Boundary Synchronization for Hyperbolic Systems (Progress in Nonlinear Differential Equations and Their Applications) [1st ed. 2019] 3030328481, 9783030328481

Table of contents :
Contents
1 Introduction and Overview
1.1 Introduction
1.2 Exact Boundary Null Controllability
1.3 Exact Boundary Synchronization
1.4 Exact Boundary Synchronization by p-Groups
1.5 Approximate Boundary Null Controllability
1.6 Approximate Boundary Synchronization
1.7 Approximate Boundary Synchronization by p-Groups
1.8 Induced Approximate Boundary Synchronization
1.9 Organization
2 Algebraic Preliminaries
2.1 Bi-orthonormality
2.2 Kalman's Criterion
2.3 Condition of Cp-Compatibility
Part I Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls: Exact Boundary Synchronization
3 Exact Boundary Controllability and Non-exact Boundary Controllability
3.1 Exact Boundary Controllability
3.2 Non-exact Boundary Controllability
4 Exact Boundary Synchronization and Non-exact Boundary Synchronization
4.1 Definition
4.2 Condition of Compatibility
4.3 Exact Boundary Synchronization and Non-exact Boundary Synchronization
5 Exactly Synchronizable States
5.1 Attainable Set of Exactly Synchronizable States
5.2 Determination of Exactly Synchronizable States
5.3 Approximation of Exactly Synchronizable States
6 Exact Boundary Synchronization by Groups
6.1 Definition
6.2 A Basic Lemma
6.3 Condition of Cp-Compatibility
6.4 Exact Boundary Synchronization by p-Groups
7 Exactly Synchronizable States by p-Groups
7.1 Introduction
7.2 Determination of Exactly Synchronizable States by p-Groups
7.3 Determination of Exactly Synchronizable States by p-Groups (Continued)
7.4 Precise Consideration on the Exact Boundary Synchronization by 2-Groups
Part II Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls: Approximate Boundary Synchronization
8 Approximate Boundary Null Controllability
8.1 Definition
8.2 D-Observability for the Adjoint Problem
8.3 Kalman'sCriterion.Total(DirectandIndirect)Controls
8.4 Sufficiency of Kalman's Criterion for T>0 Large Enough for the Nilpotent System
8.5 Sufficiency of Kalman's Criterion for T>0 Large Enough for 2times2 Systems
8.6 The Unique Continuation for Nonharmonic Series
8.7 Sufficiency of Kalman's Criterion for T>0 Large Enough in the One-Space-Dimensional Case
8.8 An Example
9 Approximate Boundary Synchronization
9.1 Definition
9.2 Condition of C1-Compatibility
9.3 Fundamental Properties
9.4 Properties Related to the Number of Total Controls
10 Approximate Boundary Synchronization by p-Groups
10.1 Definition
10.2 Fundamental Properties
10.3 Properties Related to the Number of Total Controls
10.4 Necessity of the Condition of Cp-Compatibility
10.5 Approximate Boundary Null Controllability
11 Induced Approximate Boundary Synchronization
11.1 Definition
11.2 Preliminaries
11.3 Induced Approximate Boundary Synchronization
11.4 Minimal Number of Direct Controls
11.5 Examples
Part III Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls: Exact Boundary Synchronization
12 Exact Boundary Controllability and Non-exact Boundary Controllability
12.1 Introduction
12.2 Proof of Lemma 12.2
12.3 Observability Inequality
12.4 Exact Boundary Controllability
12.5 Non-exact Boundary Controllability
13 Exact Boundary Synchronization and Non-exact Boundary Synchronization
13.1 Definition
13.2 Condition of C1-Compatibility
13.3 Exact Boundary Synchronization and Non-exact Boundary Synchronization
13.4 Attainable Set of Exactly Synchronizable States
14 Exact Boundary Synchronization by p-Groups
14.1 Definition
14.2 Condition of Cp-Compatibility
14.3 Exact Boundary Synchronization by p-Groups and Non-exact Boundary Synchronization by p-Groups
14.4 Attainable Set of Exactly Synchronizable States by p-Groups
15 Determination of Exactly Synchronizable States by p-Groups
15.1 Introduction
15.2 Determination and Estimation of Exactly Synchronizable States by p-Groups
15.3 Determination of Exactly Synchronizable States
15.4 Determination of Exactly Synchronizable States by 3-Groups
Part IV Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls: Approximate Boundary Synchronization
16 Approximate Boundary Null Controllability
16.1 Definitions
16.2 Equivalence Between the Approximate Boundary Null Controllability and the D-Observability
16.3 Kalman's Criterion. Total (Direct and Indirect) Controls
16.4 Sufficiency of Kalman's Criterion for T>0 Large Enough in the One-Space-Dimensional Case
16.5 Unique Continuation for a Cascade System of Two Wave Equations
17 Approximate Boundary Synchronization
17.1 Definition
17.2 Condition of C1-Compatibility
17.3 Fundamental Properties
17.4 Properties Related to the Number of Total Controls
17.5 An Example
18 Approximate Boundary Synchronization by p-Groups
18.1 Definition
18.2 Fundamental Properties
18.3 Properties Related to the Number of Total Controls
Part V Synchronization for a Coupled System of Wave Equations with Coupled Robin Boundary Controls: Exact Boundary Synchronization
19 Preliminaries on Problem (III) and (III0)
19.1 Regularity of Solutions with Neumann Boundary Conditions
19.2 Well-Posedness of a Coupled System of Wave Equations with Coupled Robin Boundary Conditions
19.3 Regularity of Solutions to Problem (III) and (III0)
20 Exact Boundary Controllability and Non-exact Boundary Controllability
20.1 Exact Boundary Controllability
20.2 Non-exact Boundary Controllability
21 Exact Boundary Synchronization
21.1 Exact Boundary Synchronization
21.2 Conditions of C1-Compatibility
22 Determination of Exactly Synchronizable States
22.1 Determination of Exactly Synchronizable States
22.2 Estimation of Exactly Synchronizable States
23 Exact Boundary Synchronization by p-Groups
23.1 Definition
23.2 Exact Boundary Synchronization by p-Groups
24 Necessity of the Conditions of Cp-Compatibility
24.1 Condition of Cp-Compatibility for the Internal Coupling Matrix
24.2 Condition of Cp-Compatibility for the Boundary Coupling Matrix
25 Determination of Exactly Synchronizable States by p-Groups
25.1 Determination of Exactly Synchronizable States by p-Groups
25.2 Estimation of Exactly Synchronizable States by p-Groups
Part VI Synchronization for a Coupled System of Wave Equations with Coupled Robin Boundary Controls: Approximate Boundary Synchronization
26 Some Algebraic Lemmas
27 Approximate Boundary Null Controllability
28 Unique Continuation for Robin Problems
28.1 General Consideration
28.2 Examples in Higher Dimensional Cases
28.3 One-Dimensional Case
29 Approximate Boundary Synchronization
29.1 Definitions
29.2 Fundamental Properties
30 Approximate Boundary Synchronization by p-Groups
30.1 Definitions
30.2 Fundamental Properties
31 Approximately Synchronizable States by p-Groups
32 Closing Remarks
32.1 Related Literatures
32.2 Prospects
Appendix References
Index

Citation preview

Progress in Nonlinear Differential Equations and Their Applications Subseries in Control 94

Tatsien Li Bopeng Rao

Boundary Synchronization for Hyperbolic Systems

Progress in Nonlinear Differential Equations and Their Applications PNLDE Subseries in Control Volume 94

Series Editor Jean-Michel Coron, Laboratory Jacques-Louis Lions, Pierre and Marie Curie University, Paris, France Editorial Board Viorel Barbu, Faculty of Mathematics, Alexandru Ioan Cuza University, Iaşi, Romania Piermarco Cannarsa, Department of Mathematics, University of Rome Tor Vergata, Rome, Italy Karl Kunisch, Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria Gilles Lebeau, Dieudonné Laboratory J.A., University of Nice Sophia Antipolis, Nice, Paris, France Tatsien Li, School of Mathematical Sciences, Fudan University, Shanghai, China Shige Peng, Institute of Mathematics, Shandong University, Jinan, China Eduardo Sontag, Department of Electrical & Computer Engineering, Northeastern University, Boston, MA, USA Enrique Zuazua, Department of Mathematics, Autonomous University of Madrid, Madrid, Spain

More information about this subseries at http://www.springer.com/series/15137

Tatsien Li Bopeng Rao •

Boundary Synchronization for Hyperbolic Systems

Tatsien Li School of Mathematical Sciences Fudan University Shanghai, China

Bopeng Rao Institut de Recherche Mathématique Avancée Université de Strasbourg Strasbourg, France

ISSN 1421-1750 ISSN 2374-0280 (electronic) Progress in Nonlinear Differential Equations and Their Applications PNLDE Subseries in Control ISBN 978-3-030-32848-1 ISBN 978-3-030-32849-8 (eBook) https://doi.org/10.1007/978-3-030-32849-8 © Springer Nature Switzerland AG 2019 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Exact Boundary Null Controllability . . . . . . . . . . . . . 1.3 Exact Boundary Synchronization . . . . . . . . . . . . . . . . 1.4 Exact Boundary Synchronization by p-Groups . . . . . . 1.5 Approximate Boundary Null Controllability . . . . . . . . 1.6 Approximate Boundary Synchronization . . . . . . . . . . . 1.7 Approximate Boundary Synchronization by p-Groups . 1.8 Induced Approximate Boundary Synchronization . . . . 1.9 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Algebraic Preliminaries . . . . . . . . . . 2.1 Bi-orthonormality . . . . . . . . . . 2.2 Kalman’s Criterion . . . . . . . . . 2.3 Condition of Cp-Compatibility .

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Exact Boundary Controllability and Non-exact Boundary Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Exact Boundary Controllability . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Non-exact Boundary Controllability . . . . . . . . . . . . . . . . . . . . .

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Part I

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Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls: Exact Boundary Synchronization

Exact Boundary Synchronization and Non-exact Boundary Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Condition of Compatibility . . . . . . . . . . . . . . . . . . . . . . . 4.3 Exact Boundary Synchronization and Non-exact Boundary Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Exactly Synchronizable States . . . . . . . . . . . . . . . . . . 5.1 Attainable Set of Exactly Synchronizable States . 5.2 Determination of Exactly Synchronizable States . 5.3 Approximation of Exactly Synchronizable States

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Exact 6.1 6.2 6.3 6.4

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Exactly Synchronizable States by p-Groups . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Determination of Exactly Synchronizable States by p-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Determination of Exactly Synchronizable States by p-Groups (Continued) . . . . . . . . . . . . . . . . . 7.4 Precise Consideration on the Exact Boundary Synchronization by 2-Groups . . . . . . . . . . . . .

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Part II

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Boundary Synchronization by Groups . . . . Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . A Basic Lemma . . . . . . . . . . . . . . . . . . . . . . Condition of Cp-Compatibility . . . . . . . . . . . . Exact Boundary Synchronization by p-Groups

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Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls: Approximate Boundary Synchronization . . . .

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Approximate Boundary Null Controllability . . . . . . . . . . . . . . . 8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 D-Observability for the Adjoint Problem . . . . . . . . . . . . . . 8.3 Kalman’s Criterion. Total (Direct and Indirect) Controls . . . 8.4 Sufficiency of Kalman’s Criterion for T [ 0 Large Enough for the Nilpotent System . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Sufficiency of Kalman’s Criterion for T [ 0 Large Enough for 2  2 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 The Unique Continuation for Nonharmonic Series . . . . . . . 8.7 Sufficiency of Kalman’s Criterion for T [ 0 Large Enough in the One-Space-Dimensional Case . . . . . . . . . . . . . . . . . . 8.8 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Approximate Boundary Synchronization . . . . . . . . . . . 9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Condition of C1-Compatibility . . . . . . . . . . . . . . . . 9.3 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . 9.4 Properties Related to the Number of Total Controls

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10 Approximate Boundary Synchronization by p-Groups . 10.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . 10.3 Properties Related to the Number of Total Controls 10.4 Necessity of the Condition of Cp-Compatibility . . . 10.5 Approximate Boundary Null Controllability . . . . . .

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11 Induced Approximate Boundary Synchronization . . . 11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Induced Approximate Boundary Synchronization 11.4 Minimal Number of Direct Controls . . . . . . . . . 11.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part III

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Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls: Exact Boundary Synchronization

12 Exact Boundary Controllability and Non-exact Boundary Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Proof of Lemma 12.2 . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Observability Inequality . . . . . . . . . . . . . . . . . . . . . . 12.4 Exact Boundary Controllability . . . . . . . . . . . . . . . . . 12.5 Non-exact Boundary Controllability . . . . . . . . . . . . . .

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13 Exact Boundary Synchronization and Non-exact Boundary Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Condition of C1-Compatibility . . . . . . . . . . . . . . . . . . . . . 13.3 Exact Boundary Synchronization and Non-exact Boundary Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Attainable Set of Exactly Synchronizable States . . . . . . . . Boundary Synchronization by p-Groups . . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condition of Cp-Compatibility . . . . . . . . . . . . . . . . . . . Exact Boundary Synchronization by p-Groups and Non-exact Boundary Synchronization by p-Groups . 14.4 Attainable Set of Exactly Synchronizable States by p-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 Exact 14.1 14.2 14.3

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15 Determination of Exactly Synchronizable States by p-Groups . . . . 185 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 15.2 Determination and Estimation of Exactly Synchronizable States by p-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

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15.3 Determination of Exactly Synchronizable States . . . . . . . . . . . . 192 15.4 Determination of Exactly Synchronizable States by 3-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Part IV

Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls: Approximate Boundary Synchronization

16 Approximate Boundary Null Controllability . . . . . . . . . . . . . . . 16.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Equivalence Between the Approximate Boundary Null Controllability and the D-Observability . . . . . . . . . . . . . . . 16.3 Kalman’s Criterion. Total (Direct and Indirect) Controls . . . 16.4 Sufficiency of Kalman’s Criterion for T [ 0 Large Enough in the One-Space-Dimensional Case . . . . . . . . . . . . . . . . . . 16.5 Unique Continuation for a Cascade System of Two Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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17 Approximate Boundary Synchronization . . . . . . . . . . . 17.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Condition of C1-Compatibility . . . . . . . . . . . . . . . . 17.3 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . 17.4 Properties Related to the Number of Total Controls 17.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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18 Approximate Boundary Synchronization by p-Groups . 18.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . 18.3 Properties Related to the Number of Total Controls

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Part V

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Synchronization for a Coupled System of Wave Equations with Coupled Robin Boundary Controls: Exact Boundary Synchronization

19 Preliminaries on Problem (III) and (III0) . . . . . . . . . . . . . . . 19.1 Regularity of Solutions with Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Well-Posedness of a Coupled System of Wave Equations with Coupled Robin Boundary Conditions . . . . . . . . . . . 19.3 Regularity of Solutions to Problem (III) and (III0) . . . . .

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20 Exact Boundary Controllability and Non-exact Boundary Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 20.1 Exact Boundary Controllability . . . . . . . . . . . . . . . . . . . . . . . . 239 20.2 Non-exact Boundary Controllability . . . . . . . . . . . . . . . . . . . . . 240

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21 Exact Boundary Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . 243 21.1 Exact Boundary Synchronization . . . . . . . . . . . . . . . . . . . . . . . 243 21.2 Conditions of C1-Compatibility . . . . . . . . . . . . . . . . . . . . . . . . 244 22 Determination of Exactly Synchronizable States . . . . . . . . . . . . . . . 247 22.1 Determination of Exactly Synchronizable States . . . . . . . . . . . . 247 22.2 Estimation of Exactly Synchronizable States . . . . . . . . . . . . . . . 251 23 Exact Boundary Synchronization by p-Groups . . . . . . . . . . . . . . . . 255 23.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 23.2 Exact Boundary Synchronization by p-Groups . . . . . . . . . . . . . 256 24 Necessity of the Conditions of Cp-Compatibility . . . . . . . . . . . . . . . 259 24.1 Condition of Cp-Compatibility for the Internal Coupling Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 24.2 Condition of Cp-Compatibility for the Boundary Coupling Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 25 Determination of Exactly Synchronizable States by p-Groups . . . . 267 25.1 Determination of Exactly Synchronizable States by p-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 25.2 Estimation of Exactly Synchronizable States by p-Groups . . . . . 271 Part VI

Synchronization for a Coupled System of Wave Equations with Coupled Robin Boundary Controls: Approximate Boundary Synchronization

26 Some Algebraic Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 27 Approximate Boundary Null Controllability . . . . . . . . . . . . . . . . . . 281 28 Unique Continuation for Robin Problems . . 28.1 General Consideration . . . . . . . . . . . . . 28.2 Examples in Higher Dimensional Cases 28.3 One-Dimensional Case . . . . . . . . . . . .

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29 Approximate Boundary Synchronization . . . . . . . . . . . . . . . . . . . . 297 29.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 29.2 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 30 Approximate Boundary Synchronization by p-Groups . . . . . . . . . . 305 30.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 30.2 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 31 Approximately Synchronizable States by p-Groups . . . . . . . . . . . . . 315

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Contents

32 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 32.1 Related Literatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 32.2 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Chapter 1

Introduction and Overview

An introduction and overview of the whole book can be found in this chapter.

1.1 Introduction Synchronization is a widespread natural phenomenon. Thousands of fireflies may twinkle at the same time; audiences in the theater can applaud with a rhythmic beat; pacemaker cells of the heart function simultaneously; and field crickets give out a unanimous cry. All these are phenomena of synchronization (cf. [71, 74]). In principle, synchronization happens when different individuals possess likeness in nature, that is, they conform essentially to the same governing equation, and meanwhile, the individuals should bear a certain coupled relation. The phenomenon of synchronization was first observed by Huygens [22] in 1665. The research on synchronization from a mathematical point of view dates back to Wiener [81] in the 1950s. The previous studies focused on systems described by ordinary differential equations (ODEs), such as X i = f (t, X i ) +

N 

Ai j X j (i = 1, · · · , N ),

(1.1)

j=1

where X i (i = 1, · · · , N ) are n-dimensional state vectors, “  ” stands for the time derivative, Ai j (i, j = 1, · · · , N ) are n × n matrices, and f (t, X ) is an n-dimensional vector function independent of i = 1, · · · , N (cf. [71, 74]). The righthand side of (1.1) shows that every X i (i = 1, · · · , N ) possesses two basic features: satisfying a fundamental governing equation and bearing a coupled relation among © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_1

1

2

1 Introduction and Overview

one another. If for any given initial data t =0:

X i = X i(0) (i = 1, · · · , N ),

(1.2)

the solution X = (X 1 , · · · , X N )T = X (t) to the system satisfies X i (t) − X j (t) → 0 (i, j = 1, · · · , N ) as t → +∞,

(1.3)

namely, all the states X i (t) (i = 1, · · · , N ) tend to coincide with each other as t → +∞, then we say that the system possesses the synchronization in the consensus sense, or, in particular, if the solution X = X (t) satisfies X i (t) − a(t) → 0 (i = 1, · · · , N ) as t → +∞,

(1.4)

where a(t) is a state vector which is a priori unknown, then we say that the system possesses the synchronization in the pinning sense. Obviously, the synchronization in the pinning sense implies that in the consensus sense. These kinds of synchronizations are all called the asymptotic synchronization, which should be realized on the infinite time interval [0, +∞). What the authors of this monograph have been doing in the recent years is to extend, in both concept and method, the universal phenomena of synchronization from finite-dimensional dynamical systems of ordinary differential equations to infinite-dimensional dynamical systems of partial differential equations (PDEs). This should be the first attempt in this regard, and the earliest relevant paper was published by the authors in [42] in a special issue of Chinese Annals of Mathematics in honor of the scientific heritage of Lions, the results of which were announced in 2012 in CRAS [41]. For fixing the idea, in this chapter we will only consider the following coupled system of wave equations with Dirichlet boundary conditions (cf. Part 1 and Part 2 of this monograph. The corresponding consideration with Neumann or coupled Robin boundary conditions will be given in Part 3 and Part 4 or in Part 5 and Part 6): ⎧  ⎨ U − U + AU = 0 in (0, +∞) × , U =0 on (0, +∞) × 0 , ⎩ U = DH on (0, +∞) × 1 with the initial data t =0:

0 , U  = U 1 in , U =U

(1.5)

(1.6)

where  is a bounded domain with smooth boundary  = 0 ∪ 1 such that  0 ∩  1 = ∅ and mes(1 ) > 0, U = (u (1) , · · · , u (N ) )T is the state varin  ∂2 able,  = is the n-dimensional Laplacian operator, A ∈ M N (R) is an N × N ∂x2 i=1

i

coupling matrix with constant elements D, called the boundary control matrix,

1.1 Introduction

3

is an N × M full column-rank matrix (M  N ) with constant elements, H = (h (1) , · · · , h (M) )T stands for the boundary control. Here, a boundary control matrix D is added to the boundary condition on 1 . This approach is more flexible: we will see in what follows that the introduction of D enables us to simplify the statement and the discussion to a great extent. For systems governed by PDEs, we can similarly consider the asymptotic synchronization on an infinite time interval as in the case of systems governed by ODEs, namely, we may ask the following questions: under what conditions do the system states with any given initial data possess the asymptotic synchronization in the consensus sense: u (i) (t, ·) − u ( j) (t, ·) → 0 (i, j = 1, . . . , N ) as t → +∞,

(1.7)

or, in particular, if the system states with any given initial data possess the asymptotic synchronization in the pinning sense: u (i) (t, ·) − u(t, ·) → 0 (i = 1, · · · , N ) as t → +∞,

(1.8)

where u = u(t, ·) is called the asymptotically synchronizable state, which is a priori unknown? if the answer of this question is positive, these conclusions should be realized spontaneously on an infinite time interval [0, +∞), and is a naturally developed result decided by the nature of the system itself. But for systems governed by partial differential equations, as there are boundary conditions, another possibility exists, i.e., to give artificial intervention to the evolution of state variables through appropriate boundary controls, which combines synchronization with controllability and introduces the study of synchronization to the field of control. This is also a new perspective on the investigation of synchronization for systems of partial differential equations. Here, the boundary control comes from the boundary condition U = D H on 1 . The elements in H are adjustable boundary controls, the number of which is M( N ). To put the boundary control matrix D before H will provide many possibilities for combining boundary controls. On the other hand, precisely due to the artificial intervention of control, we can make a higher demand, i.e., to meet the requirement of synchronization within a limited time, instead of waiting until t → +∞. The corresponding question is whether there is a suitably large T > 0, such that 1 ), through proper boundary controls with compact 0 , U for any given initial data (U support in [0, T ] (that is, to exert the boundary control at the time interval [0, T ], and abandon the control from the time t = T ), the solution U = U (t, x) to the corresponding problem (1.5)–(1.6) satisfies, as t  T , u (1) (t, ·) ≡ u (2) (t, ·) ≡ · · · ≡ u (N ) (t, ·) := u(t, ·),

(1.9)

that is, all state variables tend to be the same since the time t = T , while u = u(t, x) is called the corresponding exactly synchronizable state which is unknown beforehand. If the above is satisfied, we say that the system possesses the exact boundary

4

1 Introduction and Overview

synchronization. Here, “exact” means that the synchronization of state variables is exact without error, and the so-called “boundary” indicates the means or method of control, i.e., to realize the synchronization through boundary controls. In the above definition of synchronization, through boundary controls on the time interval [0, T ], we not only demand synchronization at the time t = T , but also require synchronization to continue when t  T , i.e., after all boundary controls are eliminated. This kind of synchronization is not a short-lived one, but exists once and for all, as is needed in applications. We have to point out that in Part 1 and Part 2 of this monograph we always 1 ) ∈ (L 2 ()) N × (H −1 ()) N , and the solution 0 , U assume that the initial value (U to problem (1.5)–(1.6) belongs to the corresponding function space, which will not be specified one by one in this chapter. As to the synchronization considered in the framework of classical solutions in the one-space-dimensional case, see [20, 21, 34, 57, 58].

1.2 Exact Boundary Null Controllability The exact boundary synchronization on a finite time interval is closely related to the exact boundary null controllability. 1 ), through bound0 , U If there exists T > 0, such that for any given initial data (U ary controls with compact support in [0, T ], the solution U = U (t, x) to the corresponding problem (1.5)–(1.6) satisfies, as t  T , u (1) (t, x) ≡ u (2) (t, x) ≡ · · · ≡ u (N ) (t, x) ≡ 0,

(1.10)

then we say that the system possesses the so-called exact boundary null controllability in control theory. This is of course a very special case of the abovementioned exact boundary synchronization. For a single wave equation, the exact boundary null controllability can be proved by the HUM method proposed by Lions [61, 62]. For a coupled system of wave equations, since for the purpose of studying the synchronization, the coupling matrix A should be an arbitrarily given matrix, the proof of the exact boundary null controllability cannot be simply reduced to the case of a single wave equation, however, using a compact perturbation result in [69], it is possible to establish a corresponding observability inequality for the corresponding adjoint system, and then the HUM method can be still applied. As a result, we can get the following conclusions (cf. Chap. 3): (1) Assume that M = N and let the domain  satisfy the usual multiplier geometrical condition (cf. [7, 26, 61, 62]): there exists x0 ∈ Rn , such that, setting m = x − x0 , we have (m, ν) > 0, ∀x ∈ 1 ; (m, ν)  0, ∀x ∈ 0 ,

(1.11)

where ν is the unit outward normal vector, and (·, ·) denotes the inner product in Rn .

1.2 Exact Boundary Null Controllability

5

Then through N boundary controls, when T > 0 is suitably large, the exact 1 ) ∈ (L 2 ()) N × 0 , U boundary null controllability can surely be realized for all (U (H −1 ()) N . (2) Assume that M < N , that is, if the number of boundary controls is fewer than N , then no matter how large T > 0 is, the exact boundary null controllability cannot 1 ) ∈ (L 2 ()) N × (H −1 ()) N . 0 , U be achieved for all (U Thus, in the case of partial lack of boundary controls, which kind of controllability in a weaker sense can be realized by means of fewer boundary controls? It is a significant problem from both theoretical and practical points of view, and can be discussed in many different cases as will be shown in the sequel.

1.3 Exact Boundary Synchronization Really meaningful synchronization should exclude the trivial situation of null controllability, and thus we get the following results (cf. Chap. 4): Assume that the system under consideration is exactly synchronizable, but not exactly null controllable, that is to say, assume that the system is exactly synchronizable with rank(D) < N . Then the coupling matrix A = (ai j ) should satisfy the following condition of compatibility: N 

akp = a (k = 1, · · · , N ),

(1.12)

p=1

where a is a constant independent of k = 1, · · · , N , namely, the sum of all elements in every row of A is the same (the row-sum condition). Let e1 = (1, . . . , 1)T . The condition of compatibility (1.12) is equivalent to that e1 is an eigenvector of A, corresponding to the eigenvalue a. Let ⎞ ⎛ 1 −1 ⎟ ⎜ 1 −1 ⎟ ⎜ (1.13) C1 = ⎜ ⎟ . . .. .. ⎠ ⎝ 1 −1 (N −1)×N

6

1 Introduction and Overview

be the corresponding synchronization matrix. C1 is a full row-rank matrix. Obviously, the synchronization requirement (1.9) can be written as tT :

C1 U (t, x) ≡ 0.

(1.14)

Moreover, Ker(C1 ) = Span{e1 }, and then the condition of compatibility (1.12) is equivalent to that Ker(C1 ) is a one-dimensional invariant subspace of A: AKer(C1 ) ⊆ Ker(C1 ).

(1.15)

Then, the condition of compatibility (1.12), often called the condition of C1 -compatibility in what follows, is also equivalent to that there exists a unique matrix A1 of order (N − 1), such that C 1 A = A1 C 1 .

(1.16)

Such matrix A1 is called the reduced matrix of A by C1 . Under the condition of C1 -compatibility, let W1 = (w (1) , · · · , w (N −1) )T = C1 U.

(1.17)

It is easy to see that the original system (1.5) for the variable U can be reduced to the following self-closed system for the variable W1 : ⎧ ⎨ W1 − W1 + A1 W1 = 0 in (0, +∞) × , on (0, +∞) × 0 , W =0 ⎩ 1 on (0, +∞) × 1 . W1 = C 1 D H

(1.18)

Obviously, under the condition of C1 -compatibility, the exact boundary synchronization of the original system (1.5) for U is equivalent to the exact boundary null controllability of the reduced system (1.18) for W. Hence, we have: Assume that the condition of C1 -compatibility is satisfied and the domain  satisfies the usual multiplier geometrical condition, then there exists a suitably large T > 0, such that for any boundary control matrix D with rank (C1 D) = N − 1, the exact boundary synchronization of system (1.5) can be realized at the time t = T . On the contrary, if rank(C1 D) < N − 1, in particular, if rank(D) < N − 1, i.e., the number of boundary controls is fewer than (N − 1), then no matter how large T > 0 is, the exact boundary synchronization can never be achieved for all initial data 1 ) ∈ (L 2 ()) N × (H −1 ()) N . 0 , U (U Therefore, the above condition of C1 -compatibility is not only sufficient but also necessary to ensure the exact boundary synchronization. Under this condition, for the boundary control matrix D such that rank(D) = rank(C1 D) = N − 1, appropriately chosen (N − 1) boundary controls suffice to meet the requirement.

1.3 Exact Boundary Synchronization

7

We point out that in the study of synchronization for systems governed by ODEs, the row-sum condition (1.12) is imposed on the system according to physical meanings as a reasonable sufficient condition. However, for our systems governed by PDEs and for the synchronization on a finite time interval, it is actually a necessary condition, which makes the theory of synchronization more complete for systems governed by PDEs. In the case that system (1.5) possesses the exact boundary synchronization at the time T > 0, as t  T , the exactly synchronizable state u = u(t, x) satisfies 

u  − u + au = 0 in (T, +∞) × , u=0 on (T, +∞) × ,

(1.19)

where a is given by the row-sum condition (1.12). Thus, the evolution of the exactly synchronizable state u = u(t, x) with respect to t can be uniquely determined by its initial data: t=T :

u1. u = u0, u = 

(1.20)

u 1 ) of (u, u  ) at t = T should depend However, generally speaking, the value ( u0,    on the original initial data (U0 , U1 ) as well as on the boundary controls H , which u 1 ) at t = T realize the exact boundary synchronization. Moreover, the value of ( u0,  1 ) can be determined, and in some special cases it is 0 , U for a given initial data (U independent of boundary controls H , which realize the exact boundary synchronization. u 1 ) at t = T is the whole The attainable set of all possible values of ( u0,  0 , U 1 ) vary in the space space L 2 () × H −1 (), when the original initial data (U 2 N −1 N (L ()) × (H ()) . That is to say, any given (u0 , u1 ) in L 2 () × H −1 () can be the value of an exactly synchronizable state (u, u  ) at t = T (cf. Chap. 5).

1.4 Exact Boundary Synchronization by p-Groups How will be the situation if the number of boundary controls is further reduced? Then we can only further lower the standard to be reached. One possible way is not to require synchronization of all state variables, but to divide them into several groups, e.g., p groups ( p  1), and then demand synchronization of state variables within every group, whereby we get the concept of exact boundary synchronization by p-groups. Precisely speaking, let p  1 be an integer and let 0 = n0 < n1 < n2 < · · · < n p = N

(1.21)

be integers such that n r − n r −1  2 for all 1  r  p. We rearrange the components of U into p groups

8

1 Introduction and Overview

(u (1) , · · · , u (n 1 ) ), (u (n 1 +1) , · · · , u (n 2 ) ), · · · , (u (n p−1 +1) , · · · , u (n p ) ).

(1.22)

System (1.5) is exactly synchronizable by p-groups at the time T > 0, if for any 1 ) ∈ (L 2 ()) N × (H −1 ()) N , there exist suitable boundary 0 , U given initial data (U 2 controls H ∈ L loc (0, +∞; (L 2 (1 )) M ) with compact support in [0, T ], such that the corresponding solution U = U (t, x) satisfies the following final conditions:

tT :

⎧ (1) u ≡ · · · ≡ u (n 1 ) := u 1 , ⎪ ⎪ ⎨ (n 1 +1) ≡ · · · ≡ u (n 2 ) := u 2 , u ··· ⎪ ⎪ ⎩ (n p−1 +1) ≡ · · · ≡ u (n p ) := u p , u

(1.23)

where u = (u 1 , · · · , u p )T is called the exactly synchronizable state by p-groups, which is a priori unknown. Let Sr be the following (n r − n r −1 − 1) × (n r − n r −1 ) matrix: ⎛

1 ⎜0 ⎜ Sr = ⎜ . ⎝ ..

−1 1 .. .

0 −1 .. .

··· ··· .. .

0 0 .. .

⎞ ⎟ ⎟ ⎟ ⎠

(1.24)

0 0 · · · 1 −1

and let C p be the following (N − p) × N full row-rank matrix of synchronization by p-groups: ⎞ ⎛ S1 ⎟ ⎜ S2 ⎟ ⎜ (1.25) Cp = ⎜ ⎟. .. ⎝ . ⎠ Sp The exact boundary synchronization by p-groups (1.23) is equivalent to tT :

C p U ≡ 0.

(1.26)

We can establish all the previous results in a similar way after having overcome some technical difficulties (see Chaps. 6 and 7)). For instance, if the exact boundary synchronization by p-groups can be realized by means of (N − p) boundary controls, we can get the corresponding condition of C p -compatibility: There exists a unique reduced matrix A p of order (N − p), such that C p A = A pC p,

(1.27)

which actually means that the coupling matrix A satisfies the row-sum condition by blocks.

1.4 Exact Boundary Synchronization by p-Groups

9

Table 1.1 The exact boundary synchronization by p-groups Condition of C p -compatibility Minimal number of boundary controls Exact boundary null controllability Exact boundary synchronization Exact boundary synchronization by 2-groups ········· Exact boundary synchronization by p-groups

N C 1 A = A1 C 1

N −1

C 2 A = A2 C 2

N −2

C p A = A pC p

N−p

Correspondingly, under the condition of C p -compatibility, we assume that the domain  satisfies the usual multiplier geometrical condition, then there exists a suitably large T > 0, such that for any given boundary control matrix D with rank (C p D) = N − p, the exact boundary synchronization by p-groups of system (1.5) can be realized at the time t = T . On the contrary if rank(C p D) < N − p, in particular, if rank(D) < N − p, then no matter how large T > 0 is, the exact boundary synchronization by p-groups can never be achieved for all initial data 1 ) ∈ (L 2 ()) N × (H −1 ()) N . 0 , U (U In summary, we get the following table (Table 1.1). What can we do when the number of boundary controls that can be chosen further decreases? The aforementioned controllability and synchronization should both be established in the exact sense, however, from the practical point of view, some minor error will not affect the general situation, so it also makes sense that these requirements are tenable only in the approximate sense. What we are considering here is the situation that no matter how small the error given beforehand is, we can always find suitable boundary controls so that the controllability or synchronization can be realized within the permitted range of accuracy. Since the error can be chosen progressively smaller, this corresponds actually to a limit process, which thus allows the analytical method to be applied more effectively. The corresponding controllability and synchronization are called approximate boundary null controllability and approximate boundary synchronization.

1.5 Approximate Boundary Null Controllability We first consider the approximate boundary null controllability given by the following definition:

10

1 Introduction and Overview

System (1.5) possesses the approximate boundary null controllability at the 1 ) ∈ (L 2 ()) N × (H −1 ()) N , there 0 , U time T > 0 if for any given initial data (U 2 (0, +∞; (L 2 (1 )) M ) with exists a sequence {Hn } of boundary controls, Hn ∈ L loc compact support in [0, T ], such that the corresponding sequence {Un } of solutions to problem (1.5)–(1.6) satisfies the following condition: Un → 0 as n → +∞

(1.28)

0 1 ([T, +∞); (L 2 ()) N ) ∩ Cloc ([T, +∞); (H −1 ()) N ). Cloc

(1.29)

in the space

Obviously, the exact boundary null controllability leads to the approximate boundary null controllability. However, since from the previous definition we cannot get the convergence of the sequence {Hn } of boundary controls, the approximate boundary null controllability cannot lead to the exact boundary null controllability in general. Consider the corresponding adjoint problem ⎧  in (0, +∞) × , ⎨  −  + AT  = 0 =0 on (0, +∞) × , ⎩ 1 in ,  0 ,  =  t =0: =

(1.30)

where A T denotes the transpose of A. We give the following definition: The adjoint problem (1.30) is D-observable on the interval [0, T ], if 0 ,  1 ) ≡ 0, i.e.,  ≡ 0, D T ∂ν  ≡ 0 on [0, T ] × 1 =⇒ (

(1.31)

where ∂ν denotes the outward normal derivative on the boundary. The D-observability is only a weak observability in the sense of uniqueness, which cannot guarantee the exact boundary null controllability. However, we can prove the following result (cf. Chap. 8): System (1.5) is approximately null controllable at the time T > 0 if and only if the adjoint problem (1.30) is D-observable on the interval [0, T ]. Obviously, if M = N , then by Holmgren’s uniqueness theorem (cf. [18, 62, 75]), system (1.5) is always approximately null controllable by means of N boundary controls for T > 0 large enough, even without the multiplier geometrical condition. However, it should be pointed out that the approximate boundary null controllability can be realized even if M < N , namely, by means of fewer boundary controls. To transform the approximate boundary null controllability of the original system equivalently to the D-observability of the adjoint problem cannot solve specifically the problem of judging whether a system possesses the approximate boundary null controllability, nor is it clear to what extent the weakened concept may reduce the number of boundary controls needed. But we can thereby prove the following result (cf. Chap. 8):

1.5 Approximate Boundary Null Controllability

11

Assuming that system (1.5) is approximately null controllable at the time T > 0, then we necessarily have that the following enlarged matrix composed of A and D is of full rank: rank(D, AD, · · · , A N −1 D) = N .

(1.32)

This is a necessary condition which helps to conveniently eliminate a set of systems that do not meet the requirements. Equation (1.32) is nothing but the so-called Kalman’s criterion for guaranteeing the exact controllability for the following system of ODEs (cf. [23, 73]): X  = AX + Du,

(1.33)

where u stands for the vector of control variables, however here we get it from a different point of view. Since condition (1.32) is independent of T , it is not a sufficient condition of the D-observability for the adjoint problem (1.30) in general, otherwise the D-observability should be realized almost immediately, but it is not the case because of the finite speed of wave prorogation. However, under certain assumptions on A, condition (1.32) is also sufficient for guaranteeing the approximate boundary null controllability for T > 0 large enough for some special kinds of systems, for instance, some one-space-dimensional systems, some 2 × 2 systems, the cascade system, and more generally, the nilpotent system under the multiplier geometrical condition, etc. For the exact boundary null controllability of system (1.5), the number M = rank(D), namely, the number of boundary controls, should be equal to N , the number of state variables. However, the approximate boundary null controllability of system (1.5) could be realized if the number M = rank(D) is substantially small, even if M = rank(D) = 1. Nevertheless, even if the rank of D might be small, but because of the existence and influence of the coupling matrix A, in order to realize the approximate boundary null controllability, the rank of the enlarged matrix (D, AD, · · · , A N −1 D) should be still equal to N , the number of state variables. From this point of view, we may say that the rank M of D is the number of “direct” boundary controls acting on 1 , and rank(D, AD, · · · , A N −1 D) denotes the number of “total” controls. Differently from the exact boundary null controllability, for the approximate boundary null controllability, we should consider not only the number of direct boundary controls, but also the number of total controls.

1.6 Approximate Boundary Synchronization Similar to the approximate boundary null controllability, we give the following definition:

12

1 Introduction and Overview

System (1.5) possesses the approximate boundary synchronization at the time 1 ) ∈ (L 2 ()) N × (H −1 ()) N , there exist a 0 , U T > 0 if for any given initial data (U 2 (0, +∞; (L 2 (1 )) M ) with compact sequence {Hn } of boundary controls, Hn ∈ L loc (N ) T support in [0, T ], such that the corresponding sequence {Un } = {(u (1) n , · · · , un ) } of solutions to problem (1.5)–(1.6) satisfies (l) u (k) n − u n → 0 as n → +∞

(1.34)

for all 1  k, l  N in the space 0 1 ([T, +∞); L 2 ()) ∩ Cloc ([T, +∞); H −1 ()). Cloc

(1.35)

Obviously, if system (1.5) is exactly synchronizable, then it must be approximately synchronizable; however, the inverse is not true in general. Moreover, the approximate boundary null controllability obviously leads to the approximate boundary synchronization. We should exclude this trivial situation in advance. Assume that system (1.5) is approximately synchronizable, but not approximately null controllable. Then, as in the case of exact boundary synchronization, the coupling matrix A should satisfy the same condition of C1 -compatibility (1.12) (cf. Chap. 9). Then, under the condition of C1 -compatibility, setting W1 = C1 U as in (1.17), we get again the reduced system (1.18) and its adjoint problem (the reduced adjoint problem): ⎧ T ⎨ 1 − 1 + A1 1 = 0 in (0, +∞) × , on (0, +∞) × , 1 = 0 ⎩ 0 , 1 =  1 in . t = 0 : 1 = 

(1.36)

Similarly to the D-observability, we say that the reduced adjoint problem (1.36) is C1 D-observable on the interval [0, T ], if 0 ,  1 ) ≡ 0, i.e.,  ≡ 0. (C1 D)T ∂ν 1 ≡ 0 on [0, T ] × 1 =⇒ (

(1.37)

We can prove that (cf. Chap. 9): Under the condition of C1 -compatibility, system (1.5) is approximately synchronizable at the time T > 0 if and only if the reduced adjoint problem (1.36) is C1 D-observable on the interval [0, T ]. Then, it is easy to see that under the condition of C1 -compatibility if rank(C1 D) = N − 1 (which implies M  N − 1), then, system (1.5) is always approximately synchronizable, even without the multiplier geometrical condition. We should point out that even if rank(C1 D) < N − 1, and in particular, if we essentially use fewer than (N − 1) boundary controls, it is still possible to realize the approximate boundary synchronization.

1.6 Approximate Boundary Synchronization

13

Moreover, as in the case of approximate boundary null controllability, under the condition of C1 -compatibility, we have: Assume that system (1.5) is approximately synchronizable at the time T > 0, then we necessarily have rank(C1 D, C1 AD, · · · , C1 A N −1 D) = N − 1.

(1.38)

Condition (1.38) is not sufficient in general for the approximate boundary synchronization, however, it is still sufficient for T > 0 large enough for some special systems under certain additional assumptions on A. On the other hand, we can prove: Assume that system (1.5) is approximately synchronizable under the action of a boundary control matrix D. No matter whether the condition of C1 -compatibility is satisfied or not, we necessarily have rank(D, AD, · · · , A N −1 D)  N − 1,

(1.39)

namely, at least (N − 1) total controls are needed in order to realize the approximate boundary synchronization of system (1.5). Furthermore, assume that system (1.5) is approximately synchronizable under the minimal rank condition rank(D, AD, · · · , A N −1 D) = N − 1.

(1.40)

Then the matrix A should satisfy the condition of C1 -compatibility and some algebraic properties, and there exists a scalar function u as the approximately synchronizable state, such that (1.41) u (k) n → u as n → +∞ in the space (1.35) for all 1  k  N . Moreover, the approximately synchronizable state u is independent of the sequence {Hn } of applied boundary controls. Thus the original approximate boundary synchronization in the consensus sense reduces to that in the pinning sense.

1.7 Approximate Boundary Synchronization by p-Groups Generally speaking, we can define the approximate boundary synchronization by p-groups ( p  1). Let us re-group the components of the state variable as in (1.22). We say that system (1.5) is approximately synchronizable by p-groups at the time 1 ) ∈ (L 2 ()) N × (H −1 ()) N , there exists 0 , U T > 0 if for any given initial data (U 2 (0, +∞; (L 2 (1 )) M ) with compact a sequence {Hn } of boundary controls in L loc support in [0, T ], such that the sequence {Un } of the corresponding solutions satisfies the following conditions:

14

1 Introduction and Overview (l) u (k) n − u n → 0 as n → +∞

(1.42)

in the space (1.35) for all n r −1 + 1  k, l  n r and 1  r  p, or equivalently C p Un → 0 as n → +∞

(1.43)

in the space 0 1 ([T, +∞); (L 2 ()) N − p ) ∩ Cloc ([T, +∞); (H −1 ()) N − p ), Cloc

(1.44)

where C p is given by (1.24)–(1.25). As in the case of exact boundary synchronization by p-groups, we can get the corresponding condition of C p -compatibility (cf. Chap. 10): There exists a unique reduced matrix A p of order (N − p), such that (1.27) holds. Under the condition of C p -compatibility, we necessarily get the following Kalman’s criterion: rank(C p D, C p AD, · · · , C p A N −1 D) = N − p.

