial conditions yi (t0 ) = yi0 , yi ([t0 ] N ) = yi−1 , 1 ≤ i ≤ N , y0−1 = y0 ([t0 ] N ), the solution of the system (28.3) for t ≥ t0 is determined as



N −1 yi t, t0 , y 0 , y −1 = yi0 + (t − t0 ) yi−1 − yi−1 , t ∈ [t0 , t1 ), 1 ≤ i ≤ N , τ (28.7)





N 0 −1 0 −1 0 −1 yi t j+1 , t0 , y , y yi t, t0 , y , y = yi t j , t0 , y , y + (t − t j ) τ (28.8)

−yi t j , t0 , y 0 , y −1 , t ∈ [t j , t j+1 ), j ≥ 1, 1 ≤ i ≤ N ,

where yi t j , t0 , y 0 , y −1 , 1 ≤ i ≤ N , j ≥ 1, are given by the formulas (28.4), (28.5).

28.4 Results on Exponential Stability For the system (28.2), (28.3) we can now prove the following theorem. Theorem 28.1 A necessary and sufficient condition for the system (28.2), (28.3) to be exponentially stable uniformly on N ≥ 1 is that  



    y0 t, t0 , y 0 , y −1  ≤ K e−α(t−t0 )  y 0  +  y −1  , t ≥ t0 . N N Here K ≥ 1, α > 0 do not depend upon t0 , y 0 , y −1 , N ≥ 1. Proof The necessity of the statement is obvious. Let us show its sufficiency. Considering the fact that y0 (t) is known, we obtain from Lemma 28.2 the solution of the initial value problem for the system (28.3). Let us modify the formula (28.8) in the form

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  N   τ yi [t] N + , t0 , y 0 , y −1 yi t, t0 , y 0 , y −1 = yi [t] N , t0 , y 0 , y −1 + (t − [t] N ) τ N  0 −1 , t ≥ t1 , 1 ≤ i ≤ N . −yi [t] N , t0 , y , y

Estimating the norm of the solution we have             N     −1   −1        yi−1  + yi  ≤ y 0  + 2 y −1  , yi t, t0 , y 0 , y −1  ≤ yi0  + (t − t0 ) N N τ t ∈ [t0 , t1 ) , 1 ≤ i ≤ N ,

(28.9) 

 

  yi t, t0 , y 0 , y −1  ≤  yi [t] N , t0 , y 0 , y −1  + (t − [t] N ) N τ    

 τ  0 −1  0 −1   ×  yi [t] N + , t0 , y , y  + yi [t] N , t0 , y , y N    

 τ   ≤ 2  yi [t] N , t0 , y 0 , y −1  +  yi [t] N + , t0 , y 0 , y −1  , t ≥ t1 , 1 ≤ i ≤ N . N (28.10) To expand (28.10) we use the formulas (28.4), (28.5) to estimate  the inequality

 the value  yi [t] N , t0 , y 0 , y −1 , t ≥ t1 , 1 ≤ i ≤ N . Exploiting the integer-valued     function p(t, t0 ) = t τN − t0τN , such that t p(t,t0 ) = [t] N , we get     

  0 N     −1   −1  yi [t] N , t0 , y 0 , y −1  ≤  yi− p(t,t0 )+1  + (t1 − t0 ) yi− p(t,t0 )  + yi− p(t,t0 )+1  τ     ≤  y 0  N + 2  y −1  N , t1 ≤ t < ti+1 , 1 ≤ i ≤ N ,    



   τ τ    yi [t] N , t0 , y 0 , y −1  =   ≤ K e−α ([t] N −i N −t0 )  y 0  N +  y −1  N y0 [t] N − i N   

 

  (N +1)τ  τ  y 0  +  y −1  , ≤ K e−α (t−(i+1) N −t0 )  y 0  +  y −1  ≤ K e−α(t−t0 ) eα N N

N

N

N

t ≥ ti+1 , 1 ≤ i ≤ N .

Here we applied the inequality [t] N ≥ t −

τ , N

t ≥ t1 . From (28.10) we obtain

      

    yi t, t0 , y 0 , y −1  ≤ 2  y 0  + 2  y −1  +  y 0  + 2  y −1  N N N N     = 3  y 0  N + 6  y −1  N , t ∈ [t1 , ti ), 1 ≤ i ≤ N ,   

   (N +1)τ  yi t, t0 , y 0 , y −1  ≤ 2  y 0  + 2  y −1  + K e−α (t+ Nτ −t0 ) eα N N N           ×  y 0  N +  y −1  N ≤ 2 + K e−α(t−t0 ) eατ  y 0  N + 4 + K e−α(t−t0 ) eατ  y −1  N , t ∈ [ti , ti+1 ), 1 ≤ i ≤ N ,    

 (N +1)τ  0 −1  yi t, t0 , y , y  ≤ 2 K e−α(t−t0 ) eα N  y 0  +  y −1  N N  −1 

 

   0  −α(t−t0 ) ατ  0  −α(t−t0 ) α (N +1)τ     N +K e e e y N+ y ≤ 3 Ke y N +  y −1  N , N t ≥ ti+1 , 1 ≤ i ≤ N .

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The solution estimate over the set t ≥ t0 follows from the last inequalities and the inequaity (28.9). It has the form       (N +1)τ    0    y  + y −1  , t ≥ t0 , 1 ≤ i ≤ N . yi t, t0 , y 0 , y −1  ≤ 6 K e−α(t−t0 ) eα N N

N

Therefore,   

  0   y  +  y −1   y t, t0 , y 0 , y −1  ≤ 6 K e−α(t−t0 ) eα (N +1)τ N N N N  

  ≤ 6 K e−α(t−t0 ) e2ατ  y 0  N +  y −1  N , t ≥ t0 . 

It proves the theorem.

For a given function y0 (t), t ≥ [t0 ] N ≥ 0, we will further investigate how solutions of the system (28.3) behave. Lemma 28.3 Let {yi (t)}1N be the

solution of the system (28.3) with the initial conditions yi−1 = yi0 = y0 t0 − Ni τ , 1 ≤ i ≤ N . Then, if the function y0 (t) for t ≥ t0 − τ has a measurable first derivative, such that    dy0 (t)  −α1 (t−t0 )   , t ≥ t0 − τ,  dt  ≤ K 1 e where K 1 , α1 > 0 are constants, we have the inequality y N (t) − y0 (t − τ ) ≤ 6

τ N +2 K 1 eα1 N τ e−α1 (t−t0 ) , t ≥ t0 . N

(28.11)

Proof We use the following inequalities:     y N (t) − y0 (t − τ ) ≤  y N (t) − y N0  +  y N0 − y0 (t − τ ) = y N (t) − y0 (t0 − τ ) + y0 (t0 − τ ) − y0 (t − τ ) , t ∈ [t0 , t1 ) ,

(28.12)

y N (t) − y0 (t − τ ) ≤ y N (t) − y N ([t] N ) + y N ([t] N ) − y0 (t − τ ) , t ≥ t1 . (28.13) Let us expand the first term on the right-hand side of the inequality (28.12) by using Lemma 28.2 and the special initial conditions. We obtain     N   τ  y N (t) − y0 (t0 − τ ) =  (t − t y − y t ) − τ + − τ ) (t 0 0 0 0 0   τ N       dy0 (t0 − τ + ξ )  τ τ    ≤ y0 t0 − τ + − y0 (t0 − τ ) ≤ maxτ   N N ξ ∈[0, N ]  dt τ τ N +1 ≤ K 1 eα1 τ ≤ K 1 eα1 N τ e−α1 (t−t0 ) , t ∈ [t0 , t1 ). N N

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321

Obviously, the second term on the right-hand side of (28.12) satisfies the same estimate y0 (t0 − τ ) − y0 (t − τ ) ≤

τ N +1 K 1 eα1 N τ e−α1 (t−t0 ) , t ∈ [t0 , t1 ). N

Then the inequality (28.12) implies y N (t) − y0 (t − τ ) ≤ 2

τ N +1 K 1 eα1 N τ e−α1 (t−t0 ) , t ∈ [t0 , t1 ) . N

(28.14)

To get estimates for t ≥ t1 let us note that the function y N (t), t ≥ t1 , is piecewise linear, i.e., dy N (t) N   τ = y N [t] N + − y N ([t] N ) , t ≥ t1 . dt τ N  

Therefore, we have y N (t) −y N ([t] N ) ≤  y N [t] N + Nτ − y N ([t] N ). The norm 

 y N [t] N + τ − y N ([t] N ) is estimated by using the explicit formulas for N y N ([t] N ), t ≥ t1 , from Lemma 28.1. Here we present entire investigation of the case t ≥ t N +1 . From (28.5) we derive       τ τ     − y N ([t] N ) = y0 [t] N + − τ − y0 ([t] N − τ ) y N [t] N + N N    dy0 ([t] N − τ + ξ )  τ   ≤ τ K 1 e−α1 ([t] N −τ −t0 ) ≤ max  N N ξ ∈[0, Nτ )  dt τ N +1 ≤ K 1 eα1 N τ e−α1 (t−t0 ) , t ≥ t N +1 . N (28.15) Similarly, the norm y N ([t] N ) − y0 (t − τ ), t ≥ t N +1 , satisfies y N ([t] N ) − y0 (t − τ ) = y0 ([t] N − τ ) − y0 (t − τ ) τ N +1 ≤ K 1 eα1 N τ e−α1 (t−t0 ) , t ≥ t N +1 . N

(28.16)

It follows from (28.13), (28.15), (28.16), that y N (t) − y0 (t − τ ) ≤ 2

τ N +1 K 1 eα1 N τ e−α1 (t−t0 ) , t ≥ t N +1 . N

(28.17)

The formula (28.17) describes the value y N (t) − y0 (t − τ ) as t → ∞. Because t N +1 = [t0 ] N + τ + Nτ is finite for all N ≥ 1, the norm y N (t) − y0 (t − τ ), t ∈ [t0 , t N +1 ], is bounded. Therefore, over the set t ≥ t0 the required estimate differs from (28.17) only by a constant multiplier. To specify this multiplier we state without proof the following inequality:

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y N (t) − y0 (t − τ ) ≤ 6

 τ N +2 K 1 eα1 N τ e−α1 (t−t0 ) , t ∈ t1 , t N +1 ) . N

(28.18)

As a result, from (28.14), (28.17), (28.18) we obtain the estimate y N (t) − y0 (t − τ ) ≤ 6

τ N +2 K 1 eα1 N τ e−α1 (t−t0 ) , t ≥ t0 . N 

The proof is complete. The main result of the paper is as follows.

Theorem 28.2 If for N > 6τα (a1 + a2 ) a2 K n the system (28.2), (28.3) is exponentially stable, then the system (28.1) is exponentially stable. Here a1 = supt≥0 A1 (t), a2 = supt≥0 A2 (t). Proof Let us consider a system dx0 (t) = A1 (t)x0 (t) + A2 (t)x0 (t − τ ), dt N dxi (t) = (xi−1 ([t] N ) − xi ([t] N )) , i = 1, ..., N . dt τ

(28.19)

The system (28.19) can be considered as a perturbed system with respect to the system (28.2), (28.3). For this purpose, we rewrite the system (28.19) in the form dx0 (t) = A1 (t)x0 (t) + A2 (t)x N (t) + A2 (t) (x0 (t − τ ) − x N (t)) , dt N dxi (t) = (xi−1 ([t] N ) − xi ([t] N )) , i = 1, ..., N . dt τ

(28.20)

The system (28.20) belongs to the general type of linear systems with finite lag. Its solution is given by the variation-of-constants formula [5]. Let U (t, s) =

N Ui j (t, s) 0 be the fundamental matrix of the system (28.2), (28.3). Then we have 

t

x0 (t) = y0 (t) +

U00 (t, s)A2 (s) (x0 (s − τ ) − x N (s)) ds,

(28.21)

t1

where y0 (t) is the first n-vector component of the solution of the system (28.2), (28.3). We choose values for the system (28.2), (28.3) in the form yi (t1 ) = the initial

yi ([t1 ] N ) = x0 t1 − iτN , i = 0, 1, ..., N . Exponential stability of the system (28.2), (28.3) implies the inequalities [5] y0 (t) ≤ K e−α(t−t1 ) max x0 (t) , t ≥ t1 , t∈[t1 −τ,t1 ]

U00 (t, s) ≤ K ne−α(t−s) , t ≥ s ≥ t1 .

(28.22)

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We now prove that for an arbitrarily large T there exist the constants P > K , 0 < α˜ < α, such that ˜ 0) x0 (t) < Pe−α(t−t ϕC , t0 ≤ t < T.

(28.23)

We may assume that (28.23) is violated for some t. In this case, there exists ˜ −t0 ) ϕC . Let us suppose a moment of time t = T , such that x0 (T ) = Pe−α(T t1 = t0 + τ . The first derivative of x0 (t) on [t0 , T ] satisfies the estimate    dx0 (t)   ατ ˜ ˜ 0)   ϕC , t ∈ [t0 , T ]. Pe−α(t−t  dt  ≤ a1 + a2 e Using Lemma 28.3, we obtain τ  N +2 ˜ ˜ 0) ϕC , t ∈ [t1 , T ]. a1 + a2 eατ Peα˜ N τ e−α(t−t N (28.24) From the formula (28.21), using (28.22), (28.23), (28.24), we have x N (t) − x0 (t − τ ) ≤ 6

+

6τ N (α − α) ˜

x0 (T ) ≤ K Pe−α(T −t0 −τ ) ϕC  N +2 ˜ ˜ −t0 ) ϕC . a1 + a2 eατ a2 eα˜ N τ K P ne−α(T

Let us show the existence of N , for which the following inequality holds K Pe−α(T −t0 −τ ) ϕC + ×eα˜

N +2 N τ

 6τ ˜ a1 + a2 eατ a2 N (α − α) ˜

(28.25)

˜ −t0 ) ˜ −t0 ) ϕC < Pe−α(T ϕC . K P ne−α(T

This fact will imply that the inequality (28.23) is true for all t ≥ t0 . The inequality (28.25) is equivalent to the inequality ˜ −t0 )+ατ + K e−(α−α)(T

 6τ N +2 ˜ a1 + a2 eατ a2 eα˜ N τ K n < 1. N (α − α) ˜

The last inequality holds for sufficiently large T if  6τ N +2 ˜ a1 + a2 eατ a2 eα˜ N τ K n < 1. N (α − α) ˜

(28.26)

The inequality (28.26) is satisfied for small positive α˜ if 6τ (a1 + a2 ) a2 K n < 1. Nα It completes the proof.



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References 1. Krasovskii, N.N.: On an approximation of a problem of analytical constructing regulators in a delay system. Prikl. Mat. Mekh. 28(4), 716–724 (1964) 2. Repin, U.M.: On an approximate replacement of delay systems by ordinary dynamic systems. Prikl. Mat. Mekh. 29(2), 226–235 (1965) 3. Kurjanskii, A.B.: On an approximation of linear delay differential equations. Differentsialnye Uravneniya. 3(12), 2094–2107 (1967) 4. Dolgii, Y., Sazhina, S.D.: An estimate of exponential stability of systems with delay by the method of approximating systems. Differentsialnye Uravneniya. 21(12), 2046–2052 (1985) 5. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

Chapter 29

On Stability and Stabilization with Permanently Acting Perturbations in Some Critical Cases Aleksandr Ya. Krasinskiy

Abstract Malkin’s theorem on stability under permanently acting perturbations guarantees non-asymptotic stability in the presence of permanently acting perturbations in systems in which asymptotic stability took place without such perturbations. In this case, traditionally, the conclusion about asymptotic stability in the absence of permanently acting perturbations is obtained on the basis of an analysis of the location of the roots of the characteristic equation of the system of the first approximation of the equations of perturbed motion: the real parts of all roots of the characteristic equation must be negative. In this paper, we consider the application of Malkin’s theorem on stability under permanently acting perturbations to systems whose characteristic equation without perturbations has zero roots, and the equations of perturbed motion have a special structure that allows the application of the principle of reducing the theory of critical cases with regard to additional conditions on the initial perturbations of critical variables. It is shown that this approach is applicable to the problems of stability and stabilization of steady motions of the systems with geometric connections using the Shulgin equations in the Routh variables.

29.1 Introduction In the study of the stability of systems with permanently acting perturbations, the asymptotic stability in the system in the absence of such perturbations is sufficient under the Malkin [1, p. 316] theorem for stability under the action of such perturbations. The asymptotic stability is traditionally established by the negativity of the real parts of all roots of the characteristic equation. If at least one root of the characteristic equation has a zero real part, the proof of asymptotic stability in a system A. Ya. Krasinskiy (B) Moscow State University of Food Production, Institute of Economics and Management, Volokolamskoe Shosse 11, Moscow 125080, Russia e-mail: [email protected] Moscow Aviation Institute, Faculty of Information Technologies and Applied Mathematics, Volokolamskoe Shosse 4, Moscow 125993, Russia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_29

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without constantly acting perturbations in the general case is an extremely difficult task. When the real parts of all other roots of the characteristic equation are negative, except for the roots located on the imaginary axis, a critical case takes place. For a special category of critical cases in which, besides roots with zero real parts, roots are necessarily present, the real parts of which are negative, a so-called reduction principle is developed that establishes sufficient conditions under which the conclusion about stability in the complete system is obtained as a result of consideration socalled shortened system. This system contains only those variables that correspond to the critical roots of the characteristic equation. Sufficient conditions for the applicability of the results of the reduction principle are formulated in new variables after linear (to reduce to a special kind) and nonlinear (to increase the order of freely entering of critical variables in equations corresponding to non-critical variables) variables changes, after which it is extremely difficult to reformulate in the general case these conditions for the original systems in the original variables. This is the main problem of the practical application of the achievements of the principle of reduction: sufficient conditions for stability or instability must be formulated in the initial variables. Moreover, since there is a critical case, special accuracy of the nonlinear mathematical model is needed in the form that allows us to analyze the structure of nonlinear terms of the equations of perturbed motion from the standpoint of the theory of critical cases [1–3]. In contrast to the general situation, the developed apparatus of analytical mechanics, by choosing the forms of equations of motion and types of variables, makes it possible to distinguish such classes of critical cases of stability when the above analytical transformations are not necessary, so that sufficient conditions for the applicability of the results of the reduction principle can be formulated in source variables [4]. The most general results of the principle of reduction, which can be applied and are already being applied in solving applied problems of stability, relate to special cases of critical cases. The main difficulty in applying the Kamenkov [3] theorem (on the stability in the special case of several zero roots and pairs of purely imaginary roots) is to check the sufficient conditions for the convergence of the series, which determines a nonlinear change in non-critical variables, simultaneously zeroing out freely entering critical variables in both the attached and the shortened systems. Using this theorem, sufficient conditions for non-asymptotic stability were established in systems in which the structure of the equations of perturbed motion allowed perform the above replacement only in part of non-critical variables, the latter corresponds to second-order differential equations, while the first-order equations corresponded to critical variables with special structures: the right-hand sides of equations corresponded to critical variables contain such non-critical variables as factors, for which it is not necessary to make a nonlinear replacement [4]. Such a structure arises in problems of stability of equilibrium positions of nonholonomic systems with homogeneous constraints [5–7], and the convergence of the series in a nonlinear substitution is substantially related to the nonisolation of the studied motion. In all previously published studies on the application of the Kamenkov [3] theorem in the considered system with zero roots, reducing to a special case, there

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was a non-asymptotic stability with respect to critical variables and that part of noncritical ones, for which the indicated nonlinear replacement was carried out. Due to the presence of a variety of steady motions, non-asymptotic stability takes place. This circumstance precluded the possibility of applying Malkin’s stability theorem under constantly acting perturbations. However, the special structure of the equations of perturbed motion noted above also arises [8, 9] in problems of stability of equilibrium positions of holonomic systems with geometric constraints when using (free from coupling factors) Shulgin [10] equations with differentiated equations of geometric constraints. The use of a similar algorithm with the implementation of a linear Aiserman-Gantmacher [11] replacement for extracting critical variables when the real parts of the other roots of the characteristic equation are negative formally reduces the problem to a situation similar to that for which nonholonomic systems had non-asymptotic stability. However, for systems with geometric constraints, taking into account the condition imposed by constraints on the initial perturbations of the coordinates leads to asymptotic stability. As a result, in the problems of stability and stabilization of steady motions of systems with geometric connections, the Malkin theorem on stability under constantly acting perturbations turns out to be applicable, despite the presence of zero roots of the characteristic equation.

29.2 Mathematical Models of the Dynamics of Mechanical Systems with Geometric Constraints Consider a mechanical system with the coordinate vector q  = (q1 , . . . , qn+m ), the kinetic energy n+m 1  T = a˜ ρν (q)q˙ρ q˙ν + 2 ρ,ν=1

n+m 

a˜ ρ (q)q˙ρ + T0 (q),

(29.1)

ρ=1

on which the geometric constraints are superimposed F(q) = 0;

   ∂(F1 , . . . , Fm )   = 0, F  = (F1 (q), . . . , Fm (q)); det  ∂(qn+1 , . . . , qn+m ) 

(29.2)

and the system is affected, in addition to potential forces with energy(q), nonpotential forces Q˜ ρ (q, q), ˙ corresponding to the coordinates qρ . Introduce vectors q  = (r  , s  ); r = (q1 , . . . , qn ) ; s = (qn+1 , . . . , qn+m ) ; α = (q1 , . . . , qk ) ; β = (qk+1 , . . . , ql ) ; γ = (ql+1 , . . . , qn ) ; 1 < k < l < n.

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Let the coordinates α, β be cyclic in the following sense: they are not included in the constraint equations, in the expressions for the kinetic and potential energy and non-potential forces, and there are no non-potential forces for these coordinates. The vector β is needed to extract controlled cyclic coordinates if the stabilization problem arises. Stabilizing control can be determined by the Krasovskii method [12] of solving linear-quadratic problems for controlled subsystems. We differentiate the equations of geometric constraints (29.2) with respect to time and express the dependent velocities from the obtained relations: 

s˙ = Bα (α, s) · α˙ :

∂F Bα (α, s) = − ∂s

−1   ∂F . · ∂α

(29.3)

We denote T ∗ (α, s, r˙ ) kinetic energy (29.1) after the exclusion of dependent velocities. We introduce momenta and the Routh function: R = T ∗ (α, s, r˙ ) − (α, s) − pβ β˙ − pγ γ˙ ;

pβ =

∂T ∗ ; ∂ β˙

pγ =

∂T ∗ . ∂ γ˙

Mechanical systems with geometric constraints superimposed on them have to be considered in many relevant technical problems, in particular, control of multi-link manipulators and other tasks of technical practice. This makes it impossible to use in studying the dynamics of such systems the most universal formalism based on the introduction of generalized (independent) coordinates. If the relations defining these m constraints are complex nonlinear, it is advisable to specify the configuration of a mechanical system with n degrees of freedom by n + m parameters taken in a number exceeding the number of degrees of freedom of the system. Then m of these n + m parameters are called redundant coordinates. The use of redundant coordinates requires a different formulation of the dynamics problems [13–17]. It will not be possible to use Lagrange equations of the second kind, since their derivation assumes the introduction of independent generalized coordinates, the variations of which will also be independent. For the systems under consideration, one can use the Lagrange equations of the first kind in Cartesian coordinates or the Lagrange equations with indefinite Lagrange multipliers in redundant curvilinear coordinates [13–17] (in the application to the control tasks of manipulating robots—[18]. The total number of equations becomes equal to the sum of the number of variables and the number of constraints. Direct integration of such a system of equations is a rather complicated task. If the studies do not intend to determine the reactions of constraints, then naturally the need arises to exclude undetermined factors from the obtained equations. Such an exception can be performed by various methods [13–17]. In particular, the following approach was proposed [14, pp. 328–331] to exclude indefinite Lagrange multipliers: the equations of geometric constraints were proposed to be differentiated twice and

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once the equations of differential constraints, and then substituted into these conditional equations the acceleration as linear functions of the constraints multipliers from the equations of motion. The multipliers are found from the resulting linear algebraic heterogeneous system of equations. In [15], this method of eliminating multipliers is developed in relation to a new class of problems. Analytical expressions for the indefinite Lagrange multipliers were first obtained and investigated by Lyapunov [16] and Suslov [14]. It is possible to exclude constraints multipliers with a single differentiation of the equations of geometric constraints [10, 13]. Note that this use of differentiated geometric (holonomic) constraints to exclude constraints multipliers was proposed in [13, pp. 319–321]. However, this approach was not widely used, possibly because this method was described by Lurie as applied to the Lagrange equations with multipliers for nonholonomic systems (systems with non-integrable differential constraints), and it was simply noted that among the differential constraints there could be integrable ones. The technique associated with a single differentiation of geometric relationships and the application of equations of motion obtained from the general theorems of dynamics was developed in [17]. A much simpler and more effective method for practical application of obtaining equations—multiplier-free equations of motion of systems with redundant coordinates, developed by Shulgin in [10]. The equations of motion in excess coordinates proposed by Shulgin are a special case of the equations of motion of nonholonomic systems with linear homogeneous differential constraints in the Voronets form [10, 13, 14]. For Shulgin equations, the nonholonomic terms in the Voronets equations vanish due to the integrability of the kinematic constraint equations [10, 13, 14]. Note that this allows us to apply to the study of the dynamics of systems with redundant coordinates all the methods developed for studying the dynamics of nonholonomic systems. But, at the same time, in addition to a fundamentally different structure (due to the absence of nonholonomic terms), these equations have other features that require further study and use. Earlier, on the basis of a sufficiently detailed analysis [8, 9], the choice of Shulgin equations [10] as the most suitable form of equations in the problems of stability and stabilization of the motions of holonomic systems with dependent (redundant) coordinates was substantiated:   ∂R ∂R d ∂R  − = Q α + Bα Q s + ; p˙β = 0; p˙γ = 0; s˙ = Bα · α. ˙ (29.4) dt ∂ α˙ ∂α ∂s Here, after Q α , Q s designated forces Q˜ α , Q˜ s after the exclusion of dependent speeds. Equation (29.4) allows steady motion pβ = pβ0 ,

pγ = pγ 0 , α = α0 ,

s = s0 .

(29.5)

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29.3 Application of Malkin’s Stability Theorem Under Permanently Acting Perturbations in the Problems of Stability of Steady Motions For the study the stability of the steady motion, we introduce perturbations in the neighborhood of motion (29.5) pβ = pβ0 + v;

pγ = pγ 0 + w; α = α0 + x; α˙ = x1 ; s = s0 + y.

To extract critical variables, we perform linear replacement [11] z = y − Bα (0) x.

(29.6)

Based on Eqs. (29.4) with (29.6) taken into account, we write the equations of perturbed motion with a distinguished first approximation in normal form 

ξ˙ = N ξ + V v + W w + Z z + (2) (ξ, v, w, z) ;    z˙ = Bα(1) (x, z) x1 ; v˙ = 0; w˙ = 0; ξ = x , x1 .

(29.7)

Here, the superscript in parentheses denotes the order of the lower terms in the expansion of the corresponding expression. The constant matrices of the coefficients of the first approximation are expressed in a known manner [8, 9] in terms of system parameters, acting forces, and constants of the steady motion. The methods for composing the equations of perturbed motion for various ways of introducing Routh variables both for systems without cyclic coordinates and in the presence of such coordinates are described in detail in [4, 9, 19]. If in the system (29.7) the perturbations of momenta are equal to zero and excluded from consideration the equations for perturbations of momenta v˙ = 0; w˙ = 0, we can distinguish the system ξ˙ = N ξ + Z z + (2) (ξ, 0, 0, z) ; z˙ = Bα(1) (x, z) x1 .

(29.8)

Theorem 29.1 If the real parts of all the roots of the characteristic equation of the matrix N are negative, the steady motion (29.5) is non-asymptotically stable. Proof. These Eq. (29.8) have the structure of the equations of perturbed motion in the problem of stability of the equilibrium position of a system with geometric constraints [9]. The characteristic equation of this system has m zero roots corresponding to a variable z. If the real parts of all the rest of its roots, i.e., the roots of the characteristic equation of the matrix N are negative, then the zero solution of the system (29.8)

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is asymptotically stable, according to Theorem 29.1 [9]. Returning to the complete system, integrating the equations for the momentums perturbations, we find that the variables keep constant values equal to the initial perturbations throughout the motion: v(t) = vo ; w(t) = wo . Therefore, the complete system 

ξ˙ = N ξ + V v0 + W w0 + Z z + (2) (ξ, v0 , w0 , z) ; z˙ = Bα(1) (x, z) x1 ; v(t) = vo ; w(t) = wo

(29.9)

can be obtained from system (29.8) under the application of permanently acting disturbances that depend on v0 , wo —the initial disturbances of cyclic pulses. According to Malkin’s stability theorem under permanently acting disturbances in the complete system (29.7), we have non-asymptotic stability of steady motion (29.5).

29.4 Application of Malkin’s Stability Theorem Under Permanently Acting Perturbations for the Stabilization Problems of Steady Motions As was established in the previous section, a sufficient condition for the stability of steady motion is the negativity of the real parts of the roots of the characteristic equation of the matrix N . If this condition is not fulfilled, the stabilization problem arises—to provide the required location of the roots of the characteristic equation by applying control actions. To do this, in the general case, into the system some executive mechanisms that carry out such actions must be introduced. In systems with cyclic coordinates, a method has been developed for a long time to implement such actions without introducing additional actuators [4, 9, 19, 20, 22], but with the help of suitably organized motion along cyclic coordinates, that is, by introducing control into the corresponding equations. Note that then, from the point of view of control theory, we obtain an indirect control system—controlling the change in cyclic momentums, we change the behavior in positional coordinates. We apply this approach to the problem under consideration, and, following the ideology, we will choose such impulses as controlled so that the size of the controlled subsystem is as small as possible [4]. In technical practice, to solve stabilization problems, a method for determining of the control through the solution of the tasks of analytical design of optimal controllers (ACOR, which is formulated by Letov [23]): to determine the coefficients of the linear control, which not only provide the asymptotic stability of a given motion on a first approximation, but also minimizes some integral criterion. Thus, it was proposed to consider this task as a specific problem of the optimal control. For optimal control problems, the most effective solution methods are the Pontryagin maximum principle and Bellman’s dynamic programming method.

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As noted Letov [24, p. 140], Krasovskii [25] was the first to draw attention to the relationship between the dynamic programming method and the Lyapunov function method. Based on the synthesis of these methods, Krasovskii proved [12] the theorem on optimal stabilization, on the basis of which he developed a practical method for determining stabilizing control coefficients [12] from solving nonlinear algebraic Lyapunov-Bellman-Riccati equations obtained from the Lyapunov-Bellman equation for a linear-quadratic problem. To solve the problem of stabilization of motion (29.5) in this way, in the system (29.7) we apply control u on the vector β: 

; ξ˙ = N ξ + V v + Gu + W w + Z z + (2) (ξ,v, w, z)     z˙ = Bα(1) (x, z) x1 ; v˙ = u; w˙ = 0; ξ = x , x1 .

(29.10)

A linear term with respect to the control Gu appears in this system if the coordinates α, β are gyroscopically coupled. We introduce matrices  A=

N V 0 0



 ,

B=

G El−k

 (29.11)

and formulate a sufficient condition for stabilization in this way of steady motion (29.5). Theorem 29.2 If for the pair of matrices (29.11) controllability condition   rank B AB A2 B ... Ak+l−1 B = k + l

(29.12)

is satisfied, then the steady motion (29.5) is stabilized to non-asymptotic stability by applying of linear control (29.13) u = K1ξ + K2v on the coordinates β. Moreover, the matrices K 1 , K 2 can be determined by solving the linear-quadratic problem for the subsystem ξ˙ = N ξ + V v + Gu; v˙ = u.