(1.45)

On the other hand, we can prove: Assume that system (1.5) is approximately synchronizable by p-groups under the action of a boundary control matrix D. Then, no matter whether the condition of C p -compatibility is satisfied or not, we necessarily have rank(D, AD, · · · , A N −1 D)  N − p,

(1.46)

namely, at least (N − p) total controls are needed in order to realize the approximate boundary synchronization by p-groups of system (1.5). Furthermore, assume that system (1.5) is approximately synchronizable by p-groups under the minimal rank condition rank(D, AD, · · · , A N −1 D) = N − p,

(1.47)

then the matrix A should satisfy the condition of C p -compatibility and Ker(C p ) admits a supplement V , which is also invariant for A. Moreover, there exist some linearly independent scalar functions u 1 , · · · , u p such that the approximately synchronizable state u = (u 1 , · · · , u p )T is independent of the sequence {Hn } of applied boundary controls. Thus the original approximate synchronization by p-groups in the consensus sense reduces to that in the pinning sense. In summary, we get the following table (Table 1.2).

1.8 Induced Approximate Boundary Synchronization

15

Table 1.2 The approximate boundary synchronization by p-groups Condition of C p -compatibility Minimal number of total controls Approximate boundary null controllability Approximate boundary synchronization Approximate boundary synchronization by 2-groups ········· Approximate boundary synchronization by p-groups

N C 1 A = A1 C 1

N −1

C 2 A = A2 C 2

N −2

C p A = A pC p

N−p

1.8 Induced Approximate Boundary Synchronization We cannot always realize the approximate boundary synchronization by p-groups under the minimal rank condition (1.47). In fact, let D p be the set of all boundary control matrices D which realize the approximate boundary synchronization by pgroups for system (1.5). Define the minimal number N p of total controls by N p = inf rank(D, AD, · · · , A N −1 D). D∈D p

(1.48)

Under a suitable condition on the matrix A, we can prove N p = N − q,

(1.49)

where q  p is the dimension of the largest subspace W , which is contained in Ker(C p ) and admits a supplement V , both W and V are invariant for the matrix A. So, generally speaking, (N − q) total controls are necessary for the approximate boundary synchronization by p-groups of system (1.5), while (1.43) provides only the convergence of (N − p) components of state variables. Hence, there is a loss of ( p − q) information hidden in the minimal rank condition rank(D, AD, · · · , A N −1 D) = N − q.

(1.50)

Let Cq∗ be the induced extension matrix defined by Ker(Cq∗ ) = W. Ker(Cq∗ ) is the biggest subspace of A, which is contained in Ker(C p ) and admits a supplement V , both W and V are invariant for A. Moreover, we have the following rank condition: rank(Cq∗ D, Cq∗ AD, · · · , Cq∗ A N −1 D) = N − q,

(1.51)

16

1 Introduction and Overview

which is necessary for the approximate boundary null controllability of the corresponding reduced system: ⎧  ⎨ W − W + Aq∗ W = 0 in (0, +∞) × , W =0 on (0, +∞) × 0 , ⎩ on (0, +∞) × 1 W = Cq∗ D H

(1.52)

with the initial data t =0:

0 , W  = Cq∗ U 1 in , W = Cq∗ U

(1.53)

where W = Cq∗ U and Aq∗ is given by Cq∗ A = Aq∗ Cq∗ . In some specific situations discussed in Chap. 11, the above rank condition (1.51) is indeed sufficient for the approximate boundary null controllability of the reduced system (1.52). On this occasion, we have Cq∗ Un → 0 as n → +∞

(1.54)

in the space 0 1 ([T, +∞); (L 2 ()) N −q ) ∩ Cloc ([T, +∞); (H −1 ()) N −q ). Cloc

(1.55)

In this case, we say that system (1.5) is induced approximately synchronizable. Since (1.54) provides the convergence of (N − q) components of state variables, by this way, we recover the ( p − q) missed information in (1.43). Moreover, there exist some scalar functions u 1 , · · · , u p independent of the sequence {Hn } of applied boundary controls, such that the sequence {Un } of the corresponding solutions satisfies the following conditions: u (k) n → u r as n → +∞

(1.56)

for all n r −1 + 1  k  n r and 1  r  p in the space (1.35). Thus, the approximate boundary synchronization by p-groups in the consensus sense is in fact in the pinning sense. Nevertheless, unlike the case N p = N − p, these functions u 1 , · · · , u p are linearly dependent (cf. Chap. 11).

1.9 Organization The organization of this monograph is as follows. Some preliminaries on linear algebra which are necessary in the whole book are presented in Chap. 2 We consider the exact boundary synchronization in Part 1 (from Chaps. 3 to 7) and the approximate boundary synchronization in Part 2 (from Chaps. 8 to 11) for a coupled system of wave equations with Dirichlet boundary controls.

1.9 Organization

17

In Part 3 (from Chaps. 12 to 15) and Part 4 (from Chaps. 16 to 18), we study the same subjects, respectively, for the coupled system of wave equations with the following Neumann boundary controls: ∂ν U = D H on (0, +∞) × 1 ,

(1.57)

where ∂ν denotes the outward normal derivative. Part 5 (from Chaps. 19 to 25) and Part 6 (from Chaps. 26 to 31) are devoted to the corresponding consideration for the coupled system of wave equations with the following coupled Robin boundary controls: ∂ν U + BU = D H on (0, +∞) × 1 .

(1.58)

where B = (bi j ) is a boundary coupling matrix of order N with constant elements. Although the consideration in Part 3 and Part 5 (resp. Part 4 and Part 6) seems to be similar to that in Part 1 (resp. Part 2), however, the solution to the corresponding problem with Neumann or coupled Robin boundary conditions has less regularity than that with Dirichlet boundary conditions, moreover, there is a second coupling matrix in the coupled Robin boundary conditions, technically speaking, we will encounter more difficulties and some different and new treatments are necessarily needed. We omitted the details here. In the last chapter—Chap. 32, we will give some related literature as well as certain prospects for the further study of exact and approximate boundary synchronizations.

Chapter 2

Algebraic Preliminaries

This chapter contains some algebraic preliminaries, which are useful in the whole book. In this chapter, we denote by A a matrix of order N , by D a full column-rank matrix of order N × M with M  N , and by C p a full row-rank matrix of order (N − p) × N with 0 < p < N . All these matrices are of constant entries.

2.1 Bi-orthonormality Definition 2.1 A subspace V of R N is a supplement of a subspace W of R N , if W + V = R N , W ∩ V = {0},

(2.1)

where the sum of subspaces W and V is defined as W + V = {w + v :

w ∈ W, v ∈ V }.

(2.2)

In this case, W is also a supplement of V . As a direct consequence of the dimension equality: dim(W + V ) = dim(W ) + dim(V ) − dim(W ∩ V ),

(2.3)

we have the following Proposition 2.2 A subspace V of R N is a supplement of a subspace W of R N if and only if © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_2

19

20

2 Algebraic Preliminaries

dim(W ) + dim(V ) = N , W ∩ V = {0}.

(2.4)

In particular, W and V ⊥ (or W ⊥ and V ) have the same dimension. Definition 2.3 (cf. [15, 83]) Two d-dimensional (0 < d  N ) subspaces V and W of R N are bi-orthonormal if there exist a basis (1 , . . . , d ) of V and a basis (η1 , . . . , ηd ) of W, such that (k , ηl ) = δkl , 1  k, l  d,

(2.5)

where δkl is the Kronecker symbol. Proposition 2.4 Let V and W be two subspaces in R N . Then the equality dim(V ⊥ ∩ W ) = dim(V ∩ W ⊥ )

(2.6)

holds if and only if V and W have the same dimension. Proof First, by the relationship

we have

(V ⊥ + W )⊥ = V ∩ W ⊥ ,

(2.7)

dim(V ⊥ + W ) = N − dim(V ∩ W ⊥ ).

(2.8)

Next, noting (2.3), we have dim(V ⊥ + W ) = dim(V ⊥ ) + dim(W ) − dim(V ⊥ ∩ W ).

(2.9)

Then, it follows that dim(V ⊥ ) + dim(W ) − dim(V ⊥ ∩ W ) = N − dim(V ∩ W ⊥ ), namely,

dim(W ) − dim(V ) = dim(V ⊥ ∩ W ) − dim(V ∩ W ⊥ ).

(2.10)

(2.11) 

The proof is thus complete.

Proposition 2.5 Let V and W be two nontrivial subspaces of R N . Then V and W are bi-orthonormal if and only if V ⊥ ∩ W = V ∩ W ⊥ = {0}.

(2.12)

Proof Assume that (2.12) holds. By Proposition 2.4, V and W have the same dimension d. In order to show that V and W are bi-orthonormal, we will construct, under condition (2.12), a basis (1 , . . . , d ) of V and a basis (η1 , . . . , ηd ) of W , such

2.1 Bi-orthonormality

21

that (2.5) holds. For this purpose, let (1 , . . . , d ) be a basis of V . For each i with 1  i  d we define a subspace Vi of V by Vi = Span{1 , . . . , i−1 , i+1 , . . . , d }.

(2.13)

Since it is easy to see from (2.3) that dim(W ∩ Vi⊥ )  dim(W ) + dim(Vi⊥ ) − N = d + (N − d + 1) − N = 1,

(2.14) (2.15)

there exists a nontrivial vector ηi ∈ W ∩ Vi⊥ , such that (i , ηi ) = 0. Otherwise, we have ηi ∈ V ⊥ , then ηi ∈ W ∩ V ⊥ = {0} because of (2.12), which leads to a contradiction. Thus, we can choose ηi such that (i , ηi ) = 1. Moreover, since ηi ∈ Vi⊥ , obviously we have ( j , ηi ) = 0,

j = 1, . . . , i − 1, i + 1, . . . , d.

(2.16)

In this way, we obtain a family of vectors (η1 , . . . , ηd ) of W , which satisfies the relation (2.5), therefore the vectors η1 , . . . , ηd are linearly independent. Since W has the same dimension d as V , the family of these linearly independent vectors (η1 , . . . , ηd ) is in fact a basis of W . Conversely, assume that V and W are bi-orthonormal. Let (1 , . . . , d ) and (η1 , . . . , ηd ) be the bases of V and W , respectively, such that (2.5) holds. Since V and W have the same dimension, by Proposition 2.4, it is sufficient to check the first condition of (2.12). For this purpose, let x ∈ V ⊥ ∩ W. Since x ∈ W , there exist some coefficients αk (k = 1, . . . , d) such that x=

d 

αk ηk .

(2.17)

k=1

Noting that x ∈ V ⊥ , the bi-orthonormal relationship (2.5) implies that 0 = (x, l ) =

d 

αk (ηk , l ) = αl , 1  l  d.

(2.18)

k=1

Thus x = 0, therefore V ⊥ ∩ W = {0}.



Definition 2.6 A subspace V of R N is invariant for A, if AV ⊆ V.

(2.19)

Proposition 2.7 (cf. [25]) A subspace V of R N is invariant for A if and only if its orthogonal complement V ⊥ is invariant for A T . In particular, for any given matrix C, since {Ker(C)}⊥ = Im(C T ), the subspace Ker(C) is invariant for A if and only if the

22

2 Algebraic Preliminaries

subspace Im(C T ) is invariant for A T . Furthermore, V admits a supplement W such that both V and W are invariant for A if and only if their orthogonal complements V ⊥ and W ⊥ are invariant for A T , and W ⊥ is a supplement of V ⊥ . Proof Assume that the subspace V of R N is invariant for A. By definition, we have (Ax, y)R N = (x, A T y)R N = 0, ∀x ∈ V, ∀y ∈ V ⊥ .

(2.20)

This proves the first part of the Proposition. Now let V admit a supplement W such that both V and W are invariant for A. By the first part of this Proposition, both V ⊥ and W ⊥ are invariant for A T . Moreover, the relation V ⊥ ∩ W ⊥ = {V ⊕ W }⊥ = {0}

(2.21)

implies that W ⊥ is a supplement of V ⊥ . This finishes the proof of the second part of the Proposition.  Proposition 2.8 Let W be a nontrivial invariant subspace of A T . Then W admits a supplement V ⊥ , which is also invariant for A T if and only if V is bi-orthonormal to W and invariant for A. Proof Let V be a subspace which is invariant for A and bi-orthonormal to W . By Proposition 2.7, V ⊥ is an invariant subspace for A T . By Definition 2.1, (2.12) holds. By Proposition 2.5, V and W have the same dimension, then it follows that dim(V ⊥ ) + dim(W ) = N − dim(V ) + dim(W ) = N .

(2.22)

By Proposition 2.2, V ⊥ is a supplement of W . Conversely, assume that V ⊥ is a supplement of W , and invariant for A T . By Proposition 2.7, (V ⊥ )⊥ = V is invariant for A. On the other hand, since V ⊥ is a supplement of W , by Proposition 2.2, we have N = dim(V ⊥ ) + dim(W ) = N − dim(V ) + dim(W ),

(2.23)

so V and W have the same dimension d > 0. Moreover, noting that V ⊥ ∩ W = {0}, by Proposition 2.1, (2.12) holds. Then by Proposition 2.5, V and W are bi-orthonormal. The proof is then complete.  By duality, Proposition 2.8 can be formulated as Proposition 2.9 Let V be a nontrivial invariant subspace of A. Then V admits a supplement W ⊥ , which is also invariant for A if and only if W is bi-orthonormal to V and invariant for A T . Remark 2.10 The subspace W ⊥ is a supplement of V and is invariant for A, so the matrix A is diagonalizable by blocks according to the decomposition V ⊕ W ⊥ , where ⊕ stands for the direct sum of subspaces.

2.1 Bi-orthonormality

23

Proposition 2.11 Let C and K be two matrices of order M × N and N × L, respectively. Then, the equality rank(C K ) = rank(K ) (2.24) holds if and only if Ker(C) ∩ Im(K ) = {0}.

(2.25)

Proof Define the linear map C by Cx = C x for all x ∈ Im(K ). Then we have Im(C) = Im(C K ), Ker(C) = Ker(C) ∩ Im(K ).

(2.26)

From the rank-nullity theorem: dim Im(C) + dim Ker(C) = rank(K ),

(2.27)

rank (C K ) + dim (Ker(C) ∩ Im(K )) = rank(K ),

(2.28)

namely, 

we achieve the proof of Proposition 2.11.

2.2 Kalman’s Criterion Proposition 2.12 Let d  0 be an integer. We have the following assertions: (i) The rank condition rank(D, AD, . . . , A N −1 D)  N − d

(2.29)

holds if and only if the dimension of any given invariant subspace of A T , contained in Ker(D T ), does not exceed d. (ii) The rank condition rank(D, AD, . . . , A N −1 D) = N − d

(2.30)

holds if and only if the largest dimension of invariant subspaces of A T , contained in Ker(D T ), is equal to d. Proof (i) Let

K = (D, AD, . . . , A N −1 D)

(2.31)

and V be an invariant subspace of A T , contained in Ker(D T ). Clearly, we have V ⊆ Ker(K T ).

(2.32)

24

2 Algebraic Preliminaries

Assume that (2.29) holds, then dim(V )  dim Ker(K T ) = N − rank(K T )  N − (N − d) = d.

(2.33)

Conversely, assume that (2.29) does not hold, then dim Ker(K T ) = N − rank(K T ) > N − (N − d) = d.

(2.34)

Hence, there exist (d + 1) linearly independent vectors w1 , . . . , wd , wd+1 ∈ Ker(K T ). In particular, we have Span



{wk , A T wk , . . . , (A T ) N −1 wk } ⊆ Ker(D T ).

(2.35)

1kd+1

By Cayley–Hamilton’s Theorem, the subspace 

Span

{wk , A T wk , . . . , (A T ) N −1 wk }

(2.36)

1kd+1

is invariant for A T and its dimension is greater than or equal to (d + 1). (ii) Equation (2.30) can be written as

and

rank(D, AD, . . . , A N −1 D)  N − d

(2.37)

rank(D, AD, . . . , A N −1 D)  N − d.

(2.38)

The rank condition (2.37) means that dim(V )  d for any given invariant subspace V of A T , contained in Ker(D T ). While, by (i), the rank condition (2.38) implies the existence of a subspace V0 , which is invariant for A T and is contained in Ker(D T ),  such that dim(V0 )  d. It proves (ii). The proof is complete. As a direct consequence, we get easily the following well-known Hautus test (cf. [17]). Corollary 2.13 Kalman’s rank condition rank(D, AD, . . . , A N −1 D) = N

(2.39)

is equivalent to Hautus test rank(D, A − λI ) = N , ∀λ ∈ C.

(2.40)

Ker(D, A − λI )T = Ker(D T ) ∩ Ker(A T − λI ),

(2.41)

Proof Noting that

2.2 Kalman’s Criterion

25

Equation (2.40) is equivalent to Ker(D T ) ∩ Ker(A T − λI ) = {0},

(2.42)

which means that none of eigenvectors of A T is contained in Ker(D T ). Therefore, Ker(D T ) does not contain any invariant subspace of A T , which is just what Proposition 2.12 indicates in the case d = 0. 

2.3 Condition of C p -Compatibility Definition 2.14 The matrix A satisfies the condition of C p -compatibility if there exists a unique matrix A p of order (N − p), such that C p A = A pC p.

(2.43)

The matrix A p will be called the reduced matrix of A by C p . Proposition 2.15 The matrix A satisfies the condition of C p -compatibility if and only if the kernel of C p is an invariant subspace of A: AKer(C p ) ⊆ Ker(C p ).

(2.44)

Moreover, the reduced matrix A p is given by

where

A p = C p AC + p,

(2.45)

T T −1 C+ p = C p (C p C p )

(2.46)

is the Moore–Penrose inverse of C p . Proof Assume that (2.43) holds. Then Ker(C p ) ⊆ Ker(A p C p ) = Ker(C p A).

(2.47)

Since C p Ax = 0 for any given x ∈ Ker(C p ), we get Ax ∈ Ker(C p ) for any given x ∈ Ker(C p ), namely, (2.44) holds. Conversely, assume that (2.44) holds true. Then by Proposition 2.7, we have A T Im(C Tp ) ⊆ Im(C Tp ). Then, there exists a matrix A p of order (N − p), such that

(2.48)

26

2 Algebraic Preliminaries T

A T C Tp = C Tp A p ,

(2.49)

namely, (2.43) holds. Moreover, let (e1 , . . . , e p ) be a basis of Ker(C p ). Noting (2.44), we get (C p A − A p C p )(e1 , . . . , e p , C Tp ) = (0, . . . , 0, (C p A − A p C p )C Tp ).

(2.50)

Since the N × N matrix (e1 , . . . , e p , C Tp ) is invertible, (2.43) is equivalent to (C p A − A p C p )C Tp = 0.

(2.51)

Since the matrix C p C Tp is invertible, it follows that A p = C p AC Tp (C p C Tp )−1 ,

(2.52)

which gives just (2.45)–(2.46). The proof is complete.



Proposition 2.16 Assume that the matrix A satisfies the condition of C p -compatibility. Then for given matrix D of order N × M, we have N − p−1

rank(C p D, A p C p D, . . . , A p

= rank(C p D, C p AD, . . . , C p A

N −1

C p D)

(2.53)

D).

Proof By Cayley–Hamilton’s Theorem, we have N − p−1

rank(C p D, A p C p D, . . . , A p = rank(C p D, A p C p D, . . . ,

C p D)

(2.54)

N −1 A p C p D).

l

Then, noting C p Al = A p C p for any given integer l  0, we get N −1

(C p D, A p C p D, . . . , A p

C p D)

(2.55)

= (C p D, C p AD, . . . , C p A N −1 D). The proof is complete.



Definition 2.17 A subspace V is called A-marked, if it is invariant for A and there exists a Jordan basis of A in V , which can be extended (by adding new vectors) to a Jordan basis of A in C N . V is strongly A-marked, if it is invariant for A and every Jordan basis of A in V can be extended to a Jordan basis of A in C N . Obviously, any subspace composed of eigenvectors of A is strongly A-marked. Moreover, if each eigenvalue λ of A is either semi-simple (λ has the same alge-

2.3 Condition of C p -Compatibility

27

braic and geometrical multiplicity), or dim Ker(A − λI ) = 1, then every invariant subspace of A is strongly A-marked. The existence of non-marked invariant subspaces is sometimes overlooked in linear algebra. We refer [9] for a complete discussion on this topic. Definition 2.18 Assume that the matrix A satisfies the condition of C p -compatibility (2.43). An (N − q) × N (0  q < p) full row-rank matrix Cq∗ is called the induced extension matrix of C p , related to the matrix A, if (a) Ker(Cq∗ ) is contained in Ker(C Tp ), (b) Ker(Cq∗ ) is invariant for A and admits a supplement which is also invariant for A, (c) Ker(Cq∗ ) is the largest one satisfying the previous two conditions. By duality, the above requirements are equivalent to the following ones: (i) Im(Cq∗ T ) contains Im(C Tp ), (ii) Im(Cq∗ T ) is invariant for A T and admits a supplement which is also invariant for A T , (iii) Im(Cq∗ T ) is the least one satisfying the previous two conditions. This leads to the following explicit construction for the induced extension matrix Cq∗ . Let ( j)

E0 = 0,

( j)

A T Ei

( j)

= λ j Ei

( j)

+ Ei−1 , 1  i  m j , 1  j  r

(2.56)

denote the Jordan chain of A T contained in Im(C Tp ). Assume that the subspace Im(C p T ) is A T -marked. Then for any j with 1  j  r , there exist new root vectors ( j) ( j) Em j +1 , . . . , Em ∗ which extend the Jordan chain in Im(C p T ) to a Jordan chain in C N : j

( j)

E0 = 0,

( j)

A T Ei

( j)

= λ j Ei

( j)

+ Ei−1 , 1  i  m ∗j , 1  j  r.

(2.57)

Then, we define the (N − q) × N full row-rank matrix Cq∗ by Cq∗T = (E1(1) , . . . , Em(1)∗ ; . . . . . . ; E1(r ) , . . . , Em(rr∗) ), 1

where q=N−

r 

m ∗j .

(2.58)

(2.59)

j=1

We justify the above construction by Proposition 2.19 Assume that the matrix A satisfies the condition of C p -compatibility (2.43) and that the subspace Im(C Tp ) is A T -marked. Then the matrix Cq∗ defined by (2.58) satisfies the requirements (i)–(iii).

28

2 Algebraic Preliminaries

Proof Clearly, Im(Cq∗ T ) contains Im(C Tp ). On the other hand, by Jordan’s theorem, the space C N can be decomposed into a direct sum of all the Jordan subspaces of A T , therefore, Im(Cq∗ T ), being direct sum of some Jordan subspaces, is invariant and admits a supplement which is invariant for A T . Thus, we only have to check that Im(Cq∗ T ) is the least one. If q = p, then Cq∗ = C p is obviously the least one. ( j)

Otherwise if q < p, there exists at least a root vector Em ∗ , which does not belong to j

/ Im(C Tp ). We Im(C p T ). Without loss of generality, assume that the root vector Em(1)∗ ∈ 1 ∗T remove it from Im(Cq ), and let q∗T = (E1(1) , . . . , E (1)∗ ; . . . , E1(r ) , . . . , Em(r∗) ). C m −1 r 1

(2.60)

q∗T ) does not admit any supplement, which is invariant for A T . We claim that Im(C q∗T ) admits a supplement W  , which is invariant for Otherwise, assume that Im(C T ∗T   x ∈ Im(Cq ) and  y ∈ W , such that A . Then, there exist  Em(1)∗ =  x + y.

(2.61)

1

Noting that

A T Em(1)∗ = λ1 Em(1)∗ + Em(1)∗ −1 , 1

we have

1

(2.62)

1

x + AT  y = λ1 ( x + y) + Em(1)∗ −1 . AT 

(2.63)

x − Em(1)∗ −1 + (A T − λ1 ) y = 0. (A T − λ1 )

(2.64)

1

It then follows that 1

On the other hand, noting that q∗T ) x − Em(1)∗ −1 ∈ Im(C (A T − λ1 ) 1

and

, (A T − λ1 ) y∈W

(2.65)

and

AT  y = λ1 y,

(2.66)

it follows from (2.64) that x = λ1 x + Em(1)∗ −1 AT  1

therefore,

x − Em(1)∗ ) = λ1 ( x − Em(1)∗ ). A T ( 1

1

(2.67)

Since ( x − Em(1)∗ ) ∈ Span{E1(1) , . . . , Em(1)∗ } and E1(1) is the only eigenvector of A T in 1

1

Span{E1(1) , . . . , Em(1)∗ }, there exists a ∈ R, such that  x − Em(1)∗ = aE1(1) , namely,  x− 1 1 (1) (1) (1) (1) ∗T  x and E1 belong to Im(Cq ), so does the vector  x − aE1 . But aE1 = E ∗ . Since  m1

q∗T ), we get thus a contradiction. because of (2.60), Em(1)∗ ∈ / Im(C 1



2.3 Condition of C p -Compatibility

29

Proposition 2.20 Assume that the matrix A satisfies the condition of C p -compatibility (2.43) and that Ker(C p ) is A-marked. Then, there exists a subspace Span{e1 , . . . , eq } such that (a) K er (C p ) is contained in Span{e1 , . . . , eq }. (b) A T admits an invariant subspace Span{E 1 , . . . , E q } which is bi-orthonormal to Span{e1 , . . . , eq }. (c) Span{e1 , . . . , eq } is the least one satisfying the previous two conditions. Proof By Proposition 2.9, it suffices to show that Ker(C p ) has a least extension Span{e1 , . . . , eq }, which is invariant for A and admits a supplement which is also invariant for A. Let Ker(C p ) = Span{e1 , . . . , e p }. Defining D Tp = (e1 , . . . , e p ), we have Im(D Tp ) = Ker(C p ) and Ker(D p ) = Im(C Tp ).

(2.68)

By Proposition 2.7, the condition of C p -compatibility (2.43) implies that A T Im(C Tp ) ⊆ Im(C Tp ), namely, A T Ker(D p ) ⊆ Ker(D p ). Therefore, A T satisfies the condition of D p -compatibility and Im(D Tp ) = Ker(C p ) is A-marked. Then, by Proposition 2.19, D p has a least extension Dq∗ such that Im(Dq∗ T ) is invariant for A and admits a supplement which is also invariant for A. In other words, noting (2.68), Ker(C p ) has a least extension Span{e1 , . . . , eq }, which admits a supplement such that both two subspaces are invariant for A. The proof is then complete.  Proposition 2.21 Assume that the matrix A satisfies the condition of C p -compatibility (2.43). Let {xl(k) }1kd,1lrk be a system of root vectors of the matrix A, corresponding to the eigenvalues λk (1  k  d), such that for each k(1  k  d) we have (k) , 1  l  rk . Axl(k) = λk xl(k) + xl+1

(2.69)

Define the following projected vectors by x l(k) = C p xl(k) , 1  k  d, 1  l  r k ,

(2.70)

where d(1  d  d) and r k (1  r k  rk ) are given by (2.71) below, then the system {x l(k) }1kd,1lr k forms a system of root vectors of the reduced matrix A p given by (2.43). In particular, if A is similar to a real symmetric matrix, so is A p . Proof Since Ker(C p ) is an invariant subspace of A, without loss of generality, we may assume that there exist some integers d(1  d  d) and r k (1  r k  rk ), such that {xl(k) }1lr k ,1kd forms a root system for the restriction of A to the invariant subspace Ker(C p ). Then, Ker(C p ) = Span{xl(k) : 1  k  d, 1  l  r k }.

(2.71)

30

2 Algebraic Preliminaries

In particular, we have d  (rk − r k ) = N − p.

(2.72)

k=1

Noting that C Tp (C p C Tp )−1 C p is a projection from R N onto Im(C Tp ), we have C Tp (C p C Tp )−1 C p x = x, ∀x ∈ Im(C Tp ).

(2.73)

On the other hand, by R N = Im(C Tp ) ⊕ Ker(C p ) we can write xl(k) =  xl(k) +  xl(k) with  xl(k) ∈ Im(C Tp ),  xl(k) ∈ Ker(C p ).

(2.74)

It follows from (2.70) that x l(k) = C p xl(k) , 1  k  d, 1  l  r k .

(2.75)

Then, noting (2.45)–(2.46) and (2.73), we have A p x l(k) = C p AC Tp (C p C Tp )−1 C p xl(k) = C p A xl(k) .

(2.76)

Since Ker(C p ) is invariant for A, A xl(k) ∈ Ker(C p ), then C p A xl(k) = 0. It follows that A p x l(k) = C p A( xl(k) +  xl(k) ) = C p Axl(k) . (2.77) Then, using (2.69), it is easy to see that (k) (k) A p x l(k) = C p (λk xl(k) + xl+1 ) = λk x l(k) + x l+1 .

(2.78)

(k) (k) Therefore, x (k) 1 , x 2 , . . . , x r k is a Jordan chain with length r k of the reduced matrix A p , corresponding to the eigenvalue λk . Since dim Ker(C p ) = p, the projected system {x l(k) }1kd,1lr k is of rank (N − p). On the other hand, because of (2.72), the system {x l(k) }1lr k ,1kd contains (N − p) vectors, therefore, forms a system of root vectors of the reduced matrix A p . The proof is complete. 

Part I

Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls: Exact Boundary Synchronization

We consider the following coupled system of wave equations with Dirichlet boundary controls: ⎧  ⎨ U − U + AU = 0 in (0, +∞) × , U =0 on (0, +∞) × 0 , (I ) ⎩ U = DH on (0, +∞) × 1 with the initial condition t =0:

1 in , 0 , U  = U U =U

(I0)

∪ 0 such that where  ⊂ Rn is a bounded domain with smooth boundary  = 1  2  1 ∩ 0 = ∅ and mes(1 ) > 0; “” stands for the time derivative;  = nk=1 ∂∂x 2 is the k  T T  Laplacian operator; U = u (1) , · · · , u (N ) and H = h (1) , · · · , h (M) (M  N ) denote the state variables and the boundary controls, respectively; the coupling matrix A = (ai j ) is of order N and D as the boundary control matrix is a full column-rank matrix of order N × M, both with constant elements. In this part, the exact boundary synchronization and the exact boundary Synchronization by groups for system (I) will be presented and discussed, while, correspondingly, the approximate boundary synchronization and the approximate boundary synchronization by groups for system (I) will be introduced and considered in the next part (Part 2).

Chapter 3

Exact Boundary Controllability and Non-exact Boundary Controllability

Since the exact boundary synchronization on a finite time interval is closely linked with the exact boundary null controllability, we first consider the exact boundary null controllability and the non-exact boundary null controllability for system (I) of wave equations with Dirichlet boundary controls in this chapter.

3.1 Exact Boundary Controllability Since the exact boundary synchronization on a finite time interval is closely linked with the exact boundary null controllability, we first consider the exact boundary null controllability and the non-exact boundary null controllability in this chapter. Let  ⊂ Rn be a bounded domain with smooth boundary  = 1 ∪ 0 with mes(1 ) > 0, such that  1 ∩  0 = ∅. Furthermore, we assume that there exists x0 ∈ Rn such that, setting m = x − x0 , we have the following multiplier geometrical condition (cf. [7, 26, 61, 62]): (m, ν) > 0, ∀x ∈ 1 ;

(m, ν)  0, ∀x ∈ 0 ,

(3.1)

where ν is the unit outward normal vector and (·, ·) denotes the inner product in Rn . Let (1) (M) (3.2) W = (w (1) , · · · , w (M) )T , H = (h , · · · , h )T . Consider the following coupled system of wave equations: ⎧  ⎨ W − W + AW = 0 in (0, +∞) × , W =0 on (0, +∞) × 0 , ⎩ W =H on (0, +∞) × 1 © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_3

(3.3) 33

34

3 Exact Boundary Controllability and Non-exact Boundary Controllability

with the initial condition t =0:

1 in , 0 , W  = W W =W

(3.4)

where the coupling matrix A = (a i j ) of order M has real constant elements. Definition 3.1 System (3.3) is exactly null controllable if there exists a positive constant T > 0, such that for any given initial data 0 , W 1 ) ∈ (L 2 ()) M × (H −1 ()) M , (W

(3.5)

one can find a boundary control H ∈ L 2 (0, T ; (L 2 (1 )) M ), such that problem (3.3)– (3.4) admits a unique weak solution W = W (t, x) with (W, W  ) ∈ C 0 ([0, T ]; (L 2 () × H −1 ()) M ),

(3.6)

which satisfies the final null condition t=T :

W = 0, W  = 0 in .

(3.7)

2 (0, +∞; (L 2 (1 )) M ) with compact In other words, one can find a control H ∈ L loc support in [0, T ], such that the solution W = W (t, x) to the corresponding problem (3.3)–(3.4) satisfies the following condition:

tT :

W ≡ 0 in .

(3.8)

Remark 3.2 More generally, system (3.3) will be called to be exactly controllable if the final null condition (3.7) is replaced by the following inhomogeneous final condition 1 in . 0 , W  = W (3.9) t=T : W =W However, it is well known that for a linear time-invertible system, the exact boundary null controllability with (3.7) is equivalent to the exact boundary controllability with (3.9) (cf. [61, 62, 73]). So, we will not distinguish these two notions of exact controllability later. For a single wave equation, by means of the Hilbert Uniqueness Method (HUM) proposed by J.-L. Lions in [61], the exact boundary null controllability has been studied in the literature (cf. [26, 62]), but a few were done for general coupled systems of wave equations. If the coupling matrix A is symmetric and positive definite, then the exact boundary null controllability of system (3.3) can be transformed to the case of a single wave equation (cf. [61, 62]). However, in order to study the exact boundary synchronization, we have to establish the exact boundary null controllability for the coupled system (3.3) of wave equations with any given coupling matrix A. In this chapter, we will use a result on the observability of compactly perturbed systems in

3.1 Exact Boundary Controllability

35

[69] to get the observability of the corresponding adjoint problem, and then the exact boundary null controllability follows from the standard HUM. Now let (3.10)  = (φ (1) , · · · , φ (M) )T . Consider the corresponding adjoint system: 

T

 −  + A  = 0 in (0, T ) × , =0 on (0, T ) × ,

(3.11)

T

A being the transpose of A, with the initial condition t =0:

1 in .  0 ,  =  =

(3.12)

It is well known (cf. [60, 64, 70]) that the adjoint problem (3.11)–(3.12) is well-posed in the space V × H: M  M  V = H01 () , H = L 2 () .

(3.13)

Moreover, we will prove the following direct and inverse inequalities. Theorem 3.3 Let T > 0 be suitably large. Then there exist positive constants c and 0 ,  1 ) ∈ V × H, the solution  to problem c such that for any given initial data( (3.11)–(3.12) satisfies the following inequalities:

T



c 0

1

0 2V +  1 2H  c |∂ν |2 ddt  



T 0

1

|∂ν |2 ddt,

(3.14)

where ∂ν stands for the outward normal derivative on the boundary. Before proving Theorem 3.3, we first give a unique continuation result as follows. Proposition 3.4 Let B be a matrix of order M, and let  ∈ H 2 () be a solution to the following problem:   = B in , (3.15) =0 on . Assume furthermore that ∂ν  = 0 on 1 .

(3.16)

 = P

(3.17)

Then we have  ≡ 0. Proof Let

and

36

3 Exact Boundary Controllability and Non-exact Boundary Controllability

⎛

 B = P B P −1

 b11 0 ⎜ b21  b22 =⎜ ⎝  b M2 b M1 

⎞ ··· 0 ··· 0 ⎟ ⎟, ⎠ ···  · · · bM M

(3.18)

where  B is a lower triangular matrix of complex entries. Then (3.15)–(3.16) can be reduced to ⎧  (k) = kp=1  ( p) in , bkp φ ⎨ φ (k)  =0 (3.19) φ on , ⎩ (k) ∂ν φ = 0 on 1 for k = 1, · · · , M. In particular, for k = 1 we have ⎧ (1) in , (1) =  b11 φ ⎨ φ (1)  =0 φ on , ⎩ (1) ∂ν φ = 0 on 1 .

(3.20)

Thanks to Carleman’s unique continuation (cf. [14]), we get (1) ≡ 0. φ

(3.21)

Inserting (3.21) into the second set of (3.19) leads to ⎧ (2) in , (2) =  b22 φ ⎨ φ (2)  =0 φ on , ⎩ (2) on 1 ∂ν φ = 0

(3.22)

and we can repeat the same procedure. Thus, by a simple induction, we get successively that (k) ≡ 0, k = 1, · · · , M. φ (3.23) This yields that

 ≡ 0 =⇒  ≡ 0.

(3.24) 

The proof is complete. Proof of Theorem 3.3. We rewrite system (3.11) as 

 

where



 A=

     +B , =A  

(3.25)

   0 0 0 IM , ; B= T  0 −A 0

(3.26)



3.1 Exact Boundary Controllability

37

and I M is the unit matrix of order M. It is easy to see that A is a skew-adjoint operator with compact resolvent in V × H, and B is a compact operator in V × H. Therefore, they can generate, respectively, C 0 groups SA (t) and SA+B (t) in the energy space V × H. Following a perturbation result in [69], in order to prove the observability inequalities (3.14) for a system of this kind, it is sufficient to check the following assertions: (i) The direct and inverse inequalities

T



c 0

1

0 2V +  1 2H  c |∂ν |2 ddt  



T 0

1

|∂ν |2 ddt

(3.27)

0 ,  1 ) to the decoupled problem (3.11)–(3.12) hold for the solution  = SA (t)( with A = 0. (ii) The system of root vectors of A + B forms a Riesz basis of subspaces in V × H. That is to say, there exists a family of subspaces Vi × Hi (i  1) composed of root vectors of A + B, such that for any given x ∈ V × H, there exists a unique xi ∈ Vi × Hi for each i  1, such that x=

+∞ 

xi , c1 x2 

+∞ 

i=1

xi 2  c2 x2 ,

(3.28)

i=1

where c1 , c2 are positive constants. (iii) If (, ) ∈ V × H and λ ∈ C, such that (A + B)(, ) = λ(, ) and ∂ν  = 0 on 1 ,

(3.29)

then (, ) ≡ 0. For simplification of notation, we will still denote by V × H the complex Hilbert space corresponding to V × H. Since the assertion (i) is well known under the multiplier geometrical condition (3.1), we only have to verify (ii) and (iii). Verification of (ii). Let μi2 > 0 be an eigenvalue corresponding to an eigenvector φi of − with homogeneous Dirichlet boundary condition: 

−φi = μi2 φi in , on . φi = 0

(3.30)

Let Vi × Hi = {(αφi , βφi ) :

α, β ∈ C M }.

(3.31)

Obviously, the subspaces Vi × Hi (i = 1, 2, · · · ) are mutually orthogonal and V ×H=

 i1

Vi × Hi ,

(3.32)

38

3 Exact Boundary Controllability and Non-exact Boundary Controllability

where ⊕ stands for the direct sum of subspaces. In particular, for any given x ∈ V × H, there exist xi ∈ Vi × Hi for i  1, such that x=

+∞ 

xi , x2 =

i=1

+∞ 

xi 2 .

(3.33)

i=1

On the other hand, for any given i  1, Vi × Hi is an invariant subspace of A + B and of finite dimension. Then, the restriction of A + B to the subspace Vi × Hi is a linear bounded operator, therefore its root vectors constitute a basis in the finitedimensional complex space Vi × Hi . This together with (3.32)–(3.33) implies that the system of root vectors of A + B form a Riesz basis of subspaces in V × H. Verification of (iii). Let (, ) ∈ V × H and λ ∈ C, such that (3.29) holds. We have T (3.34) = λ and  − A  = λ , namely,



T

 = (λ2 I + A ) in , =0 on .

(3.35)

It follows from the classic elliptic theory that  ∈ H 2 (). Moreover, we have ∂ν  = 0 on 1 .

(3.36)

Thus, applying Proposition 3.4 to (3.35)–(3.36), we get  ≡ 0 then ≡ 0. The proof is complete.  By a standard application of HUM, from Theorem 3.3 we get the following Theorem 3.5 The coupled system (3.3) composed of M wave equations is exactly null controllable in the space (L 2 ()) M × (H −1 ()) M by means of M boundary controls. Remark 3.6 In Theorem 3.5, we do not need any assumption on the coupling matrix A. Remark 3.7 Similar results on the exact boundary null controllability for a coupled system of 1-D wave equations in the framework of classical solutions can be found in [19, 32]. Let us denote by Uad the admissible set of all boundary controls H , which realize the exact boundary null controllability of system (3.3). Since system (3.3) is exactly null controllable, Uad is not empty. Moreover, we have the following Theorem 3.8 Assume that system (3.3) is exactly null controllable in the space (L 2 ()) M × (H −1 ()) M . Then for > 0 small enough, the values of H ∈ Uad on (T − , T ) × 1 can be arbitrarily chosen.

3.1 Exact Boundary Controllability

39

Proof First recall that there exists a positive constant T0 > 0 independent of the initial data, such that for all T > T0 system (3.3) is exactly null controllable at the time T . Next let > 0 be such that T − > T0 and let  ∈ L 2 (T − , T ; (L 2 (1 )) M ) H

(3.37)

be arbitrarily given. We solve the backward problem for system (3.3) to get a solution  and the final  on the time interval [T − , T ] with the boundary function H = H W data   = 0.  = W (3.38) t=T : W Since T − > T0 , system (3.3) is still exactly controllable on the interval [0, T − ], then we can find a boundary control  ∈ L 2 (0, T − ; (L 2 (1 )) M ), H

(3.39)

 satisfies the initial condition: such that the corresponding solution W   = W1  = W0 , W W

t =0:

(3.40)

and the final condition: t = T − : Thus, setting

 H= 

and W =

 , W   = W   .  = W W

(3.41)

 , t ∈ (T − , T ), H  , t ∈ (0, T − ) H

(3.42)

 , t ∈ (T − , T ), W  , t ∈ (0, T − ), W

(3.43)

we check easily that W is a weak solution to problem (3.3)–(3.4) and the boundary control H realizes the exact boundary null controllability. The proof is then complete. 

3.2 Non-exact Boundary Controllability We have showed the exact boundary null controllability of the coupled system (3.3) composed of M wave equations by means of M boundary controls. In this paragraph, we will show that if the number of the boundary controls is fewer than M, then we cannot realize the exact boundary null controllability for the coupled sys-

40

3 Exact Boundary Controllability and Non-exact Boundary Controllability

0 , W 1 ) in the space tem (3.3) composed of M wave equations for all initial data (W (L 2 ()) M × (H −1 ()) M . For this purpose, we further investigate the exact boundary null controllability. Now we are interested in finding a control H 0 ∈ Uad , which has the least norm among all the others: H 0  L 2 (0,T ;(L 2 (1 )) M ) = inf H  L 2 (0,T ;(L 2 (1 )) M ) . H ∈Uad

(3.44)

 ∈ L 2 (0, T ; (L 2 (1 )) M ), we solve the backward problem For any given H ⎧  V − V + AV = 0 ⎪ ⎪ ⎨ V =0  V =H ⎪ ⎪ ⎩ t = T : V = V = 0

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in 

(3.45)

and define the linear map R:

 → (V (0), V  (0)) H

(3.46)

which is continuous from L 2 (0, T ; (L 2 (1 )) M ) into (L 2 ()) M × (H −1 ()) M because of the well-posedness. Let N be the kernel of R. Problem (3.44) becomes inf H − q L 2 (0,T ;(L 2 (1 )) M ) .

q∈N

(3.47)

Let P denote the orthogonal projection from L 2 (0, T ; (L 2 (1 )) M ) to N , which is a closed subspace. Then the boundary control H 0 = (I − P)H has the least norm. Moreover, we have the following Theorem 3.9 Assume that system (3.3) is exactly null controllable in (L 2 ()) M × (H −1 ()) M . Then there exists a positive constant c > 0, such that the control H 0 with the least norm given by (3.44) satisfies the following estimate: 0 , W 1 )(L 2 ()) M ×(H −1 ()) M H 0  L 2 (0,T ;(L 2 (1 )) M )  c(W

(3.48)

1 ) ∈ (L 2 ()) M × (H −1 ()) M . 0 , W for any given (W Proof Consider the linear map R ◦ (I − P) from the quotient space L 2 (0, T ; (L 2 (1 )) M )/N into (L 2 ()) M × (H −1 ()) M . First if R ◦ (I − P)H = 0, then (I − P)H ∈ N , thus H = P H ∈ N . Therefore R ◦ (I − P) is an injective. On the other hand, the exact boundary null controllability of system (3.3) implies that R ◦ (I − P) is a surjection, therefore a bijection which is continuous from L 2 (0, T ; (L 2 (1 )) M )/N into (L 2 ()) M × (H −1 ()) M . By Banach’s theorem of closed graph, the inverse of R ◦ (I − P) is also bounded from (L 2 ()) M ×

3.2 Non-exact Boundary Controllability

41

(H −1 ()) M into L 2 (0, T ; (L 2 (1 )) M )/N . This yields the inequality (3.48). The proof is complete.  In the case of partial lack of boundary controls, we have the following negative result. Theorem 3.10 Assume that the number of boundary controls is fewer than M. Then, no matter how large T > 0 is, the coupled system (3.3) composed of M 1 ) ∈ 0 , W wave equations is not exactly null controllable for all initial data (W (L 2 ()) M × (H −1 ()) M . Proof Without loss of generality, we assume that h the special initial data as

(1)

≡ 0. Let θ ∈ D(). We choose

1 = 0. 0 = (θ, 0, · · · , 0)T , W W

(3.49)

If system (3.3) is exactly null controllable, following Theorem 3.9, the boundary control H0 with the least norm satisfies the following estimate: H 0  L 2 (0,T ;(L 2 (1 )) M )  cθ  L 2 () .