(29.14)

Proof. The fulfillment of condition (29.12) is sufficient to determine by the Krasovskii method [12] the matrixs K 1 , K 2 of the coefficients of the control (13), by which the real parts of all the roots of the characteristic equation of the subsystem ξ˙ = N ξ + V v + G(K 1 ξ + K 2 v); v˙ = K 1 ξ + K 2 v

(29.15)

are negative. We pass from system (15) to the next nonlinear system: 

ξ˙ = N ξ + V v + G(K 1 ξ + K 2 v) + Z z + (2) (ξ, v, 0, z) ; v˙ = K 1 ξ + K 2 v; z˙ = Bα(1) (x, z) x1 .

(29.16)

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The characteristic equation of the system (29.16) has m zero roots corresponding to the variable z, and the real parts of all its other roots are negative under the action of control (29.13); these nonzero roots are determined by the same equation as in the system (29.15). Note that the structure of the system (29.16) meets the conditions of Theorem 29.1 [9]. Therefore, the zero solution to the system (29.16) is asymptotically stable. Returning to the complete system (29.10), integrating the equation for the perturbations of momentum w, we find that the w(t) keep constant value equal to the initial perturbation throughout the motion w(t) = wo . After substituting this constant into the full system (29.10) we obtain the equations 

ξ˙ = N ξ + V v + G(K 1 ξ + K 2 v) + W w0 + Z z + (2) (ξ, v, w0 , z) ; z˙ = Bα(1) (x, z) x1 ; v˙ = K 1 ξ + K 2 v; w(t) = w0 ,

(29.17)

which can be considered as system (29.16) when applying permanently acting disturbances w(t) = w0 . Since the zero solution of the system (29.16) is asymptotically stable, the conditions of Malkin’s theorem on the stability of the zero solution of the system (29.17) under constantly acting perturbations w0 are satisfied. Thus, the control (29.13) stabilizes the steady motion (29.5) to non-asymptotic stability.

References 1. Malkin, I.G.: Stability Theory of Motion. Nauka, Moscow (1966) 2. Lyapunov, A.M.: The General Problem of the Stability of Motion. Kharkov Math. Society, Kharkov (1892) 3. Kamenkov, G.V.: Stability and Oscillations of Nonlinear Systems. Collected Works, v.2. Nauka, Moscow (1972) 4. Krasinskiy A.Y.: On One Method for Studying Stability and Stabilization for Non-Isolated Steady Motions of Mechanical Systems. In Proceedings of the VIII International Seminar “Stability and Oscillations of Nonlinear Control Systems”, pp. 97–103. Inst. Probl. Upravlen, Moscow (2004). http://www.ipuru/semin/arhiv/stab04 5. Karapetyan, A.V., Rumyantsev, V.V.: Stability of conservative and dissipative systems. Itogi Nauki i Tekhniki. General mechanics. V.6. VINITI, Moscow (1983) 6. Krasinskiy, A.Y.: On stability and stabilization of equilibrium positions of nonholonomic systems. J. Appl. Math. Mech. 52(2), 194–202 (1988) 7. Kalenova, V.I., Karapetjan, A.V., Morozov, V.M., Salmina, M.A.: Nonholonomic mechanical systems and stabilization of motion. Fundamentalnaya i prikladnaya matematika 11(7), 117– 158 (2005) 8. Krasinskaya, E.M., Krasinskiy, A.Y.: Stability and stabilization of equilibrium state of mechanical systems with redundant coordinates. Scientific Edition of Bauman MSTU, Science and Education (2013). https://doi.org/10.7463/0313.0541146

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9. Krasinskaya, E.M., Krasinskiy, A.Y.: On one method for studying stability and stabilization for steady motions of mechanical systems with redundant coordinates. In: Proceedings of the XII Russian Seminar on Control Problems VSPU-2014, pp. 1766–1778. Moscow, June 1–19 (2014) 10. Shulgin, M.F.: On some differential equations of analytical dynamics and their integration. In: Proceedings of the SAGU. No. 144. Sredneazitsk. Gos. Univ., Tashkent (1958) 11. Aiserman, M.A., Gantmacher, F.R.: Stabilitaet der Gleichgewichtslage in einem nichtholonomen System. Z. Angew. Math. Mech. 37(1–2), 74–75 (1957) 12. Krasovskii, N.N.: Problems of stabilization of controlled motions. In: Malkin I.G. (eds.) Stability Theory of Motion, pp. 475–514. Nauka, Moscow (1966) 13. Lurie, A.I.: Analytical Mechanics. Fizmatgiz, Moscow (1961); Springer, Berlin (2002) 14. Suslov, G.K.: Theoretical Mechanics. OGIZ, Moscow-Leningrad (1946) 15. Zegzhda, S.A., Soltakhanov, S.K.: Equations of Motion of Nonholonomic Systems and Variational Principles of Mechanics. A New Class of Control Tasks. Fizmatlit, Moscow (2005) 16. Lyapunov, A.M.: Lectures on Theoretical Mechanics. Naukova Dumka, Kiev (1982) 17. Novozhilov I.V., Zatsepin M.F.: The equations of motion of mechanical systems in an excessive set of variables. In: Collection of Scientific-Methodological Articles on Theoretical Mechanics (Mosk. Gos. Univ., Moscow, 1987), Issue 18, pp. 62–66 (1987) 18. Zenkevich, S.L., Yushchenko, A.S.: Fundamentals of Control of Manipulation Robots. Publishing House of Bauman Moscow State Technical University, Moscow (2004) 19. Krasinskaya, E.M., Krasinskiy, A.Y.: A Stabilization method for steady motions with zero roots in the closed system. Autom. Remote Control 77(8), 1386–1398 (2016). https://doi.org/ 10.1134/S0005117916080051 20. Klokov, S., Samsonov, V.A.: Stabilizability of trivial steady motions of gyroscopically coupled systems with pseudo-cyclic coordinates. J. Appl. Math. Mech. 49(2), 150–153 (1985) 21. Krasinskiy, A.Y., Il’ina, A.N., Krasinskaya, E.M.: Modeling of the ball and beam system dynamics as a nonlinear mechatronic system with geometric constraint. Vestn. Udmurt Gos. Univ. 27(3), 414–430 (2017) 22. Krasinskii, A.Y., Il’ina, A.N., Krasinskaya, E.M.: Stabilization of steady motions for systems with redundant coordinates Moscow. Univ. Mech. Bull. 74, 14 (2019). https://doi.org/10.3103/ S0027133019010035 23. Letov, A.M.: Analytical design of regulators. Automat. Telemekh. 21(4), 436–441 (1960) 24. Letov, F.M.: Mathematical Theory of Control Processes. Nauka, Moscow (1981) 25. Krasovskii, N.N.: On a problem of optimal control of nonlinear systems. J. Appl. Math. Mech. 23(2), 209–230 (1959)

Chapter 30

Control Problem with Disturbance and Unknown Moment of Change of Dynamics Viktor I. Ukhobotov and Igor’ V. Izmest’ev

Abstract A linear control problem is considered in the presence of exposure from an uncontrolled disturbance. It is known only that its values belong to the given connected compact. The payoff is the absolute value of linear function at the end time of the controlled process. It’s believed that one breakdown is possible, which leads to a change in the dynamics of the controlled process. The time of occurrence of breakdown is not known in advance. Control is constructed based on the principle of minimizing the guaranteed result. The opposite side is the disturbance and the moment of occurrence of a breakdown. The necessary and sufficient conditions are found under which an feasible control is optimal. As an example, the problem of controlling a rod, which is rigidly attached to an electric motor rotor, is considered. A rotary flywheel is attached to the other end of the rod. The voltages applied to the rotor and flywheel are control. As disturbance, we take the nonlinear addend in the equations of motion. The goal of choosing controls is to minimize the modulus of the deviation of the angle formed by the rod and the vertical axis from the given value at a fixed time moment.

30.1 Introduction The control problem with possible violations in dynamics as a result of a breakdown can be considered within the framework of an approach that is based on the principle of optimizing the guaranteed result [9]. Such an approach is natural if only the time interval is known when a breakdown may occur. One of the first papers devoted to control problems with breakdown in this formulation is the work [12].

V. I. Ukhobotov · I. V. Izmest’ev (B) N.N. Krasovskii Institute of Mathematics and Mechanics, 16 S.Kovalevskaya Str., 620108 Yekaterinburg, Russia e-mail: [email protected] V. I. Ukhobotov e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_30

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The linear differential game with a given duration, using a linear change of variables [8], we can reduce to the form, in which the dynamics of the system is determined by the sum of player controls with values that belong to time-dependent sets. In the case when in a linear differential game the quality criterion is defined as an absolute value of a linear function at a fixed time moment, using a linear change of variables, we obtain a single-type differential game, in which the vectograms of player controls are segments that depend on time (see as example [5]). In general case, in these problems, the vectograms of player controls are n-dimensional balls with time-dependent radii. Differential games that have this type of dynamics after change of variables are considered, for example, in [4] and [13]. For single-type differential games, if the target set is an n-dimensional ball of a fixed radius, the alternating integral is constructed in [13]. In [15], optimal controls of the players are found. In [17], the alternating integral for single-type differential games, in which target set is convex and closed, is constructed and appropriate optimal controls of the players are found. In present paper, linear control problem is considered in the presence of exposure from an uncontrolled disturbance. It is known only that its values belong to the given connected compact. The quality criterion is defined as the absolute value of a linear function at the end time of the control process, which is given. It is believed that one breakdown is possible, which leads to a change in the dynamics of the controlled process. The time of occurrence of breakdown is not known in advance. Control is constructed based on the principle of minimizing the guaranteed result. The opposite side is the disturbance and the moment of occurrence of a breakdown. As an example illustrating the theory, we solve the problem of controlling an oscillatory mechanical system. Laboratory pendulum installations are widely used in educational process and scientific researches from the field of automatic control systems and mechatronics. Problems of controlling the systems, which consist of a physical pendulum and a flywheel at one end of it, for example, were studied in [1, 2]. In [11, 18], numerical methods proposed by the authors were used to solve these problems, and numerical experiment was performed. In [5], the problem of controlling the rod, which is attached to the rotor of the electric motor, is considered. The control is the value of the voltage applied to the electric motor. The quality criterion is the sum of the angle deviation module that forms the rod with the vertical axis, at a given time, and the integral of the square of the voltage value. Taking the nonlinear addend in the Lagrange equation of this system as a disturbance, we obtained a linear control problem with a disturbance. In [6], we consider antagonistic differential game, where the first player controls the voltage applied to the electric motor rotor to which the rod is attached. The second player controls a rotary flywheel which is attached to other end of the rod. The first player seeks to move the rod in a given position at a fixed time. In present paper, we take as an example the modification of examples from [5] and [6]. We consider the controlled system, which consists of a rod and a rotary flywheel. One end of the rod is rigidly attached to the electric motor rotor, and a rotary flywheel is attached to the other end of the rod. The controls are the voltages applied to the rotor and the flywheel. In the process of control, a breakdown of the electric motor is

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possible. We take nonlinear addend in motion equations of the system as disturbance. The goal of choosing controls is to minimize the modulus of the deviation of the angle formed by the rod and the vertical axis from the given value at a fixed time moment.

30.2 Problem Statement We consider controlled process x˙ = A(t)x + B(t, τ )w + ξ + η, x(t0 ) = x0 , x ∈ Rn , t0 ≤ t ≤ p.

(30.1)

Here, p is the termination moment of the control process, and t0 is the initial time moment; w ∈ W ⊂ Rs and ξ ∈ M ⊂ Rm are control. Sets W and M are connected compacts, besides compact W is connected and symmetric with respect to the origin of coordinates. Disturbance η belongs to the connected compact Q ⊂ Rm . Further, B(t, τ ) = B1 (t) for t0 ≤ t < τ and B(t, τ ) = B2 (t) for τ ≤ t ≤ p. Here, A(t) is matrix with elements that are continuous on [t0 , p] functions, and Bi (t) are matrices with elements that are summable on [t0 , p] functions. Such situation can arise when a breakdown occurs at the time moment t0 ≤ τ ≤ p, and the dynamics of the process changes. The moment of breakdown τ is not known in advance. Therefore, the moment of breakdown is included in the disturbance. The quality criterion of control is value |ψ0 , x( p) − G|. Here, ψ0 ∈ Rn is the given vector, ·, · is the scalar product in Rm , G is the given number. The control is constructed using the principle of minimization of the guaranteed result [9] of the quality criterion.

30.3 Main Result Following [8, p. 160], we pass to a new controlled system, in the equations of motion of which there is no phase vector. Consider solution ψ(t) of the following Cauchy problem for t0 ≤ t ≤ p: ˙ ψ(t) = −A T (t)ψ(t), ψ( p) = ψ0 .

(30.2)

Here, A T (t) is matrix that is transposed to matrix A(t). Denote b(t, τ ) = maxB(t, τ )w, ψ(t) ≥ 0,

(30.3)

w∈W

c− (t) = minξ, ψ(t), c+ (t) = maxξ, ψ(t), ξ ∈M

ξ ∈M

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β− (t) = minη, ψ(t), β+ (t) = maxη, ψ(t). η∈Q

η∈Q

Note that these functions are summable. Using connectivity of compacts W , M, and Q, it can be shown that [10, pp. 333– 334, Theorem 4] B(t, τ )w + ξ, ψ(t) =

1 (c− (t) + c+ (t)) − a(t, τ )u, |u| ≤ 1, 2

(30.4)

1 ψ(t), η = β(t)v + (β− (t) + β+ (t)), |v| ≤ 1, 2 β(t) =

1 (β+ (t) − β− (t)) ≥ 0. 2

Here, we denote a(t, τ ) = b(t, τ ) + 21 (c+ (t) − c− (t)) ≥ 0. Let’s make a change of variables: 1 z = x, ψ(t) + 2

p (β− (r ) + β+ (r ) + c− (r ) + c+ (r ))dr − G.

(30.5)

t

Then (30.1) and (30.2) imply that z˙ = −a(t, τ )u + β(t)v, |u| ≤ 1, |v| ≤ 1. Note that z( p) = ψ0 , x( p) − G, because the quality criterion takes the form |z( p)|. Feasible control and disturbance are arbitrary functions with constraints |u(t, z)| ≤ 1, |v(t, z)| ≤ 1, z ∈ Rn , t ≤ p.

(30.6)

Following [8], the motion of the system (30.1), which corresponds to feasible control and disturbance, is determined using polygonal lines. Take the partition ω of the segment [t0 , p] with the diameter d(ω) ω : t0 < t1 < · · · < tl < t j+1 = p, d(ω) = max(ti+1 − ti ). i=0,l

On the segment [t0 , p], define polygonal line for system (30.1) t z ω (t) = z ω (ti ) −

t a(r, τ )dr u(ti , z ω (ti )) +

ti

for ti < t ≤ ti+1 . Here, z ω (t0 ) = z(t0 ).

β(r )dr v(ti , z ω (ti )) ti

(30.7)

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It can be shown that the family of the polygonal lines (30.7) is equicontinuous and uniformly bounded [16, p. 45]. Thus, it satisfies the conditions of Arzela’s theorem [7, p. 104]. The motion z(t) generated by feasible control and disturbance (30.6) with moment of breakdown τ from given z(t0 ) is defined as any uniform limit of the subsequence of the polygonal lines (30.7) with d(ω) → 0. Denote p

p β(r )dr − min

f (t) = t

φ(z) =

a(r, τ )dr,

t≤τ ≤ p

F(t) = max f (s), t≤s≤ p

t

z for |z| > 0 and φ(0) − any with constraint |φ(0)| = 1. |z|

Theorem 30.1 Control u = φ(z) guarantees the inequality |z( p)| ≤ Q(t0 , |z(t0 )|) for any disturbance |v(t, z)| ≤ 1 and for any moment of breakdown t0 ≤ τ ≤ p. Here, we denote Q(t0 , |z(t0 )|) = max(F(t0 ); |z(t0 )| + f (t0 )). Theorem 30.2 Disturbance v = φ(z) with some moment of breakdown t0 ≤ τ ≤ p guarantees the inequality |z( p)| ≥ Q(t0 , |z(t0 )|) for any control |u(t, z)| ≤ 1. The proofs of Theorems 30.1 and 30.2 are carried out by analogy with the proofs of Theorems 1.1 and 2.3 from [14], respectively.