(3.50)

Because of the well-posedness (cf. [60, 64, 70]), there exists a constant c > 0, such that W  L 2 (0,T ;(L 2 ()) M )  0 , W 1 )(L 2 ()) M ×(H −1 ()) M + H 0  L 2 (0,T ;(L 2 (1 )) M ) . c (W

(3.51)

Then it follows that W  L 2 (0,T ;(L 2 ()) M )  c (1 + c)θ  L 2 () . Now consider the first set in problem (3.3)–(3.4) with h backward problem:

(1)

(3.52)

≡ 0 into the following

⎧ (1) M ⎨ wtt − w (1) = − j=1 a 1 j w ( j) in (0, T ) × , w (1) = 0 on (0, T ) × , ⎩ t = T : w (1) = 0, ∂t w (1) = 0 in 

(3.53)

with the initial data t =0:

w(1) = θ, ∂t w (1) = 0 in .

(3.54)

Once again by the well-posedness for the backward problem (3.53), there exists a constant c > 0, such that θ  H01 ()  c W  L 2 (0,T ;(L 2 ()) M ) .

(3.55)

42

3 Exact Boundary Controllability and Non-exact Boundary Controllability

This, together with (3.52), gives a contradiction: θ  H01 ()  c c (1 + c)θ  L 2 () , ∀θ ∈ D().

(3.56) 

The proof is complete.

In order to make the control problem more flexible, we introduce a matrix D of order M with constant elements, and consider the following mixed problem for a coupled system of wave equations: ⎧  ⎨ W − W + AW = 0 in (0, T ) × , W =0 on (0, T ) × 0 , ⎩ W =DH on (0, T ) × 1

(3.57)

with the initial condition t =0:

0 , W  = W 1 in . W =W

(3.58)

Combining Theorems 3.5 and 3.10, we have the following result. Theorem 3.11 Under the multiplier geometrical condition (3.1), the coupled system (3.57) composed of M wave equations is exactly null controllable for any given initial 1 ) ∈ (L 2 ()) M × (H −1 ()) M if and only if the matrix D is of rank M. 0 , W data (W

Chapter 4

Exact Boundary Synchronization and Non-exact Boundary Synchronization

In the case of partial lack of boundary controls, we consider the exact boundary synchronization and the non-exact boundary synchronization in this chapter for system (I) with Dirichlet boundary controls.

4.1 Definition Let

U = (u (1) , · · · , u (N ) )T andH = (h (1) , · · · , h (M) )T

(4.1)

with M  N . Consider the coupled system (I) with the initial condition (I0). According to Theorem 3.11, system (I) is exactly null controllable if and only if rank(D) = N , namely, M = N and D is invertible. In the case of partial lack of boundary controls, we now give the following Definition 4.1 System (I) is exactly synchronizable, if there exists a positive con    0 , U 1 ) ∈ L 2 () N × H −1 () N , stant T > 0, such that for any given initial data (U 2 (0, +∞; (L 2 (1 )) M ) with compact there exists a suitable boundary control H ∈ L loc support in [0, T ], such that the corresponding solution U = U (t, x) to problem (I) and (I0) satisfies the following final condition tT :

u (1) ≡ u (2) ≡ · · · ≡ u (N ) := u,

(4.2)

where u = u(t, x), being unknown a priori, is called the exactly synchronizable state. In the above definition, through the boundary control acts on the time interval [0, T ], the synchronization is required not only at the time t = T , but also for all

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_4

43

44

4 Exact Boundary Synchronization and Non-exact Boundary Synchronization

t  T , namely, after all boundary controls are eliminated. Thus, this kind of synchronization is not a short-lived one, but exists once and for all, as is needed in applications. In fact, assuming that (4.2) is realized only at some moment T > 0 if we set hereafter H ≡ 0 for t > T , then, generally speaking, the corresponding solution to problem (I) and (I0) does not satisfy automatically the synchronization condition (4.2) for t > T . This is different from the exact boundary null controllability, where the solution vanishes with H ≡ 0 for t  T . To illustrate it, let us consider the following system: ⎧  u − u = 0 in (0, +∞) × , ⎪ ⎪ ⎨  v − v = u in (0, +∞) × , (4.3) u=0 on (0, +∞) × , ⎪ ⎪ ⎩ v=h on (0, +∞) × . Since the first equation is separated from the second one, for any given initial data u 1 ) we can first find a solution u. Once u is determined, we can find (cf. [85]) a ( u0,  boundary control h such that the solution v to the second equation satisfies the final conditions coming from the requirement of synchronization: t=T :

v = u, v  = u  .

(4.4)

If we set h ≡ 0 for t > T , generally speaking, we cannot get v ≡ u for t  T . So, in order to keep the synchronization for t  T , we have to maintain the boundary control h in action for t  T . However, for the sake of applications, what we want is to get the exact boundary synchronization by some boundary controls with compact support. Remark 4.2 If system (I) is exactly null controllable, then we have certainly the exact boundary synchronization. This trivial situation should be excluded. Therefore, in Definition 4.1 we should restrict ourselves to the case that the number of boundary controls is fewer than N , namely, M = rank(D) < N , so that system (I) is not exactly null controllable.

4.2 Condition of Compatibility Theorem 4.3 Assume that system (I) is exactly synchronizable but not exactly null controllable. Then the coupling matrix A = (ai j ) should satisfy the following condition of compatibility (row-sum condition): N p=1

akp := a, k = 1, · · · , N ,

(4.5)

4.2 Condition of Compatibility

45

where a is a constant independent of k = 1, · · · , N . Remark 4.4 By Theorem 3.11, the rank condition rank(D) < N

(4.6)

implies the non-exact boundary null controllability. Proof of Theorem 4.3. By synchronization (4.2), there exist a constant T > 0 and a scalar function u such that tT :

 akp u = 0 in , k = 1, · · · , N .

(4.7)

N 

 akp u = alp u in , k, l = 1, · · · , N .

(4.8)

u  − u +

N

p=1

In particular, we have tT :

N

p=1

p=1

On the other hand, since system (I) is not exactly null controllable, there exists at 0 , U 1 ) for which the corresponding synchronizable state u does least an initial data (U not identically vanish for t  T , whatever the boundary controls H would be chosen. This yields the condition of compatibility (4.5). The proof is complete.  Remark 4.5 In the study of synchronization for systems of ordinary differential equations, the row-sum condition (4.5) is imposed on the system according to physical meanings as a reasonable sufficient condition (in the most cases one takes a = 0 there). However, as we have seen before, for the synchronization of systems of partial differential equations on a finite time interval, the row-sum condition (4.5) is actually a necessary condition, which makes the theory of synchronization more complete for systems of partial differential equations. Let the synchronization matrix C1 be defined by ⎛

1 ⎜0 ⎜ C1 = ⎜ . ⎝ ..

−1 1 .. .

0 −1 .. .

··· ··· .. .

0 0 .. .

0 0 · · · 1 −1

⎞ ⎟ ⎟ ⎟ ⎠

.

(4.9)

(N −1)×N

C1 is an (N − 1) × N full row-rank matrix, and the exact boundary synchronization (4.2) can be equivalently written as

46

4 Exact Boundary Synchronization and Non-exact Boundary Synchronization

tT :

C1 U ≡ 0.

(4.10)

Let e1 = (1, · · · , 1)T .

(4.11)

Then, the condition of compatibility (4.5) is equivalent to the fact that the vector e1 is an eigenvector of A, corresponding to the eigenvalue a: Ae1 = ae1 .

(4.12)

On the other hand, since Ker(C1 )=Span{e1 }, condition (4.12) means that Ker(C1 ) is a one-dimensional invariant subspace of A: AKer(C1 ) ⊆ Ker(C1 ).

(4.13)

Moreover, by Proposition 2.15 (in which we take p = 1), the condition (4.13) is also equivalent to the existence of a unique matrix A1 of order (N − 1), such that C 1 A = A1 C 1 .

(4.14)

Such a matrix A1 = (a i j ) is called the reduced matrix of A by C1 , which can be given explicitly by A1 = C1 AC1+ , (4.15) where C1+ denotes the Moore–Penrose generalized inverse of C1 : C1+ = C1T (C1 C1T )−1 ,

(4.16)

in which C1T is the transpose of C1 . More precisely, the elements of A1 are given by j N ai j = (ai p − ai+1, p ) = (ai+1, p − ai p ) i, j = 1, · · · , N − 1. p=1

(4.17)

p= j+1

Remark 4.6 The synchronization matrix C1 given by (4.9) is not the unique choice. In fact, noting that Ker(C1 ) = Span{e1 } with e1 = (1, · · · , 1)T , any given (N − 1) × N full row-rank matrix C1 with the zero row sum, or equivalently, such that Ker(C1 ) is an invariant subspace of A (namely, (4.13) holds), can be always taken as a synchronization matrix. However, for fixing the idea, in what follows we always use the synchronization matrix C1 given by (4.9).

4.3 Exact Boundary Synchronization and Non-exact Boundary Synchronization

47

4.3 Exact Boundary Synchronization and Non-exact Boundary Synchronization We first give a rank condition for the boundary control matrix D, which is necessary for the exact boundary synchronization of system (I) in the space (L 2 ()) N × (H −1 ()) N . Theorem 4.7 Assume that system (I) is exactly synchronizable. Then we necessarily have rank(C1 D) = N − 1. (4.18) Proof If Ker(D T ) ∩ Im(C1T ) = {0}, then by Proposition 2.11, we have rank(C1 D) = rank(D T C1T ) = rank(C1T ) = N − 1.

(4.19)

Otherwise, there exists a unit vector E ∈ Ker(D T ) ∩ Im(C1T ). Assume that system (I) is exactly synchronizable, then for any given θ ∈ D(), there exists a boundary control H such that the solution U to system (I) with the following initial condition t =0:

U = Eθ, U  = 0 in 

(4.20)

satisfies the final condition (4.10). Then, applying E to problem (I) and (I0) and setting φ = (E, U ), it is easy to see that ⎧  ⎨ φ − φ = −(E, AU ) in (0, +∞) × , φ=0 on (0, +∞) × , ⎩ t =T : φ≡0 in .

(4.21)

Moreover, by a similar procedure as in the proof of Theorem 3.9, the control H can be chosen such that (4.22) H  L 2 (0,T ;(L 2 (1 )) N )  cθ L 2 () , where c > 0 is a positive constant. By the well-posedness of problem (I) and (I0), there exists a constant c > 0, such that U  L 2 (0,T ;(L 2 ()) N )  c θ L 2 () .

(4.23)

Then, by the well-posedness for the backward problem given by (4.21), there exists a constant c > 0, such that θ H01 ()  c U  L 2 (0,T ;(L 2 ()) N )  c c θ L 2 () , ∀θ ∈ D(). We get thus a contradiction which achieves the proof. As an immediate consequence of Theorem 4.7, we have the following

(4.24) 

48

4 Exact Boundary Synchronization and Non-exact Boundary Synchronization

Corollary 4.8 If rank(C1 D) < N − 1,

(4.25)

rank(D) < N − 1,

(4.26)

in particular if then, no matter how large T > 0 is, system (I) cannot be exactly synchronizable at the time T . Let

W1 = (w (1) , · · · , w (N −1) )T .

(4.27)

Then, under the condition of compatibility (4.5) and noting (4.14), the original problem (I) and (I0) for the variable U can be reduced to the following self-closed problem for the variable W = C1 U : ⎧ ⎨ W1 − W1 + A1 W1 = 0 in (0, +∞) × , (4.28) on (0, +∞) × 0 , W =0 ⎩ 1 on (0, +∞) × 1 , W1 = C 1 D H associated with the initial data 0 , W1 = C1 U 1 in , W1 = C 1 U

t =0:

(4.29)

where A1 is given by (4.15). Proposition 4.9 Under the condition of compatibility (4.5), the exact boundary synchronization of the original system (I) is equivalent to the exact boundary null controllability of the reduced system (4.28). Proof Clearly, the linear map 1 ) → (C1 U 0 , C1 U 1 ) 0 , U (U

(4.30)

is surjective from the space (L 2 ()) N × (H −1 ()) N onto the space (L 2 ()) N −1 × (H −1 ()) N −1 . Then the exact boundary synchronization of system (I) implies the exact boundary null controllability of the reduced system (4.28). On the other hand, the exact boundary synchronization of system (I) obviously follows from the exact boundary null controllability of system (4.28). The proof is complete.  By Theorem 3.11 and Proposition 4.9, we immediately get the following Theorem 4.10 Under the multiplier geometrical condition (3.1) and the condition of compatibility (4.5), system (I) is exactly synchronizable in the space (L 2 ()) N × (H −1 ()) N , provided that the rank condition (4.18) is satisfied.

Chapter 5

Exactly Synchronizable States

When system (I) possesses the exact boundary synchronization, the corresponding exactly synchronizable states will be studied in this chapter.

5.1 Attainable Set of Exactly Synchronizable States In the case that system (I) possesses the exact boundary synchronization at the time T > 0, it is easy to see that for t  T , the exactly synchronizable state u = u(t, x) defined by (4.2) satisfies the following wave equation with homogenous Dirichlet boundary condition: 

u  − u + au = 0 in (T, +∞) × , u=0 on (T, +∞) × ,

(5.1)

where a is given by (4.5). Hence, the evolution of the exactly synchronizable state u = u(t, x) with respect to t is completely determined by the values of (u, u t ) at the time t = T : u 1 in . (5.2) t = T : u = u0, u =  Theorem 5.1 Assume that the coupling matrix A satisfies the condition of compatibility (4.5). Then the attainable set of the values (u, u  ) at the time t = T of the exactly synchronizable state u = u(t, x) is actually the whole space L 2 () × H −1 () as     1 ) vary in the space L 2 () N × H −1 () N . 0 , U the initial data (U Proof For any given ( u0,  u 1 ) ∈ L 2 () × H −1 (), by solving the following backward problem   u − u + au = 0 in (0, T ) × , (5.3) u=0 on (0, T ) ×  © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_5

49

50

5 Exactly Synchronizable States

with the final condition t=T :

u 1 in , u = u0, u = 

(5.4)

we get the corresponding solution u = u(t, x). Then, under the condition of compatibility (4.5), the function U (t, x) = u(t, x)e1 ,

(5.5)

in which e1 = (1, · · · , 1)T , is the solution to problem (I) and (I0) with the null control H ≡ 0 and the initial condition t =0:

U = u(0, x)e1 , U  = u  (0, x)e1 .

(5.6)

Therefore, by solving problem (I) and (I0) with the null boundary control and the u1) initial condition (5.6), we can reach any given exactly synchronizable state ( u0,  u 1 ) ∈ L 2 () × H −1 () at the time t = T . This fact shows that any given state ( u0,  can be expected to be a exactly synchronizable state. Consequently, the set of the values (u(T ), u  (T )) of the exactly synchronizable state u = (t, x) at the time T is 0 , U 1 ) vary in the actually the whole space L 2 () × H −1 () as the initial data (U  2  N  −1 N space L () × H () . The proof is complete. 

5.2 Determination of Exactly Synchronizable States We now try to determine the exactly synchronizable state of system (I) for each given 1 ). 0 , U initial data (U As shown in Sect. 4.2, the vector e1 = (1, · · · , 1)T is an eigenvector of A, corresponding to the real eigenvalue a given by (4.5). Let 1 , · · · , r (resp. E1 , · · · , Er ) be a Jordan chain of length r of A (resp. of A T ), such that ⎧ 1  l  r, ⎨ Al = al + l+1 , A T Ek = aEk + Ek−1 , 1  k  r, (5.7) ⎩ 1  k, l  r, (Ek , l ) = δkl , where r = (1, · · · , 1)T , r +1 = 0, E0 = 0.

(5.8)

Clearly r = e1 is an eigenvector of A, respectively, E1 = E 1 is an eigenvector of A T associated with the same eigenvalue a.

5.2 Determination of Exactly Synchronizable States

51

Consider the projection P on the subspace Span{1 , · · · , r } as follows: P=

r

k ⊗ Ek ,

(5.9)

k=1

where ⊗ stands for the tensor product such that (k ⊗ Ek )U = (Ek , U )k , ∀U ∈ R N .

(5.10)

P can be represented by a matrix of order N . We can then decompose R N = Im(P) ⊕ Ker(P),

(5.11)

where ⊕ stands for the direct sum of subspaces. Moreover, we have Im(P) = Span{1 , · · · , r },

(5.12)

⊥  Ker(P) = Span{E1 , · · · , Er }

(5.13)

P A = A P.

(5.14)

and Now let U = U (t, x) be the solution to problem (I) and (I0). We define 

Uc := (I − P)U, Us := PU.

(5.15)

If system (I) is exactly synchronizable, we have tT :

U = ur ,

(5.16)

where u = u(t, x) is the exactly synchronizable state and r = (1, · · · , 1)T . Then, noting (5.15)–(5.16), we have  tT :

Uc = u(I − P)r = 0, Us = u Pr = ur .

(5.17)

Thus Uc and Us will be called the controllable part and the synchronizable part of U , respectively. Noting (5.14) and applying the projection P on problem (I) and (I0), we get immediately

52

5 Exactly Synchronizable States

Proposition 5.2 The controllable part Uc is the solution to the following problem: ⎧  U − Uc + AUc = 0 ⎪ ⎪ ⎨ c Uc = 0 Uc = (I − P)D H ⎪ ⎪ ⎩ 0 , Uc = (I − P)U 1 t = 0 : Uc = (I − P)U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in ,

(5.18)

while the synchronizable part Us is the solution to the following problem: ⎧  ⎪ ⎪ Us − Us + AUs = 0 ⎨ Us = 0 Us = P D H ⎪ ⎪ ⎩ 0 , Us = P U 1 t = 0 : Us = P U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in .

(5.19)

Remark 5.3 In fact, the boundary control H realizes the exact boundary null con0 , (I − P)U 1 ) ∈ Ker(P) × Ker(P) trollability for Uc with the initial data ((I − P)U on one hand, and the exact boundary synchronization for Us with the initial data 1 ) ∈ Im(P) × Im(P) on the other hand. 0 , P U (P U We define D N −1 = {D ∈ M N ×(N −1) : rank(D) = rank(C1 D) = N − 1}.

(5.20)

Proposition 5.4 Let a matrix D of order N × (N − 1) be defined by  ⊥ Im(D) = Span{Er } .

(5.21)

We have D ∈ D N −1 . Proof Clearly, rank(D) = N − 1. On the other hand, since (r , Er ) = 1, we have  ⊥ r ∈ / Span{Er } = Im(D). Noting that Ker(C1 ) = Span{r }, we have Im(D) ∩ Ker(C1 ) = {0}. Then, by Proposition 2.11, it follows that rank(C1 D) = rank(D) = N − 1. The proof is complete.  Theorem 5.5 When r = 1, we can take a boundary control matrix D ∈ D N −1 , such that the synchronizable part Us is independent of boundary controls H . Inversely, if the synchronizable part Us is independent of boundary controls H , then we necessarily have r = 1. Proof When r = 1, by Proposition 5.4, we can take a boundary control matrix D ∈ D N −1 as in (5.21). Noting (5.13), we have ⊥  Ker(P) = Span{E1 } = Im(D).

(5.22)

5.2 Determination of Exactly Synchronizable States

53

Then P D = 0, therefore (5.19) becomes a problem with homogeneous Dirichlet boundary condition. Consequently, the solution Us is independent of boundary controls H . Inversely, let H1 and H2 be two boundary controls which realize simultaneously the exact boundary synchronization of system (I). If the corresponding solutions Us to problem (5.19) are independent of the boundary controls H1 and H2 , then we have P D(H1 − H2 ) = 0 on (0, T ) × 1 .

(5.23)

By Theorem 3.8 and Proposition 4.9, the values of C1 D(H1 − H2 ) on (T − , T ) × 1 can be arbitrarily chosen. Since C1 D is invertible, the values of (H1 − H2 ) on (T − , T ) × 1 can be arbitrarily chosen. This yields that P D = 0. It follows that Im(D) ⊆ Ker(P).

(5.24)

Noting (5.13), we have dim Ker(P) = N − r and dim Im(D) = N − 1, then r = 1. The proof is complete.  Corollary 5.6 Assume that both Ker(C1 ) and Im(C1T ) are invariant subspaces of A. Then there exists a boundary control matrix D ∈ D N −1 , such that system (I) is exactly synchronizable and the synchronizable part Us is independent of boundary controls H . Proof Recall that Ker(C1 )=Span{e1 } with e1 = (1, · · · , 1)T . Since Im(C1T ) is an  ⊥ invariant subspace of A, by Proposition 2.7, Im(C1T ) = Ker(C1 ) is an invariant subspace of A T . Thus, e1 is also an eigenvector of A T , corresponding to the same eigenvalue a given by (4.5). Then taking E 1 = e1 /N , we have (E 1 , e1 ) = 1 and then r = 1. Thus, by Theorem 5.5 we can chose a boundary control matrix D ∈ D N −1 , such that the synchronizable part Us of system (I) is independent of boundary controls H . The proof is complete.  Remark 5.7 Under the condition r = 1, there exists an eigenvector E 1 of A T , such that (E 1 , e1 ) = 1. Clearly, if A is symmetric or A T satisfies also the condition of compatibility (4.5), then e1 is also an eigenvector of A T . Consequently, we can take E 1 = e1 /N such that (E 1 , e1 ) = 1. However, this condition is not always satisfied for any given matrix A. For example, let  2 −1 A= . 1 0 We have

 1 a = 1, e1 = , 1

(5.25) 

1 E1 = , −1

(5.26)

54

5 Exactly Synchronizable States

then (E 1 , e1 ) = 0.

(5.27)

 ⊥  ⊥ E 1 ∈ Span{e1 } = Ker (C1 ) = Im(C1T ).

(5.28)

In general, if (5.27) holds, then

This means that (E 1 , U ) is just a linear combination of the components of the vector C1 U , therefore it does not provide any new information for the synchronizable part Us .  ⊥ c N −1 ) be a basis of Span{E 1 } . Then Remark 5.8 Let ( c1 , · · · ,  a 0 , c1 , · · · , c N −1 ) = (e1 , c1 , · · · , c N −1 ) A(e1 , 0 A22

(5.29)

where A22 is a matrix of order (N − 1). Therefore, A is diagonalizable by blocks c1 , · · · , c N −1 ). under the basis (e1 , We next discuss the general case r  1. Let us denote φk = (Ek , U ), 1  k  r and write Us =

r r (Ek , U )k = φk k . k=1

(5.30)

(5.31)

k=1

Then, (φ1 , · · · , φr ) are the coordinates of Us on the bi-orthonormal basis {1 , · · · , er } and {E1 , · · · , Er }. Theorem 5.9 Let 1 , · · · , r (resp. E1 , · · · , Er ) be a Jordan chain of A (resp. A T ) corresponding to the eigenvalue a and r = (1, · · · , 1)T . Then the synchronizable part Us = (φ1 , · · · , φr ) can be determined by the solution of the following problem (1  k  r ): ⎧  φ − φk + aφk + φk−1 = 0 in (0, +∞) × , ⎪ ⎪ ⎨ k on (0, +∞) × 0 , φk = 0 (5.32) on (0, +∞) × 1 , φk = h k ⎪ ⎪ ⎩  0 ), φk = (Ek , U 1 ) in , t = 0 : φk = (Ek , U where φ0 = 0 and h k = (Ek , D H ). Moreover, the exactly synchronizable state is given by u = φr for t  T.

(5.33)

5.2 Determination of Exactly Synchronizable States

55

Proof First, for 1  k  r , we have (Ek , U ) = (Ek , Us ) = φk , (Ek , P D H ) =

r (El , D H )(Ek , l ) = (Ek , D H )

(5.34) (5.35)

l=1

and (Ek , PU0 ) = (Ek , U0 ), (Ek , PU1 ) = (Ek , U1 ).

(5.36)

Taking the inner product of (5.19) with Ek , we get (5.32)–(5.33). On the other hand, noting (5.16), we have tT :

φk = (Ek , U ) = (Ek , ur ) = uδkr

(5.37)

for 1  k  r. Thus, the exactly synchronizable state u is given by tT :

u = u(t, x) = φr (t, x).

(5.38) 

The proof is complete. In the special case r = 1, by Theorems 5.5 and 5.9, we have

Corollary 5.10 When r = 1, we can take D ∈ D N −1 , such that D T E1 = 0. Then the exactly synchronizable state u is determined by u = φ for t  T , where φ is the solution to the following problem with homogeneous Dirichlet boundary condition: ⎧  in (0, +∞) × , ⎨ φ − φ + aφ = 0 φ=0 on (0, +∞) × , ⎩ 0 ), φ = (E1 , U 1 ) in . t = 0 : φ = (E1 , U

(5.39)

Inversely if the synchronizable part Us = (φ1 , · · · , φr ) is independent of boundary controls H , then we have necessarily r = 1 and D T E1 = 0.

(5.40)

Consequently, the exactly synchronizable state u is given by u = φ for t  T , where φ is the solution to problem (5.39). In particular, if 0 ) = (E1 , U 1 ) = 0, (E1 , U 1 ). 0 , U then system (I) is exactly null controllable for such initial data (U

(5.41)

56

5 Exactly Synchronizable States

5.3 Approximation of Exactly Synchronizable States The relation (5.37) shows that only the last component φr is synchronized, while the others are steered to zero. However, in order to get φr , we have to solve the whole problem (5.32)–(5.33) for (φ1 , · · · , φr ). Therefore, except in the case r = 1, the exactly synchronizable state u depends on boundary controls which realize the exact boundary synchronization, and then, generically speaking, one cannot uniquely determine the exactly synchronizable state u. However, we have the following result. Theorem 5.11 Assume that system (I) is exactly synchronizable by means of a boundary control matrix D ∈ D N −1 . Let φ be the solution to the following homogeneous problem: ⎧  in (0, +∞) × , ⎨ φ − φ + aφ = 0 φ=0 on (0, +∞) × , ⎩ 0 ), φ = (Er , U 1 ) in . t = 0 : φ = (Er , U

(5.42)

Assume furthermore that D T Er = 0.

(5.43)

Then there exists a positive constant cT > 0, depending on T , such that the exactly synchronizable state u satisfies the following estimate: (u, u  (T ) − (φ, φ )(T ) H01 ()×L 2 () 0 , U 1 ) (L 2 ()) N −1 ×(H −1 ()) N −1 .  cT C1 (U

(5.44)

Proof By Proposition 5.4, we can take a boundary control matrix D ∈ D N −1 , such that (5.43) is satisfied. Then, considering the r th equation in (5.32), we get the following problem with homogeneous Dirichlet boundary condition: ⎧  in (0, +∞) × , ⎨ φr − φr + aφr = −φr −1 on (0, +∞) × , φr = 0 ⎩ 0 ), φr = (Er , U 1 ) in . t = 0 : φr = (Er , U

(5.45)

From (5.37) we have tT :

φr ≡ u, φr −1 ≡ 0.

(5.46)

Noting that problems (5.42) and (5.45) have the same initial data and the same homogeneous Dirichlet boundary condition, by well-posedness, there exists a positive constant c1 > 0, such that

5.3 Approximation of Exactly Synchronizable States

(u, u  )(T ) − (φ, φ )(T ) 2H 1 ()×L 2 () 0

T 2 c1 ψr −1 (s) L 2 () ds.

57

(5.47)

0

Noting that the condition (Er −1 , r ) = 0 implies that  ⊥  ⊥ Er −1 ∈ Span{r } = Ker(C1 ) = Im(C1T ),

(5.48)

Er −1 is a linear combination of the columns of C1T . Therefore, there exists a positive constant c2 > 0, such that φr −1 (s) 2L 2 () = (Er −1 , U (s)) 2L 2 ()  c2 C1 U (s) 2(L 2 ()) N −1 .

(5.49)

Recall that W = C1 U , due to the exact boundary null controllability of the reduced system (4.28), there exists a positive constant cT > 0, such that

T 0

C1 U (s) 2(L 2 ()) N −1 ds

(5.50)

0 , U 1 ) 2 2 N −1 −1 N −1 . cT C1 (U (L ()) ×(H ()) Finally, inserting (5.49)–(5.50) into (5.47), we get (5.44). The proof is complete.  Remark 5.12 When r > 1, since (E1 , r ) = 0, the exactly synchronizable state u of system (I) cannot be determined independently of applied boundary controls H . 0 , U 1 ) is suitably small, then u is closed to Nevertheless, by Theorem 5.11, if C1 (U the solution to problem (5.42), the initial data of which is given by the weighted 1 ) with the weight Er (a root vector of A T ). 0 , U average of the original initial data (U

Chapter 6

Exact Boundary Synchronization by Groups

The exact boundary synchronization by groups will be considered in this chapter for system (I) with further lack of Dirichlet boundary controls.

6.1 Definition Let

T  U = u (1) , · · · , u (N )

and

T  H = h (1) , · · · , h (M)

(6.1)

with M  N . Consider the coupled system (I) of wave equations with Dirichlet boundary controls with the initial condition (I0). It was shown in Chaps. 3 and 4 that system (I) is neither exactly null controllable nor exactly synchronizable with fewer boundary controls (M < N or M < N − 1, respectively). In order to consider the situation that the number of boundary controls is further reduced, we will investigate the exact boundary synchronization by groups for system (I). Let p  1 be an integer and let 0 = n0 < n1 < n2 < · · · < n p = N

(6.2)

be integers such that n r − n r −1  2 for all 1  r  p. We rearrange the components of U into p groups (u (1) , · · · , u (n 1 ) ), (u (n 1 +1) , · · · , u (n 2 ) ), · · · , (u (n p−1 +1) , · · · , u (n p ) ).

(6.3)

Definition 6.1 System (I) is exactly synchronizable by p-groups at the time T > 0 1 ) ∈ (L 2 ()) N × (H −1 ()) N , there exist a bound0 , U if, for any given initial data (U 2 ary control H ∈ L loc (0, +∞; (L 2 (1 )) M ) with compact support in [0, T ], such that © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_6

59

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6 Exact Boundary Synchronization by Groups

the corresponding solution U = U (t, x) to the mixed initial-boundary value problem (I) and (I0) satisfies the following final conditions:

tT :

⎧ (1) u ≡ · · · ≡ u (n 1 ) := u 1 , ⎪ ⎪ ⎨ (n 1 +1) ≡ · · · ≡ u (n 2 ) := u 2 , u ··· ⎪ ⎪ ⎩ (n p−1 +1) ≡ · · · ≡ u (n p ) := u p , u

(6.4)

where u = (u 1 , · · · , u p )T is called the exactly synchronizable state by p-groups, which is a priori unknown. Remark 6.2 The functions u 1 , · · · , u p depend on the initial data and on applied boundary controls. In the whole monograph, when we claim that u 1 , · · · , u p are linearly independent, it means that there exists at least an initial data and an applied boundary control such that the corresponding u 1 , · · · , u p are linearly independent; while, when we claim that u 1 , · · · , u p are linearly dependent, it means that they are linearly dependent for any given initial data and for any given applied boundary control. Let Sr (cf. Remark 4.6) be the following (n r − n r −1 − 1) × (n r − n r −1 ) matrix: ⎛

1 ⎜0 ⎜ Sr = ⎜ . ⎝ ..

−1 1 .. .

0 −1 .. .

··· ··· .. .

0 0 .. .

⎞ ⎟ ⎟ ⎟ ⎠

(6.5)

0 0 · · · 1 −1

and let C p be the following (N − p) × N full row-rank matrix of synchronization by p-groups: ⎞ ⎛ S1 ⎟ ⎜ S2 ⎟ ⎜ . (6.6) Cp = ⎜ . .. ⎟ ⎠ ⎝ Sp The exact boundary synchronization by p-groups (6.4) is equivalent to tT :

C p U ≡ 0.

(6.7)

1, n r −1 + 1  i  n r , 0, otherwise,

(6.8)

Ker(C p ) = Span{e1 , e2 , · · · , e p }

(6.9)

For r = 1, · · · , p, setting  (er )i = it is clear that

6.1 Definition

61

and the exact boundary synchronization by p-groups (6.4) becomes tT :

U=

p 

u r er .

(6.10)

r =1

We first observe that the exact boundary synchronization by p-groups is achieved not only at the time T by means of the action of boundary controls, but is also maintained forever. Since the control is removed from the time T , the coupling matrix A should satisfy some additional conditions in order to maintain the exact boundary synchronization by p-groups. As in Chap. 4, we should derive the corresponding condition of compatibility for the exact boundary synchronization by p-groups of system (I).

6.2 A Basic Lemma In what follows, we will prove the necessity of the condition of compatibility by a repeating reduction procedure. For clarity, this section is entirely devoted to this basic result. Lemma 6.3 Suppose that C p is a full row-rank matrix of order (N − p) × N , such that (6.11) t  T : C pU ≡ 0 for any given solution U to the system of equations U  − U + AU = 0 in (0, +∞) × ,

(6.12)

AKer(C p ) ⊆ Ker(C p ),

(6.13)

we have either p−1 of order (N − p + 1) × N , such that or C p has a full row-rank extension C p−1 U ≡ 0. tT : C

(6.14)

Proof et er ∈ R N (r = 1, · · · , p) such that Ker(C p ) = Span{e1 , · · · , e p }.

(6.15)

Noting that (6.11) implies (6.10) and applying the matrix C p to system (6.12), we get p  u r C p Aer = 0. (6.16) tT : r =1

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6 Exact Boundary Synchronization by Groups

If C p Aer = 0 (r = 1, · · · , p), then we get the inclusion (6.13). Otherwise, noting that C p Aer (r = 1, · · · , p) are constant vectors, there exist some real constant coefficients αr (r = 1, · · · , p − 1) such that up =

p−1 

αr u r .

(6.17)

u r (er + αr e p ).

(6.18)

r =1

Then (6.10) becomes tT :

p−1 

U=

r =1

Setting  er = er + αr e p , r = 1, · · · , p − 1,

(6.19)

we get tT :

U=

p−1 

u r er .

(6.20)

r =1

p−1 such that (6.14) holds. For this purpose, We next construct an enlarged matrix C let  c p+1 be a row vector defined by  αl el ep − . 2 e p  el 2 l=1 p−1

 c Tp+1 =

(6.21)

Noting (6.19) and the orthogonality of the set {e1 , · · · , e p }, it is easy to check that (e p , er )  αl (el , er ) − e p 2 el 2 l=1 p−1

er =  c p+1 + αr

(6.22)

p−1  (e , e )  αl (el , e p )  p p = −αr + αr = 0. − e p 2 el 2 l=1

p−1 by for all r with 1  r  p − 1. Then we define the enlarged matrix C p−1 = C

 Cp .  c p+1



(6.23)

Noting that  er ∈ Ker(C p ) for r = 1, · · · , p − 1 and using (6.22), we have p−1 er = C

     er 0 Cp C p  e = = , r = 1, · · · , p − 1,  c p+1 r  c p+1 er 0



(6.24)

6.2 A Basic Lemma

63

then it follows immediately from (6.20) that tT : Finally, noting that

p−1 U ≡ 0. C

 c Tp+1 ∈ Ker(C p ) = {Im(C Tp )}⊥ ,

(6.25)

(6.26)

p−1 ) = N − p + 1, namely, C p−1 is a full rowwe have c Tp+1 ∈ / Im(C Tp ), then rank(C rank matrix of order (N − p + 1) × N . The proof is complete. 

6.3 Condition of C p -Compatibility We will show that the exact boundary synchronization by p-groups of system (I) requires at least (N − p) boundary controls. In particular, if rank(D) = N − p, then we get the following condition of C p -compatibility: AKer(C p ) ⊆ Ker(C p ),

(6.27)

which is necessary for the exact boundary synchronization by p-groups of system (I). Conversely, we will show in the next section that under the condition of C p compatibility (6.27), there exists a boundary control matrix D with M = N − p, such that system (I) is exactly synchronizable by p-groups. We first give the lower bound estimate on the rank of the boundary control matrix D, which is necessary for the exact boundary synchronization by p-groups. Theorem 6.4 Assume that system (I) is exactly synchronizable by p-groups. Then we necessarily have (6.28) rank(C p D) = N − p. In particular, we have rank(D)  N − p.

(6.29)

Proof From (6.7), we have C p U ≡ 0 for t  T . If AKer(C p )  Ker(C p ), then by p−1 such that Lemma 6.3, we can construct a full row-rank (N − p + 1) × N matrix C p−1 )  Ker(C p−1 ), once again by Lemma 6.3, we p−1 U ≡ 0 for t  T . If AKer(C C p−2 U ≡ 0 p−2 such that C can construct a full row-rank (N − p + 2) × N matrix C for t  T , and so forth. The procedure should stop at some step r with 0  r  p. p−r (when r = 0, So, we get an enlarged full row-rank (N − p + r ) × N matrix C p = C p , and this special situation means that the condition of compatibility we take C (6.27) holds) such that tT :

p−r U ≡ 0 C

(6.30)

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6 Exact Boundary Synchronization by Groups

and

p−r ). p−r ) ⊆ Ker(C AKer(C

(6.31)

Then, by Proposition 2.15, there exists a unique matrix A p−r of order (N − p + r ), such that p−r . p−r A = A p−r C (6.32) C p−r U , we get the following p−r to problem (I) and (I0) and setting W = C Applying C reduced system: ⎧  ⎨ W − W + A p−r W = 0 in (0, +∞) × , W =0 on (0, +∞) × 0 , ⎩ p−r D H on (0, +∞) × 1 W =C

(6.33)

with the initial condition t =0:

p−r U 0 , W  = C 1 in . p−r U W =C

(6.34)

Moreover, (6.30) implies that tT :

W ≡ 0.

(6.35)

p−r is a (N − p + r ) × N full row-rank matrix, the On the other hand, since C linear map 1 ) → (C p−r U p−r U 0 , C 1 ) 0 , U (U

(6.36)

is surjective from the space (L 2 ()) N × (H −1 ()) N onto the space (L 2 ()) N − p+r × (H −1 ()) N − p+r . We thus get the exact boundary null controllability of the reduced system (6.33) in the space (L 2 ()) N − p+r × (H −1 ()) N − p+r . By Theorem 3.11, the p−r D satisfies rank of the corresponding boundary control matrix C p−r D) = N − p + r. rank(C

(6.37)

p−r D is of full row-rank, then the Finally, noting that the (N − p + r ) × N matrix C p−r D is of full row-rank, sub-matrix C p D composed of the first (N − p) rows of C we get thus (6.28).  Theorem 6.5 Assume that system (I) is exactly synchronizable by p-groups under the minimal rank condition M = rank(D) = N − p. Then, we necessarily have the condition of C p -compatibility (6.27).

(6.38)

6.3 Condition of C p -Compatibility

65

Proof Noting (6.37) and (6.38), it follows easily from p−r D) rank(D)  rank(C that r = 0. It means that it is not necessary to proceed the extension in the proof of  Theorem 6.4, we have already the condition of C p -compatibility (6.27). Remark 6.6 The condition of C p -compatibility (6.27) is necessary for the exact boundary synchronization by p-groups of system (I) only under the minimal rank condition (6.38), namely, under the minimal number of boundary controls. Remark 6.7 Noting (6.9), the condition of C p -compatibility (6.27) can be equivalently written as p  αsr es , 1  r  p, (6.39) Aer = s=1

where αsr are some constant coefficients. Moreover, because of the specific expression of er given in (6.8), the above expression (6.39) can be written in the form of row-sum condition by blocks: ns 

ai j = αr s

(6.40)

j=n s−1 +1

for all 1  r, s  p and n r −1 + 1  i  n r , which is a natural generalization of the row-sum condition (4.5) in the case p = 1. Remark 6.8 By Proposition 2.15, the condition of C p -compatibility (6.27) is equivalent to the existence of a unique matrix A p of order (N − p), such that C p A = A pC p.

(6.41)

A p is called the reduced matrix of A by C p .

6.4 Exact Boundary Synchronization by p-Groups Theorem 6.4 shows that the exact boundary synchronization by p-groups of system (I) requires at least (N − p) boundary controls, namely, we have the following Corollary 6.9 If rank(C p D) < N − p,

(6.42)

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6 Exact Boundary Synchronization by Groups

or, in particular, if rank(D) < N − p,

(6.43)

then, no matter how large T > 0 is, system (I) cannot be exactly synchronizable by p-groups at the time T . The following result shows that the converse also holds. Theorem 6.10 Assume that the coupling matrix A satisfies the condition of C p compatibility (6.27). Let M = N − p and let the N × (N − p) boundary control matrix D satisfy the rank condition rank(C p D) = N − p.

(6.44)

Then, under the multiplier geometrical condition (3.1), system (I) is exactly synchronizable by p-groups in the space (L 2 ()) N × (H −1 ()) N . Proof By Remark 6.8, applying C p to problem (I) and (I0) and setting W p = C p U , we get the following self-closed reduced system: ⎧  ⎨ W p − W p + A p W p = 0 in (0, +∞) × , on (0, +∞) × 0 , W =0 ⎩ p on (0, +∞) × 1 Wp = Cp D H with t =0:

0 , W p = C p U 1 in . W p = C pU

(6.45)

(6.46)

Then, by Theorem 3.11 and noting the rank condition (6.44), the reduced system (6.45) is exactly null controllable at the time T > 0 by means of controls H ∈ 2 (0, +∞; (L 2 (1 )) N − p ) with compact support in [0, T ]. Thus, we have L loc tT :

C p U ≡ W p ≡ 0. 

The proof is then complete.

Remark 6.11 The rank M of the boundary control matrix D presents the number of boundary controls applied to the original system (I), while, the rank of the matrix C p D presents the number of boundary controls effectively applied to the reduced system (6.45). Let us write D H = H0 + H1 with H0 ∈ Ker(C p )

and

H1 ∈ Im(C Tp ).

(6.47)

The part H0 will disappear in the reduced system (6.45), and is useless for the exact boundary synchronization by p-groups of the original system (I). The exact boundary null controllability of the reduced system (6.45), therefore, the exact boundary synchronization by p-groups of the original system (I) is in fact realized only by the

6.4 Exact Boundary Synchronization by p-Groups

67

part H1 . So, in order to minimize the number of boundary controls, we are interested in the matrices D such that Im(D) ∩ Ker(C p ) = {0},

(6.48)

or, by Proposition 2.11, such that rank(C p D) = rank(D) = N − p.

(6.49)

Proposition 6.12 Let D N − p be the set of N × (N − p) matrices satisfying (6.49), namely, D N − p = {D ∈ M N ×(N − p) :

rank(D) = rank(C p D) = N − p}.

(6.50)

Then, for any given boundary control matrix D ∈ D N − p , the subspaces Ker(D T ) and Ker(C p ) are bi-orthonormal. Moreover, we have D N − p = {C Tp D1 + (e1 , · · · , e p )D0 },

(6.51)

where D1 is an invertible matrix of order (N − p), D0 is a matrix of order p × (N − p), and the vectors e1 , · · · , e p are given by (6.8). Proof Noting that {Ker(D T )}⊥ = Im(D), and Ker(D T ) and Ker(C p ) have the same dimension p, by Propositions 2.4 and 2.5, to prove that Ker(D T ) and Ker(C p ) are bi-orthonormal, it suffices to show that Ker(C p ) ∩ Im(D) = {0}, which, by Proposition 2.11, is equivalent to the condition rank(C p D) = rank(D). Now we prove (6.51). Let D be a matrix of order N × (N − p). Noting that Im(C Tp ) ⊕ Ker(C p ) = R N , there exist a matrix D0 of order p × (N − p) and a square matrix D1 of order (N − p), such that D = C Tp D1 + (e1 , · · · , e p )D0 ,

(6.52)

where e1 , · · · , e p are given by (6.8). Moreover, C p D = C p C Tp D1 is of rank (N − p), if and only if D1 is invertible. On the other hand, let x ∈ R N − p , such that Dx = C Tp D1 x + (e1 , · · · , e p )D0 x = 0.

(6.53)

Since Im (C Tp ) ⊥ Ker(C p ), it follows that C Tp D1 x = (e1 , · · · , e p )D0 x = 0.