30.4 Example Consider a controlled system consisting of a rod and a rotary flywheel. Figure 30.1 shows the scheme of the considered mechanical system. The rotor axis of the first electric motor perpendicular intersects the plane of the figure at the point O.The rod O A is rigidly attached at the point O to the rotor axis.

Fig. 30.1 The problem of controlling the rod using the electric motor rotor and the rotary flywheel

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Rotary flywheel, which is symmetrical about its axis, is designed so that its center is the point A. The flywheel rotates about an axis that passes through the point A perpendicular to the plane of the figure. The rotation axis of the flywheel is the axis of the rotor of the second electric motor. The total moment of inertia of the first electric motor and rod O A relative to their rotation axis is denoted by J1 . The distance from point O to the center of mass of the rod O A is denoted by L 1 , and its mass M1 . The total moment of inertia of the second electric motor and the flywheel relative to their rotation axis is denoted by J2 , and the total mass of the flywheel and the second electric motor is denoted by M2 . Denote by L 2 the length of the rod O A and write down kinetic energy of system T =

1 J2 (M1 L 21 + J1 + M2 L 22 )φ˙ 12 + (φ˙ 1 + φ˙ 2 )2 . 2 2

(30.8)

The virtual work δ A = Q 1 δφ1 + Q 2 δφ2 of external forces is equal to δ A = −(M1 L 1 + M2 L 2 )g sin φ1 δφ1 + N1 δφ1 + N2 δφ2 .

(30.9)

Here, g denotes acceleration of gravity, Ni denotes the moment of electromagnetic forces, which is applied to the rotor of i-th electric motor from its stator, i = 1, 2. Following [2, 11, 18], we assume that Ni = ci wi − bi φ˙ i , ci > 0, bi > 0, |wi | ≤ 1.

(30.10)

Here, wi denotes the voltage applied to i-th electric motor. The motion of the system is described by Lagrange equations d dt



∂T ∂ φ˙ i

 −

∂T = Ni , i = 1, 2. ∂φi

These equations and formulas (30.8)–(30.10) imply that (M1 L 21 + J1 + M2 L 22 )φ¨ 1 + J2 (φ¨ 1 + φ¨ 2 ) = −(M1 L 1 + M2 L 2 )g sin φ1 + c1 w1 − b1 φ˙ 1 ,

(30.11)

J2 (φ¨ 1 + φ¨2 ) = c2 w2 − b2 φ˙ 2 .

(30.12)

We assume that it is possible to turn off the first electric motor at an unknown time moment τ ∈ [t0 , p]. To fix the breakdown, we need time σ > 0. After the time τ + σ , the dynamics of the first player is restored to its previous mode. Denote x1 = φ1 , x2 = φ˙ 1 , x3 = φ2 , x4 = φ˙ 2 . Using (30.11) and (30.12), for the mechanical system under consideration, we obtain the system of Eq. (30.1), where

30 Control Problem with Disturbance and Unknown Moment …

⎛

0 ⎜0 A(t) = ⎜ ⎝0 0

1 −α1 0 α1

0 0 0 0

⎞ 0 α2 ⎟ ⎟, 1 ⎠ −α3

341

⎛

⎞ 0 ⎜ γ (t, τ ) ⎟ ⎟, B(t, τ ) = ⎜ ⎝ ⎠ 0 −γ (t, τ )

⎞ ⎞ ⎛ 0 0 ⎜ −ε sin x1 ⎟ ⎜ −δ1 w2 ⎟ ⎟ ⎟. ⎜ ξ =⎜ ⎠ ⎝ 0 ⎠, η = ⎝ 0 δ2 w2 ε sin x1

(30.13)

⎛

(30.14)

Here, αi =

bi b2 c2 c2 , i = 1, 2; α3 = α2 + , δ1 = , δ2 = δ1 + , J J2 J J g ε = (M1 L 1 + M2 L 2 ) , J

γ (t, τ ) =

J = M1 L 21 + M2 L 22 ;

c1 for t0 ≤ t < τ, γ (t, τ ) = 0 for τ ≤ t ≤ min(τ + σ ; p) and J γ (t, τ ) =

c1 for min(τ + σ ; p) < t ≤ p. J

Following [3], we take nonlinear addend η (30.14) in motion equations of the system as disturbance. The quality criterion of controls w1 and w2 is |x1 ( p) − G| → min, where G is the given number. We considered similarly example in [6] as antagonistic differential game, where the first player controls voltage applied to the rotor of the first electric motor, and the second player controls voltage applied to the rotor of the second electric motor. Moreover, in example [6] we obtain same matrix A(t), though denotations αi > 0, i = 1, 3 have other means. Using results of this paper, write down solution of Cauchy problem (30.2) with initial condition ψ1 ( p) = 1, ψ2 ( p) = ψ3 ( p) = ψ4 ( p) = 0, where matrix A(t) is determined by formula (30.13): ψ1 (t) = 1, ψ2 (t) = β1 a1 e−( p−t)λ1 + β2 a1 e−( p−t)λ2 −

a3 , ψ3 (t) = 0, a1 (a3 − a2 )

ψ4 (t) = β1 (a1 − λ1 )e−( p−t)λ1 + β2 (a1 − λ2 )e−( p−t)λ2 −

a2 . a1 (a3 − a2 )

(30.15)

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Here, we denote λ1,2 = β1 =

(a1 + a3 ) ±

 (a1 − a3 )2 + 4a1 a2 , 2

a3 a3 λ2 λ1 1 1 , β2 = . − − 2 2 λ2 − λ1 a (a3 − a2 ) (λ2 − λ1 )a1 λ1 − λ2 a (a3 − a2 ) (λ1 − λ2 )a1 1 1

Formulas (30.14) imply that Q and M are symmetric sets in this example. From here we obtain that β− (t) = −β+ (t), c− (t) = −c+ (t) for all t ≤ p.

(30.16)

Therefore, a(t, τ ) = b(t, τ ) + maxξ, ψ(t); β(t) = maxη, ψ(t). ξ ∈M

η∈Q

These formulas, (30.3), (30.13), and (30.14), imply that a(t, τ ) = γ (t, τ ) |ψ4 (t) − ψ2 (t)| + |δ2 ψ4 (t) − δ1 ψ2 (t)| , β(t) = ε |ψ4 (t) − ψ2 (t)| .

(30.17) Using (30.5) and (30.16), we rewrite variable z as follows: z = ψ(t), x − G. This equality and (30.15) imply that z = x1 + (β1 a1 x2 + β1 (a1 − λ1 )x4 )e−( p−t)λ1 + +(β2 a1 x2 + β2 (a1 − λ2 )x4 )e−( p−t)λ2 −

a3 x 2 + a2 x 4 − G. a1 (a3 − a2 )

By Theorem 30.1, optimal control is u = sign z. We assume that sign 0 = 1. Using this, (30.4) and (30.16), we obtain in this example B(t, τ )w + ξ, ψ(t) = −a(t, τ )sign z. Substitute into this equality B(t, τ ), ξ and a(t, τ ) from formulas (30.13), (30.14), and (30.17), respectively: (ψ2 (t) − ψ4 (t))γ (t, τ )w1 + (−δ1 ψ2 (t) + δ2 ψ4 (t))w2 = = −(γ (t, τ ) |ψ4 (t) − ψ2 (t)| + |δ2 ψ4 (t) − δ1 ψ2 (t)|)sign z.

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From here, we find the controls w1 and w2 in this example: w1 = −sign (ψ2 (t) − ψ4 (t))sign z, w2 = −sign (δ2 ψ4 (t) − δ1 ψ2 (t))sign z.

30.5 Conclusion In present paper, we consider linear control problem with an uncontrolled disturbance. The end time of the control process is given. The payoff is the absolute value of linear function at the end time. It is believed that one breakdown is possible, which leads to a change in the dynamics of the controlled process. The time of occurrence of breakdown is not known in advance. Therefore, the moment of breakdown is included in the disturbance. Using linear change of variables, we reduce the original problem to the single-type control problem with disturbance. In this problem, we obtain necessary and sufficient conditions of control optimality. As an example, the problem of controlling a rod rigidly attached to an electric motor rotor is consider. A rotary flywheel is attached to the other end of the rod. The controls are the voltages applied to the rotor and flywheel. In the process of control, a breakdown of the electric motor is possible. It takes time σ to fix the breakdown. As disturbance, we take the nonlinear addend in the Lagrange equations. The goal of choosing controls is to minimize the modulus of the deviation of the angle formed by the rod and the vertical axis from the given value at end time moment. In the example, we find optimal voltages. Acknowledgements This work is supported by the Russian Science Foundation under grant no. 19-11-00105.

References 1. Andrievsky, B.R.: Global stabilization of the unstable reaction-wheel pendulum. Control Big Syst. 24, 258–280 (2009) (in Russian) 2. Beznos, A.V., Grishin, A.A., Lenskiy, A.V., Okhozimskiy, D.E., Formalskiy, A.M.: The control of pendulum using flywheel. In: Workshop on Theoretical and Applied Mechanics, pp. 170– 195. Publishing of Moscow State University, Moscow (2009) (in Russian) 3. Chernous’ko, F.L.: Decomposition and synthesis of control in nonlinear dynamical systems. Proc. Steklov Inst. Math. 211, 414–428 (1995) 4. Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit. Control and Optimization. Wiley, New York (1965) 5. Izmest’ev, I.V., Ukhobotov, V.I.: On a linear control problem under interference with a payoff depending on the modulus of a linear function and an integral. In: 2018 IX International Conference on Optimization and Applications (OPTIMA 2018) (Supplementary Volume), DEStech, pp. 163–173. DEStech Publications, Lancaster (2019). https://doi.org/10.12783/ dtcse/optim2018/27930

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6. Izmest’ev, I.V., Ukhobotov, V.I.: On a single-type differential game with a non-convex terminal set. In: Khachay, M., et al. (eds.) MOTOR 2019, LNCS, vol. 11548, pp. 595–606. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-030-22629-9_42 7. Kolmogorov, A.N., Fomin, S.V.: Elementy teorii funktsii i funktsional’nogo analiza (Elements of the theory of functions and functional analysis). Nauka Publ, Moscow (1972) 8. Krasovskii, N.N., Subbotin, A.I.: Pozitsionnye differentsial’nye igry (Positional differential games). Nauka Publ, Moscow (1974) 9. Krasovskii, N.N.: Upravlenie dinamicheskoi sistemoi (Control of a dynamical system). Nauka Publ, Moscow (1985) 10. Kudryavtsev, L.D.: Kurs matematicheskogo analiza (A course of mathematical analysis), vol. 1. Vysshaya shkola Publ, Moscow (1981) 11. Matviychuk, A.R., Ukhobotov, V.I., Ushakov, A.V., Ushakov, V.N.: The approach problem of a nonlinear controlled system in a finite time interval. J. Appl. Math. Mech. 81(2), 114–128 (2017). https://doi.org/10.1016/j.jappmathmech.2017.08.005 12. Nikol’skii, M.S.: The crossing problem with possible engine shutoff. Differ. Equ. 29(11), 1681–1684 (1993) 13. Pontryagin, L.S.: Linear differential games of pursuit. Math. USSR-Sbornik 40(3), 285–303 (1981). https://doi.org/10.1070/SM1981v040n03ABEH001815 14. Ukhobotov, V.I.: Domain of indifference in single-type differential games of retention in bounded time interval. J. Appl. Math. Mech. 58(6), 997–1002 (1994). https://doi.org/10.1016/ 0021-8928(94)90115-5 15. Ukhobotov, V.I.: Synthesis of control in single-type differential games with fixed time. Bull. Chelyabinsk Univ. 1, 178–184 (1996) (in Russian) 16. Ukhobotov, V.I.: Metod odnomernogo proektirovaniya v lineinykh differentsial’nykh igrakh s integral’nymi ogranicheniyami: uchebnoe posobie (Method of one-dimensional design in linear differential games with integral constraints: study guide). Publishing of Chelyabinsk State University, Chelyabinsk, Russia (2005) 17. Ukhobotov, V.I.: The same type of differential games with convex purpose. Proc. Inst. Math. Mech. Ural Branch Russ. Acad. Sci. 16(5), 196–204 (2010) (in Russian) 18. Ushakov, V.N., Ukhobotov, V.I., Ushakov, A.V., Parshikov, G.V.: On solution of control problems for nonlinear systems on finite time interval. IFAC-Pap. OnLine 49(18), 380–385 (2016). https://doi.org/10.1070/10.1016/j.ifacol.2016.10.195

Chapter 31

On an Impulse Differential Game with Mixed Payoff Igor’ V. Izmest’ev

Abstract This article considers a single-type differential game with a given duration. Reachable domains of players are balls in n-dimensional space. Control of the first player has an impulse constraint. The first player forms his control using a limited stock of resources. Control of the second player has geometrical constraints. The first player seeks to minimize the payoff, which is defined as the weighted sum of the phase vector norm at the end time and the amount of resource spent by the first player. The goal of the second player is to maximize this payoff. Under some assumptions, the price of the game is found for the problem under consideration. A solution of an example illustrating the theory is given.

31.1 Introduction Problems of controlling mechanical systems of variable composition, in which a finite amount of reaction mass may separate at certain times, can be reduced to impulse control problems [5]. The trajectories of systems with impulse control can be discontinuous, which complicates their analysis. If a mechanical system is affected by uncontrollable forces given only by known ranges of their possible values, then the control problem can be considered within the theory of differential games. Krasovskii (see [2]) proposed a method for researching pursuit game problems, which is based on the principle of absorption of reachable domains. A possible application of this method to impulse game problems was considered, for example, in [3, 4]. In [8], the alternative theorem for impulse differential games was proved under the assumption that the target coordinates of the phase vector change continuously. Impulse control problems under incomplete information on the phase state were studied in [6, 7]. In [1, 9], linear differential games of pursuit were considered in which the impulse controls are expressed in terms of the Dirac delta-function. In I. V. Izmest’ev (B) Chelyabinsk State University, Br. Kashirinykh Str. 129, 454001 Chelyabinsk, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_31

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[11–13], control adjustment procedures were applied in the construction of impulse controls of the players so that the control is constructed at each time of correction as a solution to a certain moment problem. An important class of differential games includes problems where the right-hand side of the equations contains only the sum of the controls of the first and second players with values belonging to balls with time-dependent radii. In [10], a differential game of this class with an impulse constraint on the control of the first player and a geometrical constraint on the control of the second player, in which the terminal set is a ball, was considered. In [14, 15], the terminal set of the game is determined by the condition of belonging to the norm of a phase vector to a segment with positive ends. In these papers, a set defined by this condition is called a ring. Article [16] considers a single-type differential game in which the first player chooses both an impulse control and a control subject to a geometric constraint, and the terminal set is a ball. This article considers the modification of the problem [10] in which the payoff is defined as the weighted sum of the phase vector norm at the end time and the amount of resource spent by the first player. The goal of the first player is to minimize the payoff. The goal of the second player is to maximize the payoff. Under some assumptions, the price of the game is found for the problem under consideration. As an example, we consider a differential game in which the first player controls a point of variable composition, choosing a reactive force at each time moment. The second player controls the point with a limited velocity. The first player seeks to minimize the payoff, which is defined as the weighted sum of the distance between the players at a fixed time moment and the reaction mass spent by the first player. The goal of the second player is the opposite.