(6.54)

Then, noting that C Tp D1 is of full column-rank, we have x = 0. Hence, D is of full column-rank (N − p). This proves (6.51). 

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6 Exact Boundary Synchronization by Groups

Remark 6.13 Under the multiplier geometrical condition 3.1, for any given matrix D ∈ D N − p , the condition of C p -compatibility (6.27) is necessary and sufficient for the exact boundary synchronization by p-groups of system (I) by Theorem 6.10 and Remark 6.6. In particular, if we take D = C Tp C p ∈ D N − p , we can really achieve the exact boundary synchronization by p-groups for system (I) by means of (N − p) boundary controls.

Chapter 7

Exactly Synchronizable States by p-Groups

When system (I) possesses the exact boundary synchronization by p-groups, the corresponding exactly synchronizable states by p-groups will be studied in this chapter.

7.1 Introduction Under the condition of C p -compatibility (6.27), it is easy to see that for t  T , the exactly synchronizable state by p-groups u = (u 1 , · · · u p )T satisfies the following coupled system of wave equations with homogenous Dirichlet boundary condition: 

˜ = 0 in (T, +∞) × , u  − u + Au u=0 on (T, +∞) × ,

(7.1)

where A˜ = (αr s ) is given by (6.39). Hence, the evolution of the exactly synchronizable state by p-groups u = (u 1 , · · · u p )T with respect to t is completely determined by the values of (u, u t ) at the time t = T : t=T :

u1. u = u0, u = 

(7.2)

As in Theorem 5.1 for the case p = 1, we can show that the attainable set of all possible values of (u, u  ) at t = T is the whole space (L 2 ()) p × (H −1 ()) p , when 1 ) vary in the space (L 2 ()) N × (H −1 ()) N . 0 , U the initial data (U In this chapter, we will discuss the determination of exactly synchronizable states 0 , U 1 ). Since there is an by p-groups u = (u 1 , · · · u p )T for each given initial data (U infinity of boundary controls which can realize the exact boundary synchronization by p-groups for system (I), exactly synchronizable states by p-groups u = (u 1 , · · · u p )T naturally depend on applied boundary controls H . However, for some coupling matrices A, for example, the symmetric ones, the exactly synchronizable states by © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_7

69

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7 Exactly Synchronizable States by p-Groups

p-groups u = (u 1 , · · · u p )T could be independent of applied boundary controls H . In the general case, exactly synchronizable states by p-groups u = (u 1 , · · · u p )T depend on applied boundary controls, however we can give an estimate on the difference between each exactly synchronizable state by p-groups u = (u 1 , · · · u p )T and the solution to a problem which is independent of applied boundary controls (cf. Theorems 7.1 and 7.2 below).

7.2 Determination of Exactly Synchronizable States by p-Groups Now we return to the determination of exactly synchronizable states by p-groups. The case p = 1 was considered in Sect. 5.2. We first consider the case that A T admits an invariant subspace Span{E 1 , · · · , E p }, which is bi-orthonormal to Span{e1 , · · · , e p }, namely, we have (ei , E j ) = δi j , 1  i, j  p,

(7.3)

where e1 , · · · , e p are given by (6.8)–(6.9). Theorem 7.1 Assume that the matrix A satisfies the condition of C p -compatibility (6.27). Assume furthermore that A T admits an invariant subspace Span{E 1 , · · · , E p }, which is bi-orthonormal to Ker(C p ) = Span{e1 , · · · , e p }. Then there exists a boundary control matrix D ∈ D N − p (cf. (6.51)), such that the exactly synchronizable state by p-groups u = (u 1 , · · · , u p )T is uniquely determined by tT :

u = φ,

(7.4)

where φ = (φ1 , · · · , φ p )T is the solution to the following problem independent of applied boundary controls H : for s = 1, 2, · · · , p, ⎧  p in (0, +∞) × , ⎨ φs − φs + r =1 αsr φr = 0 on (0, +∞) × , φs = 0 ⎩ 0 ), φs = (E s , U 1 ) in , t = 0 : φs = (E s , U

(7.5)

where αsr are given by (6.39). Proof Since the subspaces Span{E 1 , · · · , E p } and Span{e1 , · · · , e p } are bi-orthonormal, then taking D1 = I N − p ,

D0 = −E T C Tp with E = (E 1 , · · · , E p )

(7.6)

7.2 Determination of Exactly Synchronizable States by p-Groups

71

in (6.51), we obtain a boundary control matrix D ∈ D N − p , such that E s ∈ Ker(D T ), s = 1, · · · , p.

(7.7)

On the other hand, since Span{E 1 , · · · , E p } is invariant for A T , and noting (6.39) and (7.3), it is easy to check that AT Es =

p 

αsr Er , s = 1, · · · , p.

(7.8)

r =1

Then, taking the inner product of E s with problem (I) and (I0), and setting φs = (E s , U ), we get problem (7.5). Finally, by the exact boundary synchronization by p-groups (6.10) and the relationship (7.3), we get tT :

p  (E s , er )u r = u s , 1  s  p. φs = (E s , U ) =

(7.9)

r =1



The proof is then complete.

Theorem 7.2 Assume that the condition of C p -compatibility (6.27) holds. Then for any given boundary control matrix D ∈ D N − p , there exists a positive constant cT independent of initial data, but depending on T , such that each exactly synchronizable state by p-groups u = (u 1 , · · · , u p )T satisfies the following estimate: (u, u  )(T ) − (φ, φ )(T )(H01 ()) p ×(L 2 ()) p 0 , U 1 )(L 2 ()) N − p ×(H −1 ()) N − p , cT C p (U

(7.10)

where φ = (φ1 , · · · , φ p )T is the solution to problem (7.5), in which Span{E 1 , · · · , E p } is bi-orthonormal to Span{e1 , · · · , e p }. Proof By Proposition 6.12, the subspaces Ker(D T ) and Ker(C p ) are bi-orthonormal. Then we can chose E 1 , · · · , Er ∈ Ker(D T ), such that Span{E 1 , · · · , E p } and Span{e1 , · · · , e p } are bi-orthonormal. Moreover, noting (6.39) and (7.3), by a direct computation, we get (A T E s −

p 

αsr Er , ek )

r =1

=(E s , Aek ) −

p 

αsr (Er , ek )

r =1

=

p  l=1

αlk (E s , el ) − αsk = αsk − αsk = 0

(7.11)

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7 Exactly Synchronizable States by p-Groups

for all s, k = 1, · · · , p, and then AT Es −

p 

αsr Er ∈ {Ker(C p )}⊥ = Im(C Tp ), s = 1, · · · , p.

(7.12)

r =1

Thus, there exists a vector Rs ∈ R N − p , such that AT Es −

q 

αsr Er = −C Tp Rs .

(7.13)

r =1

Taking the inner product of E s with problem (I) and (I0), and setting ψs = (E s , U ) (s = 1, · · · , p), it is easy to see that ⎧ p  ⎪ ⎪  ⎪ αsr ψr = (Rs , C p U ) ⎨ ψs − ψs +

in (0, +∞) × ,

r =1

⎪ ψs = 0 on (0, +∞) × , ⎪ ⎪ ⎩ 0 ), ψs = (E s , U 1 ) in . t = 0 : ψs = (E s , U

(7.14)

By the well-posedness of problems (7.5) and (7.14), there exists a constant c > 0 independent of initial data, such that (ψ, ψ  )(T ) − (φ, φ )(T )2(H 1 ()) p ×(L 2 ()) p 0

T c C p U (s)2(L 2 ()) N − p ds.

(7.15)

0

Since C p U = W p , the exact boundary null controllability of the reduced system (6.45) shows that there exists another positive constant cT > 0 independent of initial data, but depending on T , such that

0

T

C p U (s)2(L 2 ()) N − p ds

(7.16)

0 , U 1 )2 2 N − p cT C p (U (L ()) ×(H −1 ()) N − p . Finally, noting that tT :

ψs = (E s , U ) =

p  (E s , er )u r = u s , s = 1, · · · , p,

(7.17)

r =1

and inserting (7.16)–(7.17) into (7.15), we get (7.10). The proof is complete.



7.2 Determination of Exactly Synchronizable States by p-Groups

73

Remark 7.3 Since {e1 , · · · , e p } is an orthonormal system, we can take E s = es /es 2 (s = 1, · · · , p) as a specific choice. Accordingly, the boundary control matrix D has the tendency to drive the initial data to the average value. Remark 7.4 Let Ker(C p ) be invariant for A and bi-orthonormal to W = Span{E 1 , E 2 , · · · , E p }. By Proposition 2.9, W ⊥ is a supplement of Ker(C p ) and invariant for the matrix A if and only if W is invariant for the matrix A T . Accordingly, the coupling matrix A can be diagonalized by blocks under the decomposition R N = K er (C p ) ⊕ W ⊥ . In particular, if A T satisfies the condition of C p -compatibility (6.27), then Ker(C p ) is also invariant for A T . By Proposition 2.7, Im(C Tp ) is an invariant subspace of A. Then, we can write  A(e1 , · · ·

, e p , C Tp )

= (e1 , · · ·

, e p , C Tp )

0 A  , 0 A

(7.18)

 is given by = (αr s ) is given by (6.40) and A where A  = (C p C Tp )−1 C p AC Tp . A

(7.19)

In that case, Theorem 7.1 shows that the exactly synchronizable state by p-groups is independent of applied boundary controls. Otherwise, it depends on applied boundary controls; however, as shown in Theorem 7.2, we can give an estimate on the difference between each exactly synchronizable state by p-groups and the solution to a problem independent of applied boundary controls.

7.3 Determination of Exactly Synchronizable States by p-Groups (Continued) We now consider the determination of exactly synchronizable states by p-groups in the case that Ker(C p ) = Span{e1 , · · · , e p } is an invariant subspace of A, but A T does not admit any invariant subspace which is bi-orthonormal to Ker(C p ). In this case, we will extend the subspace Span{e1 , · · · , e p } to an invariant subspace Span{e1 , · · · , eq } of A with q  p, so that A T admits an invariant subspace Span{E 1 , · · · , E q } that is bi-orthonormal to Span{e1 , · · · , eq }. When Ker(C p ) is A-marked, the procedure for obtaining the subspace Span{e1 , · · · , eq } is given in Proposition 2.20. When Ker(C p ) is not A-marked, let λ j (1  j  d) denote the eigenvalues of the restriction of A to Ker(C p ). We define Span{e1 , · · · , eq } =

d  j=1

Ker(A − λ j I )m j

(7.20)

74

7 Exactly Synchronizable States by p-Groups

and Span{E 1 , · · · , E q } =

d 

Ker(A T − λ j I )m j ,

(7.21)

j=1

where m j is an integer such that Ker(A T − λ j I )m j = Ker(A T − λ j I )m j +1 and q=

d 

Dim Ker(A T − λ j I )m j .

(7.22)

j=1

Clearly, the subspaces given by (7.20) and (7.21) satisfy well the first two conditions in Proposition 2.20. However, since Ker(C p ) is not A-marked, the subspace Span{e1 , · · · , eq } constructed by this way is not a priori the least one. In the above two cases, we can define the projection P on the subspace Span{e1 , · · · , eq } as follows: P=

q 

er ⊗ Er ,

(7.23)

r =1

where the tensor product ⊗ is defined by (er ⊗ Er )U = (Er , U )er , ∀U ∈ R N .

(7.24)

Im(P) = Span{e1 , · · · , eq },

(7.25)

⊥  Ker(P) = Span{E 1 , · · · , E q }

(7.26)

P A = A P.

(7.27)

Then we have

and Let U = U (t, x) be the solution to problem (I) and (I0). We may define the synchronizable part Us and the controllable part Uc by Us = PU, Uc = (I − P)U,

(7.28)

respectively. In fact, if system (I) is exactly synchronizable by p-groups, we have U ∈ Span{e1 , · · · , e p } ⊆ Span{e1 , · · · , eq } = Im(P)

(7.29)

for all t  T , so that tT :

Us = PU ≡ U, Uc ≡ 0.

(7.30)

7.3 Determination of Exactly Synchronizable States by p-Groups (Continued)

75

Moreover, noting (7.27), we have ⎧  U − Us + AUs = 0 ⎪ ⎪ ⎪ s ⎨ Us = 0 ⎪ Us = P D H ⎪ ⎪ ⎩ 0 , Us = P U 1 t = 0 : Us = P U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in .

(7.31)

The condition p = q means exactly that the matrix A satisfies the condition of C p compatibility (6.27), and A T admits an invariant subspace, which is bi-orthonormal to Ker(C p ). In this case, by Theorem 7.1, there exists a boundary control matrix D ∈ D N − p , such that the exactly synchronizable state by p-groups u = (u 1 , · · · , u p )T is independent of applied boundary controls H . Conversely, we have the following Theorem 7.5 Assume that the matrix A satisfies the condition of C p -compatibility (6.27). Assume that system (I) is exactly synchronizable by p-groups. If the synchronizable part Us is independent of applied boundary controls H , then A T admits an invariant subspace which is bi-orthonormal to Ker(C p ). Proof Let H1 and H2 be two boundary controls, which realize simultaneously the exact boundary synchronization by p-groups for system (I). If the corresponding solution Us to problem (7.31) is independent of applied boundary controls H1 and H2 , then we have P D(H1 − H2 ) = 0 on (0, T ) × 1 .

(7.32)

By Theorem 3.8, it is easy to see that the values of C p D(H1 − H2 ) on (T − , T ) × 1 can be arbitrarily chosen for  > 0 small enough. Since C p D is invertible, the values of (H1 − H2 ) on (T − , T ) × 1 can be arbitrarily chosen, too. This yields that P D = 0 so that Im(D) ⊆ Ker(P). (7.33) Noting (7.26), we have dim Ker(P) = N − q,

(7.34)

however, by Theorem 6.4, we have dim Im(D)  dim Im(C p D) = rank(C p D) = N − p.

(7.35)

Then, it follows from (7.33) that p = q. Then Span{E 1 , · · · , E p } will be the desired subspace. The proof is complete. 

76

7 Exactly Synchronizable States by p-Groups

7.4 Precise Consideration on the Exact Boundary Synchronization by 2-Groups The case p = 1 was considered in Chap. 5. The condition p = q = 1 is equivalent to the existence of an eigenvector E 1 of A T , such that (E 1 , e1 ) = 1. In this section, we give the precise consideration for the case p = 2. Let m  2 be an integer such that N − m  2. We rewrite the exact boundary synchronization by 2-groups as tT :

u (1) = · · · u (m) := u, u (m+1) = · · · = u (N ) := v. 

Let C2 =

(7.36)

Sm

(7.37)

S N −m

be the matrix of synchronization by 2-groups and let N −m

m

m

N −m

            e1 = (1, · · · , 1, 0, · · · , 0)T , e2 = (0, · · · , 0, 1, · · · , 1)T .

(7.38)

Clearly, we have Ker(C2 ) = Span{e1 , e2 }

(7.39)

and the synchronization condition by 2-groups (6.10) means tT :

U = ue1 + ve2 .

(7.40)

(i) Assume that A admits two eigenvectors r and ˜s , associated, respectively, with the eigenvalues λ and μ, contained in the invariant subspace Ker(C2 ). Let 1 , · · · , r , respectively, ˜1 , · · · , ˜s denote the corresponding Jordan chains of A: 

Ak = λk + k+1 , 1  k  r, r +1 = 0, A˜i = μ˜i + ˜i+1 , 1  i  s, ˜s+1 = 0.

(7.41)

Accordingly, let E1 , · · · , Er and E 1 · · · , E s denote the corresponding Jordan chains of A T :  T A Ek = λEk + Ek−1 , 1  k  r, E0 = 0, (7.42) A T E i = μE i + E i−1 , 1  i  s, E 0 = 0. Moreover, for any 1  k, l  r and 1  i, j  s, we have 

(k , El ) = δkl , (˜i , E j ) = δi j , (k , E i ) = 0, (˜ j , El ) = 0.

(7.43)

7.4 Precise Consideration on the Exact Boundary Synchronization by 2-Groups

77

Taking the inner product of problem (I) and (I0) with Ek and E i , and setting φk = (Ek , U ) and φ˜ i = (E i , U ), we get the subsystem ⎧  φ − φk + λφk + φk−1 = 0 in (0, +∞) × , ⎪ ⎪ ⎨ k on (0, +∞) × 0 , φk = 0 = (E , D H ) on (0, +∞) × 1 , φ ⎪ k k ⎪ ⎩ 0 ), φk = (Ek , U 1 ) in  t = 0 : ψk = (Ek , U

(7.44)

for k = 1, · · · , r , and the subsystem ⎧ ˜  φ − φ˜ i + μφ˜ i + φ˜ i−1 = 0 in (0, +∞) × , ⎪ ⎪ ⎨ ˜i φi = 0 on (0, +∞) × 0 , ⎪ φ˜ = (E i , D H ) on (0, +∞) × 1 , ⎪ ⎩ i 0 ), φ˜ i = (E i , U 1 ) in  t = 0 : φ˜ i = (E i , U

(7.45)

for i = 1, · · · , s, respectively. Once the solutions (φ1 , · · · , φr ) and (φ˜ 1 , · · · , φ˜ s ) are determined, we look for the corresponding exactly synchronizable state by 2-groups (u, v)T . Noting that r , ˜s ∈ Span{e1 , e2 }, we can write e1 = αr + β ˜s , e2 = γr + δ ˜s

(7.46)

with αδ − βγ = 0. Then the exact boundary synchronization by 2-groups (7.40) gives (7.47) t  T : U = (αu + γv)r + (βu + δv)˜s . Noting (7.43), it follows that φr = αu + γv, φ˜ s = βu + δv.

(7.48)

By solving this linear system, we get the exactly synchronizable state by 2-groups (u, v). In particular, if r = s = 1, we can choose a boundary control matrix D ∈ D N −2 , such that D T E1 = D T E 1 = 0. Then the exactly synchronizable state by 2-groups (u, v)T can be uniquely determined independently of applied boundary controls. Otherwise, we have to solve the whole systems (7.44) and (7.45) to get φ1 , · · · , φr and φ˜ 1 , · · · , φ˜ s , even though we only need φr and φ˜ s . (ii) Assume that A admits only one eigenvector r in the invariant subspace Ker(C2 ), and Ker(C2 ) is A-marked. Let 1 , · · · , r denote a Jordan chain of A. Respectively, let E1 , · · · , Er denote the corresponding Jordan chain of A T . We get again the subsystem (7.44). Since Ker(C2 ) is A-marked, then r −1 , r ∈ Span{e1 , e2 } so that we can write e1 = αr + βr −1 , e2 = γr + δr −1

(7.49)

78

7 Exactly Synchronizable States by p-Groups

with αδ − βγ = 0. The exact boundary synchronization by 2-groups (7.40) becomes tT :

U = (αu + γv)r + (βu + δv)r −1 .

(7.50)

Noting the first formula in (7.43), it follows that φr = αu + γv, φr −1 = βu + δv,

(7.51)

which gives the exactly synchronizable state by 2-groups (u, v). In particular, if r = 2, we can choose a boundary control matrix D ∈ D N −2 , such that D T E1 = D T E2 = 0. Then the exactly synchronizable state by 2-groups (u, v)T can be uniquely determined independently of applied boundary controls. Otherwise, we have to solve the whole systems (7.44) to get φ1 , · · · , φr , even though we only need φr −1 and φr . Remark 7.6 The above consideration can be used to the general case p  1 without any essential difficulties. Later in Sect. 15.4, we will give the details in the case p = 3 for Neumann boundary controls.

Part II

Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls: Approximate Boundary Synchronization From the results given in Part 1, we know that, roughly speaking, if the domain satisfies the multiplier geometrical condition, and the coupling matrix A satisfies the corresponding condition of compatibility in the case of various kinds of synchronizations, then the exact boundary null controllability, the exact boundary synchronization and the exact boundary synchronization by groups can be realized for system (I) with Dirichlet boundary controls, respectively, provided that the control time T > 0 is large enough and there are enough boundary controls. However, when the domain does not satisfy the multiplier geometrical condition or there is a lack of boundary controls, “what weakened controllability or synchronization can be obtained” becomes a very interesting and practically important problem. In this part, in order to answer this question, we will introduce the concept of approximate boundary null controllability, approximate boundary synchronization and approximate boundary synchronization by groups, and establish the corresponding theory for a coupled system of wave equations with Dirichlet boundary controls. Moreover, we will show that Kalman’s criterion of various kinds will play an important role in the discussion.

Chapter 8

Approximate Boundary Null Controllability

In this chapter we will define the approximate boundary null controllability for system (I) and the D-observability for the adjoint problem, and show that these two concepts are equivalent to each other. Moreover, the corresponding Kalman’s criterion is introduced and studied.

8.1 Definition Let

U = (u (1) , · · · , u (N ) )T

and

H = (h (1) , · · · , h (M) )T

(8.1)

with M  N . Consider the coupled system (I) with the initial condition (I0). Let 2 (0, +∞; L 2 (1 )). (8.2) H0 = L 2 (), H1 = H01 (), L = L loc The dual of H1 is denoted by H−1 = H −1 (). By Theorem 3.11, as M < N , system (I) is not exactly null controllable in the space (H0 ) N × (H−1 ) N . So we look for some weakened controllability, for example, the approximate boundary null controllability as follows. Definition 8.1 System (I) is approximately null controllable at the time T > 0 if for 0 , U 1 ) ∈ (H0 ) N × (H−1 ) N , there exists a sequence {Hn } of any given initial data (U M boundary controls in L with compact support in [0, T ], such that the corresponding sequence {Un } of solutions to problem (I) and (I0) satisfies the following condition: 0 ([T, +∞); (H0 × H−1 ) N ) as n → +∞. (Un , Un ) → (0, 0) in Cloc

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_8

(8.3)

81

82

8 Approximate Boundary Null Controllability

Remark 8.2 Since Hn has a compact support in [0, T ], the corresponding solution Un satisfies the homogeneous Dirichlet boundary condition on  for t  T . Hence, there exist positive constants c and ω such that (Un (t), Un (t))(H0 ) N ×(H−1 ) N ce

ω(t−T )

(8.4)

(Un (T ), Un (T ))(H0 ) N ×(H−1 ) N

for all t  T and for all n  0. Then the convergence (Un (T ), Un (T )) → (0, 0) in (H0 ) N × (H−1 ) N as n → +∞

(8.5)

implies the convergence (8.3). Remark 8.3 In Definition 8.1, the convergence (8.3) of the sequence {Un } of solutions does not imply the convergence of the sequence {Hn } of boundary controls. We don’t even know if the sequence {Un , Un } is bounded in C 0 ([0, T ]; (H0 × H−1 ) N ). However, since {Hn } has a compact support in [0, T ], the sequence {Un , Un } con0 ([T, +∞); (H0 × H−1 ) N ) as n → +∞. verges uniformly to (0, 0) in Cloc

8.2

D-Observability for the Adjoint Problem

Let

 = (φ(1) , · · · , φ(N ) )T .

(8.6)

⎧  in (0, +∞) × , ⎨  −  + A T  = 0 =0 on (0, +∞) × , ⎩ 1 in .  0 ,  =  t =0: =

(8.7)

Consider the adjoint problem

Definition 8.4 The adjoint problem (8.7) is D-observable on the interval [0, T ] if the observation D T ∂ν  ≡ 0 on [0, T ] × 1 (8.8) 1 ) ≡ 0, then  ≡ 0. 0 ,  implies that ( Let C be the set of all the initial states (V (0), V  (0)) given by the following backward problem: ⎧  V − V + AV = 0 ⎪ ⎪ ⎨ V =0 V = DH ⎪ ⎪ ⎩ t = T : V = V = 0

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in 

(8.9)

8.2 D-Observability for the Adjoint Problem

83

as the boundary control H varies in L M with compact support in [0, T ]. Lemma 8.5 System (I) is approximately null controllable in (H0 ) N × (H−1 ) N if and only if C = (H0 ) N × (H−1 ) N . (8.10) 1 ) ∈ (H0 ) N × (H−1 ) N 0 , U Proof Assume that (8.10) holds. Then for any given (U M there exists a sequence {Hn } of boundary controls in L , such that the corresponding sequence {Vn } of solutions to problem (8.9) satisfies 0 , U 1 ) in (H0 ) N × (H−1 ) N as n → +∞. (Vn (0), Vn (0)) → (U Let R:

1 , H ) → (U, U  ) 0 , U (U

(8.11)

(8.12)

be the resolution of problem (I) and (I0). R being linear, we have 1 , Hn ) 0 , U R(U 1 − Vn (0), 0) + R(Vn (0), Vn (0), Hn ). 0 − Vn (0), U =R(U

(8.13)

By the definition of Vn , we have

then

R(Vn (0), Vn (0), Hn )(T ) = 0,

(8.14)

1 , Hn )(T ) = R(U 0 − Vn (0), U 1 − Vn (0), 0)(T ). 0 , U R(U

(8.15)

By means of the well-posedness of problem (I) and (I0), there exists a positive constant c such that 1 , Hn )(T )(H0 ) N ×(H−1 ) N 0 , U R(U 1 − Vn (0))(H0 ) N ×(H−1 ) N . 0 − Vn (0), U c(U

(8.16)

Then, it follows from (8.11) that 1 , Hn )(T )(H0 ) N ×(H−1 ) N → 0 0 , U R(U

(8.17)

as n → +∞. This gives the approximate boundary null controllability of system (I). Inversely, assume that system (I) is approximately null controllable. Then for 1 ) ∈ (H0 ) N × (H−1 ) N , there exists a sequence {Hn } of boundary 0 , U any given (U M controls in L with compact support in [0, T ], such that the corresponding solution Un of problem (I) and (I0) satisfies

84

8 Approximate Boundary Null Controllability

(Un (T ), Un (T )) 0 , U 1 , Hn )(T ) → (0, 0) in (H0 ) N × (H−1 ) N =R(U

(8.18)

as n → +∞. Taking the boundary control H = Hn , we solve the backward problem (8.9) and denote by Vn the corresponding solution. By linearity, we have 1 , Hn ) − R(Vn (0), Vn (0), Hn ) 0 , U R(U 1 − Vn (0), 0). 0 − Vn (0), U =R(U

(8.19)

Once again, by means of the well-posedness of problem (I) and (I0) regarding as a backward problem, and noting (8.18), we get 1 − Vn (0), 0)(0)(H0 ) N ×(H−1 ) N 0 − Vn (0), U R(U

(8.20)

c(Un (T ) − Vn (T ), Un (T ) − Vn (T ))(H0 ) N ×(H−1 ) N =c(Un (T ), Un (T )(H0 ) N ×(H−1 ) N → 0 as n → +∞, which together with (8.19) implies that 1 ) − (Vn (0), Vn (0))(H0 ) N ×(H−1 ) N 0 , U (U 1 , Hn )(0) − R(Vn (0), Vn (0), Hn )(0)(H0 ) N ×(H−1 ) N 0 , U =R(U

(8.21)

0 − Vn (0), U 1 − Vn (0), 0)(0)(H0 ) N ×(H−1 ) N → 0 cR(U as n → +∞. This shows C = (H0 ) N × (H−1 ) N . The proof is complete.



Theorem 8.6 System (I) is approximately null controllable at the time T > 0 if and only if the adjoint problem (8.7) is D-observable on the interval [0, T ]. Proof Assume that system (I) is not approximately null controllable. Then, by 1 ,  0 ) ∈ C ⊥ . Here, the orthogoLemma 8.5, there exists a nontrivial element (− 0 ) ∈ (H0 ) N × (H1 ) N . 1 ,  nality is defined in the sense of duality, therefore (−   Taking (0 , 1 ) as the initial data, we solve the adjoint problem (8.7) to get a solution . Next, multiplying the backward problem (8.9) by  and integrating by parts, we get  

1 )d x − (V (0), 

 

0 )d x = (V  (0), 



T 0

 1

(D H, ∂ν )ddt.

(8.22)

1 ,  0 ) ∈ C ⊥ , it follows that Noting that (− 

T 0

 1

(D H, ∂ν )ddt = 0

(8.23)

8.2 D-Observability for the Adjoint Problem

85

for all H ∈ L M . This gives the observation (8.8) with  ≡ 0, therefore, contradicts the D-observation of the adjoint problem (8.7). Inversely, assume that the adjoint problem (8.7) is not D-observable. Then there 1 ) ∈ (H1 ) N × (H0 ) N , such that the correspond0 ,  exists a nontrivial initial data ( ing solution  to the adjoint problem (8.7) satisfies the observation (8.8). Now for any 1 ) ∈ C, by the definition of C, there exists a sequence {Hn } in L M with 0 , U given (U compact support in [0, T ], such that the corresponding solution Vn to the backward problem (8.9) satisfies 0 , U 1 ) in (H0 ) N × (H−1 ) N as n → +∞. (Vn (0), Vn (0)) → (U

(8.24)

Noting (8.8), the relation (8.22) becomes  

1 )d x − (Vn (0), 

 

0 )d x = 0. (Vn (0), 

(8.25)

Passing to the limit as n → +∞, we get 1 ), (− 0 , U 1 ,  0 )(H0 ) N ×(H−1 ) N ;(H0 ) N ×(H1 ) N = 0

(U

(8.26)

⊥

1 ) ∈ C. In particular, we get (− 0 , U 1 ,  0 ) ∈ C . Therefore C = (H0 ) N × for all (U N  (H−1 ) . Corollary 8.7 If M = N , then system (I) is always approximately null controllable. Proof Since M = N , D is invertible, then the observation (8.8) gives that ∂ν  ≡ 0 on [0, T ] × 1 .

(8.27)

By means of Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]), we deduce the D-observability of the adjoint problem (8.7). Then by means of Theorem 8.6, we get the approximate boundary null controllability of system (I). The proof is complete.  Remark 8.8 Corollary 8.7 is a unique continuation result, which is not sufficient for the exact boundary null controllability. By Theorem 3.11, under the multiplier geometrical condition (3.1), system (I) is exactly null controllable by means of N boundary controls, however, without the multiplier geometrical condition (3.1), generally speaking, we cannot conclude the exact boundary null controllability even though we have applied N boundary controls for system (I) of N wave equations. We will discuss later the approximate boundary null controllability with fewer than N boundary controls.

86

8 Approximate Boundary Null Controllability

8.3 Kalman’s Criterion. Total (Direct and Indirect) Controls Theorem 8.9 Assume that the adjoint problem (8.7) is D-observable. Then, we have necessarily the following Kalman’s criterion: rank(D, AD, · · · , A N −1 D) = N .

(8.28)

Proof By (ii) of Proposition 2.12, it is easy to see that we only need to prove that Ker (D T ) does not contain a nontrivial subspace V which is invariant for A T . Let φn be the solution to the following eigenvalue problem with μn > 0: 

−φn = μ2n φn in , on . φn = 0

(8.29)

Assume that A T possesses a nontrivial invariant subspace V ⊆ K er (D T ). For any fixed integer n > 0, we define W = {φn w :

w ∈ V }.

(8.30)

Clearly, W is a finite-dimensional invariant subspace of − + A T . Therefore, we can solve the adjoint problem (8.7) in W and the corresponding solutions are given by  = φn w(t), where w(t) ∈ V satisfies 

w  + (μ2n I + A T )w = 0, 0 < t < ∞, 1 ∈ V. t =0: w=w 0 ∈ V, w  = w

(8.31)

Since w(t) ∈ V for all t  0, it follows that D T ∂ν  = ∂ν φn D T w(t) ≡ 0 on [0, T ] × 1 . Clearly,  ≡ 0, then we get a contradiction. The proof is complete.

(8.32) 

Thus, by Theorem 8.6, if system (I) is approximately null controllable, then we have necessarily the Kalman’s criterion (8.28). For the exact boundary null controllability, by Theorem 3.11 the number M = rank(D), namely, the number of boundary controls, should be equal to N , the number of state variables. However, as we will see in what follows, the approximate boundary null controllability of system (I) could be realized if the number M = rank(D) is very small, even if M = rank(D) = 1. Nevertheless, Theorems 8.6 and 8.9 show that if system (I) is approximately null controllable, then the enlarged matrix (D, AD, · · · , A N −1 D), composed of the coupling matrix A and the boundary control matrix D, should be of full row-rank. That is to say, even if the rank of D might be small, but because of the existence and influence of the coupling matrix

8.3 Kalman’s Criterion. Total (Direct and Indirect) Controls

87

A, in order to realize the approximate boundary null controllability, the rank of the enlarged matrix (D, AD, · · · , A N −1 D) should be still equal to N , the number of state variables. From this point of view, we may say that the rank M of D is the number of “direct” boundary controls acting on 1 , and rank(D, AD, · · · , A N −1 D) denotes the “total” number of direct and indirect controls, while the number of “indirect” controls is given by the difference: rank(D, AD, · · · , A N −1 D) − rank(D), which is equal to (N − M) in the case of approximate boundary null controllability. It is different from the exact boundary null controllability, in which only the number rank(D) of direct boundary controls is concerned and M = rank(D) should be equal to N , that for the approximate boundary null controllability, we should consider not only the number of direct boundary controls, but also the number of indirect controls, namely, the number of the total (direct and indirect) controls. Remark 8.10 It is well known that Kalman’s criterion (8.28) is necessary and sufficient for the exact controllability of systems of ODEs (cf. [23, 73]). But, the situation is more complicated for hyperbolic distributed parameter systems. Because of the finite speed of wave propagation, it is natural to consider Kalman’s criterion only in the case that T > 0 is sufficiently large. However, the following theorem precisely shows that even on the infinite observation interval [0, +∞), the sufficiency of Kalman’s criterion may fail. So, some algebraic assumptions on the coupling matrix A should be required to guarantee the sufficiency of Kalman’s criterion. Theorem 8.11 Let μ2n and φn be defined by (8.29). Assume that the set  = {(m, n) :

μm = μn , ∂ν φm = ∂ν φn on 1 }

(8.33)

is not empty. For any given (m, n) ∈ , let =

μ2m − μ2n . 2

(8.34)

Then the adjoint system ⎧  ⎨ φ − φ + ψ = 0 in (0, +∞) × , ψ  − ψ + φ = 0 in (0, +∞) × , ⎩ φ=ψ=0 on (0, +∞) × 

(8.35)

admits a nontrivial solution (φ, ψ) such that ∂ν φ ≡ 0 on [0, +∞) × 1 ,

(8.36)

hence the corresponding adjoint problem (8.7) is not D-observable with D = (1, 0)T . Proof Let φλ = (φn − φm ), ψλ = (φn + φm ), λ2 =

μ2m + μ2n . 2

(8.37)

88

8 Approximate Boundary Null Controllability

We check easily that (φλ , ψλ ) satisfies the following eigensystem: ⎧ 2 ⎨ λ φλ + φλ − ψλ = 0 in , λ2 ψλ + ψλ − φλ = 0 in , ⎩ on . φλ = ψ λ = 0

(8.38)

Moreover, noting the definition (8.33) of , we have ∂ν φλ ≡ 0 on 1 .

(8.39)

φ = eiλt φλ , ψ = eiλt ψλ .

(8.40)

Now, let It is easy to see that (φ, ψ) is a nontrivial solution to system (8.35) and satisfies the observation (8.36). In order to complete the proof, we examine the following situations (there are many others!), in which the set  is indeed not empty. (i)  = (0, π) with 1 = {0}. In this case, we have μn = n, φn =

1 sin nx and φn (0) = 1. n

(8.41)

So, (m, n) ∈  for all m = n. (ii)  = (0, π) × (0, π) with 1 = {0} × [0, π]. Setting μm,n =



m 2 + n 2 , φm,n =

1 sin mx sin ny, m

(8.42)

we have

∂ ∂ φm,n (0, y) = φm  ,n (0, y) = sin ny, 0  y  π. ∂x ∂x

 So {m, n}, {m  , n} ∈  for all m = m  and n  1. The proof is thus complete.

Remark 8.12 Note that 1 D= , 0

A=

0 10 , (D, AD) = 0 0

(8.43)



(8.44)

correspond to the adjoint system (8.35) with the observation (8.36). Since the matrices A and D satisfy well the corresponding Kalman’s criterion (8.28), Theorem 8.11 shows that even in the infinite interval of observation, Kalman’s criterion is not sufficient for the D-observability of the adjoint problem (8.35) at least in the two cases mentioned above. Therefore, in order to get the D-observability, some additional algebraic assumptions on A should be imposed.

8.4 Sufficiency of Kalman’s Criterion for T > 0 Large …

89

8.4 Sufficiency of Kalman’s Criterion for T > 0 Large Enough for the Nilpotent System A matrix A of order N is called to be nilpotent if there exists an integer k with 1  k  N , such that Ak = 0. Then it is easy to see that A is nilpotent if and only if all the eigenvalues of A are equal to zero. Thus, under a suitable basis B, a nilpotent matrix A can be written in a diagonal form of Jordan blocks: ⎛ ⎜ B −1 AB = ⎝

⎞

Jp Jq

..

⎟ ⎠,

(8.45)

.

where J p is the following Jordan block of order p: ⎛ ⎞ 01 ⎜ 01 ⎟ ⎜ ⎟ ⎜ · · ⎟ Jp = ⎜ ⎟. ⎝ 0 1⎠ 0

(8.46)

When A is a sole Jordan block, so-called the cascade matrix, it was shown in [2] that the observation on the last component of adjoint variable of the adjoint problem (8.7) is sufficient for the corresponding D-observability. In this section, we will generalize this result to the nilpotent system. We first consider a very special case. Proposition 8.13 Let A = a I , where a is a real number. If Kalman’s criterion (8.28) holds, then the adjoint problem (8.7) is D-observable, provided that T > 0 is large enough. Proof In this case, Kalman’s criterion (8.28) implies that M = N . Consequently, the observation (8.8) implies that ∂ν  ≡ 0 on [0, T ] × 1 .

(8.47)

Thus, by the classical Holmgren’s uniqueness theorem, we get  ≡ 0, provided that T > 0 is large enough. In this situation, we don’t need any multiplier geometrical condition on the domain . The proof is complete.  Lemma 8.14 Assume that there exists an invertible matrix P such that P A = A P. Then the adjoint problem (8.7) is D-observable if and only if it is P D-observable.  satisfies the same  = P −T . Noting P A = A P, the new variable  Proof Let  system as in problem (8.7). On the other hand, since  on 1 , D T ∂ν  = (P D)T ∂ν 

(8.48)

90

8 Approximate Boundary Null Controllability

. The proof is the D-observability on  is equivalent to the P D-observability on  complete.  Proposition 8.15 Let P be an invertible matrix. Define A˜ = P A P −1

and

 = P D. D

Then the matrices A and D satisfy Kalman’s criterion (8.28) if and only if the matrices  do so. A˜ and D Proof It is sufficient to note that  A˜ D,  · · · , A˜ N −1 D  = P[D, AD, · · · , A N −1 D] [ D, 

and that P is invertible.

Theorem 8.16 Assume that  ⊂ Rn satisfies the multiplier geometrical condition (3.1). Let T > 0 be large enough. Assume that the coupling matrix A is nilpotent. Then Kalman’s criterion (8.28) is sufficient for the D-observability of the adjoint system (8.7). Proof (i) Case that A is a Jordan block (the cascade matrix): ⎛ ⎞ 01 ⎜ 01 ⎟ ⎜ ⎟ ⎜ · · ⎟ A=⎜ ⎟ =: JN . ⎝ 0 1⎠ 0

(8.49)

Noting that E = (0, · · · , 0, 1)T is the only eigenvector of A T , by Proposition 2.12 (ii), A and D satisfy Kalman’s criterion (8.28) if and only if D T E = 0,

(8.50)

namely, if and only if the last row of D is not a zero vector. We denote by d = (d1 , d2 , · · · , d N )T a column of D with d N = 0 and let ⎛

d N d N −1 ⎜ 0 dN ⎜ · P=⎜ ⎜ · ⎝0 0 0 0

⎞ · d1 · d2 ⎟ ⎟ · · ⎟ ⎟. d N d N −1 ⎠ 0 dN

(8.51)

Obviously, P is invertible. Noting that P = d N I + d N −1 JN + · · · + d1 JNN −1 ,

(8.52)

8.4 Sufficiency of Kalman’s Criterion for T > 0 Large …

91

it is easy to see that P A = A P. On the other hand, under the multiplier geometrical condition (3.1) on , it was shown in [2] that the adjoint system (8.7) with the coupling matrix (8.49) is D0 -observable with D0 = (0, · · · , 0, 1)T .

(8.53)

Thus, by Lemma 8.14, the same system is P D0 -observable, then, noting that P D0 = (d1 , · · · , d N )T

(8.54)

is a submatrix of D, problem (8.7) must be D-observable. (ii) Case that A is composed of two Jordan blocks of the same size: A=

Jp 0 , 0 Jp

(8.55)

where J p is the Jordan block of order p. First, let i be defined by (i)

i = (0, · · · , 0, 1 , 0, · · · , 0)T

(8.56)

for i = 1, · · · , 2 p. Consider the special boundary control matrix D0 = ( p , 2 p ).

(8.57)

Noting that in this situation the adjoint system (8.7) and the observation (8.8) are entirely decoupled into two independent subsystems, each of them satisfies Kalman’s criterion (8.28) with N = p. Then, by the consideration in the previous case, the adjoint system (8.7) is D0 -observable. Now, let us consider the general boundary control matrix of order 2 p × M : ⎛

a1 ⎜ .. ⎜. ⎜ ⎜a p D=⎜ ⎜ b1 ⎜ ⎜. ⎝ ..

⎞ c1 · · · · · · ⎟ .. ⎟ . ⎟ c p · · · · · ·⎟ ⎟. d1 · · · · · ·⎟ ⎟ ⎟ .. ⎠ .

(8.58)

bp dp · · · · · ·

Noting that  p and 2 p are the only two eigenvectors of A T , associated with the same eigenvalue zero, then for any given real numbers α and β with α2 + β 2 = 0, α p + β2 p is also an eigenvector of A T , by Proposition 2.12 (ii), Kalman’s criterion (8.28) holds if and only if D T  p and D T 2 p , namely, the row vectors

92

8 Approximate Boundary Null Controllability

(a p , c p , · · · · · · ), (b p , d p , · · · · · · ),

(8.59)

are linearly independent. Without loss of generality, we may assume that a p d p − b p c p = 0.

(8.60)

Let the matrix P of order 2 p be defined by ⎛

ap ⎜0 ⎜ ⎜ .. ⎜ . ⎜ ⎜0 P=⎜ ⎜ cp ⎜ ⎜0 ⎜ ⎜ . ⎝ .. 0

a p−1 ap .. . 0 c p−1 cp .. . 0

··· ··· .. .

a1 a2 .. .

· · · ap · · · c1 · · · c2 . . .. . .

b p b p−1 0 bp .. .. . . 0 0 d p d p−1 0 dp .. .. . .

· · · cp 0

0

··· ··· .. .

··· ··· ··· .. .