31.2 Problem Statement Consider the differential game ˙ z˙ = −a(t)φ(t)u + b(t)v, t ≤ p, z ∈ Rn .

(31.1)

Here a : (−∞, p] → R+ is a continuous function and b : (−∞, p] → R+ is a function integrable on any interval of the half-line (−∞, p]. The set of non-negative real numbers is denoted by R+ . The control of the first player is a pair of functions [12, p. 74] φ(t) ∈ R and u(t, z) ∈ Rn . An impulse constraint is imposed on the choice of the function φ(t) [5] t μ(t) = μ(t0 ) −

|dφ(r )| ≥ 0, t0 ≤ t ≤ p, t0

(31.2)

31 On an Impulse Differential Game with Mixed Payoff

347

where t0 < p is the initial time and μ(t0 ) ≥ 0 is the initial amount of resource, which can be used by the first player to form the function φ(t). Arbitrary function u : (−∞, p] × Rn → Rn satisfies the geometric constraints u(t, z) = 1. Here  ·  is the norm in Rn . The admissible control is called the control, in which φ(t) satisfies (31.2). The control of the second player is an arbitrary function v : (−∞, p] × Rn → Rn satisfying the constraint v(t, z) ≤ 1. When choosing the function φ(t), the first player can correct it at certain times [12, p. 74]. The correction is as follows. The player chooses the times of corrections t0 = τ0 < τ1 < · · · < τq < p. At a time τi , knowing the realized values z(τi ) and μ(τi ) ≥ 0 the first player chooses an absolutely continuous nondecreasing function φi : [τi , τi+1 ] → R and a number i ≥ 0 such that t μ(t) = μ(τi ) − i −

φ˙ i (r )dr ≥ 0, τi < t ≤ τi+1 .

τi

The motion of system (31.1) generated by the chosen controls is defined on the segment [τi , τi+1 ] by means of polygonal lines [12, p. 75]. Take a partition ω with diameter d(ω): ω : τi = t (0) < t (1) < · · · < t (k+1) = τi+1 , d(ω) = max (t ( j+1) − t ( j) ). 0≤ j≤k

We construct the polygonal line z ω (t (0) ) = z(τi ) − i a(τi )u(τi , z(τi )), ⎛ ⎜ z ω (t) = z ω (t ( j) ) − ⎝

t

⎞

⎛

⎟ ⎜ a(r )φ˙ i (r )dr ⎠ u(t ( j) , z(t ( j) )) + ⎝

t ( j)

t

⎞ ⎟ b(r )dr ⎠ v(t ( j) , z(t ( j) )).

t ( j)

for t ( j) ≤ t ≤ t ( j+1) , j = 0, k. See more details in [16]. Define payoff as follows p z( p)α +

|dφ(r )| → min max . φ,u

v

(31.3)

t0

Here, α > 0 is a weight coefficient; control φ(t), u(t, z) is admissible. Definition 31.1 The solution of the problem (31.1), (31.3) is called the admissible control of the first player φ∗ (t), u ∗ (t, z) and the number G 0 such that (1) for any control v(t, z) of the second player and for any motion z(t) with initial condition z(t0 ) = z 0 , which corresponds to φ∗ (t), u ∗ (t, z), and v(t, z), inequality

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p z( p)α +

|dφ∗ (r )| ≤ G 0 t0

holds; (2) for any admissible control φ(t), u(t, z) and for any number G < G 0 there exists control of the second player v(t, z) such that for any motion z(t) with initial condition z(t0 ) = z 0 generated by φ(t), u(t, z) and v(t, z), inequality p z( p)α +

|dφ(r )| > G t0

holds.

31.3 Main Result Fix ε ≥ 0, μ(t0 ) ≥ 0 and consider differential game (31.1) with terminal condition z( p) ≤ ε and an impulse constraint (31.2). Necessary and sufficient conditions of termination in this problem can be written as [10] f (ε) ≤ μ(t0 ),

(31.4)

where ⎞ ⎛ p 1 ⎝ f (ε) = z(t0  + b(r )dr − ε⎠ for t (ε) ≤ t0 ≤ p, m(t0 ) t0

z(t0  + f (ε) = m(t0 )

t (ε) t0

b(r ) dr for t0 < t (ε). m(r )

Here m(t) = max a(r ), t (ε) = inf t≤r ≤ p

⎧ ⎨ ⎩

p t≤p: t

⎫ ⎬

b(r )dr ≤ ε . ⎭

Further, we assume that m(t) > 0 for t < p. Note that function m(t) is continuous and satisfies the monotonicity condition: m(t1 ) ≤ m(t2 ) for t2 ≤ t1 ≤ p. Note also that function t (ε) satisfies the monotonicity condition: t (ε1 ) ≤ t (ε2 ) for 0 ≤ ε2 ≤ ε1 . Consider problem

31 On an Impulse Differential Game with Mixed Payoff

αε + μ → min,

349

f (ε) ≤ μ, 0 ≤ ε, 0 ≤ μ ≤ μ(t0 ).

ε, μ

(31.5)

Theorem 31.1 Let ε0 ≥ 0 and μ0 ≥ 0 are solution of the problem (31.5). Then G 0 = ε0 α + μ0 . Proof Numbers ε0 and μ0 satisfy inequality (31.4). Because there exists control φ∗ (t), u ∗ (t, z) such that p |dφ∗ (t)| ≤ μ0 , t0

which guarantees the fulfillment of the inequality z( p) ≤ ε0 for any control v(t, z) ≤ 1 and for any realized motion z(t). Thus, p z( p)α +

|dφ∗ (r )| ≤ ε0 α + μ0 = G 0 . t0

Assume that there exists number G < G 0 and admissible control φ(t), u(t, z), which guarantees fulfillment of inequality p z( p)α +

|dφ(r )| ≤ G t0

for any function v(t, z) ≤ 1 and for any realized motion z(t). Then this admissible control guarantees the fulfillment of inequality ⎛ z( p) ≤

1⎝ G− α

p

⎞ |dφ(r )|⎠ = ε

(31.6)

t0

for any function v(t, z) ≤ 1 and for any realized motion z(t). Therefore, these ε ≥ 0 and p μ = |dφ(r )| t0

satisfy inequality (31.4) and, therefore, the constraints in problem (31.5). Because G 0 ≤ εα + μ. From this and from the right-hand side of (31.6) we obtain a contradiction G 0 ≤ G.

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Remark 31.1 Control φ∗ (t), u ∗ (t, z) of the first player in the original problem is the optimal control of the first player in differential game (31.1) with terminal condition z( p) ≤ ε0 and an impulse constraint (31.2) with μ(t0 ) = μ0 (see [12, pp. 76–84]). Conjecture 31.1 Function b(t) is continuous for t ∈ [t0 , p]. Conjecture 31.2 Function b(t) > 0 for t ∈ [t0 , p]. Further, we assume that Conjectures 31.1 and 31.2 are satisfied. Definition of t (ε) implies equality p b(r )dr = ε. t (ε)

We differentiate both sides of this equality with respect to ε. We obtain the following: t˙(ε) = −

1 . b(t (ε))

Using this equality, we differentiate f (ε): ⎧ 1 ⎪ ⎨− for t (ε) ≤ t0 ≤ p, m(t 0) f˙(ε) = 1 ⎪ ⎩− for t0 ≤ t (ε). m(t (ε)) Using the monotonicity conditions of functions t (ε) and m(t), obtain that f˙(ε) is a nondecreasing function with respect to ε. Therefore, the function f (ε) is convex. Thus, the problem (31.5) is a convex programming problem. For μ(t0 ) > 0 it can be shown that the Slater regularity condition is satisfied in this problem. Using the Lagrange multiplier method, the following solutions found in problem (31.5) for various values of α > 0 and μ(t0 ) ≥ 0. Case 1 Let μ(t0 ) = 0, then p ε0 = z(t0 ) +

b(r )dr, μ0 = 0. t0

Case 2 Let 0 < μ(t0 ) ≤

z(t0 ) . m(t0 )

Case 2.1 Let m(t0 )α < 1, then solution ε0 , μ0 is the same as in Case 1. Case 2.2 Let m(t0 )α = 1, then (ε0 , μ0 ) is any point from segment with ends

31 On an Impulse Differential Game with Mixed Payoff

⎛ ⎝z(t0 ) +

p

⎞

⎛

b(r )dr, 0⎠ and ⎝z(t0 ) +

t0

351

p

⎞ b(r )dr − μ(t0 )m(t0 ), μ(t0 )⎠ .

t0

Case 2.3 Let m(t0 )α > 1, then p ε0 = z(t0 ) +

b(r )dr − μ(t0 )m(t0 ), μ0 = μ(t0 ). t0

Case 3 Let z(t0 ) z(t0 ) < μ(t0 ) < + m(t0 ) m(t0 )

p t0

b(r ) dr. m(r )

Denote by ε∗ the solution of the equation z(t0 ) + m(t0 )

t (ε) t0

b(r ) dr = μ(t0 ). m(r )

Case 3.1 Let m(t0 )α < 1, then solution ε0 , μ0 is the same as in Case 1. Case 3.2 Let m(t0 )α = 1, then (ε0 , μ0 ) is any point from segment with ends ⎛ ⎝z(t0 ) +

p t0

Case 3.3 Let

⎞

⎛ p ⎞  z(t0 ) ⎠ b(r )dr, 0⎠ and ⎝ b(r )dr, . m(t0 ) t0

1 1 0 is a weight coefficient; control φ(t), u(t, z) is admissible. After the change of variables z(t) = y(t) − x1 (t) − ( p − t)x2 (t), the problem under consideration takes the form ˙ z˙ = −( p − t)φ(t)u + bv, t ≤ p. Since z( p) = y( p) − x1 ( p), the payoff (31.7) takes the form (31.3). Note also that in example ε t (ε) = p − . b

(31.8)

31 On an Impulse Differential Game with Mixed Payoff

353

Using the results of the previous section, in this example, for various values of α > 0 and μ(t0 ) ≥ 0, the following solutions were calculated. Case 1 Let μ(t0 ) = 0 then ε0 = z(t0 ) + ( p − t0 )b, μ0 = 0. Case 2 Let 0 < μ(t0 ) ≤

z(t0 ) . p − t0

Case 2.1 Let ( p − t0 )α < 1, then solution ε0 , μ0 is the same as in Case 1. Case 2.2 Let ( p − t0 )α = 1, then (ε0 , μ0 ) is any point from segment with ends (z(t0 ) + ( p − t0 )b, 0) and (z(t0 ) + (b − μ(t0 ))( p − t0 ), μ(t0 )) . Case 2.3 Let ( p − t0 )α > 1, then ε0 = z(t0 ) + (b − μ(t0 ))( p − t0 ), μ0 = μ(t0 ). Case 3 Let

z(t0 ) < μ(t0 ). p − t0

Using (31.8), compute ε∗ as solution of the equation z(t0 ) + p − t0

t (ε) t0

z(t0 ) b ( p − t0 )b dr = = μ(t0 ). + b ln p−r p − t0 ε

Thus, we obtain   z(t0 ) μ(t0 ) + ( p − t0 )b. ε∗ = exp − b ( p − t0 )b Case 3.1 Let ( p − t0 )α < 1, then solution ε0 , μ0 is the same as in Case 1. Case 3.2 Let ( p − t0 )α = 1, then (ε0 , μ0 ) is any point from segment with ends   z(t0 ) . ( p − t0 )b, (z(t0 ) + ( p − t0 )b, 0) and p − t0 Case 3.3 Let

1 b 0 is the vector of main production factors, and Rn>0 is the set of real valued n-dimensional vectors with positive components, i.e., Rn>0 := {x ∈ Rn , s.t. xi > 0, i = 1, 2, . . . , n}. and the vector-function The functional matrix F(x) = { f i j (x)}i,n,m j=1 , n are twice continuously differentiable functions. Vector u = G(x) = {gi (x)}i=1 (u 1 , . . . , u m ) stands for the regulating parameter (or control). The quality of the control process is estimated by the functional of the form 

+∞

J (·) =

e−ρt ln c(x(t), u(t))dt,

(32.2)

0

where c(x, u) is determined by the equality c(t) =

m  i=1

(1 − u i (t) − wi (x(t))) f (x(t)).

(32.3)

32 Structure of a Stabilizer for the Hamiltonian Systems

359

Here f (x) and wi (x) (i = 1, . . . , m) are twice continuously differentiable functions. In the economic growth models, function f (x) is called production function. Due to the structure of the quality functional (32.2), we impose an additional restriction of the controls u i in (32.3), specifically, 0
0 and nonzero conjugate components ψi∗ = 0, i ∈ {1, . . . , n}. Taking into account Assumption 1, we linearize the system (32.8) at the neighborhood Oδ∗ of the steady state P ∗ and obtain the following system:

˙ ,  x = A x + Bψ

 ˙  , ψ = C x + ρEn − A ψ

 x (t) = x(t) − x ∗ (t) = ψ(t) − ψ ∗ ψ

(32.9)

using the Jacoby matrix

J∗ =





  ⎞ ∂ 2H x ∗, ψ ∗ ∂ 2H x ∗, ψ ∗  ⎜ ⎟ ∂ψ∂ x ∂ψ 2

 ⎟ := ⎜ ∗ ∗ ⎠ 2 ⎝ ∂ 2 H x ∗ , ψ ∗  ∂ H x ,ψ − ρEn − 2 ∂x ∂ x∂ψ (32.10) ⎛

A B C ρEn − A

Next section is devoted to the stabilizability conditions of the Hamiltonian system (32.8) that are based on the properties of the Jacobian (32.10).

32.4 Stabilizability Conditions The stabilization problem of the linearized dynamics (32.9) can be reduced to the search of a such matrix X that establishes the linear relation between phase and  = X conjugate variables ψ x at the vicinity Oδ∗ and makes the following system asymptotically stable (see [9])  = X  x˙ = (A − B X ) x, ψ x.

(32.11)

In order to prove the existence of the matrix X , we introduce an auxiliary linear system

32 Structure of a Stabilizer for the Hamiltonian Systems

⎧  ρ  ⎪ x e−ρ/2t ⎨ ξ˙ = A − En ξ + Bz, ξ =  2 ,  ρ  ⎪  e−ρ/2t ⎩ z˙ = Cξ − A − En z, z = ψ 2

361

(32.12)

and suppose that eigenvalues of the Jacobi matrix M of the system (32.12) satisfy the condition ⎛ ⎞ ρ A − En B ρ 2 ⎠. |Re (λ(M))| > , where M = ⎝ (32.13) ρ 2 En − A C 2 As it is shown in [17], the Jacobian J ∗ and matrix M are related by the equality J∗ = M +

ρ E2n . 2

(32.14)

Remark 32.1 Condition (32.13) is very important for the existence of the desired matrix X . However, in many models this condition is satisfied for almost all values of the model parameters. The corresponding sensitivity analysis is provided, for example, in [12, 15]. The following theorem establishes the existence of the matrix X . Theorem 32.1 Matrix X stabilizing dynamics (32.9) does exist and can be found as a solution of the following matrix Riccati equation:   ρ  ρ En − A T X − X B X = 0, C − X A − En + 2 2

(32.15)

if the condition (32.13) holds. Proof Solution of the Riccati equation (32.15) is constructed by the eigenvectors of the matrix M that is a Hamiltonian matrix, whose spectrum is symmetric with respect to the imaginary axis (see [10]). Moreover, due to the condition (32.13), matrix M does not have pure imaginary eigenvalues. Solution of the matrix Riccati equation has the form (see, for example, [9, 10]) X = V2 V1−1 ,

(32.16)

where V = {v1 , v2 , . . . , vn } = (V1 , V2 ) ∈ R2n×n are eigenvectors corresponding to the eigenvalues of the matrix M with negative real parts. Finally, we get z = X ξ and the stabilized system is   ρ ξ˙ (t) = A − En + B X ξ(t), 2    ρ where Re λ A − En + B X < 0. 2

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By the condition (32.13), matrix J ∗ that is equal to (M + ρ/2En ) has the same number of eigenvalues with negative real parts as matrix M. Moreover, solution X of the Riccati equation (32.15) corresponds to the formula (32.16), since eigenvectors of matrices M and J ∗ coincide. Besides, the equalities take place 

 = ze(ρ/2)t = X ξ e(ρ/2)t = X ψ x. Consequently, the stabilized system (32.9) has the structure  x˙ (t) = (A + B X )  x (t), where Re (λ (A + B X )) < 0. Thus, proof is completed. The theorem allows to design such controller u = u(x, ψ(x)) (see (32.3)) for (x, ψ(x)) ∈ Oδ∗ and ψ(x) = ψ ∗ + X (x − x ∗ ), that ensures the asymptotic stability of the Hamiltonian system (32.8) at a steady-state vicinity.