⎞ b1 b2 ⎟ ⎟ .. ⎟ . ⎟ ⎟

bp ⎟ ⎟ = P11 P12 . P21 P22 d1 ⎟ ⎟ d2 ⎟ ⎟ .. ⎟ . ⎠

(8.61)

· · · dp

Since P11 , P12 , P21 , and P22 have a similar structure as (8.52) for P, we check easily that



J p P11 J p P12 P11 J p P12 J p = = A P. (8.62) PA = P21 J p P22 J p J p P21 J p P22 Moreover, P is invertible under condition (8.60). Since the adjoint system (8.7) is D0 -observable, by Lemma 8.14, it is P D0 -observable, then, D-observable since P D0 is composed of the first two columns of D. (iii) Case that A is composed of two Jordan blocks with different sizes: A=

Jp 0 0 Jq

(8.63)

with q < p. In this case, the adjoint system (8.7) is composed of the first subsystem  i = 1, · · · , p :

φi − φi + φi−1 = 0 in (0, +∞) × , on (0, +∞) ×  φi = 0

(8.64)

with φ0 = 0, and of the second subsystem  j = p − q + 1, · · · , p :

ψ j − ψ j + ψ j−1 = 0 in (0, +∞) × , on (0, +∞) ×  ψj = 0

(8.65)

with ψ p−q = 0. These two subsystems (8.64) and (8.65) are coupled together by the D-observations:

8.4 Sufficiency of Kalman’s Criterion for T > 0 Large …

93

⎧ p p   ⎪ ⎪ ⎪ a ∂ φ + b j ∂ν ψ j = 0 on (0, T ) × 1 , i ν i ⎪ ⎪ ⎪ ⎪ i=1 j= p−q+1 ⎪ ⎪ p p ⎨  ci ∂ν φi + d j ∂ν ψ j = 0 on (0, T ) × 1 , ⎪ ⎪ i=1 j= p−q+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪· · · · · · · · · · · · ⎪ ⎩ ············

(8.66)

In order to transfer the problem to the case p = q, we expand the second subsystem (8.65) from { p − q + 1, · · · , p} to {1, · · · , p}:  j = 1, · · · , p :

ψ j − ψ j + ψ j−1 = 0 in (0, +∞) × , on (0, +∞) ×  ψj = 0

(8.67)

with ψ0 = 0, so that the two subsystems (8.64) and (8.67) have the same size.  Accordingly, the D-observations (8.66) can be extended to the D-observations: ⎧ p p   ⎪ ⎪ ⎪ a ∂ φ + b j ∂ν ψ j = 0 on (0, T ) × 1 , i ν i ⎪ ⎪ ⎪ ⎪ i=1 j=1 ⎪ ⎪ p p ⎨  ci ∂ν φi + d j ∂ν ψ j = 0 on (0, T ) × 1 , ⎪ ⎪ i=1 j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ············ ⎪ ⎪ ⎩ ············

(8.68)

with arbitrarily given b j and d j for j = 1, · · · , p − q. Let us write the matrix D of order ( p + q) × M of the observations (8.66) as ⎛

a1 c1 ⎜ .. .. ⎜ . . ⎜ ⎜ ap c p D=⎜ ⎜b p−q+1 d p−q+1 ⎜ ⎜ .. .. ⎝ . . dp bp

······

⎞

⎟ ⎟ ⎟ · · · · · ·⎟ ⎟. · · · · · ·⎟ ⎟ ⎟ ⎠ ······

(8.69)

Similarly to the case (ii),  p and  p+q are the only two eigenvectors of A T , associated with the same eigenvalue zero, then for any given real numbers α and β with α2 + β 2 = 0, α p + β p+q is also an eigenvector of A T . By Proposition 2.12 (ii), Kalman’s criterion (8.28) holds if and only if D T  p and D T  p+q , namely, the row vectors (a p , c p , · · · · · · ), (b p , d p , · · · · · · ),

(8.70)

94

8 Approximate Boundary Null Controllability

are linearly independent. Without loss of generality, we may assume that (8.60) holds.  of order 2 p × M of the extended observations Similarly, we write the matrix D (8.68) as ⎞ ⎛ a1 c1 · · · · · · ⎟ ⎜ .. .. ⎟ ⎜ . . ⎟ ⎜ ⎜ ap c p · · · · · ·⎟ ⎟ ⎜ ⎜ b1 d1 · · · · · ·⎟ ⎟ ⎜ ⎟ . .. =⎜ (8.71) D ⎟. ⎜ .. . ⎟ ⎜ ⎟ ⎜ b p−q d p−q ⎟ ⎜ ⎜b p−q+1 d p−q+1 · · · · · ·⎟ ⎟ ⎜ ⎟ ⎜ . .. ⎠ ⎝ .. . dp · · · · · · bp  of order 2 p of the extended adjoint system, composed of (8.64) and The matrix A  and D  satisfy the corresponding (8.67), is the same as given in (8.55). Then, A Kalman’s criterion if and only if the condition (8.60) holds. Then by the conclusion  of (ii), the extended adjoint system composed of (8.64) and (8.67) is D-observable in the space (H01 ())2 p × (L 2 ())2 p . In particular, if we choose the specific initial data such that t =0:

ψ1 = · · · = ψ p−q = 0 and ψ1 = · · · = ψ p−q = 0

(8.72)

for the extended subsystem (8.67), then by well-posedness we have ψ1 ≡ · · · ≡ ψ p−q ≡ 0 in (0, +∞) × .

(8.73)

 Thus, the extended adjoint system composed of (8.64) and (8.67) with the Dobservations (8.68) reduces to the original adjoint system composed of (8.64) and (8.65) with the D-observations (8.66), hence we get that the original adjoint system is D-observable in the space (H01 ()) p+q × (L 2 ()) p+q . The case that A is composed of several Jordan blocks can be carried on similarly. Since any given nilpotent matrix can be decomposed in a diagonal form of Jordan blocks under a suitable basis, by Proposition 8.15, the previous conclusion is still valid for any given nilpotent matrix A. The proof is complete.  Theorem 8.17 Assume that  ⊂ Rn satisfies the multiplier geometrical condition (3.1). Let T > 0 be large enough. Assume furthermore that the coupling matrix A admits one sole eigenvalue λ  0. Then the adjoint problem (8.7) is D-observable if and only if the matrix D satisfies Kalman’s criterion (8.28). Proof In fact, the operator − + λI is still self-adjoint and coercive in L 2 () and A − λI is nilpotent. So, Theorem 8.16 remains true as − is replaced by − + λI . 

8.5 Sufficiency of Kalman’s Criterion for T > 0 Large …

95

8.5 Sufficiency of Kalman’s Criterion for T > 0 Large Enough for 2 × 2 Systems Theorem 8.18 Let  ⊂ Rn satisfy the multiplier geometrical condition (3.1). Assume that the 2 × 2 matrix A has real eigenvalues λ  0 and μ  0 such that |λ − μ|  0 ,

(8.74)

where 0 > 0 is small enough. Then the adjoint problem (8.7) is D-observable for T > 0 large enough if and only if the matrix D satisfies Kalman’s criterion (8.28). Proof By Theorem 8.9, we only need to prove the sufficiency. If D is invertible, then the observation (8.8) implies that ∂ν  ≡ 0 on [0, T ] × 1

(8.75)

and the classical Holmgren’s uniqueness theorem, implies that  ≡ 0, provided that T > 0 is large enough. Noting that in this case, we do not need the multiplier geometrical condition (3.1) on . Thus, we only need to consider the case that D is of rank one. Without loss of generality, we may consider the following three cases. (a) If λ = μ with two linearly independent eigenvectors, then any nontrivial vector is an eigenvector of A T . By (ii) of Lemma 2.12 with d = 0, Ker(D T ) is reduced to {0}, then the matrix D is invertible, which gives a contradiction. (b) If λ = μ with one eigenvector, then A∼

λ1 . 0λ

Since λ  0, it suffices to apply Theorem 8.17. (c) If λ = μ, then



λ+μ 0 0 λ−μ 2 2 + . A∼ λ−μ 0 0 λ+μ 2 2

(8.76)

(8.77)

Since λ + μ  0, the operator − + λ+μ I is still coercive in H01 (), then it is 2 sufficient to consider the adjoint problem (8.7), in which − is replaced by − + λ+μ I , with the matrix 2

0 λ−μ (8.78) A = λ−μ 2 . 0 2 Since  satisfies the multiplier geometrical condition (3.1), following a result in [1] (cf. also [65] in the case of different speeds of wave propagation), the corresponding adjoint problem (8.7) is D0 -observable with

96

8 Approximate Boundary Null Controllability

D0 =

1 . 0

On the other hand, since D is of rank one, we have

a 0 a b D= , A= , (D, AD) = b 0 b a

(8.79)

(8.80)

. So, Kalman’s criterion (8.28) is satisfied if and only if a 2 = b2 . Then with  = λ−μ 2 the matrix ab P= (8.81) ba is invertible and commutes with A. Thus, by Lemma 8.14, the adjoint problem (8.7) is also P D0 -observable, but a = D. (8.82) P D0 = b 

The proof is complete.

Remark 8.19 Theorem 8.11 shows that the condition “0 > 0 is small enough” in Theorem 8.18 is actually necessary. Proposition 8.20 Let  ⊂ Rn satisfy the multiplier geometrical condition (3.1) and || > 0 be small enough. Let T > 0 be large enough. Then the adjoint system (8.35) is D-observable if and only if the matrix D satisfies the corresponding Kalman’s criterion. Proof Since  satisfies the multiplier geometrical condition (3.1), following a result in [1], the adjoint system (8.35) is D0 -observable with 1 D0 = . 0 Noting that

a D= , b



0 a b A= , (D, AD) = , 0 b a

(8.83)

(8.84)

Kalman’s criterion (8.28) is satisfied if and only if a 2 = b2 . On the other hand, the matrix ab P= (8.85) ba is invertible and commutes with A. Thus, by Lemma 8.14, the adjoint problem (8.35) is also P D0 -observable with a P D0 = = D. (8.86) b

8.5 Sufficiency of Kalman’s Criterion for T > 0 Large …

97

The proof is complete.



Remark 8.21 Because of the equivalence between the approximate boundary null controllability and the D-observability (cf. Theorem 8.6), in the cases discussed in Theorems 8.16 and 8.18, Kalman’s criterion (8.28) is indeed necessary and sufficient to the approximate boundary null controllability for the corresponding system (I). In the one-space-dimensional case, some more general results can be obtained in this direction, and the following two sections are devoted to this end.

8.6 The Unique Continuation for Nonharmonic Series In this section, we will establish the unique continuation for nonharmonic series, which will be used in the next section to prove the sufficiency of Kalman’s criterion (8.28) to the D-observability for T > 0 large enough for diagonalizable systems in one-space-dimensional case. The study is based on a generalized Ingham’s inequality (cf. [28]). Let Z∗ denote the set of all nonzero integers and {βn(l) }1lm,n∈Z∗ be a strictly increasing real sequence: (1) (m) < · · · < β−1 < β1(1) < · · · < β1(m) < · · · . · · · β−1

(8.87)

We want to show that the following condition m 

(l)

an(l) eiβn t = 0 on [0, T ]

(8.88)

n∈Z∗ l=1

with

m  n∈Z∗

implies that

|an(l) |2 < +∞

(8.89)

l=1

an(l) = 0, n ∈ Z∗ , 1  l  m,

(8.90)

provided that T > 0 is large enough. If this is true, we say that the sequence (l) {eiβn t }1lm;n∈Z∗ is ω-linearly independent in L 2 (0, T ). Theorem 8.22 Assume that (8.87) holds and that there exist positive constants c, s, and γ such that (l) βn+1 − βn(l)  mγ, (8.91) c  βn(l+1) − βn(l)  γ |n|s

(8.92)

98

8 Approximate Boundary Null Controllability

for all 1  l  m and all n ∈ Z∗ with |n| large enough. Then the sequence (l) {eiβn t }1lm;n∈Z∗ is ω-linearly independent in L 2 (0, T ), provided that T > 2π D + , where D + is the upper density of the sequence {βn(l) }1lm;n∈Z∗ , defined by D + = lim sup R→+∞

N (R) , 2R

(8.93)

where N (R) denotes the number of {βn(l) } contained in the interval [−R, R]. Proof We first define the sequence of difference quotients as follows: en(l) (t) =

l  

 ( p) (βn( p) − βn(q) )−1 eiβn t

l 

(8.94)

q=1,q = p

p=1

for l = 1, · · · , m and n ∈ Z∗ . Since the sequence {βn(l) }1lm,n∈Z∗ can be asymptotically close at the rate |n|1 s , the classical Ingham’s theorem does not work. We will use a generalized Ingham-type theorem based on the divided differences, which tolerates asymptotically close frequencies {βn(l) }1lm,n∈Z∗ (cf. Theorem 9.4 in [28]). Namely, under conditions (8.87) and (8.91)–(8.92), the sequence of difference quotients {en(l) }1lm,n∈Z∗ is a Riesz sequence in L 2 (0, T ), provided that T > 2π D + . (l, p) Let the lower triangular matrix An = (an ) be defined by l 

an(1,1) = 1; an(l, p) =

(βn( p) − βn(q) )−1

(8.95)

q=1,q = p

for 1 < l  m and 1  p  l. Since the diagonals an(l,l) are positive for all 1  l  m, (l, p) An is invertible. Then, setting A−1 ), we write (8.94) as n = Bn = (bn (l)

eiβn t =

l 

bn(l, p) en( p) (t), l = 1, · · · , m.

(8.96)

p=1

Inserting (8.96) into (8.88), we get m  n∈Z∗

 an( p) en( p) (t) = 0 on [0, T ],

(8.97)

p=1

where  an( p) =

m  l= p

Assume for the time being that

bn(l, p) an(l) .

(8.98)

8.6 The Unique Continuation for Nonharmonic Series m  n∈Z∗

99

| an( p) |2 < +∞.

(8.99)

p=1

Then by means of the property of Riesz sequence, it follows from (8.97) and (8.99) that  an( p) = 0, 1  p  m, n ∈ Z∗ , (8.100) which implies (8.90). Now we return to the verification of (8.99). From (8.98), it is sufficient to show that the matrix Bn is uniformly bounded for all n. From (8.92) and (8.95), we have bn(l,l) =

1 an(l,l)

=

l−1 

(βn(l) − βn(q) )  c1 γ l−1 , 1  l  m,

(8.101)

q=1

where c1 is a positive constant independent of n. Since Bn is also a lower triangular matrix, without loss of generality, we assume that γ < 1. Then, it follows that the spectral radius ρ(Bn )  c1 . It is well known that for any given ˜ > 0, there exists a vector norm in Rm , such that the subordinate matrix norm satisfies Bn   (ρ(Bn ) + ˜)  c1 + 1, ∀n ∈ Z∗ .

(8.102) 

The proof is then complete. Corollary 8.23 For δ1 < δ2 < · · · < δm , we define



βn(l) =

n 2 + δl , l = 1, 2, · · · , m, n  1,

(l) β−n = −βn(l) ,

l = 1, 2, · · · , m, n  1.

(8.103)

(8.104)

(l)

Then, for || > 0 small enough, the sequence {eiβn t }1lm;n∈Z∗ is ω-linearly independent in L 2 (0, T ), provided that T > 2mπ.

(8.105)

Proof First, for || > 0 small enough, the sequence {βn(l) }1lm;n∈Z∗ satisfies (8.87). On the other hand, a straightforward computation gives that (l) − βn(l) = O(1) βn+1

and

   (δl+1 − δl )   =O   βn(l+1) − βn(l) = 2 2 n n + δl+1  + n + δl 

(8.106)

(8.107)

100

8 Approximate Boundary Null Controllability

for |n| large enough. Then the sequence {βn(l) }1lm;n∈Z∗ satisfies all the requirements of Theorem 8.22 with s = 1 and D + = m. Consequently, the sequence (l) {eiβn t }1lm;n∈Z∗ is ω-linearly independent in L 2 (0, T ), provided that (8.105) holds. 

8.7 Sufficiency of Kalman’s Criterion for T > 0 Large Enough in the One-Space-Dimensional Case In this section, under suitable conditions on the coupling matrix A and for || > 0 small enough, we will first establish the sufficiency of Kalman’s criterion (8.28) for the following one-space-dimensional problem: ⎧  in (0, +∞) × (0, π), ⎨  − x x + A T  = 0 (t, 0) = (t, π) = 0 on (0, +∞), ⎩ 1 in (0, π)  0 ,  =  t =0: =

(8.108)

with the observation at the end x = 0: D T x (t, 0) ≡ 0 on [0, T ].

(8.109)

We will next give the optimal time of observation for the coupling matrix A, which has distinct real eigenvalues. First assume that A T is diagonalizable with the real eigenvalues: δ 1 < δ2 < · · · < δm

(8.110)

and the corresponding eigenvectors w(l,μ) such that A T w (l,μ) = δl w (l,μ) , 1  l  m, 1  μ  μl , in which

m 

μl = N .

(8.111)

(8.112)

l=1

Let en = sin nx, n  1

(8.113)

be the eigenfunctions of − in H01 (0, π). Then en w (l,μ) is an eigenvector of − + A T corresponding to the eigenvalue n 2 + δl . Furthermore, we define {βn(l) }1lm;n∈Z∗ as in (8.104) and the corresponding eigenvectors of the system given in (8.108) by

8.7 Sufficiency of Kalman’s Criterion for T > 0 Large Enough …

⎛

en w (l,μ)

101

⎞

E n(l,μ) = ⎝ iβn(l) ⎠ , 1  l  m, 1  μ  μl , n ∈ Z∗ , en w (l,μ)

(8.114)

in which we define e−n = en for all n  1. Since the eigenfunctions en (n  1) are (l,μ) orthogonal in L 2 (0, π) as well as in H01 (0, π), the linear hull Span{E n }1lm,1μμl of finite dimension N are mutually orthogonal for all n  1. On the other hand, the system of eigenvectors (8.114) is complete in (H01 (0, π)) N × (L 2 (0, π)) N , therefore it forms a Hilbert basis of subspaces, then a Riesz basis in (H01 (0, π)) N × (L 2 (0, π)) N (cf. [15] for the basis of subspaces). For any given initial data  μl m  0  = αn(l,μ) E n(l,μ) , 1  ∗

(8.115)

n∈Z l=1 μ=1

the corresponding solution of problem (8.108) is given by

 

=

μl m  

(l)

αn(l,μ) eiβn t E n(l,μ) .

(8.116)

n∈Z∗ l=1 μ=1

In particular, we have =

μl m  (l,μ)  αn (l) n∈Z∗ l=1 μ=1 iβn

(l)

eiβn t en w (l,μ) ,

(8.117)

and the observation (8.109) becomes m 

DT

n∈Z∗ l=1

μl (l,μ)  nα μ=1

n iβn(l)

 (l) w (l,μ) eiβn t ≡ 0 on [0, T ].

(8.118)

Theorem 8.24 Assume that A and D satisfy Kalman’s criterion (8.28). Assume furthermore that A T is real diagonalizable with (8.110)–(8.111). Then problem (8.108) is D-observable for || > 0 small enough, provided that T > 2mπ.

(8.119)

Proof Applying Corollary 8.23 to each line of (8.118), we get DT

μl (l,μ)  nα μ=1

n iβn(l)

 w (l,μ) = 0, 1  l  m, n ∈ Z∗ .

(8.120)

102

8 Approximate Boundary Null Controllability

By virtue of Proposition 2.12 (ii), because of Kalman’s criterion (8.28), Ker(D T ) does not contain any nontrivial invariant subspace of A T , then it follows that μl (l,μ)  nαn μ=1

Hence

iβn(l)

w (l,μ) = 0, 1  l  m, n ∈ Z∗ .

αn(l,μ) = 0, 1  μ  μl , 1  l  m, n ∈ Z∗ .

(8.121)

(8.122) 

The proof is complete.

We now improve the estimate (8.119) on the observation time in the case that the eigenvalues of A are distinct. Theorem 8.25 Under the assumptions of Theorem 8.24, assume furthermore that A T possesses N distinct real eigenvalues: δ1 < δ2 < · · · < δ N .

(8.123)

Then problem (8.108) is D-observable for || > 0 small enough, provided that T > 2π(N − rank(D) + 1).

(8.124)

Proof Let w(1) , w (1) , · · · , w (N ) be the corresponding eigenvectors of A T . Accordingly, (8.118) becomes N 

DT

n∈Z l=1

nαn(l) iβn(l)

(l)

w (l) eiβn t ≡ 0 on [0, T ].

(8.125)

Then, setting r = rank(D), without loss of generality, we assume that D T w (1) , · · · , D T w (r ) are linearly independent. There exists an invertible matrix S of order N , such that (8.126) S D T (w (1) , · · · , w (r ) ) = (e1 , · · · er ), where e1 , · · · er are the canonic basis vectors in R N . Since S is invertible, (8.125) can be equivalently rewritten as r   nα(l) n∈Z∗

n (l) iβ n l=1

(l)

el eiβn t +

N  nαn(l) l=r +1

iβn(l)

(l)

S D T w (l) eiβn

t



≡0

(8.127)

for all 0  t  T . Once again, we apply Corollary 8.23 to each equation of (8.127), but this time, the upper density of the sequence {βn(1) , βn(l) }r +1lN ;n∈Z∗ is equal to (N − r + 1). Therefore, we get again (8.122), provided that (8.124) holds. The proof is complete. 

8.8 An Example

103

8.8 An Example Let || > 0 be small enough. By Corollary 8.20, we know that the following system ⎧  ⎨ φ − φ + ψ = 0 in (0, +∞) × , ψ  − ψ + φ = 0 in (0, +∞) × , ⎩ φ=ψ=0 on (0, +∞) × 

(8.128)

is observable by means of the trace ∂ν φ|1 or ∂ν ψ|1 on all the interval [0, T ] for T > 0 large enough. We now consider the following example which is approximately null controllable, but not exactly null controllable: ⎧  u − u + v = 0 ⎪ ⎪ ⎨  v − v + u = 0 u=v=0 ⎪ ⎪ ⎩ u = h, v = 0

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 on (0, +∞) × 1 ,

(8.129)

1 and h is a boundary control. 0 First, by means of Theorem 3.10, system (8.129) is never exactly null controllable in the space (H0 )2 × (H−1 )2 because of the lack of boundary controls. On the other hand, since its adjoint problem (8.128) is D-observable via the observation of the trace ∂ν φ|1 , then, applying Theorem 8.6, system (8.129) is approximately null controllable in the space (L 2 ())2 × (H −1 ())2 via only one boundary control h ∈ L 2 (0, T ; L 2 (1 )), provided that T is large enough. where N = 2, M = 1, D =

Chapter 9

Approximate Boundary Synchronization

The approximate boundary synchronization is defined and studied in this chapter for system (I) with Dirichlet boundary controls.

9.1 Definition We now give the definition on the approximate boundary synchronization as follows. Definition 9.1 System (I) is approximately synchronizable at the time T > 0, if for 0 , U 1 ) ∈ (H0 ) N × (H−1 ) N , there exists a sequence {Hn } of any given initial data (U M boundary controls in L with compact support in [0, T ], such that the corresponding sequence {Un } of solutions to problem (I) and (I0) satisfies (l) u (k) n − un → 0

as n → +∞

(9.1)

for all 1  k, l  N in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(9.2)

Let C1 be the synchronization matrix of order (N − 1) × N , defined by ⎛ ⎞ 1 −1 ⎜ 1 −1 ⎟ ⎟. C1 = ⎜ ⎝ ⎠ · · 1 −1 © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_9

(9.3)

105

106

9 Approximate Boundary Synchronization

Obviously, the approximate boundary synchronization (9.1) can be equivalently rewritten as (9.4) C1 Un → 0 as n → +∞ in the space 0 1 ([T, +∞); (H0 ) N −1 ) ∩ Cloc ([T, +∞); (H−1 ) N −1 ). Cloc

(9.5)

9.2 Condition of C1 -Compatibility Theorem 9.2 Assume that system (I) is approximately synchronizable, but not approximately null controllable. Then the coupling matrix A = (ai j ) should satisfy the following condition of compatibility (the row-sum condition): N

akp := a, k = 1, · · · , N ,

(9.6)

p=1

where a is a constant independent of k = 1, · · · , N . 0 , U 1 ) ∈ (H0 ) N × (H−1 ) N , and let {Hn } be a sequence of boundary Proof Let (U controls, which realize the approximate boundary synchronization of system (I). Let {Un } be the sequence of the corresponding solutions. We have Un − Un + AUn = 0 in (T, +∞) × .

(9.7)

(N ) T Noting Un = (u (1) n , · · · , u n ) , we have  u (k) n

−

u (k) n

+

N

akp u n( p) = 0 in (T, +∞) × , 1  k  N .

(9.8)

p=1 (N ) Let wn(k) = u (k) for 1  k  N − 1. It follows from (9.8) that n − un 

wn(k) − wn(k) +

N −1

akp wn( p) +

p=1

N



 ) akp − a u (N = 0. n

(9.9)

p=1

If (9.6) fails, then, noting (9.1), it follows that

 ) u (N → 0 in D (T, +∞) ×  as n → +∞. n

(9.10)

In order to avoid the technical details of the proof, here we claim (cf. Corollary 10.15 below) that the convergence (9.10) holds actually in the usual space

9.2 Condition of C1 -Compatibility

107

0 1 Cloc ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ),

(9.11)

which contradicts the non-approximate boundary null controllability. The proof is complete.  Remark 9.3 The condition of compatibility (9.6) is just the same as (4.5) for the exact boundary synchronization. The condition of compatibility (9.6) indicates that e1 = (1, · · · , 1)T is an eigenvector of the coupling matrix A, associated with the eigenvalue a given by (9.6). Moreover, since Ker(C1 ) = Span{e1 }, this condition is equivalent to AKer(C1 ) ⊆ Ker(C1 ),

(9.12)

namely, Ker(C1 ) is an invariant subspace of A. We call (9.12) the condition of C1 -compatibility, which is also equivalent to the existence of a unique matrix A1 of order (N − 1), such that (9.13) C 1 A = A1 C 1 , where the matrix A1 is called the reduced matrix of A by C1 .

9.3 Fundamental Properties Under the condition of C1 -compatibility (9.12), let W1 = (w (1) , · · · , w (N −1) )T = C1 U.

(9.14)

The original problem (I) and (I0) for the variable U can be reduced to the following self-closed problem for the variable W : ⎧ ⎨ W1 − W1 + A1 W1 = 0 in (0, +∞) × , on (0, +∞) × 0 , W =0 ⎩ 1 on (0, +∞) × 1 W1 = C 1 D H

(9.15)

with the initial data t =0: Accordingly, let

0 , W1 = C1 U 1 in . W1 = C 1 U 1 = (ψ (1) , · · · , ψ (N −1) )T .

Consider the adjoint problem of the reduced system (9.15)

(9.16)

(9.17)

108

9 Approximate Boundary Synchronization

⎧ T ⎨ 1 − 1 + A1 1 = 0 in (0, +∞) × , on (0, +∞) × , 1 = 0 ⎩ 0 , 1 =  1 in . t = 0 : 1 = 

(9.18)

In what follows, problem (9.18) will be called the reduced adjoint problem of system (I). From Definitions 8.1 and 9.1, we immediately get the following Lemma 9.4 Assume that the coupling matrix A satisfies the condition of C1 -compatibility (9.12). Then system (I) is approximately synchronizable at the time T > 0 if and only if the reduced system (9.15) is approximately null controllable at the time T > 0, or equivalently (cf. Theorem 8.6), the reduced adjoint problem (9.18) is C1 D-observable on the time interval [0, T ]. Moreover, we have Lemma 9.5 Under the condition of C1 -compatibility (9.12), assume that system (I) is approximately synchronizable, then we necessarily have rank(C1 D, C1 AD, · · · , C1 A N −1 D) = N − 1.

(9.19)

Proof By Lemma 9.4 and Theorem 8.9, we have N −2

rank(C1 D, A1 C1 D, · · · , A1

C1 D) = N − 1.

Then, noting (9.13) and using Proposition 2.16, we get immediately (9.19).

(9.20) 

9.4 Properties Related to the Number of Total Controls As we have already explained in Sect. 8.3, the rank of the enlarged matrix (D, AD, · · · , A N −1 D) measures the number of total (direct and indirect) controls. So, it is significant to determine the minimal number of total controls, which is necessary to the approximate boundary synchronization of system (I), no matter whether the condition of C1 -compatibility (9.12) is satisfied or not. Theorem 9.6 Assume that system (I) is approximately synchronizable under the action of a boundary control matrix D. Then we necessarily have rank(D, AD, · · · , A N −1 D)  N − 1.

(9.21)

In other words, at least (N − 1) total controls are needed in order to realize the approximate boundary synchronization of system (I). Proof First, assume that A satisfies the condition of C1 -compatibility (9.12). By Lemma 9.5, we have (9.19). Next, assume that A does not satisfy the condition of

9.4 Properties Related to the Number of Total Controls

109

C (9.12). By Proposition 2.15, we have C1 Ae1 = 0. So, the matrix 1 -compatibility  C1 is of full column-rank. By (9.4), it is easy to see from problem (I) and (I0) C1 A with U = Un and H = Hn that C1 Un → 0 and C1 AUn → 0 in (D ((T, +∞) × )) N −1

(9.22)

as n → +∞. Then we have Un → 0 in (D ((T, +∞) × )) N as n → +∞.

(9.23)

We claim that in this case we have rank(D, AD, · · · , A N −1 D) = N .

(9.24)

Otherwise, let d  1 be such that rank(D, AD, · · · , A N −1 D) = N − d.

(9.25)

By the assertion (ii) of Proposition 2.12, there exists a nontrivial subspace of A T , which is contained in Ker(D T ) and invariant for A T , therefore, there exists a nontrivial vector E and a number λ ∈ C, such that A T E 1 = λE 1 and D T E 1 = 0.

(9.26)

Applying E to problem (I) and (I0) with U = Un and H = Hn , and noting φ = (E, Un ), it follows that ⎧  in (0, +∞) × , ⎨ φ − φ + λφ = 0 φ=0 on (0, +∞) × , (9.27) ⎩ 0 ), φ = (E 1 , U 1 ) in . t = 0 : φ = (E 1 , U Noting that φ is obviously independent of n if the initial data is chosen so that 0 ) = 0 or (E 1 , U 1 ) = 0, then φ ≡ 0 which contradicts (9.23). (E 1 , U Finally, combining (9.19) and (9.24), we get the rank condition (9.21). The proof is then achieved.  According to Theorem 9.6, it is natural to consider the approximate boundary synchronization of system (I) with the minimal number (N − 1) of total controls. In this case, the coupling matrix A should possess some fundamental properties related to the synchronization matrix C1 . Theorem 9.7 Assume that system (I) is approximately synchronizable under the minimal rank condition rank(D, AD, · · · , A N −1 D) = N − 1.

(9.28)

110

9 Approximate Boundary Synchronization

Then we have the following assertions: (i) The coupling matrix A satisfies the condition of C1 -compatibility (9.12). (ii) There exists a scalar function u as the approximately synchronizable state, such that (9.29) u (k) n → u as n → +∞ for all 1  k  N in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(9.30)

Moreover, the approximately synchronizable state u is independent of the sequence {Hn } of applied controls. (iii) The transpose A T of the coupling matrix A admits an eigenvector E 1 such that (E 1 , e1 ) = 1, where e1 = (1, · · · , 1)T is the eigenvector of A, associated with the eigenvalue a given by (9.6). Proof (i) If A does not satisfy the condition of C1 -compatibility (9.13), then the minimal rank condition (9.28) makes the rank condition (9.24) impossible. (ii) By the assertion (ii) of Proposition 2.12 with d = 1, the rank condition (9.28) implies the existence of one-dimensional invariant subspace of A T , contained in Ker(D T ), therefore, there exist a vector E 1 ∈ Ker(D T ) and a number b ∈ C, such that (9.31) E 1T D = 0 and A T E 1 = bE 1 . Applying E 1 to problem (I) and (I0) with U = Un and H = Hn , and setting φ = (E 1 , Un ), it follows that ⎧  in (0, +∞) × , ⎨ φ − φ + bφ = 0 φ=0 on (0, +∞) × , (9.32) ⎩ 0 ), φ = (E 1 , U 1 ) in . t = 0 : φ = (E 1 , U Clearly, φ is independent of n and of applied control Hn . Moreover, by (9.4) we have 

     C1 C 1 Un 0 U = → as n → +∞ n E 1T (E 1 , Un ) φ

(9.33)

in the space 0 1 ([T, +∞); (H0 ) N ) ∩ Cloc ([T, +∞); (H−1 ) N ). Cloc

 We will show that the matrix  Un →

C1 E 1T

C1 E 1T

(9.34)

 is invertible, then it follows that

−1   0 =: U as n → +∞ φ

(9.35)

9.4 Properties Related to the Number of Total Controls

111

in the space 0 1 ([T, +∞); (H0 ) N ) ∩ Cloc ([T, +∞); (H−1 ) N ). Cloc

(9.36)

Ker(C1 ) = Span(e1 ) with e1 = (1, · · · , 1)T ,

(9.37)

Noting that it follows from (9.35) that there exists a scalar function u such that U = ue1 , thus, we get (9.29). Since φ is independent of applied control Hn , so is u. Moreover, 1 ) such that (E 1 , U 0 ) ≡ 0 or 0 , U noting (9.35), we have u ≡ 0 for all initial data (U 1 ) ≡ 0. (E 1 , U   C1 is invertible. In fact, assume that there exists We now show that the matrix E 1T a vector x ∈ C N , such that   C1 = 0. (9.38) xT E 1T Then, applying x to (9.33), it is easy to see that xT

  0 = 0. φ

(9.39)

1 ). Then the last component of x must 0 , U Clearly, φ ≡ 0 at least for an initial data (U be zero. We can thus write    x (9.40) x= with  x ∈ C N −1 . 0 Then, it follows from (9.38) that  x T C1 = 0. But C1 is of full row-rank, so  x = 0, therefore x = 0. (iii) By (9.33) and noting U = ue1 , we get φ = (E 1 , e1 )u.

(9.41)

1 ), we get (E 1 , e1 ) = 0. Without loss 0 , U Since φ ≡ 0 at least for an initial data (U of generality, we may take (E 1 , e1 ) = 1. It follows that a(E 1 , e1 ) = (E 1 , Ae1 ) = (A T E 1 , e1 ) = b(E 1 , e1 ).

(9.42)

So b = a is a real number, and E 1 is a real eigenvector of A T , associated with the eigenvalue a given by (9.6). The proof is then complete.  Remark 9.8 What is surprising is the existence of the approximately synchronizable state u under the minimal rank condition (9.28). Moreover, by Theorem 8.9, system 0 , U 1 ), (I) is not approximately null controllable. Then, at least for an initial data (U the approximately synchronizable state u ≡ 0. In this case, system (I) is called to be

112

9 Approximate Boundary Synchronization

approximately synchronizable in the pinning sense, while that originally given by Definition 9.1 is in the consensus sense. Remark 9.9 The rank condition (9.28) indicates that the number of total controls is equal to (N − 1), but the state variable U of system (I) has N components, so, there exists a direction E 1 , on which the projection (E 1 , Un ) of the solution Un is independent of (N − 1) controls, therefore, (E 1 , Un ) converges in the space 0 1 ([0, +∞); H0 ) ∩ Cloc ([0, +∞); H−1 ) as n → +∞. Moreover, the necessity of Cloc the condition of C1 -compatibility (9.12) becomes also a consequence of the minimal value of the number of applied total controls. Remark 9.10 By Lemma 9.5, under the condition of C1 -compatibility (9.12), for the approximate boundary synchronization of system (I), we have (9.19), which shows that the reduced system (9.15) is still submitted to (N − 1) total controls. In other words, as we transform system (I) into system (9.15), only the number of equations is reduced from N to (N − 1), but the number (N − 1) of the total controls remains always unchanged, so that we get a reduced system of (N − 1) equations still submitted to (N − 1) total controls, that is just what we want. Remark 9.11 By Definition 2.3, the assertion (iii) means that the subspace Span{E 1 } is bi-orthonormal to the subspace Span{e1 }. Then, by Proposition 2.5 we have Span{e1 } ∩ (Span{E 1 })⊥ = {0}.

(9.43)

Thus, it follows from Proposition 2.2 that Span{E 1 }⊥ is a supplement of Span{e1 }. Moreover, since Span{E 1 } is an invariant subspace of A T , by Proposition 2.7, Span{E 1 }⊥ is invariant for A. Therefore, A is diagonalizable by blocks according to the decomposition Span{e1 } ⊕ Span{E 1 }⊥ . Theorem 9.12 Let A satisfy the condition of C1 -compatibility (9.12). Assume that A T admits an eigenvector E 1 such that (E 1 , e1 ) = 1 with e1 = (1, · · · , 1)T . Then there exists a boundary control matrix D satisfying the minimal rank condition (9.28), which realizes the approximate boundary synchronization of system (I). Proof By Lemma 9.4, under the condition of C1 -compatibility (9.12), the approximate boundary synchronization of system (I) is equivalent to the C1 D-observability of the reduced adjoint problem (9.18). Let D be the following full column-rank matrix defined by Ker(D T ) = Span{E 1 }.

(9.44)

Since Span{E 1 } is the sole subspace contained in Ker(D T ) and invariant for A T , applying the assertion (ii) of Lemma 2.12 with d = 1, we get the rank condition (9.28). On the other hand, the assumption (E 1 , e1 ) = 1 implies that Ker(C1 ) ∩ Im(D) = {0}, therefore, by Proposition 2.4 we have rank(C1 D) = rank(D) = N − 1.

(9.45)

9.4 Properties Related to the Number of Total Controls

113

Thus, the C1 D-observation of the reduced adjoint problem (9.18) becomes the complete observation: (9.46) ∂ν  ≡ 0 on [0, T ] × 1 , which implies well  ≡ 0 because of Holmgren’s uniqueness Theorem (cf. Theorem 8.2 in [62]), provided that T > 0 is large enough. The proof is then complete.  Remark 9.13 Since the matrix D given by (9.44) is of rank (N − 1), Theorem 9.12 shows the approximate boundary synchronization of system (I) by means of (N − 1) direct boundary controls. However, we are more interested in using fewer direct boundary controls in practice. We will give later in Theorem 10.10 a matrix D with the minimal rank, such that Kalman’s criteria (9.28) and (9.19) are simultaneously satisfied. We have pointed out in Chap. 8 that Kalman’s criterion (9.19) is indeed sufficient to the approximate boundary null controllability of the reduced system (9.15), then to the approximate boundary synchronization of the original system (I), for some special reduced systems such as the nilpotent system, 2 × 2 systems with the small spectral gap condition (8.74) and some one-space-dimensional systems, provided that T > 0 is large enough. Remark 9.14 Let D1 be the set of all boundary control matrices D, which realize the approximate boundary synchronization of system (I). Define the minimal number N1 of total controls for the approximate boundary synchronization of system (I): N1 = inf rank(D, AD, · · · , A N −1 D). D∈D1

(9.47)

Let Va denote the subspace of all the eigenvectors of A T , associated to the eigenvalue a given by (9.6). By the results obtained in Theorems 9.6, 9.7 and 9.12, we have  N1 =

N − 1, if A is C1 -compatible and (E 1 , e1 ) = 1, N , if A is C1 -compatible but (E 1 , e1 ) = 0, ∀E 1 ∈ Va .

(9.48)

Example 9.15 Consider the following system: ⎧  u − u + v = 0 ⎪ ⎪ ⎨  v − v − u + 2v = 0 u=v=0 ⎪ ⎪ ⎩ u = αh, v = βh Let



 0 1 A= , −1 2

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 .

(9.49)

  α D= β

(9.50)

and C1 = (1, −1), Ker(C1 ) = Span{e1 } with e1 = (1, 1)T .

(9.51)

114

9 Approximate Boundary Synchronization

Clearly, A satisfies the condition of C1 -compatibility with A1 = 1. We point out that the only eigenvector E 1 = (1, −1)T of A T satisfies (E 1 , e1 ) = 0. By Theorems 9.6 and 9.7, we have to use two (instead of one!) total controls to realize the approximate boundary synchronization of system (9.49). More precisely, we have 

 α β (D, AD) = , det(D, AD) = −(α − β)2 β 2β − α

(9.52)

(C1 D, C1 AD) = (α − β, α − β).

(9.53)

rank(C1 D, C1 AD) = 1 ⇐⇒ rank(D, AD) = 2.

(9.54)

and

It is easy to see that

By Corollary 8.7, the corresponding reduced system (9.15) (with A1 = 1): ⎧  ⎨ w − w + w = 0 in (0, +∞) × , w=0 on (0, +∞) × 0 , ⎩ w = (α − β)h on (0, +∞) × 1

(9.55)

is approximately null controllable as α = β, so the original system (9.49) is approximately synchronizable, provided that α = β and T > 0 is large enough. On the other hand, system (9.49) satisfies well the gap condition (8.74). By Theorem 8.18, Kalman’s criterion rank(D, C D) = 2 is sufficient for its approximate boundary null controllability, provided that T > 0 is large enough. So, when α = β, we are required to use 2 total controls and then to get a better result that system (9.49) is actually approximately null controllable under the action of the same boundary controls.

Chapter 10

Approximate Boundary Synchronization by p-Groups

The approximate boundary synchronization by p-groups is introduced and studied in this chapter for system (I) with Dirichlet boundary controls.

10.1 Definition In this chapter, we will consider the approximate boundary synchronization by p-groups. Let p  1 be an integer and let 0 = n0 < n1 < n2 < · · · < n p = N

(10.1)

be integers such that n r − n r −1  2 for all 1  r  p. We divide the components of U = (u (1) , · · · , u (N ) )T into p groups as (u (1) , · · · , u (n 1 ) ), (u (n 1 +1) , · · · , u (n 2 ) ), · · · , (u (n p−1 +1) , · · · , u (n p ) ).

(10.2)

Definition 10.1 System (I) is approximately synchronizable by p-groups at the 1 ) ∈ (H0 ) N × (H−1 ) N , there exists 0 , U time T > 0 if for any given initial data (U M a sequence {Hn } of boundary controls in L with compact support in [0, T ], such that the corresponding sequence {Un } of solutions to problem (I) and (I0) satisfies the following condition: (l) 0 1 u (k) n − u n → 0 in Cloc ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 )

(10.3)

as n → +∞ for all n r −1 + 1  k, l  n r and 1  r  p. © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_10

115

116

10 Approximate Boundary Synchronization by p-Groups

Let Sr be the following (n r − n r −1 − 1) × (n r − n r −1 ) full row-rank matrix: ⎞ ⎛ 1 −1 ⎟ ⎜ 1 −1 ⎟ ⎜ Sr = ⎜ ⎟, 1  r  p . . .. .. ⎠ ⎝ 1 −1

(10.4)

and let C p be the following (N − p) × N matrix of synchronization by p-groups: ⎛ ⎜ ⎜ Cp = ⎜ ⎝

⎞

S1 S2

..

⎟ ⎟ ⎟. ⎠

.

(10.5)

Sp Clearly, the approximate boundary synchronization by p-groups (10.3) can be equivalently rewritten in the following form: C p Un → 0 as n → +∞

(10.6)

0 1 ([T, +∞); (H0 ) N − p ) ∩ Cloc ([T, +∞); (H−1 ) N − p ). Cloc

(10.7)

in the space

10.2 Fundamental Properties For r = 1, · · · , p, setting

1, n r −1 + 1  i  n r , 0, otherwise,

(10.8)

Ker(C p ) = Span{e1 , e2 , · · · , e p }.

(10.9)

(er )i = it is clear that

We will say that A satisfies the condition of C p -compatibility if Ker(C p ) is an invariant subspace of A: (10.10) AKer(C p ) ⊆ Ker(C p ), or equivalently, by Proposition 2.15, there exists a matrix A p of order (N − p), such that (10.11) C p A = A pC p. A p is called the reduced matrix of A by C p .

10.2 Fundamental Properties

117

Under the condition of C p -compatibility, setting W p = C pU

(10.12)

and noting (10.11), from problem (I) and (I0) we get the following self-closed reduced system: ⎧  ⎪ ⎨W p − W p + A p W p = 0 in (0, +∞) × , on (0, +∞) × 0 , Wp = 0 ⎪ ⎩ on (0, +∞) × 1 Wp = Cp D H

(10.13)

with the initial data: t =0:

0 , W p = C p U 1 in . W p = C pU

(10.14)

Obviously, we have the following Lemma 10.2 Under the condition of C p -compatibility (10.10), system (I) is approximately synchronizable by p-groups at the time T > 0 if and only if the reduced system (10.13) is approximately null controllable at the time T > 0, or equivalently (cf. Theorem 8.6) if and only if the adjoint problem of the reduced system (10.13) ⎧ T ⎨  p −  p + A p  p = 0 in (0, +∞) × , on (0, +∞) × , p = 0 ⎩  p0 ,  p =   p1 in , t = 0 : p = 

(10.15)

called the reduced adjoint problem, is C p D-observable on the time interval [0, T ], namely, the partial observation (C p D)T ∂ν  p ≡ 0 on [0, T ] × 1

(10.16)

 p1 ≡ 0, then  p ≡ 0.  p0 =  implies  Thus, we have Lemma 10.3 Under the condition of C p -compatibility (10.10), assume that system (I) is approximately synchronizable by p-groups, we necessarily have the following Kalman’s criterion: rank(C p D, C p AD, · · · , C p A N −1 D) = N − p.

(10.17)

Proof From Lemma 10.2 and Theorem 8.9, we get N − p−1

rank(C p D, A p C p D, · · · , A p

C p A N −1 D) = N − p.

Thus, noting (10.11), (10.17) follows from Proposition 2.16.

(10.18) 

118

10 Approximate Boundary Synchronization by p-Groups

10.3 Properties Related to the Number of Total Controls The following result indicates the lower bound on the number of total controls necessary to the approximate boundary synchronization by p-groups for system (I), no matter whether the condition of C p -compatibility (10.10) is satisfied or not. Theorem 10.4 Assume that system (I) is approximately synchronizable by p-groups. Then we necessarily have rank(D, AD, · · · , A N −1 D)  N − p.