32.5 Nonlinear Stabilizer Structure Under conditions of the Theorem 32.1, there exists a nonlinear stabilizer that can be constructed by the algorithm 1. First of all, we need to find the steady state P ∗ = (x ∗ , ψ ∗ ) (see Assumption 1) of the Hamiltonian system (32.8). 2. Next, we determine the domain where the steady state P ∗ is located. 3. Following the Theorem 32.1, we express the conjugate variables ψ through the phase components x, using matrix X (32.16), namely, ψ(x) = ψ ∗ + X (x − x ∗ ).

(32.17)

4. Finally, we substitute the found relation (32.17) into the formula (32.3) of the optimal control u(x, ψ) corresponding to the steady-state domain. As a results, the nonlinear stabilizer has the form  u(x) := u(x, ψ(x)).

(32.18)

For example, under the condition that steady state belongs to the domain of variable controls, each component of the nonlinear stabilizer has the form  u j (x) := 1 − w j (x) −  j (x, ψ(x)),

j ∈ {1, . . . , m}.

(32.19)

The next proposition proves that the nonlinear stabilizer (32.18) makes the original system (32.1) stable in the first approximation.

32 Structure of a Stabilizer for the Hamiltonian Systems

363

Theorem 32.2 The nonlinear stabilizer of the form (32.18) ensures stability in the first approximation of the original system (32.1). Proof Let us substitute the nonlinear stabilizer (32.18) to the original dynamic equation (32.1) x˙ = F(x) u(x) + G(x). By the definition of the Hamiltonian function H(x, ψ), the right-hand part of the last equation can be rewritten as follows: x˙ =

∂H(x, ψ(x)) . ∂ψ

Next, we derive the Jacobian of the obtained equation calculated for x = x ∗ , using the notations introduced in (32.10) J =







 ∂ 2 H(x ∗ , ψ x ∗ ) ∂ψ x ∗ ∂ 2H x ∗, ψ x ∗ + = A + B X. ∂ψ∂ x ∂ψ 2 ∂x

~

0.52

v1 = v1 + v2

0.51 0.50

~

v2 = i(v1 - v2)

0.49 0.48 0.47 0.46

x3 v3

0.45 0.44 0.43 0.42 7.0 6.5

x2

6.0

2.20

x1

5.5

2.15 2.10

5.0

2.05

4.5 4.0 1.95

2.00

Fig. 32.1 The stabilized solution in the case of the focal type of the steady state

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According to the Theorem 32.1, matrix X is designed in such a way that the eigenvalues of the matrix A + B X =: Jhave negative real parts. It implies the asymptotic stability of the linearized system  x˙ (t) = (A + B X ) x (t). Therefore, the nonlinear regulator (32.18) stabilizes the original system (32.1) in the first approximation. Remark 32.2 The steady state may have the saddle type and the focal type. Nevertheless, in all cases, the nonlinear regulator (32.18) ensures stability for the system (32.1). For example, as it is shown in the paper [13, 18], for the three-factors optimization model of the resource productivity, the phase portrait of the stabilized solution demonstrates the cyclic behavior (see Fig. 32.1). Another example can be found in [16], where for other values of the model parameters the model, mentioned in the paper [13], behaves in a different way and stabilizes at the steady state of the saddle type (see Fig. 32.2).

0.382

v2

0.38

v3

x3

0.378

v1

0.376 0.374 0.372 0.37

0.955

1.82 1.84

0.96 1.86

x2

0.965 1.88 0.97

1.9 1.92

x1

0.975

Fig. 32.2 The stabilized solution in the case of the saddle type of the steady state

32 Structure of a Stabilizer for the Hamiltonian Systems

365

32.6 Conclusion The paper discusses issues devoted to the stabilization of the Hamiltonian systems by means of the nonlinear regulator of the special type. We derive stabilizability conditions using the solution of the Riccati equation constructed for the auxiliary system that can be obtained from the original one using special change of variables. The proposed approach for design of the nonlinear regulator can be applied if two important conditions are satisfied. The first condition is the existence of the unique steady state of the Hamiltonian system. The second condition imposes constraints on the eigenvalues of the Jacobian calculated at the steady state of the Hamiltonian system. The future research presumes the detailed analysis of Jacobi matrices. According to the numerical experiments, the proposed method for stabilizing the Hamiltonian systems is applicable for almost all values of model parameters. Therefore, the condition on eigenvalues of the Jacobian is, probably, satisfied almost everywhere for the optimal control problems considered in the paper. Acknowledgements This work was supported by the Russian Science Foundation (project no. 1911-00105).

References 1. Arrow, K.: Production and Capital. vol. 5. Collected Papers, The Belknap Press of Harvard University Press (1985) 2. Aseev, S.M., Kryazhimskiy, A.V.: The Pontryagin maximum principle and optimal economic growth problems. Proc. Steklov Inst. Math. 257 (2007) 3. Ayres, R., Warr, B.: The Economic Growth Engine: How Energy and Work Drive Material Prosperity. Edward Elgar Publishing, Cheltenham UK (2009) 4. Balder, E.: An existence result for optimal economic growth problems. J. Math. Anal. Appl. 95(1), 195–213 (1983) 5. Grass, D., Caulkins, J., Feichtinger, G., et al.: Optimal Control of Nonlinear Processes. Springer, Berlin (2008) 6. Grossman, G., Helpman, E.: Innovation and Growth in the Global Economy. Press, MIT (1991) 7. Krasovskii, A., Tarasyev, A.: Dynamic optimization of investments in the economic growth models. Autom. Remote Control 68(10), 1765–1777 (2007) 8. Krasovskii, A., Kryazhimskiy, A., Tarasyev, A.: Optimal control design in models of economic growth. Evolutionary Methods for Design, Optimization and Control, pp. 70–75. CIMNE, Barcelona (2008) 9. Ledyaev, Y.: On analytical solutions of matrix Riccati equations. Proc. Steklov Inst. Math. 273(1), 214–228 (2011) 10. Paige, C., Loan, C.V.: A Schur decomposition for Hamiltonian matrices. Linear Algebr. Appl. 41, 11–32 (1981) 11. Pontryagin, L., Boltyanskii, V., et al.: The Mathematical Theory of Optimal Processes. Interscience, New York (1962) 12. Russkikh, O.V., Usova, A.A., Tarasyev, A.M.: Sensitivity of the qualitative behavior of the Hamiltonian trajectories with respect to growth model parameters. In: proceedings: CEUR Workshop Proceedings, vol. 1662, pp. 110–120 (2016)

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13. Tarasyev, A., Usova, A., Wang, W.: Hamiltonian trajectories in a heterogeneous economic growth model for optimization resource productivity. In: IFAC-PapersOnLine, vol. 48, Issue 25, pp. 74–79. ISSN 2405-8963, https://doi.org/10.1016/j.ifacol.2015.11.062 (2015) 14. Tarasyev, A., Usova, A.: Construction of a regulator for the Hamiltonian system in a two-sector economic growth model. Proc. Steklov Inst. Math. 271, 1–21 (2010) 15. Tarasyev, A., Usova, A.: Cyclic behaviour of optimal trajectories in growth models. In: Proceedings of the 10th IFAC Symposium NOLCOS 2016, vol. 49, Issue 18, pp. 1048–1053. CA, USA (2016) 16. Tarasyev, A., Usova, A.: Robust methods for stabilization of Hamiltonian systems in economic growth models. In: IFAC-PapersOnLine, vol. 51, Issue 32, pp. 7–12. ISSN 2405-8963, https:// doi.org/10.1016/j.ifacol.2018.11.344 (2018) 17. Tarasyev, A., Usova, A.: Structure of the Jacobian in economic growth models. In: Proceedings of the 16th IFAC Workshop CAO 2015, vol. 48, Issue 25, pp. 191–196. Germany (2015) 18. Tarasyev, A., Zhu, B.: Optimal proportions in growth trends of resource productivity. In: Proceedings of the 15th IFAC Workshop CAO 12, vol. 45, Issue 25, pp. 182–187. Italy (2012)

Chapter 33

Calculus of Variations in Solutions of Dynamic Reconstruction Problems Nina N. Subbotina

Abstract Dynamic reconstruction problems are studied for controlled systems linear relative to controls. It is assumed that information about inaccurate current measurements of real states of the systems comes at discrete instants. The control generating this motion has to be reconstructed in real time. A new method is suggested to solve this inverse problem. This method relies on necessary optimality conditions in auxiliary variational problems with integral convex–concave cost functional.

33.1 Introduction Inverse problems were always actual and important for the analysis and synthesis of optimal controls in dynamical systems theory (See, [1–6]). We outline the approach for solving dynamic reconstruction (DR) problems which was suggested by Yu. S. Osipov and A. V. Kryazhimskii and developed their coworkers [7, 8] This approach is based on extreme aiming on measurements of the realizing states. These ideas source in the differential game theory developed in N. N. Krasovskii’s school [9]. The new method, suggested by N. N. Subbotina, T. B. Tokmantsev and E. A. Krupennukov [10, 11], is based on solutions of the system of state variables and conjugate variables. This coupled system arises in necessary optimality conditions for auxiliary problems of calculus of variations (CV) with convex–concave Lagrange function.

N. N. Subbotina (B) Krasovskii Institute of Mathematics and Mechanics UB RAS, 16 S. Kovalevskaya Str., Yekaterinburg 620108, Russia e-mail: [email protected] Ural Federal University, 19 Mira Str., Yekaterinburg 620002, Russia © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_33

367

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33.2 Dynamic Reconstruction Problem We consider controlled systems of the form dx(t) = f (t) + G(t, x(t))u(t), t ∈ [0, T ], dt

(33.1)

where x ∈ R n are state variables, parameters u ∈ R m , m ≤ n are controls, constrained by the geometric restrictions: u ∈ U = {u i ∈ [ai− , ai+ ], ai− < ai+ , i = 1, 2, . . . , m}.

(33.2)

Information about current inaccurate measurements y(tk ) of the real trajectory x ∗ (t) of system (33.1)–(33.2) comes in real time at discrete instants tk = t0 + kt; k = 0, . . . , N , t ∈ (0, 0 ], t0 = 0, t N = T . The estimation δ > 0 of inaccuracy is known (33.3) y(tk ) − x ∗ (tk ) ≤ δ, k ∈ 0, N , where δ ∈ (0, δ0 ]. The symbol z denotes the Euclidean norm of the vector z ∈ R n . The inverse problem arises: to reconstruct in real time the control u ∗ (·) generating the real trajectory x ∗ (·). We consider the case when control u ∗ (·) : [0, T ] → U can be a measurable function.

33.2.1 Assumptions It is assumed that A1. For any δ ∈ (0, δ0 ], a piecewise smooth functions y δ () : [0, T ] → R n (namely, the vector-function with coordinates yiδ (t), i ∈ 1, n, which have piecewise continuous second derivatives) can be defined step-by-step up to the current instants tk ≥ t2 , k = 2, . . . , N as an interpolation of measurements y(tk ) (33.3) of the real trajectory x ∗ (t), such that

|

y δ (t) − x ∗ (t) ≤ 2δ, ∀t ∈ [0, T ],

(33.4)

d2 yiδ (t) | ≤ K 1 , i ∈ 1, n, ∀t ∈ [0, T ] \ δ , d2 t

(33.5)

y δ (tk )−y δ (tr ) , T2 k,r ∈{0,N }

where the positive constant K 1 > max

the sets δ have the measure

β δ = β(δ ) and β δ → 0, as δ → 0. A2. Coordinates f i (t) of the vector-function f (t) and coordinates gi, j (t, x), i = 1, . . . , n, j = 1, . . . , m, of the matrix function G(t, x) are defined and continuous in the domain T = [0, T ] × R n , they are non oscillate in t. The functions gi, j (t, x),

33 Calculus of Variations in Solutions of Dynamic Reconstruction Problems

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i ∈ 1, n, j ∈ 1, m, are continuously differentiable in domain (0, T ) × R n and their ∂g j (t,x) ∂gi, j (t,x) partial derivatives i,∂t , , i, l ∈ 1, n, j ∈ 1, m, are extendable on any ∂ xl compact set D ⊂ T . A3. There exist positive numbers δ ∗ , α ∗ d0 and a compact set D0 ⊂ T , containing all measurements (tk , y(tk )), such that δ ∗ ∈ (0, δ0 ], 4δ ∗ + α ∗ < d0 ,

(33.6)

where K 2 > 0 is a constant defined by the set D0 and input data of the control problem (33.1)–(33.2) under assumptions A2, and the for any δ ∈ (0, δ ∗ ] relations δ = {(t, x) ∈ T : x − y δ (t) ≤ d0 , t ∈ [0, T ]} ⊂ D0

(33.7)

are satisfied. A4. The rank of the m × m -matrix {gi, j (t, x)}, i, j ∈ 1, m is equal to m for all (t, x) ∈ D0 . Note that the piecewise continuous interpolations y δ (t) that satisfy (33.4)–(33.5) can be constructed as polynomials which values at nodes tk are defined by measurements y(tk ).

33.2.2 Statement of Dynamic Reconstruction Problem We consider the following dynamic reconstruction problem (DR): After receiving in real time, at the current discrete instants tk ≥ t2 , information about a continuous interpolation y δ (·) : [0, tk ] → R n of the current measurements, we need to reconstruct such the control u δ (·) : [0, tk − t] → U , which generates the trajectory x δ (·) : [0, tk − t] → R n of system (33.1)–(33.2), that, at the end of the reconstruction process, the following relations hold: (t, x δ (t)) ∈ D0 , ∀t ∈ [0, T ], x δ (·) − x ∗ (·)C =

δ

u (·) − u

∗

x δ (τ ) − x ∗ (τ ) → 0,

(33.9)

u δ (τ ) − u ∗ (τ )2 dτ → 0,

(33.10)

max

τ ∈[0,T −t] T−t

(·)2L 2

=

(33.8)

0

as δ → 0, t → 0. Here the symbol x(·)C denotes the norm in the space C of continuous vectorfunctions x(·) : [0, T − t] → R n ; the notion u(·) L 2 means the norm in the space L 2 of vector-functions u(·) : [0, T − t] → R n .

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As follows from A4., the admissible control u ∗ (·), which generates the real trajectory x ∗ (·) of system (33.1)–(33.2), is unique.

33.3 Solution of Dynamic Reconstruction Problem To solve the dynamic reconstruction problem (DR) (33.8)–(33.10), we consider the following variational problems.

33.3.1 Auxiliary Constructions Consider the interval [tk−2 , tk ], k = 2, . . . , N , a fixed parameter δ, and a continuous interpolation y δ (τ ), τ ∈ [tk−2 , tk ]. We introduce the following integral discrepancy cost functional of the form: tk   α2 x(τ ˆ ) − y δ (τ )2 − v(τ )2 dτ, Itk−2 ,xk−2 (u(·)) = 2 2

(33.11)

tk−2

where α > 0 is a small parameter of regularization, the initial state (tk−2 , xk−2 ) ∈ D0 . The measurable vector-function v(·)=(v1 (·), . . . vn (·)) : [tk−2 , tk ] → R n is an admissible control in the following modified controlled system dx(τ ˆ ) ˆ = f (t) + G(τ, x(τ ˆ ))v(τ ), τ ∈ [tk−2 , tk ], dτ

(33.12)

ˆ where xˆ ∈ R n are state variables, controls v ∈ R n . The n × n-matrix G(τ, x) ˆ = ˆ i ∈ 1, n, j ∈ 1, n has the following structure: {gˆ i, j (τ, x)},

ˆ = gˆ i, j (τ, x)

⎧ gi, j (τ, x), ˆ i ∈ 1, n, j ∈ 1, m, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0, i ∈ 1, m, j ∈ m + 1, n, ⎪ ⎪ 0, i ∈ m + 1, n, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ γ , i ∈ m + 1, n,

(33.13) j ∈ m + 1, n, i = j j ∈ m + 1, n, i = j.

The function x(τ ˆ ), τ ∈ [tk−2 , tk ] in representation of the cost functional (33.11) means the solution of system (33.12)–(33.13) which was generated in the modified controlled system under the admissible control v(·) on the interval [tk−2 , tk ], and x(t ˆ k−2 ) = xk−2 . The symbol γ > 0 denotes a small parameter of approximation.