(10.19)

In other words, at least (N − p) total controls are needed to realize the approximate boundary synchronization by p-groups for system (I). Proof Keeping in mind that the condition of C p -compatibility (10.10) is not assumed to be satisfied a priori, we define an (N − p) ˜ × N full row-rank matrix p˜ (0 p ˜ p) by C Tp˜ ) = Span(C Tp , A T C Tp , · · · , (A T ) N −1 C Tp ). Im(C

(10.20)

T ) ⊆ Im(C T ), or equivalently, By Cayley–Hamilton’s theorem, we have A T Im(C p˜ p˜ p˜ ) ⊆ Ker(C p˜ ). AKer(C

(10.21)

p˜ of order (N − p), ˜ such that By Proposition 2.15, there exists a matrix A p˜ C p˜ A = A p˜ . C

(10.22)

Then, applying C p to the equations in system (I) with U = Un and H = Hn , by (10.6) it is easy to see that for any given integer l  0, we have C p Al Un → 0 in (D ((T, +∞) × )) N − p as n → +∞, then

p˜ Un → 0 in (D ((T, +∞) × )) N − p˜ C

as n → +∞.

(10.23)

(10.24)

We claim that ˜ p˜ C N − p−1 p˜ D, · · · , A p˜ D)  N − p. p˜ D, A ˜ C rank(C p˜

(10.25)

p˜ is of order (N − p), Otherwise, noting that A ˜ then by the assertion (i) of Proposition T , contained in 2.12 with d = 0, there exists a nontrivial invariant subspace of A p˜ p˜ D)T . Therefore, there exists a nonzero vector E ∈ C N − p˜ and a number λ ∈ C, Ker(C such that p˜ D)T E = 0. Tp˜ E = λE and (C (10.26) A

10.3 Properties Related to the Number of Total Controls

119

T E to problem (I) and (I0) with U = Un and H = Hn , and noting Applying C p˜ p˜ Un ), it is easy to see that φ = (E, C ⎧  in (0, +∞) × , ⎨ φ − φ + λφ = 0 φ=0 on (0, +∞) × , ⎩ p˜ U 0 ), φ  = (E, C 1 ) in . p˜ U t = 0 : φ = (E, C

(10.27)

p˜ U p˜ U 0 ) = 0 or (E, C 1 ) = 0, then φ ≡ 0. If the initial data are chosen such that (E, C Noting that φ is independent of n, this contradicts (10.24). p˜ -compatibility (10.22), the Finally, by Proposition 2.16, under the condition of C rank condition (10.25) yields rank(D, AD, · · · , A N −1 D) p˜ D, C p˜ AD, · · · , C p˜ A N −1 D) rank(C

(10.28)

˜ p˜ D, A p˜ C N − p−1 p˜ D, · · · , A p˜ D)  N − p, =rank(C ˜ C p˜

which leads to (10.19) because of p˜  p. The proof is then complete.



According to Theorem 10.4, it is natural to consider the approximate boundary synchronization by p-groups for system (I) with the minimal number (N − p) of total controls. In this case, the coupling matrix A should possess some basic properties related to C p , the matrix of synchronization by p-groups. Theorem 10.5 Assume that system (I) is approximately synchronizable by p-groups under the minimal rank condition rank(D, AD, · · · , A N −1 D) = N − p.

(10.29)

Then we have the following assertions: (i) There exist some linearly independent scalar functions u 1 , u 2 , · · · , u p such that (10.30) u (k) n → u r as n → +∞ for all nr −1 + 1  k  n r and 1  r  p in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(10.31)

(ii) The coupling matrix A satisfies the condition of C p -compatibility (10.10). (iii) A T admits an invariant subspace, which is contained in Ker(D T ) and biorthonormal to Ker(C p ). Proof (i) By the assertion (ii) of Proposition 2.12 with d = p, the rank condition (10.29) guarantees the existence of an invariant subspace V of A T , with dimension p and contained in Ker(D T ). Let {E 1 , · · · , E p } be a basis of V , such that

120

10 Approximate Boundary Synchronization by p-Groups

A T Er =

p 

αr s E s and D T Er = 0, 1  r  p.

(10.32)

s=1

Applying Er to problem (I) and (I0) with U = Un and H = Hn , and setting φr = (Er , Un ) for r = 1, · · · , p, we get

φr − φr + φr = 0

p s=1

αr s φs = 0 in (0, +∞) × , on (0, +∞) × 

(10.33)

with the initial data t =0:

0 ), φr = (Er , U 1 ) in . φr = (Er , U

(10.34)

Clearly, φ1 , · · · , φr are independent of n and of applied controls Hn . Moreover, by (10.6) we get ⎛ ⎞ ⎞ ⎞ ⎛ Cp 0 C p Un ⎜ φ1 ⎟ ⎜ E 1T ⎟ ⎜ (E 1 , Un ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ .. ⎟ Un = ⎜ ⎟ → ⎜ .. ⎟ as n → +∞ .. ⎝ . ⎠ ⎝ . ⎠ ⎠ ⎝ . E Tp φp (E p , Un ) ⎛

0 1 in the space Cloc ([T, +∞); (H0 ) N ) ∩ Cloc ([T, +∞); (H−1 ) N ). We claim that the matrix ⎛ ⎞ Cp ⎜ E 1T ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎝ . ⎠

(10.35)

(10.36)

E Tp

is invertible. Then,

⎛

⎞−1 ⎛ ⎞ Cp 0 ⎜ E 1T ⎟ ⎜ φ1 ⎟ ⎜ ⎟ ⎜ ⎟ Un → ⎜ .. ⎟ ⎜ . ⎟ := U ⎝ . ⎠ ⎝ .. ⎠

(10.37)

φp

E Tp

0 1 in Cloc ([T, +∞); (H0 ) N ) ∩ Cloc ([T, +∞); (H−1 ) N ) as n → +∞. Since C p U = 0 because of (10.35) and (10.37), noting (10.9), there exist u r (r = 1, · · · , p) such that

U=

p 

u r er .

(10.38)

r =1

Then, by the expression of er given in (10.8), the convergence (10.30) follows from (10.37).

10.3 Properties Related to the Number of Total Controls

121

Now we go back to show that the matrix (10.36) is indeed invertible. In fact, assume that there exists a vector x ∈ R N , such that ⎛

⎞ Cp ⎜ E 1T ⎟ ⎜ ⎟ x T ⎜ .. ⎟ = 0. ⎝ . ⎠

(10.39)

E Tp

Let x=

   x with  x ∈ C N − p and  x ∈ Cp.  x

(10.40)

Applying x to (10.35), it is easy to see that ⎛

⎞ φ1 ⎜ ⎟ tT :  x T ⎝ ... ⎠ = 0.

(10.41)

φp

Now for r = 1, · · · , p, let us take the initial data (10.34) as t =0:

φr = θr , φr = 0 in .

(10.42)

Then, it follows from problem (10.33) and (10.34) that (θ1 , · · · , θ p ) → (φ1 (T ), · · · , φ p (T ))

(10.43)

defines an isomorphism of (H0 ) p . Thus, as (θ1 , · · · , θ p ) vary in (H0 ) p , the state variable (φ1 (T ), · · · , φ p (T )) will run through the space (H0 ) p . Thus, it follows from x T C p = 0. But C p is of full row-rank, (10.41) that  x T = 0, then by (10.39) we get  we have  x = 0, then x = 0. (ii) Applying C p to the equations in system (I) with U = Un and H = Hn , and passing to the limit as n → +∞, it follows easily from (10.38) that tT :

p 

u r C p Aer = 0.

(10.44)

r =1

On the other hand, noting (10.35) and (10.38), we have tT :

φr =

p  (Er , es )u s , r = 1, · · · , p.

(10.45)

s=1

Since the state variable (φ1 (T ), · · · , φ p (T )) of system (10.33) runs through the space (H0 ) p as (θ1 , · · · , θ p ) given in the initial data (10.42) vary in (H0 ) p , so is the

122

10 Approximate Boundary Synchronization by p-Groups

approximately synchronizable state by p-groups (u 1 , · · · , u p )T . In particular, the functions u 1 , · · · , u p are linearly independent, then C p Aer = 0, 1  r  p,

(10.46)

namely, AKer(C p ) ⊆ AKer(C p ). We get thus (10.10). (iii) Noting that Span{E 1 , · · · , E p } and Ker(C p ) have the same dimension, and that {Ker(C p )}⊥ = Im(C Tp ), by Propositions 2.4 and 2.5, in order to show that Ker(C p ) is bi-orthonormal to Span{E 1 , · · · , E p }, it is sufficient to show that Span{E 1 , · · · , E p } ∩ Im(C Tp ) = {0}. Let E ∈ Span{E 1 , · · · , E p } ∩ Im(C Tp ) be a nonzero vector. There exist some coefficients α1 , · · · , α p and a nonzero vector r ∈ R N − p , such that E=

p 

αr Er and E = C Tp r.

(10.47)

r =1

Let φ = (E, Un ) =

p  r =1

αr (Er , Un ) =

p 

αr φr ,

(10.48)

r =1

where (φ1 , · · · , φ p ) is the solution to problem (10.33) and (10.34) with the homogeneous boundary condition, therefore, independent of n. In particular, for any given nonzero initial data (θ1 , · · · , θ p ), the corresponding solution φ ≡ 0. On the other hand, we have (10.49) φ = (C Tp r, Un ) = (r, C p Un ). Then, the convergence (10.6) implies that φ goes to zero as n → +∞ in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(10.50)

We get thus a contradiction, which confirms that Span{E 1 , · · · , E p } ∩ Im(C Tp ) = {0}. The proof is then complete.  Remark 10.6 Under the rank condition (10.29), the requirement (10.6) actually implies the existence of the approximately synchronizable state by p-groups (u 1 , · · · , u p )T , which are independent of applied boundary controls. In this case, system (I) is approximately synchronizable by p-groups in the pinning sense, while that originally given by Definition 10.1 is in the consensus sense. Remark 10.7 The rank condition (10.29) indicates that the number of total controls is equal to (N − p), but the state variable U of system (I) has N independent components, so if system (I) is approximately synchronizable by p-groups, there should exist p directions E 1 , · · · , E p , on which the projections (E 1 , Un ), · · · , (E p , Un ) of the solution Un to problem (I) and (I0) are independent of the (N − p) controls (cf.

10.3 Properties Related to the Number of Total Controls

123

(10.33)), therefore, converge as n → +∞. Consequently, we get the necessity of the condition of C p -compatibility (10.10) in this situation. Remark 10.8 Noting that the rank condition (10.29) indicates that the reduced system (10.13) of (N − p) equations is still submitted to (N − p) total controls, which is necessary for the corresponding approximate boundary null controllability. In other words, the number of total controls is not reduced. The procedure of reduction from system (I) to the reduced system (10.13) reduces only the number of equations, but not the number of total controls, so that we get a reduced system of (N − p) equations submitted to (N − p) total controls, that is just what we want. Remark 10.9 Since the invariant subspace Span{E 1 , · · · , E p } of A T is bi-orthonormal to the invariant subspace Span{e1 , · · · , e p } of A, by Proposition 2.8, the invariant subspace Span{E 1 , · · · , E p }⊥ of A is a supplement of Span{e1 , · · · , e p }. Therefore, A is diagonalizable by blocks under the decomposition Span {e1 , · · · , e p } ⊕ Span{E 1 , · · · , E p }⊥ . Theorem 10.10 Assume that the coupling matrix A satisfies the condition of C p compatibility (10.10) and that Ker(C p ) admits a supplement which is also invariant for A. Then there exists a boundary control matrix D which satisfies the minimal rank condition (10.29), and realizes the approximate boundary synchronization by p-groups for system (I) in the pinning sense. Proof By Lemma 10.2, under the condition of C p -compatibility (10.10), the approximate boundary synchronization by p-groups of system (I) is equivalent to the C p Dobservability of the reduced adjoint problem (10.15). Let W ⊥ be a supplement of Ker(C p ), which is invariant for A. By Proposition 2.8, the subspace W is invariant for A T and bi-orthonormal to Ker(C p ). Define the boundary control matrix D by Ker(D T ) = W.

(10.51)

Clearly, W is the only invariant subspace of A T , with the maximal dimension p and contained in Ker(D T ). Then, by the assertion (ii) of Proposition 2.12 with d = p, the rank condition (10.29) holds for this choice of D. On the other hand, the biorthonormality of W with Ker(C p ) implies that Ker(C p ) ∩ Im(D) = {0}. Therefore, by Proposition 2.11 we have rank(D) = rank(C p D) = N − p.

(10.52)

Then the C p D-observation (10.16) becomes the complete observation: ∂ν  ≡ 0 on [0, T ] × 1 ,

(10.53)

which implies well  ≡ 0 because of Holmgren’s uniqueness Theorem (cf. Theorem 8.2 in [62]). The proof is thus complete. 

124

10 Approximate Boundary Synchronization by p-Groups

Remark 10.11 The matrix D defined by (10.51) is of rank (N − p). So, we realize the approximate boundary synchronization by p-groups for system (I) by means of (N − p) direct boundary controls. But we are more interested in using fewer direct boundary controls to realize the approximate boundary synchronization by p-groups for system (I). We will give later in Proposition 11.14 a matrix D with the minimal rank, such that the rank conditions (10.29) and (10.17) are simultaneously satisfied. We point out that, by the results given in Chap. 8, Kalman’s criterion (10.17) is indeed sufficient for the approximate boundary null controllability of the reduced system (10.13), then to the approximate boundary synchronization by p-groups for system (I), for some special reduced systems such as the nilpotent system, 2 × 2 systems with the gap condition (8.7.4) and some one-space-dimensional systems. Remark 10.12 Let D p be the set of all matrices D, which realize the approximate boundary synchronization by p-groups for system (I). Define the minimal number N p of total controls for the approximate boundary synchronization by p-groups of system (I) by (10.54) N p = inf rank(D, AD, · · · , A N −1 D). D∈D p

Then, summarizing the results obtained in Theorems 10.4, 10.5, and 10.10, we have Np = N − p

(10.55)

if and only if Ker(C p ) is an invariant subspace of A, and A T admits an invariant subspace which is bi-orthonormal to Ker(C p ), or equivalently, by Proposition 2.8 if and only if Ker(C p ) admits a supplement V , and both Ker(C p ) and V are invariant for A. The following result matches the rank conditions (10.29) and (10.17) with a certain algebraic property of A with respect to the matrix C p . Proposition 10.13 Let C p be the matrix of synchronization by p-groups, given by (10.5). Then the rank conditions (10.29) and (10.17) simultaneously hold for some boundary control matrix D if and only if A T admits an invariant subspace W which is bi-orthonormal to Ker(C p ). Proof By the assertion (ii) of Proposition 2.12 with d = p, the rank condition (10.29) implies the existence of an invariant subspace W of A T , with the dimension p and contained in Ker(D T ). It is easy to see that

and

then we have

W ⊆ Ker(D, AD, · · · , A N −1 D)T

(10.56)

dim(W ) = dim Ker(D, AD, · · · , A N −1 D)T = p,

(10.57)

W = Ker(D, AD, · · · , A N −1 D)T .

(10.58)

10.3 Properties Related to the Number of Total Controls

125

By Proposition 2.11, the rank conditions (10.29) and (10.17) imply that Ker(C p ) ∩ W ⊥ = Ker(C p ) ∩ Im(D, AD, · · · , A N −1 D) = {0}.

(10.59)

Since dim(W ) = dim Ker(C p ), by Propositions 2.4 and 2.5, W is bi-orthonormal to Ker(C p ). Conversely, assume that W is an invariant subspace of A T , and bi-orthonormal to Ker(C p ), therefore with dimension p. Define a full column-rank matrix D of order N × (N − p) by (10.60) Ker(D T ) = W. Clearly, W is an invariant subspace of A T , with dimension p and contained in Ker(D T ). Moreover, the dimension of Ker(D T ) is equal to p. Then by the assertion (ii) of Proposition 2.12 with d = p, the rank condition (10.29) holds. Keeping in mind that (10.58) remains true in the present situation, it follows that Ker(C p ) ∩ Im(D, AD, · · · , A N −1 D) = Ker(C p ) ∩ W ⊥ = {0},

(10.61)

which, by Proposition 2.12, implies the rank condition (10.17). The proof is then complete. 

10.4 Necessity of the Condition of C p -Compatibility We will continue to discuss the approximate boundary synchronization by p-groups of system (I) and clarify the necessity of the condition of C p -compatibility. In Definition 10.1, we should exclude some trivial situations. For this purpose, we assume that each group is not approximately null controllable. p˜ introduced by (10.20) satisfies well the Let us recall that the extension matrix C  condition of C p˜ -compatibility (10.21). Moreover, the convergence p˜ Un → 0 in (D ((T, +∞) × )) N − p˜ as n → +∞ C

(10.62)

p˜ has only (N − p) ˜ rows, we have implies (10.25), then, noting that C ˜ p˜ D, A p˜ C N − p−1 p˜ D, · · · , A p˜ D) = N − p. rank(C ˜ C p˜

(10.63)

Based on these results, we will introduce a stronger version of the approximate boundary synchronization by p-groups. Theorem 10.14 Assume that system (I) is approximately synchronizable by p1 ) in the space (H0 ) N × (H−1 ) N , 0 , U groups. Then, for any given initial data (U there exists a sequence of boundary controls {Hn } in L 2 (0, +∞; (L 2 (1 )) M ) with compact support in [0, T ], such that the corresponding sequence {Un } of solutions

126

10 Approximate Boundary Synchronization by p-Groups

to problem (I) and (I0) satisfies p˜ Un → 0 as n → +∞ C

(10.64)

in the space 0 1 ([T, +∞); (H0 ) N − p˜ ) ∩ Cloc ([T, +∞); (H−1 ) N − p˜ ). Cloc

(10.65)

Since the proof of this result is quite long, we will give it at the end of this section. We first consider the special case p = 1. Corollary 10.15 In the case p = 1, assume that A does not satisfy the condition of C1 -compatibility. If system (I) is approximately synchronizable, then it is approximately null controllable. Consequently if system (I) is approximately synchronizable, but not approximately null controllable, then we necessarily have the condition of C1 -compatibility. Proof Since p = 1, the extension matrix C˜ p˜ is of order N and of full row-rank, therefore invertible. Then the convergence (10.64) gives the approximate boundary null controllability.  Now we consider the general case p  1. Theorem 10.16 Let p˜ be the integer given by (10.20). If system (I) is approximately synchronizable by p-groups, then it is approximately synchronizable by p-groups ˜ under a suitable basis. Proof Let 0 = n˜ 0 < n˜ 1 < n˜ 2 < · · · < n˜ p˜

(10.66)

be an arbitrarily given partition of the integer set {0, 1, · · · , N } such that ˜ n˜ r − n˜ r −1  2 for all 1  r  p. ˜ with Denote by C p˜ the corresponding matrix of synchronization by p-groups Ker(C p˜ ) = {e˜1 , · · · , e˜ p˜ }.

(10.67)

p˜ have the same rank, there exists an invertible matrix P of order N , Since C p˜ and C such that p˜ = C p˜ P. (10.68) C Then, it is easy to see that

Now, setting

p˜ ) = Ker(C p˜ ). PKer(C

(10.69)

 = PU and A  = P A P −1 , U

(10.70)

problem (I) and (I0) can be written into the following form:

10.4 Necessity of the Condition of C p -Compatibility

127

⎧  − U + A U  = 0 in (0, +∞) × , ⎪ ⎨U  U =0 on (0, +∞) × 0 , ⎪ ⎩ U = P DH on (0, +∞) × 1

(10.71)

with the initial data t =0:

 = P U 1 in .  = PU 0 , U U

(10.72)

By Theorem 10.14 and noting (10.68), we have p˜ Un → 0 n = C C p˜ U

as n → +∞

(10.73)

in the space 0 1 ([T, +∞); (H0 ) N − p˜ ) ∩ Cloc ([T, +∞); (H−1 ) N − p˜ ). Cloc

(10.74)

In other words, under the basis P, system (10.71) is approximately synchronizable by p-groups. ˜  As a direct consequence, we have Corollary 10.17 Assume that system (I) is approximately synchronizable by pgroups, but not approximately synchronizable by p-groups ˜ with p˜ < p under a basis. Then, we necessarily have the condition of C p -compatibility (10.10). Now we return to the proof of Theorem 10.14. We will perceive that the stronger convergence (10.64) follows from the convergence (10.6) and the assumption that A does not satisfy the condition of C p -compatibility (10.10). We point out that only the rank condition (10.63) can not guarantee this convergence. Let us first generalize Definition 8.1 to weaker initial data. Definition 10.18 Let m  0 be an integer. System (I) is approximately null controllable in the space (H−2m ) N × (H−(2m+1) ) N at the time T > 0, if for any given initial 1 ) ∈ (H−2m ) N × (H−(2m+1) ) N , there exists a sequence {Hn } of bound0 , U data (U ary controls in L 2 (0, +∞; (L 2 (1 ) M ) with compact support in [0, T ], such that the corresponding sequence {Un } of solutions to problem (I) and (I0) satisfies Un → 0

as n → +∞

(10.75)

in the space 0 1 ([T, +∞); (H−2m ) N ) ∩ Cloc ([T, +∞); (H−(2m+1) ) N ). Cloc

Correspondingly, we give

(10.76)

128

10 Approximate Boundary Synchronization by p-Groups

Definition 10.19 The adjoint problem (8.7) is D-observable in the space 0 , 1 ) ∈ (H2m+1 ) N × (H2m+1 ) N × (H2m ) N on the time interval [0, T ] if for ( N 0 ≡ 1 ≡ 0, then ≡ 0. (H2m ) , the observation (8.8) implies Similarly to Theorem 8.6 for the case m = 0, we can establish the following Proposition 10.20 System (I) is approximately null controllable in the space (H−2m ) N × (H−(2m+1) ) N at the time T > 0 if and only if the adjoint problem (8.7) is D-observable in the space (H2m+1 ) N × (H2m ) N on the time interval [0, T ]. Proposition 10.21 Let m  0 be an integer. System (I) is approximately null controllable in the space (H−2m ) N × (H−(2m+1) ) N if and only if it is approximately null controllable in the space (H0 ) N × (H−1 ) N . Proof By Proposition 10.20, it is sufficient to show that adjoint problem (8.7) is D-observable in the space (H2m+1 ) N × (H2m ) N if and only if it is D-observable in the space (H1 ) N × (H0 ) N . Assume that adjoint problem (8.7) is D-observable in the space (H2m+1 ) N × (H2m ) N , then the following expression 0 , 1 ) 2F = (



T 0

 1

|D T ∂ν |2 ddt

(10.77)

defines a Hilbert norm in the space (H2m+1 ) N × (H2m ) N . Let F be the closure of (H2m+1 ) N × (H2m ) N with respect to the F-norm. By the hidden regularity obtained for problem (8.7) (cf. [62]), we have  0

T

 1

0 , 1 ) 2 N |D T ∂ν |2 ddt  c ( (H1 ) ×(H0 ) N ,

(10.78)

then (H1 ) N × (H0 ) N ⊆ F.

(10.79)

Since problem (8.7) is D-observable in F, therefore, it is still D-observable in its  subspace (H1 ) N × (H0 ) N . The converse is trivial. The proof is complete. Remark 10.22 Similar results on the exact boundary controllability can be found in [12]. Proposition 10.23 Assume that system (I) is approximately synchronizable by p1 ) ∈ 0 , U groups. Then, for any given integer l  0 and any given initial data (U (H0 ) N × (H−1 ) N , we have 0 ([T, +∞); (H−2l ) N − p ) as n → +∞, C p Al Un → 0 in Cloc

(10.80)

respectively, 0 ([T, +∞); (H−(2l+1) ) N − p ) as n → +∞. C p Al Un → 0 in Cloc

(10.81)

10.4 Necessity of the Condition of C p -Compatibility

129

Proof Note that Un satisfies the homogeneous system

Un − Un + AUn = 0 in (T, +∞) × , on (T, +∞) × . Un = 0

(10.82)

Applying C p Al−1 to system (10.82), it follows that 0 C p Al Un Cloc ([T,+∞);(H−2l ) N − p )

(10.83)

0  C p A Un Cloc ([T,+∞);(H−2l ) N − p ) l−1 0 + C p A Un Cloc ([T,+∞);(H−2l ) N − p )

l−1

0 c C p Al−1 Un Cloc ([T,+∞);(H−2(l−1) ) N − p ) 0 cl C p Un Cloc ([T,+∞);(H0 ) N − p ) ,

where c > 0 is a positive constant. Similar result can be shown for C p Al Un . The proof is complete.  We now give the proof of Theorem 10.14. 0 , U 1 ) ∈ (H0 ) N × (H−1 ) N . Then, by (10.20) and noting (10.80)–(10.81) (i) Let (U with 0  l  N − 1, we get 0 p˜ Un → 0 in Cloc ([T, +∞); (H−2(N −1) ) N − p˜ ) as n → +∞, C

(10.84)

respectively, 0 p˜ Un → 0 in Cloc ([T, +∞); (H−(2N −1) ) N − p˜ ) as n → +∞. C

(10.85)

1 ) ∈ (H−2(N −1) ) N × (H−(2N −1) ) N . By density, there exists a 0 , U (ii) Let (U m m  sequence {(U0 , U1 )}m0 in (H0 ) N × (H−1 ) N , such that 1m ) → (U 0 , U 1 ) in (H−2(N −1) ) N × (H−(2N −1) ) N 0m , U (U

(10.86)

as m → +∞. For each fixed m, there exists a sequence of boundary controls {Hnm }n0 , such that the corresponding sequence {Unm }n0 of solutions to problem 1m ) satisfies 0m , U (I) and (I0) with the initial data (U 0 p˜ Unm → 0 in Cloc ([T, +∞); (H−2(N −1) ) N − p˜ ) as n → +∞. C

(10.87)

(iii) Let R denote the resolution of problem (I) and (I0) : R: which is continuous from

1 ; Hn ) → (Un , Un ), 0 , U (U

(10.88)

130

10 Approximate Boundary Synchronization by p-Groups

(H−2(N −1) ) N × (H−(2N −1) ) N × L M

(10.89)

into 0 1 ([T, +∞); (H−2(N −1) ) N ) ∩ Cloc ([T, +∞); (H−(2N −1) ) N ). Cloc

(10.90)

1 ) ∈ (H−2(N −1) ) N × (H−(2N −1) ) N , we write 0 , U Now, for any given (U 1 ; Hnm ) = R(U 0m , U 1m ; Hnm ) + R(U 0 − U 0m , U 1 − U 1m ; 0). 0 , U R(U

(10.91)

By well-posedness, for all 0  t  T  S we have 0m , U 1 − U 1m ; 0)(t)  c S (U 0 − U 0m , U1 − U1m ) 0 − U R(U

(10.92)

with respect to the norm of (H−2(N −1) ) N × (H−(2N −1) ) N , where c S is a positive constant depending only on S. Then, noting (10.86) and (10.87), we can chose a diagonal subsequence {Hnmk k }k0 such that 0 p˜ R(U 0 , U 1 ; Hnm k ) → 0 in Cloc C ([T, +∞); (H−2(N −1) ) N − p˜ ) k

(10.93)

as k → +∞. Hence, the reduced system (10.71) is approximately null controllable in the space (H−2(N −1) ) N − p˜ × (H−(2N −1) ) N − p˜ , therefore, by Proposition 10.21, it is also approximately null controllable in the space (H0 ) N − p˜ × (H−1 ) N − p˜ . The proof is complete. 

10.5 Approximate Boundary Null Controllability Let d be a column vector in D, or more generally, a linear combination of the column vectors of D, namely, d ∈ Im(D). If d ∈ Ker(C p ), then it will be canceled in the product matrix C p D, therefore it cannot give any effect to the reduced system (10.13). However, the vectors in Ker(C p ) may play an essential role in the approximate boundary null controllability. More precisely, we have the following Theorem 10.24 Let the coupling matrix A satisfy the condition of C p -compatibility (10.10). Assume that (10.94) e1 , · · · , e p ∈ Im(D), where Ker(C p ) = Span{e1 , · · · , e p }. If system (I) is approximately synchronizable by p-groups, then it is in fact approximately null controllable. Proof To prove this theorem, by Theorem 8.6, it is sufficient to show that the adjoint problem (8.7) is D-observable.

10.5 Approximate Boundary Null Controllability

131

For 1  r  p, applying er to the adjoint problem (8.7) and noting φr = (er , ), it follows from the condition of C p -compatibility (10.10) that  p φr − φr + s=1 βr s φs = 0 in (0, T ) × , on (0, T ) × , φr = 0

(10.95)

where βsr are given by Aer =

p 

βr s es , 1  r  p.

(10.96)

s=1

Noting er ∈ Im(D), there exists xr ∈ R M , such that er = Dxr . Then, the D-observation (8.8) gives ∂ν φr = (er , ∂ν ) = (xr , D T ∂ν ) ≡ 0 on (0, T ) × 1 .

(10.97)

Hence, by Holmgren’s uniqueness theorem, φr ≡ 0 for all 1  r  p. Consequently, we have (10.98) ∈ {Ker(C p )}⊥ = Im(C Tp ). Hence, the solution of the adjoint problem (8.7) can be written as = C Tp ,

(10.99)

and it follows from adjoint problem (8.7) that  C Tp   − C Tp  + A T C Tp  = 0 in (0, T ) × , on (0, T ) × . C Tp  = 0

(10.100)

Noting the condition of C p -compatibility (10.11), it follows that  T C Tp   − C Tp  + C Tp A p  = 0 in (0, T ) × , on (0, T ) × . C Tp  = 0

(10.101)

Since the matrix C Tp is of full column-rank, we get the reduced adjoint problem (10.15). Accordingly, the D-observation gives D T ∂ν = D T C Tp ∂ν  ≡ 0 on (0, T ) × 1 .

(10.102)

By Lemma 10.2, the reduced adjoint problem (10.15) is C p D-observable, it follows that  ≡ 0. Then, from (10.99), we have ≡ 0. So, the adjoint problem (8.7) is D-observable. The proof is thus complete. 

132

10 Approximate Boundary Synchronization by p-Groups

p˜ defined by (10.20) satisfies the condition Remark 10.25 The extension matrix C p˜ -compatibility (10.21), and, by Theorem 10.14, the corresponding reduced of C system (similar to (10.13)) is approximately null controllable. Moreover, we have p˜ ) ⊆ K er (C p ). So, if A does not satisfy the condition of C p -compatibility K er (C p˜ , and Theorem 10.24 remains valid in this case. (10.10), we can replace C p by C

Chapter 11

Induced Approximate Boundary Synchronization

We introduce the induced approximate boundary synchronization for system (I) with Dirichlet boundary controls and give some examples in this chapter.

11.1 Definition We have precisely studied the approximate boundary synchronization by p-groups for system (I) under the minimal rank condition (10.29). In this situation, the convergence of the sequence {Un } of solutions to problem (I) and (I0) and the necessity of the condition of C p -compatibility (10.10) are essentially the consequence of the minimal number of applied total controls. The objective of this chapter is to investigate the approximate boundary synchronization by p-groups in the case that N p > N − p,

(11.1)

where N p is defined by (10.54). By the condition of C p -compatibility (10.10), Ker(C p ) is invariant for A. However, by Remark 10.12, the situation (11.1) occurs if and only if Ker(C p ) does not admit any supplement which is invariant for A. So, in order to determine the minimal number N p of total controls, which are necessary for the approximate boundary synchronization by p-groups of system (I), a natural thought is to extend the matrix C p of synchronization by p-groups to the induced extension matrix Cq∗ given in Definition 2.18. Thus, we can bring this situation to the framework of Chap. 10. Accordingly, we introduce the following Definition 11.1 System (I) is induced approximately synchronizable by Cq∗ at the 0 , U 1 ) ∈ (H0 ) N × (H−1 ) N , there exists a time T > 0, if for any given initial data (U © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_11

133

134

11 Induced Approximate Boundary Synchronization

sequence {Hn } of boundary controls in L M with compact support in [0, T ], such that the corresponding sequence {Un } of solutions to problem (I) and (I0) satisfies Cq∗ Un → 0 as n → +∞

(11.2)

0 1 in the space Cloc ([T, +∞); (H0 ) N −q ) ∩ Cloc ([T, +∞); (H−1 ) N −q ).

In general, the induced approximate boundary synchronization does not occur. To see this aspect, let E1 , · · · , Em ∗ be a Jordan chain of A T associated with an eigenvalue λ: (11.3) E0 = 0, A T Ei = λEi + Ei−1 , i = 1, · · · , m. Let m with 1  m < m ∗ be an integer such that E1 , · · · Em ∈ Im(C Tp ). Then, applying E1 , · · · , Em to system (I) with U = Un and H = Hn , and setting φi = (Ei , Un )(i = 1, · · · , m), we get ⎧  φ1,n − φ1,n + λφ1,n = 0 in (0, +∞) × , ⎪ ⎪ ⎪ ⎪ ⎨······ φm,n − φm,n + λφm,n + φm−1,n = 0 in (0, +∞) × , ⎪ ⎪ ······ ⎪ ⎪ ⎩  φm,n − φm,n + λφm,n + φm−1,n = 0 in (0, +∞) × .

(11.4)

Since E1 , · · · , Em ∈ Im(C Tp ), there exist some vectors ri ∈ R N − p for i = 1, · · · , m, such that (11.5) Ei = C Tp ri , i = 1, · · · , m. By (10.6), for all i = 1, · · · , m we have φi,n = (ri , C p Un ) → 0 as n → +∞

(11.6)

0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(11.7)

in the space Thus, the first m components φ1,n , · · · , φm,n converge to zero as n → +∞. However, except in some specific situations (cf. Theorems 11.11 and 11.12 below), we don’t know if the other components φm+1,n , · · · , φm,n converge also to zero or not. However, when system (I) is induced approximately synchronizable, it is not only approximately synchronizable by p-groups, but also possesses some additional properties, which are hidden in the extension matrix Cq∗ . That is to say, in this case, when we realize the approximate boundary synchronization by p-groups for system (I), we could additionally get some unexpected results, which would improve the desired approximate boundary synchronization by p-groups!

11.1 Definition

135

Since Im(Cq∗ ) admits a supplement which is invariant for A T if system (I) is induced approximately synchronizable, by Remark 10.12, we immediately perceive that (11.8) N p = N − q, namely, (N − q) total controls are necessary to realize the approximate boundary synchronization by p-groups for system (I). We will show the minimal number (11.8) in the general case. This result improves the estimate (10.19) and deeply reveals that the minimal number of total controls necessary to the approximate boundary synchronization by p-groups depends not only on the number p of groups, but also on the algebraic structure of the coupling matrix A with respect to the matrix C p of synchronization by p-groups.

11.2 Preliminaries In what follows, for better understanding the driving idea, we always assume that Im(C Tp ) is A T -marked. In this framework, the procedure for obtaining the matrix Cq∗ is given in Chap. 2. In Remark 11.3 below, we will explain that the non-marked case is senseless for the induced approximate synchronization. Proposition 11.2 Assume that the coupling matrix A satisfies the condition of C p compatibility (10.10). Let D satisfy rank(C p D, C p AD, · · · , C p A N −1 D) = N − p.

(11.9)

Then, we necessarily have rank(D, AD, · · · , A N −1 D)  N − q,

(11.10)

where q is given by (2.59). Proof By the assertion (i) of Proposition 2.12, it is sufficient to show that the dimension of any invariant subspace W of A T , contained in Ker(D T ), does not exceed q. By the condition of C p -compatibility (10.10), there exists a matrix A p of order (N − p), such that C p A = A p C p . By Proposition 2.16, the rank condition (11.9) is equivalent to N − p−1

rank(C p D, A p C p D, · · · , A p

C p D) = N − p,

(11.11)

which, by the assertion (ii) of Proposition 2.12, implies that Ker(C p D)T does not T contain any nontrivial invariant space of A p .

136

11 Induced Approximate Boundary Synchronization

Now let W be any given invariant subspace of A T contained in Im(C Tp ). The projected subspace W = (C p C Tp )−1 C p W = {x :

C Tp x = x, ∀x ∈ W }

(11.12)

T

is an invariant subspace of A p . In particular, we have W ∩ Ker(C p D)T = {0}.

(11.13)

For any given x ∈ W , there exists x ∈ W , such that x = C Tp x, then we have D T x = D T C Tp x = (C p D)T x.

(11.14)

W ∩ Ker(D T ) = {0}

(11.15)

Thus we get for any given invariant subspace W of A T , contained in Im(C Tp ). Now let W ∗ be an invariant subspace of A T contained in Im(Cq∗T ) ∩ Ker(D T ). Since W ∗ ∩ Im(C Tp ) is also an invariant subspace of A T , contained in Im(C Tp ) ∩ Ker(D T ), we have W ∗ ∩ Im(C Tp ) = {0} because of (11.15). Then, it follows that W ∗ ⊆ Im(Cq∗T ) \ Im(C Tp ).

(11.16)

Since Im(C Tp ) is A T -marked, by the construction, Im(Cq∗T ) \ Im(C Tp ) does not contain any eigenvector of A T , then it follows that W ∗ = {0}.

(11.17)

Finally, let W be an invariant subspace of A T contained in Ker(D T ). Since W ∩ Im(Cq∗T ) is an invariant subspace of A T , contained in Im(Cq∗T ) ∩ Ker(D T ), it follows from (11.17) that W ∩ Im(Cq∗T ) = {0}.

(11.18)

dim Im(Cq∗T ) + dim(W ) = N − q + dim (W )  N ,

(11.19)

Then, we get

which implies that dim(W )  q. This achieves the proof.



Remark 11.3 In the case that Im(C Tp ) is not A T -marked, similarly as in §7.3, we can also construct the extension matrix Cq∗ . But the set Im(Cq∗T ) \ Im(C Tp ) possibly

11.2 Preliminaries

137

contains eigenvectors of A T , then (11.17) will not be satisfied. So, instead of (11.10), we can only get a weaker estimation such as rank(D, AD, · · · , A N −1 D)  N − q 

(11.20)

with q  > q. In the case of equality, we have rank(D, AD, · · · A N −1 D) = N − q  < N − q.

(11.21)

So, system (I) is never induced approximately synchronizable. This explains why we only study the A T -marked case at the beginning of the section. Proposition 11.4 Assume that the coupling matrix A satisfies the condition of C p compatibility (10.10). If rank(D, AD, · · · , A N −1 D) = N − q,

(11.22)

then we necessarily have the rank condition rank(Cq∗ D, Cq∗ AD, · · · Cq∗ A N −1 D) = N − q.

(11.23)

Proof First, by (ii) of Proposition 2.12, the rank condition (11.22) implies the existence of an invariant subspace W of A T contained in Ker(D T ) with dimension q. On the other hand, noting (11.18), we have dim Im(Cq∗T ) + dim(W ) = (N − q) + q = N ,

(11.24)

then Im(Cq∗T ) is a supplement of W , which is also invariant for A T . Furthermore, by the definition of W , it is easy to see that

or equivalently,

W ⊆ Ker(D, AD, · · · , A N −1 D)T ,

(11.25)

Im(D, AD, · · · , A N −1 D) ⊆ W ⊥ ,

(11.26)

which together with the rank condition (11.22) imply Im(D, AD, · · · , A N −1 D) = W ⊥ .

(11.27)

But W ⊥ is a supplement of Ker(Cq∗ ), which is invariant for A, then Im(D, AD, · · · , A N −1 D) ∩ Ker(Cq∗ ) = {0}.

(11.28)

138

11 Induced Approximate Boundary Synchronization

By Proposition 2.11, we get rank(Cq∗ D, Cq∗ AD, · · · , Cq∗ A N −1 D) =rank(D, AD, · · · , A

N −1

(11.29)

D). 

This achieves the proof.

11.3 Induced Approximate Boundary Synchronization We first give a lower bound estimate on the number of total controls which are necessary for the approximate boundary synchronization by p-groups of system (I). Proposition 11.5 Assume that the coupling matrix A satisfies the condition of C p compatibility (10.10). If system (I) is approximately synchronizable by p-groups under the action of the boundary control matrix D, then we necessarily have rank(D, AD, · · · , A N −1 D)  N − q,

(11.30)

where q is given by (2.59). Proof By Lemma 10.3, we have the following Kalman’s criterion rank(C p D, C p AD, · · · , C p A N −1 D) = N − p.

(11.31) 

Then, the rank condition (11.30) follows from Proposition 11.2. Now we go back to the induced approximate boundary synchronization.

Theorem 11.6 Assume that the coupling matrix A satisfies the condition of C p compatibility (10.10). If system (I) is induced approximately synchronizable with the minimal number of total controls (11.22), then, there exist scalar functions u ∗1 , · · · , u q∗ such that q  u ∗s es∗ as n → +∞ (11.32) Un → s=1

in the space 0 1 Cloc ([T, +∞); (H0 ) N ) ∩ (Cloc ([T, +∞); (H−1 ) N ),

(11.33)

where Span{e1∗ , · · · eq∗ } = Ker(Cq∗ ). In particular, system (I) is approximately synchronizable by p-groups in the pinning sense. Proof The proof is similar to that of Theorem 10.5. Here, we only give its sketch as follows.

11.3 Induced Approximate Boundary Synchronization

139

By Proposition 2.12, the rank condition (11.22) guarantees the existence of a subspace Span{E 1 , · · · , E q }, which is invariant for A T and contained in Ker(D T ). Thus, it is easy to see that the projection of Un on Span{E 1 , · · · , E q } is independent of boundary controls Hn . Similarly as in Theorem 10.5, we can show that Span{E 1 , · · · , E q } is a supplement of Im(Cq∗ T ). We have Un =

q 

u r∗ er∗ + Cq∗ T Rn ,

(11.34)

r =1

where Rn ∈ R N −q . Since Ker(Cq∗ ) = Span{e1∗ , · · · eq∗ }, the induced approximate boundary synchronization (11.2) implies that Cq∗ Cq∗ T Rn → 0. Noting that Cq∗ Cq∗ T is invertible, we get Rn → 0, which yields (11.32). Since Ker(Cq∗ ) ⊆ Ker(C p ), there exist some coefficients αsr such that es∗ =

p 

αsr er , s = 1, · · · , q.

(11.35)

αsr u ∗s , r = 1, · · · , p

(11.36)

r =1

Then, setting ur =

q  s=1

in (11.32), we get Un →

p 

u r er as n → +∞

(11.37)

r =1

in the space 0 0 ([T, +∞); (H0 ) N ) ∩ Cloc ([T, +∞); (H−1 ) N ). Cloc

(11.38)

Thus, system (I) is approximately synchronizable by p-groups in the pinning sense.  Remark 11.7 Because of the minimal rank condition (11.22), the functions u ∗1 , · · · , u q∗ in (11.32) are linearly independent, while, since p > q, the functions u 1 , · · · , u p given in (11.36) are linearly dependent. The relationship among u 1 , · · · , u p , which depends on the structure of Ker(Cq∗ ), will be illustrated in Examples 11.16 and 11.17 below. Theorem 11.8 Assume that the coupling matrix A satisfies the condition of C p compatibility (10.10). There exists a boundary control matrix D such that the corresponding system (I) is induced approximately synchronizable with the minimal number of total controls (11.22). Proof Since Ker(Cq∗ ) is an invariant subspace of A, by Proposition 2.15, there exists a matrix Aq∗ of order (N − q), such that Cq∗ A = Aq∗ Cq∗ . Setting W = Cq∗ U in problem

140

11 Induced Approximate Boundary Synchronization

(I) and (I0), we get the following reduced system: ⎧  ⎨ W − W + Aq∗ W = 0 in (0, +∞) × , W =0 on (0, +∞) × 0 , ⎩ on (0, +∞) × 1 W = Cq∗ D H

(11.39)

with the initial data: t =0:

0 , W  = Cq∗ U 1 in . W = Cq∗ U

(11.40)

Clearly, the induced approximate boundary synchronization by Cq∗ for system (I) is equivalent to the approximate boundary null controllability of the reduced system (11.39), then equivalent to the Cq∗ D-observability of the corresponding reduced adjoint problem ⎧  in (0, +∞) × , ⎨  −  + Aq∗T  = 0 =0 on (0, +∞) × , ⎩ 1 in . 0 ,   =  t =0: =

(11.41)

Let W be a subspace which is invariant for A T and bi-orthonormal to Ker(Cq∗ ). Clearly, W and Ker(Cq∗ ) have the same dimension q. Setting Ker(D T ) = W,

(11.42)

W is the largest invariant subspace of A T contained in Ker(D T ). By the assertion (ii) of Proposition 2.12 with d = q, we have the rank condition (11.22). On the other hand, since W is bi-orthonormal to Ker(Cq∗ ), by Proposition 2.5, we have (11.43) Im(D) ∩ Ker(Cq∗ ) = W ⊥ ∩ Ker(Cq∗ ) = {0}. Then by Proposition 2.11, we have rank(Cq∗ D) = rank(D) = N − q.

(11.44)

Therefore, the Cq∗ D-observation becomes the complete observation ∂ν  ≡ 0 on [0, T ] × 1 ,

(11.45)

which, because of Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]), implies the Cq∗ D-observability of the reduced adjoint problem (11.41). We get thus the convergence (11.2). The proof is complete. 

11.3 Induced Approximate Boundary Synchronization

141

As a direct consequence of Theorem 11.8, we have Corollary 11.9 Let q with 0  q < p be given by (2.59). Then we have N p = N − q.

(11.46)

Remark 11.10 Noting that rank(D) = N − q for the boundary control matrix D given by (11.42), Theorem 11.8 means that the approximate boundary synchronization by p-groups can be realized by means of the minimal number (N − q) of direct controls. However, in practice, we prefer to use a boundary control matrix D which provides (N − q) total controls, and has the minimal rank(D). To this end, we will replace the rank condition (11.44) by a weaker rank condition (11.23), which is only necessary in general for the induced approximate boundary synchronization. The following results give some confirmative answers under suitable additional conditions. Theorem 11.11 Let  ⊂ Rn satisfy the multiplier geometrical condition (3.1). Assume that the coupling matrix A satisfies the condition of C p -compatibility (10.10) and is similar to a nilpotent matrix. If system (I) is approximately synchronizable by p-groups under the rank condition (11.22), then it is actually induced approximately synchronizable. Proof As explained at the beginning of the proof of Theorem 11.8, the induced approximate boundary synchronization of system (I) is equivalent to the Cq∗ Dobservability of the reduced adjoint problem (11.41). First, by Proposition 11.4, we have the rank condition (11.23). So, by Proposition 2.16, we have rank(Cq∗ D, Aq∗ Cq∗ D, · · · , Aq∗ N −q−1 Cq∗ D) = N − q.

(11.47)

On the other hand, the matrix A is similar to a nilpotent matrix. By Proposition 2.21, the projection of the system of root vectors of A provides the system of root vectors of the reduced matrix Aq∗ . Thus, it is easy to see that Aq∗ is also similar to a nilpotent matrix. Then by Theorem 8.16, the reduced adjoint problem (11.41) is  Cq∗ D-observable. The proof is thus complete. Theorem 11.12 Let the coupling matrix A satisfy the condition of C p -compatibility (10.10). Assume that system (I) is approximately synchronizable by p-groups by means of a boundary control matrix D. Let  = (e˜1 , · · · , e˜ p−q , D), D

(11.48)

where e˜1 , · · · , e˜ p−q ∈ Ker(C p ) ∩ Im(Cq∗ T ). Then system (I) is induced approxi mately synchronizable under the action of the new boundary control matrix D.

142

11 Induced Approximate Boundary Synchronization

 Proof It is sufficient to show the Cq∗ D-observability of the reduced adjoint problem ∗T N −q (11.41). Since Cq is a bijection from R onto Im(Cq∗ T ), we can set  = Cq∗ T . ∗ ∗ ∗ Then, noting Cq A = Aq Cq , it follows from system (11.41) that

with the initial  Cq∗ D-observation

 −  + A T  = 0 in (0, +∞) × , =0 on (0, +∞) ×  0 ,  1 ) ∈ Im(Cq∗ T ) × Im(Cq∗ T ). (

data

T Cq∗ T  ≡ 0 on (0, T ) ×  D

(11.49) Accordingly,

the

(11.50)

 becomes the D-observation T  ≡ 0 on (0, T ) × . D

(11.51)

  Then, the Cq∗ D-observability of (11.41) becomes the D-observability of system T ∗ 0 ,  1 ) ∈ Im(Cq ) × Im(Cq∗ T ). (11.49) with the initial data ( Now let C p denote the restriction of C p to the subspace Im(Cq∗ T ). We have Im(Cq∗ T ) = Im(C Tp ) ⊕ Ker(C p ).

(11.52)

Since the induced approximate boundary synchronization can be considered as the approximate boundary null controllability in the subspace Im(Cq∗ T ) × Im(Cq∗ T ), so, applying Theorem 10.24, the proof is achieved. 

11.4 Minimal Number of Direct Controls In this section, we first clarify the relation between the number of total controls and that of direct controls. Then, we give an explicit construction for the boundary control matrix D which has the minimal rank. Proposition 11.13 Let A be a matrix of order N and D be an N × M matrix such that rank(D, AD, · · · , A N −1 D) = N . (11.53) Then we have the following sharp lower bound estimate rank(D)  μ,

(11.54)

11.4 Minimal Number of Direct Controls

143

where μ = max dim Ker(A T − λI )

(11.55)

λ∈Sp(A T )

is the largest geometrical multiplicity of the eigenvalues of A T . Proof Let λ be an eigenvalue of A T , the geometrical multiplicity of which is equal to μ, and let V be the subspace composed of all the eigenvectors associated with the eigenvalue λ. By the assertion (ii) of Proposition 2.12, the rank condition (11.53) implies that there does not exist any nontrivial invariant subspace of A T , contained in Ker(D T ). Therefore, we have dim Ker(D T ) + dim (V )  N ,

(11.56)

dim (V )  N − dim Ker(D T ) = rank(D),

(11.57)

namely,

which yields the lower bound estimate (11.54). In order to show the sharpness of (11.54), we will construct a matrix D0 of rank μ, which does not contain any nontrivial invariant subspace of A T , therefore, satisfies the rank condition (11.53) by Proposition 2.12. Let λ1 , · · · , λd be the distinct eigenvalues of A, with geometrical multiplicity μ1 , · · · , μd , respectively. For any given r = 1, · · · , d and k = 1, · · · , μr , we denote by (xr,(k)p ) the following Jordan chain of length dr(k) : x (k)(k)

r,dr +1

= 0,

Axr,(k)p = λr xr,(k)p + xr,(k)p+1

(11.58)

for p = 1, · · · , dr(k) . (l) ) be the Accordingly, for any given s = 1, · · · , d and l = 1, · · · , μs , let (ys,q (l) corresponding Jordan chain of length ds : (l) ys,0 = 0,

We can chose

(l) (l) (l) A T ys,q = λs ys,q + ys,q−1 , q = 1, · · · , ds(l) .

(l) ) = δkl δr s δ pq . (xr,(k)p , ys,q

(11.59)

(11.60)

Then we define the matrix D0 by (μ )

(μ )

(1) (1) D0 = (x1,1 , · · · , x1,11 , · · · , xd,1 , · · · , xd,1d )D,

where D is a (μ1 + · · · + μd ) × μ matrix defined by

(11.61)

144

11 Induced Approximate Boundary Synchronization

⎛

Iμ1 ⎜ Iμ2 ⎜ D=⎜ . ⎝ ..

⎞ 0 0⎟ ⎟ .. ⎟ , .⎠

(11.62)

Iμd 0 in which, if μr = μ, then the r th zero sub-matrix will disappear. Clearly, rank(D0 ) = μ. Now for any given s = 1, · · · , d, let ys =

μs 

(l) αl ys,1 with (α1 , · · · , αμs ) = 0

(11.63)

l=1

be an eigenvector of A T , corresponding to the eigenvalue λs . Noting (11.60), it is easy to see that ysT D0 = (0, · · · , 0, α1 , α2 , · · · , αμs , 0, · · · , 0) = 0.

(11.64)

Thus, Ker(D0T ) does not contain any eigenvector of A T , then, any nontrivial invariant subspace of A T either. By Proposition 2.12, the matrix D0 satisfies well the rank condition (11.53). The proof is complete.  Proposition 11.14 There exists a boundary control matrix D with the minimal rank μ, such that the rank conditions (11.22) and (11.23) simultaneously hold. Proof Since the coupling matrix A always satisfies the condition of Cq∗ -compatibility, there exists a matrix Aq∗ of order (N − q), such that Cq∗ A = Aq∗ Cq∗ . Then, by Proposition 2.16, the rank condition (11.23) is equivalent to rank(Cq∗ D, Aq∗ Cq∗ D, · · · , Aq∗N −q−1 Cq∗ D) = N − q.

(11.65)

Noting that Aq∗ is of order (N − q), as in the proof of Proposition 11.13, there exists a matrix D ∗ of order (N − q) × μ∗ with the minimal rank μ∗ , such that rank(D ∗ , Aq∗ D ∗ , · · · , Aq∗N −q−1 D ∗ ) = N − q,

(11.66)

where μ∗ is the largest geometrical multiplicity of the eigenvalues of the reduced matrix Aq∗ T . On the other hand, A T admits an invariant subspace W which is bi-orthonormal to Ker(Cq∗ ). Then {Ker(Cq∗ )}⊥ = Im(Cq∗T ) is an invariant subspace of A T , and W ⊥ = Span{eq+1 , · · · , e N } is an invariant subspace of A and bi-orthonormal to Im(Cq∗T ), namely, we have Cq∗ (eq+1 , · · · , e N ) = I N −q .

(11.67)

11.4 Minimal Number of Direct Controls

145

Then, from (11.66) we get that D = (eq+1 , · · · , e N )D ∗

(11.68)

satisfies (11.65), therefore, (11.23) holds. Finally, since Im(D) ⊆ Span{eq+1 , · · · , e N } which is invariant for A, so Im(Ak D) ⊆ Span{eq+1 , · · · , e N }, ∀k  0.

(11.69)

Noting that Span{eq+1 , · · · , e N } is a supplement of Ker(Cq∗ ), it follows that Ker(Cq∗ ) ∩ Im(D, AD, · · · , A N −1 D) = {0},

(11.70)

which, thanks to Proposition 2.11 and noting (11.23), implies the equality (11.22) and the sharpness of the rank of D: rank(Cq∗ D) = rank(D) = μ∗ .

(11.71) 

The proof is thus complete.

11.5 Examples Example 11.15 As shown in Proposition 11.13, a “good” coupling matrix A should have distinct eigenvalues, or the geometrical multiplicity of its eigenvalues should be as small as possible! In particular, if all the eigenvalues λi (i = 1, · · · , N ) of A are simple, we can take a boundary control matrix D of rank one: D = x, where x=

N 

(11.72)

xi with Axi = λi xi (i = 1, · · · , N ).

(11.73)

i=1

Let y j ( j = 1, · · · , N ) be the eigenvectors of A T , such that (xi , y j ) = δi j . Then for all j = 1, · · · N , we have DT y j = x T y j =

N 

(xi , y j ) = 1.

(11.74)

i=1

So, Ker(D T ) does not contain any eigenvector of A T , by (ii) of Proposition 2.12 (with d = 0), D satisfies Kalman’s criterion (11.53).

146

11 Induced Approximate Boundary Synchronization

If A has one double eigenvalue: λ1 = λ2 < λ3 · · · < λ N with Axi = λi xi (i = 1, · · · , N ),

(11.75)

we can take a boundary control matrix D of rank two: D = (x1 , x), where x=

N 

(11.76)

xi .

(11.77)

i=2

Let y j ( j = 1, · · · N ) be the eigenvectors of A T , such that (xi , y j ) = δi j . Then we have     1 (x1 , y1 ) = , (11.78) D T y1 = 0 (x, y1 ) and for j = 2, · · · , N , we have DT y j =

    0 (x1 , y j ) = . 1 (x, y j )

(11.79)

So, Ker(D T ) does not contain any eigenvector of A T . Once again, by (ii) of Proposition 2.12 (with d = 0), D satisfies Kalman’s criterion (11.53). Example 11.16 Let N = 4, M = 1, p = 2, ⎛

2 ⎜1 A=⎜ ⎝1 0 and  C2 =

1 −1 0 0 0 0 1 −1



−2 −1 −1 0

−1 0 −1 0

⎞ 1 0⎟ ⎟ 1⎠ 0

⎛ ⎞ ⎛ ⎞ 1 0 ⎜ 1⎟ ⎜0⎟ ⎜ ⎟ ⎟ with e1 = ⎜ ⎝0⎠ , e2 = ⎝1⎠ . 0 1

(11.80)

(11.81)

First, noting that Ae1 = Ae2 = 0, we have the condition of C2 -compatibility: AKer(C2 ) ⊆ Ker(C2 ), and the corresponding reduced matrix   1 −1 A2 = 1 −1  is similar to the cascade matrix

 01 of order 2. 00

(11.82)

11.5 Examples

147

In order to determine the minimal number of total controls necessary to the approximate boundary synchronization by 2-groups, we exhibit the system of root vectors of the matrix A T : ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 0 0 ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ 0 −1 0 (2) (2) (2) ⎜ ⎟ ⎜ ⎟ ⎜1⎟ ⎟ E1(1) = ⎜ (11.83) ⎝0⎠ , E1 = ⎝−1⎠ , E2 = ⎝ 1 ⎠ , E3 = ⎝−1⎠ . 1 1 −1 0 Since E1(2) , E2(2) ∈ Im(C2T ), Im(C2T ) is A T -marked. Then, the induced matrix C1∗ can be chosen as ⎛ ⎞ 1 −1 0 0 C1∗ = C1 = ⎝0 1 −1 0 ⎠ . (11.84) 0 0 1 −1 This justifies well that we have to use 3 (instead of 2!) total controls to realize the approximate boundary synchronization by 2-groups. On the other hand, all the one-column matrices D satisfying the following rank conditions: rank(C2 D, A2 C2 D) = 2 and rank(D, AD, A2 D, A3 D) = 3

(11.85)

are given either by ⎛ ⎞ α+β ⎜ α ⎟ ⎟ D=⎜ ⎝ β ⎠ , ∀α, β ∈ R 1 or by

⎛ ⎞ γ ⎜α ⎟ ⎟ D=⎜ ⎝β ⎠ , ∀α, β, γ ∈ R such that γ = α + β. 0

(11.86)

(11.87)

Since the above one-column matrix D provides only 3 total controls, by Theorems 8.6 and 8.9, system (I) is not approximately boundary null controllable. However, the corresponding Kalman’s criterion rank(C2 D, A2 C2 D) = 2 is sufficient for the approximate boundary null controllability of the corresponding reduced system (10.13) of cascade type, then sufficient for the approximate boundary synchronization by 2-groups of the original system (I) in the consensus sense: (2) (3) (4) u (1) n − u n → 0 and u n − u n → 0 as n → +∞

(11.88)

148

11 Induced Approximate Boundary Synchronization

in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(11.89)

Furthermore, the reduced matrix A∗1 given by ⎛

⎞ 1 0 −1 A∗1 = ⎝0 0 1 ⎠ 1 0 −1

(11.90)

is similar to a cascade matrix. Moreover, noting that q = 1 and C1∗ A = A∗1 C1∗ , by Proposition 11.4, we necessarily have the Kalman’s criterion rank(C1∗ D, A∗1 C1∗ D, A∗1 2 C1∗ D, A∗1 3 C1∗ D) = 3

(11.91)

for the boundary control matrix D in the both cases (11.86) and (11.87). Since the rank condition (11.91) is sufficient for the approximate boundary null controllability of the corresponding reduced system (11.39), system (I) is induced approximately synchronizable. By Theorem 11.6, there exists a nontrivial scalar function u ∗ such that (11.92) Un → u ∗ e1∗ as n → +∞, where Ker(C1∗ ) = {e1∗ } with e1∗ = (1, 1, 1, 1)T , or equivalently, (2) (3) (4) u (1) n → u, u n → u, u n → u, u n → u as n → +∞

(11.93)

in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(11.94)

The induced approximate boundary synchronization (11.93) reveals the hidden information and makes clear the situation of the approximate boundary synchronization by 2-groups (11.88) in the consensus sense. Example 11.17 Let N = 4, M = 1, p = 2, ⎛ 0 ⎜0 A=⎜ ⎝1 1

0 0 −1 −1

1 −1 0 0

and  C2 =

1 −1 0 0 0 0 1 −1



⎞ −1 1⎟ ⎟, 0⎠ 0

⎛

⎞ 0 ⎜0⎟ ⎟ D=⎜ ⎝1⎠ −1

⎛ ⎞ ⎛ ⎞ 1 0 ⎜1⎟ ⎜0 ⎟ ⎜ ⎟ ⎟ with e1 = ⎜ ⎝0⎠ , e2 = ⎝1⎠ . 0 1

(11.95)

(11.96)

11.5 Examples

149

First, the rank of Kalman’s matrix ⎛

0 ⎜0 2 3 (D, AD, A D, A D) = ⎜ ⎝1 −1

2 −2 0 0

0 0 4 4

⎞ 0 0⎟ ⎟ 0⎠ 0

(11.97)

is equal to 3. So, by Theorems 8.6 and 8.9, system (I) is not approximately null controllable under the action of the boundary control matrix D. Next, noting Ae1 = Ae2 = 0, we have the condition of C2 -compatibility (10.10): AKer(C2 ) ⊆ Ker(C2 ). The reduced matrix is A2 =

C2 AC2T (C2 C2T )−1

Moreover,

  02 = . 00

 (C2 D, A2 C2 D) =

 04 . 20

(11.98)

(11.99)

The reduced matrix A2 is of cascade type and then Kalman’s criterion: rank(C2 D, A2 C2 D) = 2 is sufficient for the approximate boundary null controllability of the corresponding reduced system (10.13), hence, the original system (I) is approximately synchronizable by 2-groups in the consensus sense: (2) (3) (4) u (1) n − u n → 0 and u n − u n → 0

(11.100)

as n → +∞ in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(11.101)

Now, we exhibit the system of root vectors of the matrix A T :

E1(1)

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 0 ⎜1⎟ ⎜0⎟ ⎜−1⎟ ⎜0⎟ (2) (2) (2) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ =⎜ ⎝0⎠ , E1 = ⎝ 2 ⎠ , E2 = ⎝ 0 ⎠ , E3 = ⎝0⎠ . 0 −2 0 1

(11.102)

Since E1(2) , E2(2) ∈ Im(C2T ), Im(C2T ) is A T -marked. Then, the induced matrix C1∗ can be chosen as ⎛ ⎞ 0 0 1 −1 C1∗ = ⎝1 −1 0 0 ⎠ . (11.103) 0 0 0 1

150

11 Induced Approximate Boundary Synchronization

Furthermore, the reduced matrix A∗1 given by ⎛ ⎞ 000 A∗1 = ⎝1 0 0⎠ 010

(11.104)

is a cascade matrix. Moreover, by Proposition 11.4, we necessarily have the following Kalman’s criterion rank(C1∗ D, A∗1 C1∗ D, A∗1 2 C1∗ D, A∗1 3 C1∗ D) = 3, which is sufficient for the approximate boundary null controllability of the corresponding reduced system (11.39). Then system (I) is induced approximately synchronizable. By Theorem 11.6, there exists a nontrivial scalar function u ∗ , such that Un → u ∗ e1∗ as n → +∞,

(11.105)

where Ker(C1∗ ) = {e1∗ } with e1∗ = (1, 1, 0, 0)T , or equivalently, ∗ (2) ∗ (4) and u (3) u (1) n → u , un → u n → 0, u n → 0

(11.106)

as n → +∞ in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(11.107)

Once again, the induced approximate boundary synchronization (11.106) clarifies the approximate boundary synchronization by 2-groups (11.100) in the consensus sense. Remark 11.18 The nature of the induced approximate boundary synchronization is determined by the structure of Ker(Cq∗ ). In Example 11.16, since C1∗ = C1 , the induced approximate boundary synchronization becomes the approximate boundary synchronization. This is purely a matter of chance. In fact, in Example 11.17, since C1∗ = C1 , the induced approximate boundary synchronization implies the approximate boundary synchronization for the first group and the approximate boundary null controllability for the second one, namely, the approximate boundary null controllability and synchronization by 2-groups, which can be treated in a way similar to the approximate boundary synchronization by groups (cf. Chap. 10). Other examples can be constructed to illustrate more complicated situations. However, the next example shows that the induced approximate boundary synchronization can fail in the general case.

11.5 Examples

151

Example 11.19 Let N = 2, M = 1, p = 1, and  A=

 0

,

0

D=

  1 , 0

(11.108)

where = 0 is a real number. We consider the following system: ⎧  u − u + v = 0 ⎪ ⎪ ⎨  v − v + u = 0 ⎪u = v = 0 ⎪ ⎩ u = h, v = 0

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 .

(11.109)

Setting w = u − v, we get the reduced system ⎧  ⎨ w − w − w = 0 in (0, +∞) × , w=0 on (0, +∞) × 0 , ⎩ w=h on (0, +∞) × 1 ,

(11.110)

which is approximately null controllable, therefore system (11.109) is approximately synchronizable. Moreover, it is easy to see that the Kalman’s criterion rank(D, AD) = 2 holds for all = 0, namely, the number of total controls is equal to 2 > N − p = 1. However, as shown in Theorem 8.11, the adjoint system (8.35) is not D-observable for some values of . So, system (11.109) is not approximately null controllable in general. This example shows that for the approximate boundary synchronization by pgroups, even rank(D, AD, · · · , A N −1 D), the number of total controls, is bigger than N − p, we cannot always get more information from the point of view of the induced approximate boundary synchronization.

Part III

Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls: Exact Boundary Synchronization

We consider the following coupled system of wave equations with Neumann boundary controls: ⎧  ⎨ U − U + AU = 0 in (0, +∞) × , U =0 on (0, +∞) × 0 , (II) ⎩ on (0, +∞) × 1 ∂ν U = D H with the initial condition t =0:

1 in , 0 , U  = U U =U

(II0)

where  ⊂ Rn is a bounded domain with smooth boundary  = 1 ∪  0 such that 2  1 ∩  0 = ∅ and mes(1 ) > 0; “” stands for the time derivative;  = nk=1 ∂∂x 2 is k the Laplacian operator; ∂ν denotes the outward normal derivative on the boundary;     (1) T T U = u , · · · , u (N ) and H = h (1) , · · · , h (M) (M  N ) stand for the state variables and the boundary controls, respectively; the coupling matrix A = (ai j ) is of order N, and D as the boundary control matrix is a full column-rank matrix of order N × M, both with constant elements. The exact boundary synchronization and the exact boundary synchronization by groups for system (II) will be presented and discussed in this part, while, correspondingly, the approximate boundary synchronization and the approximate boundary synchronization by groups for system (II) will be introduced and considered in the next part (Part IV).

Chapter 12

Exact Boundary Controllability and Non-exact Boundary Controllability

In this chapter, we will consider the exact boundary controllability and the non-exact boundary controllability for system (II) of wave equations with Neumann boundary controls.

12.1 Introduction In this part, we will consider system (II) of wave equations with Neumann boundary controls, together with the initial data (II0). Let us denote (12.1) H0 = L 2 () and H1 = H10 (), where H10 () is the subspace of H 1 (), composed of functions with the null trace on 0 . We denote by H−1 the dual space of H1 . When mes(0 ) = 0, namely, 1 = , instead of (12.1) we denote  ⎧   2 ⎪ ⎪ H = φ ∈ L (), φd x = 0 , 0 ⎨  (12.2)    ⎪ ⎪ ⎩ H1 = φ ∈ H 1 (), φd x = 0 . 

We will show the exact boundary controllability of system (II) for any given initial 0 , U 1 ) ∈ (H1 ) N × (H0 ) N via the HUM approach. data (U To this end, let  = (φ(1) , · · · , φ(N ) )T

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_12

(12.3)

155

156

12 Exact Boundary Controllability and Non-exact Boundary Controllability

denote the adjoint variables. We consider the following adjoint problem: ⎧   −  + A T  = 0 ⎪ ⎪ ⎨ =0 ∂ ⎪ ν = 0 ⎪ ⎩ 1 0 ,  =  t =0: =

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in .

(12.4)

We will establish the following Theorem 12.1 There exist positive constants T > 0 and C > 0, independent of initial data, such that the following observability inequality 0 ,  1 )2 N ( (H0 ) ×(H−1 ) N  C



T 0

 1

||2 ddt

(12.5)

0 ,  1 ) holds for the solution  to the adjoint problem (12.4) with any given ( N N belonging to a subspace F ⊂ (H0 ) × (H−1 ) (cf. (12.64) below). Recall that without any assumption on the coupling matrix A, the usual multiplier method cannot be applied directly. The absorption of coupling lower terms is a delicate issue even for a single wave equation (cf. [62] and [26]). In order to deal with the lower order terms, we propose a method based on the compactness-uniqueness argument that we formulate in the following Lemma 12.2 Let F be a Hilbert space endowed with the p-norm. Assume that F =N



L,

(12.6)

where ⊕ denotes the direct sum and L is a finite co-dimensional closed subspace of F. Assume that there is another norm—the q-norm in F, such that the projection from F into N is continuous with respect to the q-norm. Assume furthermore that q(y)  p(y), ∀y ∈ L.

(12.7)

Then there exists a positive constant C > 0, such that q(z)  C p(z), ∀z ∈ F.

(12.8)

Following the above Lemma, we have to first show the observability inequality for the initial data with higher frequencies in L. In order to extend this inequality to the whole space F, it is sufficient to verify the continuity of the projection from F into N for the q-norm. In many situations, it often occurs that the subspaces N and L are mutually orthogonal with respect to the q-inner product, and this is true in the

12.1 Introduction

157

present case. This new approach turns out to be particularly simple and efficient for getting the observability of some distributed systems with lower order terms. As for the problem with Dirichlet boundary controls in Chap. 3, we show the exact boundary controllability and the non-exact boundary controllability for system (II) with Neumann boundary controls in the case M = N or with fewer boundary controls (M < N ) (cf. Theorems 12.14 and 12.15), respectively. Roughly speaking, in the framework that all the components of initial data are in the same energy space, a coupled system of wave equations with Dirichlet or Neumann boundary controls is exactly controllable if and only if one applies the same number of boundary controls as the number of state variables of wave equations.

12.2 Proof of Lemma 12.2 Assume that (12.8) fails, then there exists a sequence z n ∈ F, such that q(z n ) = 1 and p(z n ) → 0 as n → +∞.

(12.9)

Using (12.6), we write z n = xn + yn with xn ∈ N and yn ∈ L. Since the projection from F into N is continuous with respect to the q-norm, there exists a positive constant c > 0, such that q(xn )  cq(z n ) = c, ∀n ≥ 1.

(12.10)

Noting that N is of finite dimension, we may assume that there exists x ∈ N , such that xn → x in N . Then, since the second relation of (12.9) means that z n → 0 in F, we deduce that yn → −x in L for the p-norm. Therefore, we get x ∈ L ∩ N , which leads to x = 0. Then, we have q(xn ) → 0 and p(yn ) → 0

(12.11)

as n → +∞. Then using (12.7), we get a contradiction: 1 = q(z n )  q(xn ) + q(yn )  q(xn ) + p(yn ) → 0

(12.12)

as n → +∞. The proof is then complete. Remark 12.3 Noting that L is not necessarily closed with respect to the weaker q-norm, so, a priori, the projection z → x is not continuous with respect to the q-norm (cf. [8]).

158

12 Exact Boundary Controllability and Non-exact Boundary Controllability

12.3 Observability Inequality In order to give the proof of Theorem 12.1, we first start with some useful preliminary results. Assume that  ⊂ Rn is a bounded domain with smooth boundary . Let  = 1 ∪ 0 be a partition of  such that  1 ∩  0 = ∅. Throughout this chapter, we assume that  satisfies the usual multiplier geometrical condition (cf. [7, 26, 62]). More precisely, assume that there exists x0 ∈ Rn , such that setting m = x − x0 , we have (12.13) (m, ν)  0, ∀x ∈ 0 ; (m, ν) > 0, ∀x ∈ 1 , where (·, ·) denotes the inner product in Rn . We define the linear unbounded operator − in H0 by D(−) = {φ ∈ H 2 () :

φ|0 = 0, ∂ν φ|1 = 0}.

(12.14)

Clearly, − is a densely defined self-adjoint and coercive operator with a compact resolvent in H0 . Then we can define the power operator (−)s/2 for any given s ∈ R (cf. [64]). Moreover, the domain Hs = D((−)s/2 ) endowed with the norm φs = (−)s/2 φH0 is a Hilbert space, and its dual space with respect to the pivot space H0 is Hs = H−s . In particular, we have √ H1 = D( −) = {φ ∈ H 1 () :

φ = 0 on 0 }.

(12.15)

Then we formulate the adjoint problem (12.4) into an abstract evolution problem in the space (Hs ) N × (Hs−1 ) N for any given s ∈ R: 

 −  + A T  = 0, 1 . 0 ,   =  t =0: =

(12.16)

Moreover, we have the following result (cf. [60, 64, 70]). 0 ,  1 ) ∈ (Hs ) N × (Hs−1 ) N with s ∈ Proposition 12.4 For any given initial data ( R, the adjoint problem (12.16) admits a unique weak solution  in the sense of C 0 -semigroups, such that  ∈ C 0 ([0, +∞); (Hs ) N ) ∩ C 1 ([0, +∞); (Hs−1 ) N ).

(12.17)

Now let em be the normalized eigenfunction defined by ⎧ ⎨ −em = μ2m em in , on 0 , em = 0 ⎩ on 1 , ∂ν em = 0

(12.18)

where the positive sequence {μm }m1 is increasing such that μm → +∞ as m → +∞.

12.3 Observability Inequality

159

For each m  1, we define the subspace Z m by Z m = {αem :

α ∈ R N }.

(12.19)

Since A is a matrix with constant coefficients, for any given m  1 the subspace Z m is invariant for A T . Moreover, for any given integers m, n with m = n and any given vectors α, β ∈ R N , we have (αem , βen )(Hs ) N

=(α, β)R N (−)s/2 em , (−)s/2 en H0

=(α, β)R N μsm μsn em , en H0

(12.20)

=(α, β)R N μsm μsn δmn . Then, the subspaces Z m (m  1) are mutually orthogonal in the Hilbert space (Hs ) N for any given s ∈ R, and in particular, we have 1 (Hs+1 ) N , ∀ ∈ Z m . (12.21) μm  Let m 0  1 be an integer. We denote by mm 0 (Z m × Z m ) the linear hull of the subspaces Z m × Z m for m  m 0 . In other words, mm 0 (Z m × Z m ) is composed of all finite linear combinations of elements of Z m × Z m for m  m 0 . (Hs ) N =

Proposition 12.5Let  be the solution to the adjoint problem (12.16) with the initial 0 ,  1 ) ∈ m1 (Z m × Z m ), and satisfy an additional condition data (  ≡ 0 on [0, T ] × 1

(12.22)

1 ≡ 0. 0 ≡  for T > 0 large enough. Then, we have  Proof By Schur’s Theorem, we may assume that A = (ai j ) is an upper triangular matrix so that problem (12.16) with the additional condition (12.22) can be rewritten as for k = 1, · · · , N , ⎧ (k)   (φ ) − φ(k) + kp=1 a pk φ( p) = 0 ⎪ ⎪ ⎨ (k) φ =0 ⎪ ∂ν φ(k) = 0 ⎪ ⎩ (k) , (φ(k) ) = φ (k) t = 0 : φ(k) = φ 0 1

in (0, +∞) × , on (0, +∞) × , on (0, +∞) × 1 , in ,

(12.23)

where corresponding to (12.3) we have (1) , · · · , φ (N ) )T and  (1) , · · · , φ (N ) )T . 0 = (φ 1 = (φ  0 0 1 1

(12.24)

Then, using Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]), there exists a (1) ), (1) , φ positive constant T > 0 large enough and independent of the initial data (φ 0 1

160

12 Exact Boundary Controllability and Non-exact Boundary Controllability

such that φ(1) ≡ 0. Then, we get successively φ(k) ≡ 0 for k = 1, · · · , N . The proof is then complete.  Proposition 12.6 Let m 0  1 be an integer and  be the solution to the adjoint prob 1 ) ∈ mm (Z m × Z m ). Define the energy 0 ,  lem (12.16) with the initial data ( 0 by  1 E(t) = (| |2 + |∇|2 )d x. (12.25) 2  Let σ denote the Euclidian norm of the matrix A. Then we have the following energy estimates: −σt σt (12.26) e μm0 E(0)  E(t)  e μm0 E(0), t  0 1 ) ∈ 0 ,  for all ( problem (12.18).



mm 0 (Z m

× Z m ), where the sequence (μm )m1 is defined by

Proof First, a straightforward computation yields E  (t) = −

 

(A , )d x.

(12.27)

Then, using (12.21) we get      (A , )d x  

σ H0 H0  It then follows that −

(12.28) σ σ  H0 H1  E(t). μm 0 μm 0

σ σ E(t)  E  (t)  E(t). μm 0 μm 0

(12.29)

σt

Therefore, the function E(t)e μm0 is increasing with respect to the variable t; while, −σt the function E(t)e μm0 is decreasing with respect to the variable t. Thus we get (12.26) and then the proof is complete.  Proposition 12.7 There exist an integer m 0  1 and positive constants T > 0 and C > 0 independent of initial data, such that the following observability inequality 1 )2 N 0 ,  ( (H1 ) ×(H0 ) N  C



T 0

 1

| |2 ddt

(12.30)

0 ,  1 ) ∈ holds for all solutions  to adjoint problem (12.16) with the initial data (  mm 0 (Z m × Z m ).

12.3 Observability Inequality

161

Proof First we write the adjoint problem (12.16) as ⎧ (k)   (φ ) − φ(k) + Np=1 a pk φ( p) = 0 ⎪ ⎪ ⎨ (k) φ =0 ∂ ⎪ ν φ(k) = 0 ⎪ ⎩ (k) , (φ(k) ) = φ (k) t = 0 : φ(k) = φ 0 1

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in 

(12.31)

for k = 1, 2, · · · , N . Then, multiplying the k-th equation of (12.31) by M (k) := 2m · ∇φ(k) + (N − 1)φ(k) with m = x − x0

(12.32)

and integrating by parts, we get easily the following identities (cf. [26, 62]): 





∂ν φ(k) M (k) + (m, ν)(|(φ(k) ) |2 − |∇φ(k) |2 ddt 0  T  T  

(φ(k) ) M (k) d x + |(φ(k) ) |2 + |∇φ(k) |2 d xdt = +

T

 N  T 

0



akp φ( p) M (k) d xdt, k = 1, · · · , N .



0

p=1

0



(12.33)

Noting the multiplier geometrical condition (12.13), we have ∂ν φ(k) M (k) + (m, ν)(|(φ(k) ) |2 − |∇φ(k) |2 ) (k) 2

=(m, ν)|∂ν φ |  0

(12.34)

on (0, T ) × 0

and ∂ν φ(k) M (k) + (m, ν)(|(φ(k) ) |2 − |∇φ(k) |2 ) (k)  2

(12.35)

(k) 2

=(m, ν)(|(φ ) | − |∇φ | ) (m, ν)|φ(k) |2

on (0, T ) × 1 .

Then, it follows from (12.33) that 

T

0

 

T

0

−



(k)  2

|(φ ) | + |∇φ(k) |2 d xdt    T (k)  2 (m, ν)|(φ ) | ddt − (φ(k) ) M (k) d x

N   p=1

1

0

T

 



akp φ( p) M (k) d xdt, k = 1, · · · , N .

0

(12.36)

162

12 Exact Boundary Controllability and Non-exact Boundary Controllability

Taking the summation of (12.36) with respect to k = 1, · · · , N , we get 



T

E(t)dt 

2 





0

1

( , M)d x

−

0

−

T

T 0

(m, ν)| |2 ddt 

T 0

(12.37)

 

(, AM)d xdt,

where M is the vector composed of M (k) (k = 1, · · · , N ) given by (12.32). Next, we estimate the last two terms on the right-hand side of (12.37). First, it follows from (12.32) that M(H0 ) N  2R

N 

∇φ(k) (H0 )n

(12.38)

k=1

+ (N − 1)(H0 ) N  γ(H1 ) N , where R = m∞ is the diameter of  and γ=

 4R 2 + (N − 1)2 .

(12.39)

T  On the other hand, since Z m is invariant for A , for any given (0 , 1 ) ∈ mm 0 (Z m × Z m ), the corresponding solution  to the adjoint problem (12.16) belongs to mm 0 Z m for any given t  0. Thus, using (12.21) and (12.38), we have

     (, AM)d x   σ(H0 ) N M(H0 ) N

(12.40)



γσ(H0 ) N (H1 ) N 

2γσ E(t). μm 0

Similarly, we have      ( , M)d x    (H0 ) N M(H0 ) N

(12.41)



γ (H0 ) N (H1 ) N  γ E(t). Thus, setting T =

μm 0 σ

(12.42)

(12.43)

and noting (12.26), we get   T     Md x   γ(E(T ) + E(0))  γ(1 + e)E(0).  

0

(12.44)

12.3 Observability Inequality

163

Inserting (12.40) and (12.44) into (12.37) gives 

T

2



T

E(t)dt 

0



0

+γ(1 + e)E(0) +

2σγ μm 0

1

(m, ν)| |2 ddt



(12.45)

T

E(t)dt. 0

Thus, we have 

T



T

E(t)dt  R

0

 1

0

| |2 ddt + γ(1 + e)E(0),

(12.46)

provided that m 0 is so large that μm 0  2σγ.

(12.47)

Now, integrating the inequality on the left-hand side of (12.26) over [0, T ], we get  T −σT μm 0 1 − e μm0 E(0)  E(t)dt, (12.48) σ 0 then, noting (12.26), we get 

−1

T (1 − e )E(0) 

T

E(t)dt.

(12.49)

0

Thus, it follows from (12.46) and (12.49) that E(0) 

R T (1 − e−1 ) − γ(1 + e)

0 ,  1 ) ∈ holds for any given (



mm 0 (Z m

T >



T 0

 1

| |2 ddt

(12.50)

× Z m ), provided that

γe(1 + e) , e−1

(12.51)

which is guaranteed by the following choice: μm 0 >

2σγe(1 + e) e−1

(cf. (12.43), (12.47), and (12.51)). The proof is then complete.

(12.52) 

164

12 Exact Boundary Controllability and Non-exact Boundary Controllability

Proposition 12.8 There exist an integer m 0  1 and positive constants T > 0 and C > 0 independent of initial data, such that the following observability inequality 1 )2 N 0 ,  ( (H0 ) ×(H−1 ) N  C



T 0

 1

||2 ddt

(12.53)

holds for all solutions  to the adjoint problem (12.16) with the initial data 0 ,  1 ) ∈ mm (Z m × Z m ). ( 0 Proof Noting that Ker(− + A T ) is of finite dimension, there exists an integer m 0  1 so large that Ker(− + A T )



Z m = {0}.

(12.54)

mm 0

Let W={



Zm }

(H0 ) N

⊆ (H0 ) N .

(12.55)

mm 0

 Since mm 0 Z m is an invariant subspace of (− + A T ), by Fredholm’s alternative, (− + A T )−1 is an isomorphism from W onto its dual space W  . Moreover, we have (− + A T )−1 2(H0 ) N ∼ 2(H−1 ) N , ∀ ∈ W. 1 ) ∈ 0 ,  For any given (

We have



mm 0 (Z m

(12.56)

× Z m ), let

1 , 1 =  0 . 0 = ( − A T )−1 

(12.57)

1 2 2 0 2(H1 ) N + 1 2(H0 ) N ∼  (H−1 ) N + 0 (H0 ) N .

(12.58)

Next let  be the solution to the adjoint problem (12.16) with the initial data (0 , 1 ) given by (12.57). We have t =0: By well-posedness, we get

0 ,   =  1 .  =    = .

On the other hand, since the subspace have (0 , 1 ) ∈

 mm 0

mm 0

(12.59)

(12.60) Z m is invariant for (− + A T ), we

(Z m × Z m ).

(12.61)

12.3 Observability Inequality

165

Then, applying (12.30) to , we get  (0 , 1 )2(H1 ) N ×(H0 ) N

T

C 0

 1

|  |2 ddt.

(12.62)

Thus, using (12.58) and (12.60), we get immediately (12.53). The proof is finished.  We are now ready to give the  proof of Theorem 12.1. 0 ,  1 ) ∈ m1 (Z m × Z m ), define For any given ( 0 ,  1 ) = p(



T 0

 1

||2 ddt,

(12.63)

where  is the solution to the adjoint problem (12.16). By Proposition 12.5, for T > 0 large enough, p(·, ·) defines well a norm in m1 (Z m × Z m ). Then, we denote by  F the completion of m1 (Z m × Z m ) with respect to the p-norm. Clearly, F is a Hilbert space. We next write

F =N L (12.64) with N =



(Z m × Z m ), L =

1m 1/2, noting (12.63), we get the following continuous embeddings: (Hs ) N × (Hs−1 ) N ⊂ F ⊂ (H0 ) N × (H−1 ) N , s > 1/2.

(12.69)

Multiplying the equations in system (II) by a solution  to the adjoint problem (12.4) and integrating by parts, we get (12.70) (U  (t), (t))(H0 ) N − (U (t),  (t))(H0 ) N  t 0 ,  1 ,  0 )(H0 ) N − (U 1 )(H0 ) N + (D H (τ ), (τ ))ddτ . = (U 0

1

12.4 Exact Boundary Controllability

167

Taking (H0 ) N as the pivot space and noting (12.69), (12.70) can be written as (U  (t), −U (t)), ((t),  (t))  t (D H (τ ), (τ ))ddτ , = (U1 , −U0 ),(0 , 1 ) + 0

(12.71)

1

where ·, · denotes the duality between the spaces (H−s ) N × (H1−s ) N and (Hs ) N × (Hs−1 ) N . Definition 12.11 U is a weak solution to problem (II) and (II0), if (U, U  ) ∈ C 0 ([0, T ]; (H1−s ) N × (H−s ) N )

(12.72)

1 ) ∈ (Hs ) N × 0 ,  such that the variational Eq. (12.71) holds for any given ( N (Hs−1 ) with s > 1/2. Proposition 12.12 For any given H ∈ L 2 (0, T ; (L 2 (1 )) M ) and any given 1 ) ∈ (H1−s ) N × (H−s ) N with s > 1/2, problem (II) and (II0) admits a unique 0 , U (U weak solution U . Moreover, the linear map R:

1 , H ) → (U, U  ) 0 , U (U

(12.73)

is continuous with respect to the corresponding topologies. Proof Define the linear form 0 ,  1 ) L t (  t 0 ), ( 1 , −U 0 ,  1 ) + (D H (τ ), (τ ))ddτ . = (U 0

(12.74)

1

By the definition (12.63) of the p-norm and the continuous embedding (12.69), the linear form L t is bounded in (Hs ) N × (Hs−1 ) N for any given t  0. Let St be the semigroup associated to the adjoint problem (12.16) on the Hilbert space (Hs ) N × (Hs−1 ) N , which is an isomorphism on (Hs ) N × (Hs−1 ) N . The composed linear form L t ◦ St−1 is bounded in (Hs ) N × (Hs−1 ) N . Then, by Riesz–Fréchet’s representation theorem, there exists a unique element (U  (t), −U (t)) ∈ (H−s ) N × (H1−s ) N , such that (12.75) L t ◦ St−1 ((t),  (t)) = (U  (t), −U (t)), ((t),  (t)) 1 ) ∈ (Hs ) N × (Hs−1 ) N . Since 0 ,  for any given ( 0 ,  1 ), L t ◦ St−1 ((t),  (t)) = L t (

(12.76)

168

12 Exact Boundary Controllability and Non-exact Boundary Controllability

0 ,  1 ) ∈ (Hs ) N × (Hs−1 ) N . Moreover, we have we get (12.71) for any given ( (U  (t), −U (t))(H−s ) N ×(H1−s ) N

1 ,U 0 )(H−s ) N ×(H1−s ) N + H  L 2 (0,T ;(L 2 (1 )) M ) .  C T (U

(12.77)

Then, by a classic argument of density, we get the regularity (12.72). The proof is thus complete.  Definition 12.13 System (II) is exactly null controllable at the time T in the space (H1 ) N × (H0 ) N , if there exists a positive constant T > 0, such that for any 0 ) ∈ (H1 ) N × (H0 ) N , there exists a boundary control H ∈ L 2 (0, T ; 1 , U given (U 2 M (L (1 )) ), such that problem (II) and (II0) admits a unique weak solution U satisfying the final condition (12.78) t = T : U = U  = 0. Theorem 12.14 Assume that M = N . Then there exists a positive constant T > 0, such that system (II) is exactly null controllable at the time T for any given initial 1 ) ∈ (H1 ) N × (H0 ) N . Moreover, we have the continuous dependence 0 , U data (U 0 , U 1 )(H1 ) N ×(H0 ) N . H  L 2 (0,T ;(L 2 (1 )) N )  c(U

(12.79)

Proof Let  be the solution to the adjoint problem (12.4) in (Hs ) N × (Hs−1 ) N with s > 1/2. Let (12.80) H = D −1 |1 . Because of the first inclusion in (12.69), we have H ∈ L 2 (0, T ; (L 2 (1 )) N ). Then, by Proposition 12.12, the corresponding backward problem ⎧  U − U + AU = 0 ⎪ ⎪ ⎨ U =0 ∂ν U =  ⎪ ⎪ ⎩ t = T : U = U = 0

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in 

(12.81)

admits a unique weak solution U with (12.72). Accordingly, we define the linear map 1 ) = (−U  (0), U (0)). 0 ,  (12.82) ( Clearly,  is a continuous map from (Hs ) N × (Hs−1 ) N into (H−s ) N × (H1−s ) N . Next, noting (12.78), it follows from (12.71) that 0 ,  1 ), ( 0 ,  1 ) = (

 0

T

 1

(τ )(τ )ddτ ,

(12.83)

12.4 Exact Boundary Controllability

169

0 ,  1 ). where  is the solution to the adjoint problem (12.16) with the initial data ( It then follows that 1 ), ( 0 ,  1 )  ( 0 ,  1 )F ( 0 ,  1 )F . 0 ,  (

(12.84)

By definition, (Hs ) N × (Hs−1 ) N is dense in F, then the linear form 1 ) → ( 0 ,  1 ), ( 0 ,  1 ) 0 ,  (

(12.85)

can be continuously extended to F, so that (0 , 1 ) ∈ F  . Moreover, we have 1 )F   ( 0 ,  1 )F . 0 ,  (

(12.86)

Then,  is a continuous linear map from F to F  . Therefore, the symmetric bilinear form 1 ), ( 0 ,  1 ), 0 ,  (12.87) ( where ·, · denotes the duality between the spaces (H−s ) N × (H1−s ) N and (Hs ) N × (Hs−1 ) N , is continuous and coercive in the product space F × F. By Lax–Milgram’s 1 , U 0 ) ∈ F  , Lemma,  is an isomorphism from F onto F  . Then for any given (−U there exists an element (0 , 1 ) ∈ F, such that 1 , U 0 ). 1 ) = (−U 0 ,  (

(12.88)

This is precisely the exact boundary controllability of system (II) for any given 0 ) ∈ F  , in particular, for any given initial data (U 1 , −U 0 ) ∈ 1 , −U initial data (U N N  (H0 ) × (H1 ) ⊂ F , because of the second inclusion in (12.69). Finally, from the definitions (12.80) and (12.63), we have 0 ,  1 )F H  L 2 (0,T ;(L 2 (1 )) N )  C L 2 (0,T ;(L 2 (1 )) N ) = C(

(12.89)

which together with (12.88) implies the continuous dependence: 0 , U 1 )(H1 ) N ×(H0 ) N . H  L 2 (0,T ;(L 2 (1 )) N )  C−1 L(F  ,F ) (U

(12.90)

The proof is thus complete.