33 Calculus of Variations in Solutions of Dynamic Reconstruction Problems

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33.3.2 Calculus of Variations Problem Now, for fixed δ ∈ (0, δ0 ], α > 0, we solve the following calculus of variations problem (CV). We have to minimize the cost functional (33.11) over all continuously differentiable control functions v(·) : [tk−2 , tk ] → R n and continuously differentiable functions x(·) ˆ : [tk−2 , tk ] → R n —corresponding solutions of system (33.12)–(33.13) which satisfy the boundary conditions x(t ˆ k−2 ) = xk−2 = y δ (tk−2 ),

dy δ (tk−2 ) dx(t ˆ k−2 ) = . dt dt

(33.14)

Note, that the necessary optimality conditions in the auxiliary CV problems will ensure the restoration of the control and states of system (33.12)–(33.13)on each time interval [tk−2 , tk−1 ] in the proposed below algorithm.

33.3.3 Necessary Optimality Conditions Necessary optimality conditions for x(·) in CV (33.11)–(33.13) have the form dx(t) ˆ ˆ x)v = f (t) + G(t, ˆ 0, dt (33.15) dˆs (t)  ˆ x)v ˆ x)D = (x(t) ˆ − y δ (t)) − s  Dx G(t, ˆ 0 − s  G(t, ˆ x v 0 + α 2 v 0 Dx v 0 , dt (33.16) ˆ sˆ ) = − v0 = v0 (t, x,

1 ˆ ˆ s. G (t, x)ˆ α2

(33.17)

t ∈ [tk−2 , tk ], k = 2, . . . , N , and the following boundary conditions hold (in accordance with (33.12), (33.14)): (33.18) x(t ˆ k−2 ) = y δ (tk−2 ), δ dy (tk−2 )

− f (tk−2 ) . (33.19) sˆ (tk−2 ) = −α 2 Qˆ −1 (tk−2 , y δ (tk−2 )) dt  ˆ x) ˆ x) ˆ x), ˆ x) Here Q(t, x) ˆ = G(t, ˆ Gˆ  (t, x). ˆ Dx G(t, ˆ = Dx1 G(t, ˆ . . . , Dxn G(t, ˆ ,  ˆ sˆ ), . . . , Dxn v0 (t, x, ˆ sˆ ) , Note, that assumptions A1.–A3. imply Dx v0 = Dx1 v0 (t, x, existence of the unique solution x(·), ˆ sˆ (·) for system (33.15)–(33.19).

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33.4 An Algorithm for Constructing DR Solution So, following to the dynamic arrival of measurements, we construct interpolations y δ (t) on each time interval [tk−2 , tk ], k = 2, . . . , N as smooth functions. We will consider the value y δ (tk−1 ) instead of the measurement y(tk−1 ) for the next step on the time interval [tk−1 , tk+1 ] to provide continuity of interpolations on the whole time interval [0, T ]. Let us fix δ > 0, denote z = x − y δ (t), and consider the following system for variables (¯z , s¯ ), which is system (33.15)–(33.19) linearized at point (tk−2 , y δ (tk−2 )): d¯z (t) 1 dy δ (t) = f (tk−2 ) − 2 Q s¯ − ; dt α dt d¯s (t) = z¯ ; dt

(33.20) (33.21)

t ∈ [tk−2 , tk ], k = 2, . . . , N with the boundary conditions: z¯ (tk−2 ) = 0, s¯ (t0 ) = 0, s¯ (tk−2 ) = s¯ k−1 (tk−2 ), ∀ k ≥ 3.

(33.22) (33.23)

Here s¯ k−1 (·) is the solution of system (33.20) on the interval [tk−3 , tk−1 ]; ˆ k−2 , y δ (tk−2 ))Gˆ  (tk−2 , y δ (tk−2 )). Q = Q(tk−2 , y δ (tk−2 )) = G(t dy δ (tk−2 ) 1 = f (tk−2 ) − 2 Q(tk−2 , y δ (tk−2 ))¯s (tk−2 ). dt α

(33.24) (33.25)

Note (see, [12]), that matrixes Q(t, x) (33.24) are symmetrical, and positively defined ξ  Q(t, x)ξ > 0, ∀(t, x) ∈ D0 , ξ ∈ R n , ξ  = 0.

(33.26)

According to the matrixes theory, the representation Q = H  H

(33.27)

is true, where n × n-matrix H  = H −1 , is the diagonal n × n-matrix, which elements λi , i ∈ 1, n, eigenvalues of the matrix Q, are positive numbers. The fundamental 2n × 2n-matrix W (t, t∗ ) of solutions w(t) = (¯z (t), s¯ (t)), t ∈ [tk−2 , tk ], t∗ = tk−2 of the homogeneous system corresponding to nonhomogeneous system (33.20)–(33.25) has the form W¯ (t, t∗ ) = Hˆ −1 (t, t∗ ) Hˆ , where Hˆ =

H 0 0 H

 ,

(33.28)

33 Calculus of Variations in Solutions of Dynamic Reconstruction Problems

373

H is the n × n-matrix from (33.27), 0 is zero n × n-matrix;

(t, t∗ ) =

 1 2 3 4

 .

(33.29)

Here 1 = 4 is the diagonal n × n-matrix, which diagonal elements φii1 have the form √ λi φii1 = cos( (t − t∗ )); α 2 is the diagonal n × n-matrix, which diagonal elements φii2 have the form √ φii2 = −

√ λi λi sin( (t − t∗ )); α α

3 is the diagonal n × n-matrix, which diagonal elements φii3 have the form √ λi α φii3 = √ sin( (t − t∗ )); α λi λi > 0, i ∈ 1, n, from in (33.27).

33.4.1 Estimates We consider new variables z˘ = H z¯ , z˘ = H s¯ , which satisfy the system  1 dy δ (t) d˘z (t) = − 2 ˘s + H f (tk−2 )) − , dt α dt d˘s (t) = z˘ (t), dt

(33.30) (33.31)

and the boundary condition z˘ (tk−2 ) = o, s˘ (tk−2 ) = H s¯ (tk−2 ).

(33.32)

Applying the Cauchy formula for solutions of nonhomogeneous system (33.30)– (33.32) on the interval [tk−2 , tk ], and using (33.28)–(33.29), one can obtain the estimations for all i = 1, . . . , n  λ∗ Q| |˘si (tk−2 )| + r (t, α, γ )[r0 (t) + |˘si (tk−2 )|] ; (33.33) |˘z i (tk−1 )| ≤ H  α α2

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N. N. Subbotina



α Q| |˘si (tk−1 )| ≤ H  |˘si (tk−2 )| + r (t, α, γ )[r0 (t) + |˘si (tk−2 )|] , (33.34) 2 γ α where r (t, α, γ ) =

α 1/2 (t + √ ) , 2 2 γ

t

r0 (t) = K 1 t + H  =

max i, j∈1,n,(t,x)∈D0

λ∗ = λ∗ (γ ) =

(33.35)

β(t) , N

(33.36)

|h i, j (t, x)|, Q = maxi, j∈1,n,(t,x)∈D0 |qi, j (t, x)|

min i∈1,m,(t,x)∈D0

λi (t, x) = γ 2 , λ∗ =

max

i∈1,m,(t,x)∈D0 ,γ ∈[0,γ0 ]

λi (t, x),

ˆ x) = Q(t, ˆ x, γ ), for where λi (t, x) are positive eigenvalues of the matrixes Q(t, γ > 0, (t, x) ∈ D0 . We choose γ 2 = α, α 3 = t, α/γ < 1.

(33.37)

33.4.2 Constructions of Approximations for Control Repeat the process for any time interval [tk−2 , tk ], k = 2, ..., N and get the resulting piecewise continuous feedback vα,δ (t) vα,δ (t) = −

1 ˆ G (tk−2 , y δ (tk−2 ))`s (t), ∀t ∈ [tk−2 , tk−1 ], k = 2, ..., N , α2

(33.38)

which we put to modified system (33.12). The following estimates can be obtained from (33.33)–(33.36): ˆ vα,δ (t) ≤ G where



˘s (t) ˆ · H  4T K 1 t (t + √α ) 1/2 , ≤ G 2 α γα 2 2 λ∗ ˆ = G

max i, j∈1,n,(t,x)∈D0

(33.39)

|gˆ i j (t, x)|,

Control (33.38)generates in modified system (33.12) the solution x(·). ˆ We estimate x(t) ˆ − x(t), ¯ t ∈ [0, T − t] using Gronwall–Bellman lemma and get ¯ ≤ R(t)T e L T , xˆ α,δ (t) − x(t) where

(33.40)

33 Calculus of Variations in Solutions of Dynamic Reconstruction Problems L = Gˆ x  ·

max

t∈[0,T −t]

v α,δ (t) = |Gˆ x v α,δ (·)C , Gˆ x  =

max

(i, j,l)∈1,n,(t,x)∈D0

375 |

∂ gˆi j (t, x) |, ∂ xl

R(t) = ω f (t) + L K 1 (t) + ω(t)vα,δ (·)C ,

(33.41)

where ω f (t) is the modulus of continuity for function f (·), ω(t) is the modulus of continuity for functions gi, j (·), Gˆ t (·) =

max

(i, j)∈1,n,(t,x)∈D0

|

∂ gˆ i j (t, x) |. ∂t

Relations (33.40)and (33.33) imply the estimates xˆ α,δ (·) − y δ (·)C ≤ R(t)T e L T + H  · ˘z C . Hence, the estimate xˆ α,δ (·) − x ∗ (·)C ≤ 2δ + R(t)T e L T + r (t, α, γ ) r (t, α, γ )

+ r (t, α, γ )(K 1 t + β(t) + 4T Q ) H  4T λ∗ α αγ αγ (33.42) implies that the convergence x α,δ (·) − x ∗ (·)C → 0 takes place, as γ → 0,

α → 0,

(33.43)

t → 0 in concordance with (33.37).

33.4.3 Solution of DR We choose [tk−2 , )tk−1 and put u 0 = u 0 (t) = (u 01 (t), . . . , u 0m (t)), vi0 (t, x(t), s(t)) = vα,δ (tk−2 ), y δ (tk−2 ), s˘ (t)) ⎧ 0 u i = vi0 (t, x(t), s(t)), ∀ i ∈ 1, m, v0 (t, x(t), s(t)) ∈ U, ⎪ ⎪ ⎪ ⎪ ⎨ u 0 = u i0∗ = ai+∗ , vi0∗ (t, x(t), s(t)) ≥ ai+∗ , ∃ i ∗ ∈ 1, m, ⎪ ⎪ ⎪ ⎪ ⎩ 0 u j∗ = a −j∗ , u 0j∗ (t, x(t), s(t)) ≤ a −j∗ , ∃ j∗ ∈ 1, m,

(33.44)

and consider u 0 (τ ) as approximations for control u ∗ (τ ) reconstructed with a small delay t on the intervals [tk−2 , tk−1 ], k ∈ 2, N for basic system (33.1)–(33.2). Estimates (33.40) imply that the controls vα,δ (·) (33.38) are restricted in the space of

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continuous functions. The first m coordinates give the vectors u α,δ (t) ∈ U , the last n − m coordinates tend to zero. The sequence vα,δ (·) converges to a restricted control v∗∗ (·) in the space L 2 , and the relations hold ∗

t

x (t) =

ˆ f (τ ) + G(τ, x (τ ))v (τ )dτ =

t0

∗

∗∗

t

f (τ ) + G(τ, x ∗ (τ ))u ∗∗ (τ )dτ

t0

(33.45) for all τ ∈ [t0 , T ]. Here u ∗∗ = (u i∗∗ (τ ) = vi∗∗ (τ ), i ∈ 1, m). We assume that assumption A.3 is true. Hence, the unique control is generating x ∗ (·) is u ∗ (·). So, we can use the relations (33.37) to construct the controls u α,δ (t), which are the first m coordinates of controls vα,δ (·) (33.38) satisfied the relations (33.8)–(33.10). It means that the controls solve DR. We have got the following assertion. Theorem 33.1 If assumptions A.1–A.4 are true for problems DR and CV, then the concordance of parameters δ, t γ and α can be established via (33.37), such that controls u 0 (τ ) of the form (33.44) satisfy the relations (33.8)–(33.10), i.e., they solve DR.

33.5 Comparison with Another Approach The mentioned above method suggested in for solving dynamic control reconstruction problems can be considered as the development and application of N. N. Krasovskii’s extremal aiming to incorrect measurements of real motion. The authors minimize convex regularized discrepancy. The key feature of new method suggested in the paper is the use of convex–concave regularized discrepancy. It provides stable oscillating character of the solutions. The more accurate comparison will be investigated in the nearest future.

33.6 Conclusion In this paper a development of the new approach to solutions of inverse problem is suggested and verified for solutions of control reconstruction problems in real time. Connections of the method with convex–concave variational problems and questions of effectiveness of the approach are subjects of future researches. Acknowledgements This work was supported by the Russian Foundation for Basic Research (Project No. 20-01-00362).

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References 1. Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957) 2. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Nauka, Moscow (1961) 3. Krasovskii, N.N.: Theory of Motion Control. Nauka, Moscow (1968) 4. Leitmann, G. (ed.): Optimization Techniques: With Applications to Aerospace Systems, vol. 5. Academic Press (1962) 5. Letov, A.M.: Dynamics of Flight and Control. Nauka, Moscow (1969) 6. Michel, A.: Deterministic and stochastic optimal control. IEEE Trans. Automat. Control 22(6), 997–998 (1977) 7. Kryazhimskij, A.V., Osipov, YuS: Modelling of a control in a dynamic system. Eng. Cybern. 21(2), 38–47 (1983) 8. Osipov, YuS, Kryazhimskii, A.V., Maksimov, V.I.: Some algorithms for the dynamic reconstruction of inputs. P. Steklov Inst. Math. (2011). https://doi.org/10.1134/S0081543811090082 9. Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, NY (1988) 10. Subbotina, N.N., Tokmantsev, T.B.: The method of characteristics in inverse problems of dynamics. Univers. J. Contr. Automat. NY, Horizon Research Publishing, 1(3), 79–85 (2013) 11. Subbotina, N.N., Tokmantsev, T.B., Krupennikov, E.A.: On the solution of inverse problems of dynamics of linearly controlled systems by the negative discrepancy method. P. Steklov Inst. Math. 291, 253–262 (2015). https://doi.org/10.1134/S0081543815080209 12. Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, New York (2019)

Chapter 34

Control Problems for Set-Valued Motions of Systems with Uncertainty and Nonlinearity Tatiana F. Filippova and Oxana G. Matviychuk

Abstract The control and estimation problems are considered for a nonlinear control system with uncertainty in the initial data and with quadratic nonlinearity in vectors of system velocities. It is assumed also that the values of unknown initial states and admissible controls are constrained by related ellipsoids. Basing on the application of the results and methods of the theory of ellipsoidal calculus applied for estimation of set-valued motions of studied systems, the problems of guaranteed control for the tubes of trajectories of a nonlinear system with uncertainty are investigated. Algorithms for guaranteed moving of the set-valued state of the control system to the smallest neighborhood of a given target set are proposed, the results are illustrated by a model example.

34.1 Introduction The nonlinear problems for dynamical control systems under uncertainty considered in the set-membership framework [25–28, 32] are studied here. The guaranteed simulation accuracy for such models is of great importance in all applications where the related error bounds are used for robust decision-making problems. The approaches and results which are developed in this paper and in the previous work of the authors take into account state estimation errors and noises from disturbances and parametric uncertainties. The studies are also motivated by recent developments in the theory of dynamical systems with impulse controls and fractional derivatives [1, 7, 10], as well as by new results of the model predictive control theory [12, 20]. Examples include numerous applications in physics, cybernetics, biology, economics resource planning, operations scheduling, predictive control, and other areas [2–6, 29, 31, T. F. Filippova (B) · O. G. Matviychuk Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences and Ural Federal University, Yekaterinburg, Russian Federation e-mail: [email protected] O. G. Matviychuk e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Tarasyev et al. (eds.), Stability, Control and Differential Games, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-42831-0_34

379

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T. F. Filippova and O. G. Matviychuk

33, 34]. Further intensive development of the theory and analytical and numerical methods for studying control systems of this class is very important not only for the internal development of the control and optimization theory, but also for applications, including the classes indicated above. The paper develops previous studies [9–11, 14, 16–18] and complements fundamental researches [22–24] and recent achievements in related fields of mathematical control theory [8, 13–15, 19–21, 30]. It should be noted that one of the central places in the control and estimation theory is a reachable set of the control system, which incorporates all structural features of the dynamical control system under study including uncertainty, nonlinearity, etc. As a similar construction, one can also consider the so-called integral funnels in the theory of differential inclusions, generalizing the corresponding notions of controlled differential equations. This approach is also substantially used in this work. It is well known that the shape of reachable sets may be quite complicated. In such cases, the approximation of reachable sets by special canonical geometrical figures is of interest. As such canonical figures, the most natural are ellipsoids, parallelepipeds, polyhedra, and some other special sets. Approximately begun from the 1970 s to the present, a complete theory has been developed for constructing various set-valued estimates of reachable sets mainly for linear control systems with uncertainties and for a number of classes of nonlinear dynamical systems. We note that the indicated results were largely based on the ellipsoidal calculus technique developed in [25, 27] and later on a more developed theory [28]. The approximation of reachable sets by domains of different canonical forms is steel an object of constant interest for researchers, especially considering the approximations by parallelepipeds, parallelotops, polyhedra, and some other canonical figures [21]. In this paper, an approach based on the ideas of ellipsoidal calculus is developed for solving control problems and estimating trajectory tubes of nonlinear controlled dynamic systems with uncertainty in the initial data and quadratic type nonlinearity, with the main emphasis on solving the problem of bringing the set-valued state of a system into a minimal neighborhood of a given target set. Based on the results related to corresponding necessary optimality conditions in the form of the maximum principle, the properties of tubes of optimal motions that solve the problem are investigated. The results of computer modeling are presented, illustrating the proposed methods and algorithms for control and estimation problems under study.

34.2 Main Assumptions and Problem Formulation We formulate in this section the main problem of the paper and also introduce necessary terms and notations.

34 Control Problems for Set-Valued Motions of Systems with Uncertainty …

381

34.2.1 Basic Notations Let IRn mean the n-dimensional Euclidean space, with the inner product x  y of vectors x, y ∈ IRn where a prime is a transpose and with a norm x = (x  x)1/2 . The symbol comp IRn will be used for the variety of all compact subsets A ⊂ IRn and the symbol conv IRn will denote a collection of all compact convex subsets A ⊂ IRn . The notation ρ(l|M) is used for the support function of a set M ∈ conv IRn , that is, ρ(l|M) = max{l  m | m ∈ M}, ∀l ∈ IRn , we denote by d(x, M) the distance from a vector x ∈ IRn to a set M ⊂ IRn , d(x, M) = inf{x − m | m ∈ M}. Denote also by I ∈ IRn×n the identity matrix of related dimension. By B(a, r ) = {x ∈ IRn : x − a ≤ r } we denote the ball in IRn (with a center a ∈ IRn and of a radius r > 0) and denote by E(a, Q) = {x ∈ IRn : (Q −1 (x − a), (x − a)) ≤ 1} the ellipsoid in IRn (with a center a ∈ IRn and with a symmetric positive definite n × n-matrix Q).

34.2.2 Description of the Main Problem Consider the following nonlinear control system under uncertainty conditions x˙ = A(t)x + f (x)d + u(t), x0 ∈ X 0 , t ∈ [t0 , T ].

(34.1)

Here we assume that x, d ∈ IRn , x ≤ K (K > 0), the function f (x) is nonlinear and we assume that it is quadratic in x, that is we have the equality f (x) = x  Bx, where B is a given symmetric and positive definite n × n-matrix. Admissible control function u(t) is assumed to be Lebesgue measurable on [t0 , T ] and the following constraint should be satisfied: u(t) ∈ U for a.e. t ∈ [t0 , T ],

(34.2)

here U is a given set, U ∈ comp IRn . More precisely, we will assume that U = ˆ E(a, ˆ Q). We take the ellipsoid for the initial set X 0 in (34.1), namely, we assume that X 0 = E(a0 , Q 0 ),

(34.3)

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n with a symmetric and positive definite matrix Q 0 ∈ IRn×n  a center a0 ∈ IR .  and with Let the absolutely continuous function x(t) = x t; u(·), x0 be a solution to dynamical system (34.1) with initial state x0 ∈ X 0 and with admissible control u(·).

Definition 34.1 The reachable set X (t) at time t (t0 < t ≤ T ) of system (34.1)– (34.3) is defined as the set  X (t) = x ∈ IRn : ∃ x0 ∈X 0 , ∃ u(·)∈U,   x = x(t) = x t; u(·), x0 .

(34.4)

The problem of exact description of reachable sets (34.4) of control systems of type (34.1)–(34.3) is enough complicated even for linear systems. The various estimation approaches were proposed and studied in many researches [8, 25–28, 32]. The main problem studied here is to find the solution of the optimization problem formulated below using the effective external ellipsoidal estimates (with respect to inclusion of sets) for reachable sets X (t) (t0 < t ≤ T ). However we need to slightly modify the above definition and define an additional trajectory tube X (t; u(·)) (t0 < t ≤ T, u(·) ∈ U) which depends on a control u(·). Definition 34.2 Let u(·) be an admissible control. The set X (t; u(·)) at time t (t0 < t ≤ T ) of system (34.1)–(34.3) is defined as the set    X (t; u(·)) = x ∈ IRn : ∃ x0 ∈X 0 , x = x(t) = x t; u(·), x0 .

(34.5)

Note that for each fix t (t0 < t ≤ T ) and a fixed control u(·) (u(·) ∈ U) the set X (t; u(·)) defined by (34.5) represents the reachable set of system (34.1)–(34.3) taken with respect to x0 ∈ X 0 only. Thus, the main two problems solved in this paper are as follows. Problem 34.1 For each feasible control u(·) ∈ U, find the optimal (closest with ˆ T, u(·)) of the respect to inclusion of sets) external ellipsoidal estimate E(a, ˆ Q; reachable set X (T ; u(·)) of the dynamical system (34.1), ˆ T, u(·)). X (T ; u(·)) ⊂ E(a, ˆ Q;

(34.6)

Problem 34.2 Given a vector x ∗ ∈ IRn , find the feasible control u ∗ (·) ∈ U and a number  ∗ > 0 such that d(x ∗ , E(aˆ∗ , Qˆ ∗ ; T, u ∗ (·))) = inf d(x ∗ , E(aˆ∗ , Qˆ ∗ ; T, u(·))) =  ∗ . u(·)∈U

(34.7)

Remark 34.1 Problem 34.2 with the criterion (34.7) may be considered as an auxiliary one for solving the more complicated Problem 34.1 where we would like to claim the inclusion (34.6) for any feasible control u(·). However, the approximation scheme for nonlinear control systems under uncertainty used in Problem 34.2 generates very tight estimates of reachable sets of nonlinear systems (see related numerical

34 Control Problems for Set-Valued Motions of Systems with Uncertainty …

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simulations for a model example in the next section). Therefore, this technique may be sufficient or at least useful in solving control and estimation problems of the type under study.

34.3 Main Constructions and Problems Solution In this section, we introduce the basic necessary constructions and describe the solution to the problem under study. Note first that in a number of studies (e.g., [8, 11, 13, 34]) the differential equations were founded for the dynamics of external (and internal, only in some special cases of systems) ellipsoidal estimates of reachable sets for control system with uncertainty. In these papers, the authors considered the systems with uncertain matrices involved in dynamical equations, but additional nonlinear terms (e.g., of quadratic type) were not present in the right-hand parts of related differential equations. Differential equations for ellipsoidal estimates of reachable sets of nonlinear dynamical control systems were derived also earlier in [9] but only for the case when system state velocities may contain quadratic forms but matrix coefficients of linear terms were fixed there (precisely known). Thus, only several situations were studied, and in the general statement the problem was not considered before. We remind that here the initial set X 0 is an ellipsoid: X 0 = E(a0 , Q 0 ),

(34.8)

with a center a0 ∈ IRn and with a symmetric and positive definite n × n-matrix Q 0 . Define max l  B 1/2 Q 0 B 1/2 l. (34.9) (k0+ )2 = n l∈{l∈IR :||l||=1}

It is easy to check that the following estimate is true E(a0 , Q 0 ) ⊆ E(a0 , (k0+ )2 B −1 ),

(34.10)

and k0+ is the smallest number ( k0+ > 0) for which the above inclusion is true. Theorem 34.1 The following inclusion is true X (t; t0 , X 0 ) ⊆ E(a + (t), Q + (t); t, u(·)), t0 ≤ t ≤ T,

(34.11)

here Q + (t) = r + (t)B −1 and a + (t), r + (t) are defined as the solutions of the following system of differential equations:

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da + (t) = Aa + (t) + a + (t)Ba + (t)d + r + (t)d + u(t), dt + dr (t)  = max{l∈IRn :l=1} {l (2r + (t) B˜ + (t)l}, dt B˜ + (t) = B 1/2 (A + 2da + (t)B)B −1/2 ,

(34.12)

with k0+ defined by (34.8)–(34.10) and a + (t0 ) = a0 ,

r + (t0 ) = (k0+ )2 .

(34.13)

Proof The validity of relations (34.11), (34.12), and (34.13) is established following the main lines and ideas presented in [26]. However we need to emphasize that, in contrast to previous investigations (e.g., in [11]), we look here for ellipsoidal estimates of the tube X (T ; u(·)) for each fixed control u(·), therefore, the parameters a + (t), r + (t) of the estimating ellipsoids may depend on u(·), that is generally speaking we have a + (t) = a + (t; u(·)) and r + (t) = r + (t; u(·)). Remark 34.2 Some modifications of Theorem 34.1 which are close to the above statement but differ in a number of additional conditions and assumptions on the problem description may be done also in the context of ideas presented earlier in [11, 13]. Using this estimation technique, we come to the following result. Theorem 34.2 The optimal values  ∗ , u ∗ (·) of the Problem 34.2 satisfy the following equations:  ∗ = min max {l  (a + (T ; u(·)) − x ∗ ) + r + (T ; u(·))(l  B −1l)1/2 } = u(·)∈U ||l||=1

max {l  (a + (T ; u ∗ (·)) − x ∗ ) + r + (T ; u ∗ (·))(l  B −1 l)1/2 }.

||l||=1

(34.14)

Proof According to the problem formulation, we have to find the minimal positive number  such that the inclusion is true E(a + (T ), Q + (T ); T, u(·)) ⊆ B(x ∗ , ), or equivalently ρ(l|E(a + (T ), Q + (T ); T, u(·)) ≤ ρ(l|B(x ∗ , )), ∀l ∈ IRn . After calculations and taking into account results of Theorem 34.1, we come to the relation l  a + (T ) + (l  Q + (T )l)1/2 ≤ l  x ∗ + ||l||, and therefore we have

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 ∗ = min max (l  a + (T ) + (l  Q + (T )l)1/2 − l  x ∗ ), u(·) ||l||=1

and substituting here Q + (T ) = r + (T )B −1 we come to the equations (34.14).

34.4 Example and Numerical Simulations Consider an example which illustrates the theoretical results and shows that even in nonlinear case it is possible to find the estimates of reachable sets of the studied control system where both a nonlinearity and an uncertainty present simultaneously. We see in these examples that the reachable sets may lose the convexity property with increasing time t > t0 . Nevertheless, the related external estimates calculated on the basis of above ideas and results are ellipsoids (and therefore convex) and may be used as the convenient auxiliary tool in studying and solving approximately related optimization problems. Moreover, the examples given below show that the ellipsoidal estimates are tight in some directions, so they are optimal in their class because it is not possible to further reduced them keeping the property to be the outer ellipsoidal estimates for reachable sets of the systems of the studied type. We also note that the performed numerical simulation is based on new approaches and principles and is focused on algorithms of a higher dimension than was done in earlier papers on related topics, where illustrative examples with a dimension of at most two were considered. Example 34.1 Consider the following control system ⎧ ⎨ x˙1 = 2x1 + u 1 (t), x˙2 = x2 + u 2 (t), ⎩ x˙3 = x3 + x12 + x22 + 0.5x32 + u 3 (t).

(34.15)

Here we take x0 ∈ X 0 = B(0, 1), 0 ≤ t ≤ T = 0.4 and U = B(0, 0.1). The projections of estimating ellipsoids E + (t) = E(a + (t), Q + (t)) and the reachable sets X (t) onto the planes of state coordinates are presented in Figs. 34.1, 34.2 and 34.3 for time moments t = 0.1; 0.15; 0.2; 0.25; 0.3; 0.35; 0.4 (here for simplicity we take u(t) = 0), in other cases the pictures are similar). The last 3d-picture (Fig. 34.4) shows the upper estimating ellipsoid E + (t) = E(a + (t), Q + (t)) and the reachable set X (t) for t = 0.4.

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2.5 2

+

E (t)

Xt

+

E (t)

1.5

Xt

0.4

1

x

0.3

0.5

t

20

0.2

-0.5 0.1

-1

2

-1.5

1

-2 -2.5

2

x2 -2 -1.5 -1 -0.5 0

0.5

1

1.5

1

0

0

-1

2

-1

-2

x1

-2

x1

b)

a)

Fig. 34.1 Projections of ellipsoids E + (t) = E(a + (t), Q + (t)) (blue color) and reachable sets (black color) X (t) at the plane of {x1 , x2 }-coordinates (left picture) and at the plane of {x1 , x2 , t}coordinates (right picture)

Xt

+

3

E (t)

Xt

+

E (t)

0.4

2.5 2

0.3

1.5

t

x3 1

0.2

0.5 0.1

0

3

-0.5

2

-1 -2.5 -2 -1.5 -1 -0.5 0

x1

a)

0.5 1

1.5 2

2.5

x3

1 0 -1

-2

-1

1

0

2

x1

b)

Fig. 34.2 Projections of ellipsoids E + (t) = E(a + (t), Q + (t)) (blue lines) and reachable sets (black lines) X (t) at the plane of {x1 , x3 }-coordinates (left picture) and at the plane of {x1 , x3 , t}coordinates (right picture)

34.5 Conclusion In this paper, it was shown that the method of ellipsoidal estimation of reachable sets for dynamical systems with nonlinearities, proposed earlier, can be successfully used in solving control problems for a class of uncertain systems with dynamics of a similar type. The need to use such methodology arose earlier, however, difficulties caused by a significant nonlinearity of state velocities prevented its solution.

34 Control Problems for Set-Valued Motions of Systems with Uncertainty … 3.5 3

+

E (t)

X t

+

E (t)

387

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X t

0.4

2

0.3

1.5

x3

t

1

0.2

0.5 0

0.1 3

-0.5

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-1

x3

-1.5 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2

1

1

0

0 -1

x2

-1

x2

-2

b)

a)

Fig. 34.3 The projections of estimating ellipsoids E + (t) = E(a + (t), Q + (t)) (indicated in blue lines) and also the reachable sets (indicated in black lines) X (t) at the plane of {x2 , x3 }-coordinates (left picture) and at the plane of {x2 , x3 , t}-coordinates (right picture) Fig. 34.4 Reachable set X (t) and its upper ellipsoidal estimate E + (t) = E(a + (t), Q + (t)) for different time moments (3d-picture in the plane of state variables {x1 , x2 , x3 })

4

+

E (t)

X t

3

2

x3 1 0

-1

2 1.5

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0

-0.5

-1

-1

-0.5

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1

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1.5

2

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The differential equations for the dynamics of the upper ellipsoidal estimates of reachable sets for the systems under consideration were derived. The illustrative example and results of numerical simulations based on the proposed estimating procedures for reachable sets are given. The results presented here may be used in further theoretical and applied researches in optimal control and estimation problems for dynamical systems with more complicated classes of uncertainty and nonlinearity. The approach and the results given in the paper may be applied to solve a number of model predictive control problems both in an independent setting or as a part of more complicated projects.

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