12.5 Non-exact Boundary Controllability In the case of fewer boundary controls, we have the following negative result. Theorem 12.15 Assume that M < N . Then system (II) is not exactly boundary con 0 , U 1 ) ∈ (H1 ) N × (H0 ) N . trollable for all initial data (U

170

12 Exact Boundary Controllability and Non-exact Boundary Controllability

Proof Since M < N , there exists a non-null vector e ∈ R N , such that D T e = 0. We choose a special initial data as 1 = 0, 0 = θe, U U

(12.91)

where θ ∈ D() is arbitrarily given. If system (II) is exactly boundary controllable, noting that Theorem 3.9 is still valid here, there exists a boundary control H such that (12.92) H  L 2 (0,T ;(L 2 (1 )) M )  Cθ H 1 () . Then, by Proposition 12.12 we have U  L 2 (0,T ;(H1−s ()) N )  Cθ H 1 () , ∀s > 1/2.

(12.93)

Now, taking the inner product of e with problem (II) and (II0) and noting φ = (e, U ), we get ⎧  φ − φ = −(e, AU ) in (0, T ) × , ⎪ ⎪ ⎪ ⎪ on (0, T ) × 0 , ⎨φ = 0 on (0, T ) × 1 , ∂ν φ = 0 (12.94) ⎪  ⎪ = 0 in , t = 0 : φ = θ, φ ⎪ ⎪ ⎩ t = T : φ = 0, φ = 0 in . Noting (12.68), by well-posedness and noting (12.93), we get θ H 2−s ()  CU  L 2 (0,T ;(H1−s ()) N )  C  θ H 1 ()

(12.95)

Choosing s such that 1 > s > 1/2, then 2 − s > 1, which gives a contradiction. The proof is then complete.  Remark 12.16 As shown in the proof of Theorem 12.14, a weaker regularity such as (U, U  ) ∈ C 0 ([0, T ]; (H0 ) N × (H−1 ) N ) is sufficient to make sense to the value (U (0), U  (0)), therefore, sufficient for proving the exact boundary controllability. At this stage, it is not necessary to pay much attention to the regularity of the weak solution with respect to the space variable. However, in order to establish the non-exact boundary controllability in Theorem 12.15, this regularity becomes indispensable for the argument of compact perturbation. In the case with Dirichlet boundary controls, the weak solution has the same regularity as the controllable initial data. This regularity yields the non-exact boundary controllability in the case of fewer boundary controls (cf. Chap. 3). However, for Neumann boundary controls, the direct inequality is much weaker than the inverse inequality. For example, in Proposition 12.12, we can get only (U, U  ) ∈ C 0 ([0, T ]; (H1−s ) N × (H−s ) N ) for any given s > 1/2, 1 ) lies in the space (H1 ) N × (H0 ) N . Even 0 , U while, the controllable initial data (U though this regularity is not sharp in general, it is already sufficient for the proof of the non-exact boundary controllability of system (II). Nevertheless, the gap of regularity between the solution and its initial data could cause some problems for

12.5 Non-exact Boundary Controllability

171

the non-exact boundary controllability with coupled Robin boundary controls (cf. Part V and Part VI below). Remark 12.17 As for the system with Dirichlet boundary controls discussed in Part 1, we have shown in Theorems 12.14 and 12.15 that system (II) with Neumann boundary controls is exactly boundary controllable if and only if the boundary controls have the same number as the state variables or the wave equations. Of course, the non-exact boundary controllability is valid only in the framework that all the components of the initial data are in the same energy space. For example, the authors of [65] considered the exact boundary controllability by means of only one boundary control for a system of two wave equations with initial data of different levels of finite energy. More specifically, the exact boundary controllability by means of only one boundary control for the a cascade system of N wave equations was established in [1, 2]. On the other hand, in contrast with the exact boundary controllability, the approximate boundary controllability is more flexible with respect to the number of boundary controls, and is closely related to the so-called Kalman’s criterion on the rank of an enlarged matrix composed of the coupling matrix A and the boundary control matrix D (cf. Part IV below).

Chapter 13

Exact Boundary Synchronization and Non-exact Boundary Synchronization

In the case of partial lack of boundary controls, we consider the exact boundary synchronization and the non-exact boundary synchronization in this chapter for system (II) with Neumann boundary controls.

13.1 Definition Let

 T T  U = u (1) , · · · , u (N ) and h (1) , · · · , h (M)

(13.1)

with M  N . Consider the coupled system (II) with the initial condition (II0). For simplifying the statement and without loss of generality, in what follows we always suppose that mes(0 ) = 0 and denote 2 (0, +∞; L 2 (1 )), H0 = L 2 (), H1 = H10 (), L = L loc

(13.2)

where H10 () is the subspace of H 1 () composed of all the functions with the null trace on 0 . According to Theorem 12.14, when M = N , there exists a constant T > 0, such that system (II) is exactly null controllable at the time T for any given initial data 1 ) ∈ (H1 ) N × (H0 ) N . 0 , U (U On the other hand, according to Theorem 12.15 if there is a lack of boundary controls, namely, when M < N , no matter how large T > 0 is, system (II) is not 1 ) ∈ (H1 ) N × 0 , U exactly null controllable at the time T for any given initial data (U N (H0 ) . Thus, it is necessary to discuss whether system (II) is controllable in some weaker senses when there is a lack of boundary controls, namely, when M < N . Although the results are similar to those for the coupled system of wave equations with Dirichlet © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_13

173

174

13 Exact Boundary Synchronization and Non-exact Boundary Synchronization

boundary controls, discussed in Part I, since the solution to a coupled system of wave equations with Neumann boundary conditions has a relatively weaker regularity, in order to realize the desired result, we need stronger function spaces, and the corresponding adjoint problem is also different. First, we give the following Definition 13.1 System (II) is exactly synchronizable at the time T > 0 in the 0 , U 1 ) ∈ (H1 ) N × (H0 ) N , there space (H1 ) N × (H0 ) N if for any given initial data (U M exists a boundary control H ∈ L with compact support in [0, T ], such that the weak solution U = U (t, x) to the mixed initial-boundary value problem (II) and (II0) satisfies tT :

u (1) ≡ · · · ≡ u (N ) := u,

(13.3)

where u = u(t, x), being unknown a priori, is called the corresponding exactly synchronizable state. The above definition requires that system (II) maintains the exactly synchronizable state even though the boundary control is canceled after the time T . Let ⎞ ⎛ 1 −1 ⎟ ⎜ 1 −1 ⎟ ⎜ C1 = ⎜ (13.4) ⎟ .. .. ⎠ ⎝ . . 1 −1

(N −1)×N

be the corresponding matrix of synchronization. C1 is a full row-rank matrix, and Ker(C1 ) = Span{e1 }, where e1 = (1, · · · , 1)T . Clearly, the exact boundary synchronization (13.3) can be equivalently written as tT :

C1 U (t, x) ≡ 0 in .

(13.5)

13.2 Condition of C1 -Compatibility We have Theorem 13.2 Assume that M < N . If system (II) is exactly synchronizable in the space (H1 ) N × (H0 ) N , then the coupling matrix A = (ai j ) should satisfy the following condition of compatibility (row-sum condition: the sum of elements in every row is equal to each other):

13.2 Condition of C1 -Compatibility

175 N 

ai j := a,

(13.6)

j=1

where a is a constant independent of i = 1, · · · , N . Proof By Theorem 12.15, since M < N , system (II) is not exactly null controllable, 0 , U 1 ) ∈ (H1 ) N × (H0 ) N , such that for any given then there exists an initial data (U boundary control H , the corresponding exactly synchronizable state u(t, x) ≡ 0. Then, noting (13.3), the solution to problem (II) and (II0) corresponding to this initial data satisfies u  − u +

N 

ai j u = 0 in D ((T, +∞) × )

(13.7)

j=1

for all i = 1, · · · , N . Then, we have N 

ak j −

j=1

N 

ai j u = 0 in D ((T, +∞) × )

(13.8)

j=1

for i, k = 1, · · · , N . It follows that N  j=1

ak j =

N 

ai j , i, k = 1, · · · , N ,

(13.9)

j=1

which is just the required condition of compatibility (13.6).



By Proposition 2.15, it is easy to get Lemma 13.3 The following properties are equivalent: (i) The condition of compatibility (13.6) holds; (ii) e = (1, · · · , 1)T is an eigenvector of A corresponding to the eigenvalue a given by (13.6); (iii) Ker(C1 ) is a one-dimensional invariant subspace of A: AKer(C1 ) ⊆ Ker(C1 );

(13.10)

(iv) There exists a unique matrix A1 of order (N − 1), such that C 1 A = A1 C 1 . A1 = (a i j ) is called the reduced matrix of A by C1 , where

(13.11)

176

13 Exact Boundary Synchronization and Non-exact Boundary Synchronization N 

ai j =

p= j+1

(ai+1, p − ai p ) =

j  (ai p − ai+1, p )

(13.12)

p=1

for i, j = 1, · · · , N − 1. In what follows we will call the row-sum condition (13.6) to be the condition of C1 -compatibility.

13.3 Exact Boundary Synchronization and Non-exact Boundary Synchronization We have Theorem 13.4 Under the condition of C1 -compatibility (13.10), system (II) is exactly synchronizable at some time T > 0 in the space (H1 ) N × (H0 ) N if and only if rank(C1 D) = N − 1. Moreover, we have the following continuous dependence: 0 , U 1 )(H1 ) N ×(H0 ) N −1 , H  L 2 (0,T ;(L 2 (1 )) N −1 )  cC1 (U

(13.13)

where c is a positive constant. Proof Under the condition of C1 -compatibility (13.10), let 0 = C1 U 1 = C1 U 0 , W 1 . W = C1 U, W

(13.14)

Noting (13.11), it is easy to see that the original problem (II) and (II0) for U can be reduced to the following self-closed system for W = (w(1) , · · · , w (N −1) )T : ⎧  ⎨ W − W + A1 W = 0 in (0, +∞) × , W =0 on (0, +∞) × 0 , ⎩ ∂ν W = D H on (0, +∞) × 1

(13.15)

with the initial condition t =0:

0 , W  = W 1 in , W =W

(13.16)

where D = C1 D. Noting that C1 is a surjection from (H1 ) N × (H0 ) N onto (H1 ) N −1 × (H0 ) N −1 , we easily check that the exact boundary synchronization of system (II) for U is equivalent to the exact boundary null controllability of the reduced system (13.15) for W . Thus, by means of Theorems 12.14 and 12.15, the exact boundary synchronization of system (II) is equivalent to the rank condition rank(D) = rank(C1 D) = N − 1. Moreover, the continuous dependence (13.13) comes directly from (12.79). 

13.4 Attainable Set of Exactly Synchronizable States

177

13.4 Attainable Set of Exactly Synchronizable States In the case that system (II) possesses the exact boundary synchronization at the time T > 0, under the condition of C1 -compatibility (13.6), it is easy to see that for t  T , the exactly synchronizable state u = u(t, x) defined by (13.3) satisfies the following wave equation with homogenous boundary conditions: ⎧  ⎨ u − u + au = 0 in (T, +∞) × , u=0 on (T, +∞) × 0 , ⎩ on (T, +∞) × 1 , ∂ν u = 0

(13.17)

where a is given by (13.3). Hence, the evolution of the exactly synchronizable state u = u(t, x) with respect to t is completely determined by the values of (u, u  ) at the time t = T : u 1 in . (13.18) t = T : u = u0, u =  We have Theorem 13.5 Under the condition of C1 -compatibility (13.6), the attainable set of the values (u, u  ) at the time t = T of the exactly synchronizable state u = u(t, x) 0 , U 1 ) vary in the space (H1 ) N × is the whole space H1 × H0 as the initial data (U N (H0 ) . Proof For any given ( u0,  u 1 ) ∈ H1 × H0 , by solving the following backward problem ⎧  ⎨ u − u + au = 0 in (0, T ) × , u=0 on (0, T ) × 0 , (13.19) ⎩ on (0, T ) × 1 ∂ν u = 0 with the final condition t=T :

u = u0, u =  u 1 in ,

(13.20)

we get the corresponding solution u = u(t, x). Then, under the condition of C1 -compatibility (13.6), the function U (t, x) = u(t, x)e1

(13.21)

with e1 = (1, · · · , 1)T is the solution to problem (II) and (II0) with the null boundary control H ≡ 0 and the initial condition t =0:

U = u(0, x)e1 , U  = u  (0, x)e1 .

(13.22)

Therefore, by solving problem (II) and (II0) with the null boundary control and the u1) initial condition (13.22), we can reach any given exactly synchronizable state ( u0, 

178

13 Exact Boundary Synchronization and Non-exact Boundary Synchronization

at the time t = T . This fact shows that any given state ( u0,  u 1 ) ∈ H1 × H0 can be expected to be a exactly synchronizable state. Consequently, the set of the values (u(T ), u  (T )) of the exactly synchronizable state u = (t, x) at the time T is the 0 , U 1 ) vary in the space (H1 ) N × (H0 ) N . whole space H1 × H0 as the initial data (U The proof is complete.  The determination of the exactly synchronizable state u for each given initial data 0 , U 1 ) will be considered in Chap. 15. (U

Chapter 14

Exact Boundary Synchronization by p-Groups

The exact boundary synchronization by p-groups will be considered in this chapter for system (II) with further lack of Neumann boundary controls.

14.1 Definition When there is a further lack of boundary controls, we consider the exact boundary synchronization by p-groups with p  1. This indicates that the components of U are divided into p groups: (u (1) , · · · , u (n 1 ) ), (u (n 1 +1) , · · · , u (n 2 ) ), · · · , (u (n p−1 +1) , · · · , u (n p ) ),

(14.1)

where 0 = n 0 < n 1 < n 2 < · · · < n p = N are integers such that n r − n r −1  2 for all 1  r  p. Definition 14.1 System (II) is exactly synchronizable by p-groups at the time T > 0 0 , U 1 ) ∈ (H1 ) N × (H0 ) N , in the space (H1 ) N × (H0 ) N if for any given initial data (U there exists a boundary control H ∈ L M with compact support in [0, T ], such that the weak solution U = U (t, x) to problem (II) and (II0) satisfies tT :

u (i) = u r , n r −1 + 1  i  n r , 1  r  p,

(14.2)

where, u = (u 1 , · · · , u p )T , being unknown a priori, is called the corresponding exactly synchronizable state by p-groups. Let Sr be a (n r − n r −1 − 1) × (n r − n r −1 ) full row-rank matrix:

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_14

179

180

14 Exact Boundary Synchronization by p-Groups

⎞ 1 −1 ⎟ ⎜ 1 −1 ⎟ ⎜ Sr = ⎜ ⎟ , 1  r  p, .. .. ⎠ ⎝ . . 1 −1 ⎛

(14.3)

and let C p be the following (N − p) × N matrix of synchronization by p-groups: ⎛ ⎜ ⎜ Cp = ⎜ ⎝

⎞

S1 S2

..

⎟ ⎟ ⎟. ⎠

.

(14.4)

Sp Obviously, we have Ker(C p ) = Span{e1 , · · · , e p },

(14.5)

where for 1  r  p, (er )i =

1, n r −1 + 1  i  n r , 0, otherwise.

(14.6)

Thus, the exact boundary synchronization by p-groups (14.2) can be equivalently written as tT :

C p U ≡ 0 in ,

(14.7)

or equivalently, tT :

U=

p

u r er in .

(14.8)

r =1

14.2 Condition of C p -Compatibility We have Theorem 14.2 Assume that system (II) is exactly synchronizable by p-groups. Then we necessarily have M  N − p. In particular, when M = N − p, the coupling matrix A = (ai j ) should satisfy the following condition of C p -compatibility: AKer(C p ) ⊆ Ker(C p ).

(14.9)

14.2 Condition of C p -Compatibility

181

Proof Noting that Lemma 6.3 is proved in a way independent of applied boundary conditions, it can be still used in the case with Neumann boundary controls. Since we have (14.7), if AKer(C p )  Ker(C p ), by Lemma 6.3, we can construct p−1 such that an enlarged full row-rank (N − p + 1) × N matrix C tT :

p−1 U ≡ 0 in . C

p−1 ), still by Lemma 6.3, we can construct another enlarged p−1 )  Ker(C If AKer(C p−2 such that full row-rank (N − p + 2) × N matrix C tT :

p−2 U ≡ 0 in , C

· · · · · · . This procedure should stop at the r th step with 0  r  p. Thus, we get an p−r such that enlarged full row-rank (N − p + r ) × N matrix C p−r U ≡ 0 in  C

(14.10)

p−r ). p−r ) ⊆ Ker(C AKer(C

(14.11)

tT : and

Then, by Proposition 2.15, there exists a unique matrix A˜ of order (N − p + r ), such that p−r . p−r A = A˜ C C

(14.12)

p−r U in problem (II) and (II0), we get the following reduced Setting W = C problem for W = (w (1) , · · · , w (N − p+r ) )T : ⎧  ˜ = 0 in (0, +∞) × , ⎨ W − W + AW W =0 on (0, +∞) × 0 , ⎩ ∂ν W = D˜ H on (0, +∞) × 1

(14.13)

with the initial condition t =0:

p−r U 0 , W  = C 1 in , p−r U W =C

(14.14)

p−r D. Moreover, by (14.10) we have where D˜ = C tT :

W ≡ 0.

(14.15)

p−r is a (N − p + r ) × N full row-rank matrix, the linear mapping Noting that C 1 ) → (C p−r U p−r U 0 , C 1 ) 0 , U (U

(14.16)

182

14 Exact Boundary Synchronization by p-Groups

is a surjection from (H1 ) N × (H0 ) N onto (H1 ) N − p+r × (H0 ) N − p+r , then system (14.13) is exactly null controllable at the time T in the space (H1 ) N − p+r × (H0 ) N − p+r . By Theorems 12.14 and 12.15, we necessarily have p−r D) = N − p + r, rank(C then we get p−r D) = N − p + r  N − p. M = rank(D)  rank(C

(14.17)

In particular, when M = N − p, we have r = 0, namely, the condition of  C p -compatibility (14.9) holds. Remark 14.3 The condition of C p -compatibility (14.9) is equivalent to the fact that there exist some constants αr s (1  r, s  p) such that Aer =

p

αsr es , 1  r  p,

(14.18)

s=1

or, noting (14.6), A satisfies the following row-sum condition by blocks: ns

ai j = αr s , n r −1 + 1  i  n r , 1  r, s  p.

(14.19)

j=n s−1 +1

Especially, this condition of compatibility becomes the row-sum condition 13.6 when p = 1.

14.3 Exact Boundary Synchronization by p-Groups and Non-exact Boundary Synchronization by p-Groups Theorem 14.4 Assume that the condition of C p -compatibility (14.9) holds. Then system (II) is exactly synchronizable by p-groups if and only if rank(C p D) = N − p. Moreover, we have the continuous dependence: 0 , U 1 )(H1 ) N − p ×(H0 ) N − p , H  L 2 (0;T ;(L 2 (1 )) N − p )  cC p (U

(14.20)

where c is a positive constant. Proof Assume that the coupling matrix A = (ai j ) satisfies the condition of C p compatibility (14.9). By Proposition 2.15, there exists a unique matrix A p of order (N − p), such that

14.3 Exact Boundary Synchronization by p-Groups …

C p A = A pC p.

183

(14.21)

Setting W = C p U,

D = C p D,

(14.22)

we get the following reduced system for W = (w(1) , · · · , w (N − p) )T : ⎧  ⎨ W − W + A p W = 0 in (0, +∞) × , W =0 on (0, +∞) × 0 , ⎩ ∂ν W = D H on (0, +∞) × 1

(14.23)

with the initial condition t =0:

0 , W  = C p U 1 in . W = C pU

(14.24)

Noting that C p is an (N − p) × N full row-rank matrix, it is easy to see that system (II) is exactly synchronizable by p-groups if and only if the reduced system (14.23) is exactly null controllable, or equivalently, by Theorem 12.14 and Theorem 12.15 if and only if rank(C p D) = N − p. Moreover, the continuous dependence (14.20) comes from (12.79). 

14.4 Attainable Set of Exactly Synchronizable States by p-Groups Under the condition of C p -compatibility (14.9), if system (II) is exactly synchronizable by p-groups at the time T > 0, then it is easy to see that for t  T , the exactly synchronizable state by p-groups u = (u 1 , · · · u p )T satisfies the following coupled system of wave equations with homogenous boundary conditions: ⎧   = 0 in (T, +∞) × , ⎨ u − u + Au u=0 on (T, +∞) × 0 , ⎩ on (T, +∞) × 1 , ∂ν u = 0

(14.25)

 = (αr s ) is given by (14.18). Hence, the evolution of the exactly synchronizwhere A able state by p-groups u = (u 1 , · · · u p )T with respect to t is completely determined by the values of (u, u  ) at the time t = T : t=T :

u1. u = u0, u = 

(14.26)

184

14 Exact Boundary Synchronization by p-Groups

As in Theorem 13.5 for the case p = 1, we can show that the attainable set of all possible values of (u, u  ) at t = T is the whole space (H1 ) p × (H0 ) p , when the 1 ) vary in the space (H1 ) N × (H0 ) N . 0 , U initial data (U The determination of the exactly synchronizable state by p-groups u = 0 , U 1 ) will be considered in Chap. 15. (u 1 , · · · , u p )T for each given initial data (U

Chapter 15

Determination of Exactly Synchronizable States by p-Groups

When system (II) possesses the exact boundary synchronization by p-groups, the corresponding exactly synchronizable states by p-groups will be studied in this chapter.

15.1 Introduction Now, under the condition of C p -compatibility (14.9), we are going to discuss the determination of exactly synchronizable states by p-groups u = (u 1 , · · · , u p )T with p  1 for system (II). Generally speaking, exactly synchronizable states by p-groups 1 ) and applied boundary controls H . However, 0 , U should depend on the initial data (U when the coupling matrix A possesses some good properties, exactly synchronizable states by p-groups are independent of applied boundary controls, and can be determined entirely by the solution to a system of wave equations with homogeneous 1 ). 0 , U boundary conditions for any given initial data (U First, as in the case of Dirichlet boundary controls, we have the following Theorem 15.1 Let Uad denote the set of all boundary controls H = (h (1) , · · · , h (N − p) )T which can realize the exact boundary synchronization by p-groups at the time T > 0 for system (II). Assume that the condition of C p -compatibility (14.9) holds. Then for  > 0 small enough, the value of H ∈ Uad on (0, ) × 1 can be arbitrarily chosen. Proof Under the condition of C p -compatibility (14.9), the exact boundary synchronization by p-groups of system (II) is equivalent to the exact boundary null controllability of the reduced system (14.23). Let T0 > 0 be a number independent of the initial data, such that the reduced system (14.23) is exactly null controllable at any given time T with T > T0 . Therefore, taking an  > 0 so small that T −  > T0 , © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_15

185

186

15 Determination of Exactly Synchronizable States by p-Groups

system (II) is still exactly synchronizable by p-groups at the time T for the initial data given at t = . 1 ) ∈ (H1 ) N × (H0 ) N , we arbitrarily give 0 , U For any given initial data (U  ∈ (C0∞ ([0, ] × 1 )) N − p H

(15.1)

 . Let (U  , U  ) and solve problem (II) and (II0) on the time interval [0, ] with H = H denote the corresponding solution. We check easily that  ) ∈ C 0 ([0, ]; (H1 ) N × (H0 ) N ).  , U (U

(15.2)

By Theorem 14.4, we can find a boundary control  ∈ L 2 (, T ; (L 2 (1 )) N − p ) H

(15.3)

which realizes the exact boundary synchronization by p-groups at the time t = T for system (II) with the initial condition t =:

 , U  = U  .  = U U

(15.4)

Let  H=

 , t ∈ (0, ), H  H , t ∈ (, T ),

  , t ∈ (0, ), U U=  , t ∈ (, T ). U

(15.5)

It is easy to check that U is the solution to problem (II) and (II0) with the boundary control H . Then, system (II) is exactly synchronizable by p-groups still at the time 1 ) given at t = 0. 0 , U  T for the initial data (U

15.2 Determination and Estimation of Exactly Synchronizable States by p-Groups Exactly synchronizable states by p-groups are closely related to the properties of the coupling matrix A. Let D N − p = {D ∈ M N ×(N − p) (R) : rank(D) = rank(C p D) = N − p}.

(15.6)

By Proposition 6.12, D ∈ D N − p if and only if it can be expressed by D = C Tp D1 + (e1 , · · · , e p )D0 ,

(15.7)

15.2 Determination and Estimation of Exactly Synchronizable States by p-Groups

187

where e1 , · · · , e p are given by (14.6), D1 is an invertible matrix of order (N − p), and D0 is a p × (N − p) matrix. We have Theorem 15.2 Under the condition of C p -compatibility (14.9), assume that A T possesses an invariant subspace Span{E 1 , · · · , E p } which is bi-orthonormal to Ker(C p ) = Span{e1 , · · · , e p }: (Er , es ) = δr s , 1  r, s  p.

(15.8)

Then there exists a boundary control matrix D ∈ D N − p , such that each exactly synchronizable state by p-groups u = (u 1 , · · · , u p )T is independent of applied boundary controls, and can be determined as follows: t  T : u = ψ,

(15.9)

where ψ = (ψ1 , · · · , ψ p )T is the solution to the following problem with homogeneous boundary conditions for r = 1, · · · , p: ⎧ p ⎪ ⎪  ⎪ ψ − ψ + αr s ψs = 0 ⎪ r r ⎪ ⎪ ⎨ s=1 ψr = 0 ⎪ ⎪ ⎪ ⎪∂ν ψr = 0 ⎪ ⎪ ⎩ 0 ), ψr = (Er , U 1 ) t = 0 : ψr = (Er , U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in ,

(15.10)

where αr s (1  r, s  p) are given by (14.19). Proof Noting that Span{E 1 , · · · , E p } is bi-orthonormal to Ker(C p ) = Span {e1 , · · · , e p }, and taking D1 = I N − p ,

D0 = −E T C Tp with E = (E 1 , · · · , E p )

(15.11)

in (15.7), we get a boundary control matrix D ∈ D N − p and it is easy to see that E s ∈ Ker(D T ), 1  s  p.

(15.12)

On the other hand, since Span{E 1 , · · · , E p } is an invariant subspace of A T , noting (14.18) and the bi-orthonormality (15.8), we get easily that AT Es =

p

αsr Er , 1  s  p.

(15.13)

r =1

Let ψs = (E s , U ) for s = 1, · · · , p. Taking the inner product with E s on both sides of problem (II) and (II0), we get (15.10). Finally, for the exactly synchronizable

188

15 Determination of Exactly Synchronizable States by p-Groups

state by p-groups u = (u 1 , · · · , u p )T , by (14.8) we have tT :

ψs = (E s , U ) =

p (E s , er )u r = u s , 1  s  p.

(15.14)

r =1



The proof is complete.

When A does not possess any invariant subspace Span{E 1 , · · · , E p } which is bi-orthonormal to Ker(C p ) = Span{e1 , · · · , e p }, we can use the solution of problem (15.10) to give an estimate on each exactly synchronizable state by p-groups. T

Theorem 15.3 Under the condition of C p -compatibility (14.9), assume that there exists a subspace Span{E 1 , · · · , E p } that is bi-orthonormal to the subspace Span {e1 , · · · , e p }. Then there exist a boundary control matrix D ∈ D N − p , such that each exactly synchronizable state by p-groups u = (u 1 , · · · , u p )T satisfies the following estimate: (u, u  )(t) − (ψ, ψ  )(t)(H2−s ) p ×(H1−s ) p 0 , U 1 )(H1 ) N − p ×(H0 ) N − p , cC p (U

(15.15)

for all t  T , where ψ = (ψ1 , · · · , ψ p )T is the solution to problem (15.10), s > 21 , and c is a positive constant independent of the initial data. Proof Since Span{E 1 , · · · , E p } is bi-orthonormal to Span{e1 , · · · , e p }, still by (15.11), there exists a boundary control matrix D ∈ D N − p , such that (15.12) holds. Noting (14.18), it is easy to see that AT Es −

p

αsr Er ∈ {K er (C p )}⊥ = Im(C Tp ),

r =1

then there exists a vector Rs ∈ R N − p , such that AT Es −

p

αsr Er = C Tp Rs .

(15.16)

r =1

Thus, taking the inner product with E s on both sides of problem (II) and (II0), and setting φs = (E s , U ) for s = 1, · · · , p, we have ⎧ p ⎪ ⎪  ⎪ φs − φs + αsr φr = −(Rs , C p U ) ⎪ ⎪ ⎪ ⎨ r =1 φs = 0 ⎪ ⎪ ⎪ ⎪ ∂ν φ s = 0 ⎪ ⎪ ⎩ 0 ), φs = (E s , U 1 ) t = 0 : φr = (E s , U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in ,

(15.17)

15.2 Determination and Estimation of Exactly Synchronizable States by p-Groups

189

where αsr (1  r, s  p) are defined by (14.19), and U = U (t, x) ∈ Cloc (0, +∞; 1 (0, +∞; (H−s ) N ) with s > 1/2 is the solution to problem (II) and (H1−s ) N ) ∩ Cloc (II0). Moreover, we have tT :

φs = (E s , U ) =

p

(E s , er )u r = u s

(15.18)

r =1

for s = 1, · · · , p. Noting that (15.10) and (15.17) possess the same initial data and the same boundary conditions, by the well-posedness for a system of wave equations with Neumann boundary condition, we have (cf. Chap. 3 in [70]) that (ψ, ψ  )(t) − (φ, φ )(t)2(H2−s ) p ×(H1−s ) p

T C p U 2(H1−s ) N − p ds, ∀ 0  t  T, c

(15.19)

0

where c is a positive constant independent of the initial data. Noting that W = C p U , by the well-posedness of the reduced problem (14.23), it is easy to get that



T

C p U 2(H1−s ) N − p ds 0 2 0 , U 1 )2 N − p c(C p (U (H1 ) ×(H0 ) N − p + D H  L 2 (0;T ;(L 2 (1 )) N − p ) ).

(15.20)

Moreover, by (14.20) we have 0 , U 1 )(H1 ) N − p ×(H0 ) N − p . D H  L 2 (0;T ;(L 2 (1 )) N − p )  cC p (U

(15.21)

Substituting it into (15.20), we have

0

T

0 , U 1 )2 N − p C p U 2(H1−s ) N − p ds  cC p (U (H1 ) ×(H0 ) N − p ,

(15.22)

then, by (15.19) we get (ψ, ψ  )(t) − (φ, φ )(t)2(H2−s ) p ×(H1−s ) p 0 , U 1 )2 N − p cC p (U N−p . (H1 )

(15.23)

×(H0 )

Finally, substituting (15.18) into (15.23), we get (15.15).



Remark 15.4 Differently from the case with Dirichlet boundary controls, although the solution to problem (II) and (II0) with Neumann boundary controls possesses a weaker regularity, the solution to problem (15.10), which is used to estimate the exactly synchronizable state by p-groups, possesses a higher regularity than the original problem (II) and (II0) itself. This improved regularity makes it possible to

190

15 Determination of Exactly Synchronizable States by p-Groups

approach the exactly synchronizable state by p-groups by a solution to a relatively smoother problem. If there does not exist any subspace Span{E 1 , · · · , E p }, which is invariant for A T and bi-orthonormal to Ker(C p ) = Span{e1 , · · · , e p }, in order to express the exactly synchronizable state by p-groups of system (II), by the same procedure as in Sect. 7.3, we can always extend the subspace Span{e1 , · · · , e p } to the minimal invariant subspace Span{e1 , · · · , eq } of A with q  p, so that A T possesses an invariant subspace Span{E 1 , · · · , E q }, which is bi-orthonormal to Span{e1 , · · · , eq }. Let P be the projection on the subspace Span{e1 , · · · , eq } given by P=

q

es ⊗ E s ,

(15.24)

s=1

in which the tensor product ⊗ is defined by (e ⊗ E)U = (E, U )e = E T U e, ∀ U ∈ R N . P can be represented by a matrix of order N , such that Im(P) = Span{e1 , · · · , eq }

(15.25)

⊥  Ker(P) = Span{E 1 , · · · , E q } .

(15.26)

and

Moreover, it is easy to check that P A = A P.

(15.27)

Let U = U (t, x) be the solution to problem (II) and (II0). We define its synchronizable part Us and controllable part Uc , respectively, as follows: Us := PU, Uc := (I − P)U.

(15.28)

If system (II) is exactly synchronizable by p-groups, then U ∈ Span{e1 , · · · , e p } ⊆ Span{e1 , · · · , eq } = Im(P),

(15.29)

hence we have tT :

Us = PU = U, Uc = (I − P)U = 0.

(15.30)

Noting (15.27), multiplying P and (I − P) from the left on both sides of problem (II) and (II0), respectively, we see that the synchronizable part Us of U satisfies the

15.2 Determination and Estimation of Exactly Synchronizable States by p-Groups

191

following problem: ⎧  Us − Us + AUs = 0 ⎪ ⎪ ⎪ ⎨U = 0 s ⎪ ∂ν U s = P D H ⎪ ⎪ ⎩ 0 , Us = P U 1 t = 0 : Us = P U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in ,

(15.31)

while the controllable part Uc of U satisfies the following problem: ⎧  Uc − Uc + AUc = 0 ⎪ ⎪ ⎪ ⎨U = 0 c ⎪ ∂ν Uc = (I − P)D H ⎪ ⎪ ⎩ 0 , Uc = (I − P)U 1 t = 0 : Uc = (I − P)U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in .

(15.32)

In fact, by (15.30), under the boundary control H , Uc with the initial data ((I − 1 ) ∈ Ker(P) × Ker(P) is exactly null controllable, while Us with 0 , (I − P)U P)U 1 ) ∈ Im(P) × Im(P) is exactly synchronizable. 0 , P U the initial data (P U Theorem 15.5 Assume that the condition of C p -compatibility (14.9) holds. If system (II) is exactly synchronizable by p-groups, and the synchronizable part Us is independent of applied boundary controls H for t  0, then A T possesses an invariant subspace which is bi-orthonormal to Ker(C p ). Proof It is sufficient to show that p = q. Then Span{E 1 , · · · , E p } will be the desired subspace. Suppose that the synchronizable part Us is independent of applied boundary controls H for all t  0. Let H1 and H2 be two boundary controls which realize simultaneously the exact boundary synchronization by p-groups for system (II). It follows from (15.31) that P D(H1 − H2 ) = 0 on (0, ) × 1 . By Theorem 15.1, the value of H1 on (0, ) × 1 can be arbitrarily taken, then P D = 0, hence Im(D) ⊆ Ker(P). Noting (15.25), we have dim Ker(P) = N − q and dim Im(D) = N − p, then p = q. The proof is complete.



192

15 Determination of Exactly Synchronizable States by p-Groups

0 , U 1 ) such that P U 0 = P U 1 = 0, Remark 15.6 In particular, for the initial data (U system (II) is exactly null controllable.

15.3 Determination of Exactly Synchronizable States In the case of exact boundary synchronization, by the condition of C1 -compatibility (13.10), e1 = (1, · · · , 1)T is an eigenvector of A, corresponding to the eigenvalue a defined by (13.6). Let 1 , · · · , r and E 1 , · · · , Er with r  1 be the Jordan chains of A and A T , respectively, corresponding to the eigenvalue a, and Span{1 , · · · , r } is bi-orthonormal to Span{E 1 , · · · , Er }. Thus, we have ⎧ ⎪ 1  l  r, ⎨ Al = al + l+1 , T A E k = a E k + E k−1 , 1  k  r, ⎪ ⎩ 1  k, l  r (E k , l ) = δkl ,

(15.33)

with r = e1 = (1, · · · , 1)T , r +1 = 0,

E 0 = 0.

(15.34)

Let U = U (t, x) be the solution to problem (II) and (II0). If system (II) is exactly synchronizable, then tT :

U = ur ,

(15.35)

where u = u(t, x) is the corresponding exactly synchronizable state. For the synchronizable part Us and the controllable part Uc , we have tT :

Us = ur , Uc = 0.

(15.36)

Setting ψk = (E k , U ), 1  k  r,

(15.37)

and noting (15.24) and (15.28), similarly we have Us =

r k=1

(E k , U )k =

r

ψk k .

(15.38)

k=1

Thus, (ψ1 , · · · , ψr ) can be regarded as the coordinates of Us under the basis {1 , · · · , r }. Taking the inner product with E k on both sides of (15.31), and noting (15.35), we have

15.3 Determination of Exactly Synchronizable States

tT :

193

ψk = (E k , U ) = (E k , r ) = δkr , 1  k  r,

(15.39)

then we easily get the following Theorem 15.7 Let 1 , · · · , r and E 1 , · · · , Er be the Jordan chains of A and A T , respectively, corresponding to the eigenvalue a given by (13.6), in which r = e1 = (1, · · · , 1)T . Then, the exactly synchronizable state u is determined by tT :

u = ψr ,

(15.40)

where the synchronizable part Us = (ψ1 , · · · , ψr ) is determined by the following system (1  k  r ): ⎧  ⎪ ⎪ψk − ψk + aψk + ψk−1 = 0, ⎪ ⎨ψ = 0 k ⎪ ψk = h k ∂ ν ⎪ ⎪ ⎩ 0 ), ψk = (E k , U 1 ) t = 0 : ψk = (E k , U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in ,

(15.41)

where ψ0 = 0, h k = E kT P D H.

(15.42)

Remark 15.8 When r = 1, A T possesses an eigenvector E 1 such that (E 1 , 1 ) = 1.

(15.43)

By Theorem 15.2, we can choose a boundary control matrix D such that the exactly synchronizable state is independent of applied boundary controls H . When r  1, the exactly synchronizable state depends on applied boundary controls H . Moreover, in order to get the exactly synchronizable state, we must solve the whole coupled problem (15.41)–(15.42).

15.4 Determination of Exactly Synchronizable States by 3-Groups In this section, for an example, we will give the details on the determination of exactly synchronizable states by 3-groups for system (II). Here, we always assume that the condition of C3 -compatibility (14.9) is satisfied and that Ker(C3 ) is A-marked. All the exactly synchronizable states by p-groups for general p  1 can be discussed in a similar way.

194

15 Determination of Exactly Synchronizable States by p-Groups

By the exact boundary synchronization by 3-groups, when t  T , we have u (1) ≡ · · · ≡ u (n 1 ) := u 1 , u u

(n 1 +1) (n 2 +1)

(15.44)

≡ ··· ≡ u

(n 2 )

:= u 2 ,

(15.45)

≡ ··· ≡ u

(N )

:= u 3 .

(15.46)

Let ⎧ n1 n 2 −n 1 N −n 2 ⎪ ⎪

      ⎪ ⎪ ⎪ e1 = (1, · · · , 1, 0, · · · , 0, 0, · · · , 0)T , ⎪ ⎪ ⎨ n1 n 2 −n 1 N −n 2

      e 0, · · · , 0, 1, · · · , 1, 0, · · · , 0)T , = ( ⎪ 2 ⎪ ⎪ ⎪ n n −n N −n 2 1 2 1 ⎪ ⎪      ⎪ ⎩e = ( 0, · · · , 0, 0, · · · , 0, 1, · · · , 1)T . 3

(15.47)

We have Ker(C3 ) = Span{e1 , e2 , e3 }

(15.48)

and the exact boundary synchronization by 3-groups given by (15.44)–(15.46) means that tT :

U = u 1 e1 + u 2 e2 + u 3 e3 .

(15.49)

Since the invariant subspace Span{e1 , e2 , e3 } of A is of dimension 3, it may contain one, two, or three eigenvectors of A, and thus we can distinguish the following three cases. (i) A admits three eigenvectors fr , gs , and h t contained in Span{e1 , e2 , e3 }, corresponding to eigenvalues λ, μ and ν, respectively. Let f 1 , · · · , fr ; g1 , · · · , gs and h 1 , · · · , h t be the Jordan chains corresponding to these eigenvectors of A: ⎧ ⎪ ⎨ A f i = λ f i + f i+1 , 1  i  r, fr +1 = 0, Ag j = μg j + g j+1 , 1  j  s, gs+1 = 0, ⎪ ⎩ Ah k = νh k + h k+1 , 1  k  t, h t+1 = 0

(15.50)

and let ξ1 , · · · , ξr ; η1 , · · · , ηs and ζ1 , · · · , ζt be the Jordan chains corresponding to the related eigenvectors of A T : ⎧ T ⎪ ⎨ A ξi = λξi + ξi−1 , 1  i  r, ξ0 = 0, A T η j = μη j + η j−1 , 1  j  s, η0 = 0, ⎪ ⎩ T A ζk = νζk + ζk−1 , 1  k  t, ζ0 = 0,

(15.51)

15.4 Determination of Exactly Synchronizable States by 3-Groups

195

such that ( f i , ξl ) = δil , (g j , ηm ) = δ jm , (h k , ζn ) = δkn

(15.52)

and ( f i , ηm ) = ( f i , ζn ) = (g j , ξl ) = (g j , ζn ) = (h k , ξl ) = (h k , ηm ) = 0

(15.53)

for all i, l = 1, · · · , r ; j, m = 1, · · · , s; and k, n = 1, · · · , t. Taking the inner product with ξi , η j , and ζk on both sides of problem (II) and (II0), respectively, and denoting φi = (U, ξi ), ψ j = (U, η j ), θk = (U, ζk )

(15.54)

for i = 1, · · · , r ; j = 1, · · · , s; and k = 1, · · · t, we get ⎧  φi − φi + λφi + φi−1 = 0 ⎪ ⎪ ⎪ ⎨φ = 0 i T ⎪ ∂ ν φi = ξi D H ⎪ ⎪ ⎩ 0 ), φi = (ξi , U 1 ) t = 0 : φi = (ξi , U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in ,

(15.55)

⎧  ψ j − ψ j + μψ j + ψ j−1 = 0 ⎪ ⎪ ⎪ ⎨ψ = 0 j ⎪ ψ j = η Tj D H ∂ ν ⎪ ⎪ ⎩ 0 ), ψ j = (η j , U 1 ) t = 0 : ψ j = (η j , U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in 

(15.56)

⎧  θk − θk + νθk + θk−1 = 0 ⎪ ⎪ ⎪ ⎨θ = 0 k ⎪ ∂ν θk = ζkT D H ⎪ ⎪ ⎩ 0 ), θk = (ζk , U 1 ) t = 0 : θk = (ζk , U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in 

(15.57)

and

with φ0 = ψ0 = θ0 = 0.

(15.58)

Solving the reduced problems (15.55)–(15.57), we get φ1 , · · · , φr ; ψ1 , · · · , ψs ; and θ1 , · · · , θt . Noting that fr , gs , and h t are linearly independent eigenvectors and contained in Span{e1 , e2 , e3 }, we have

196

15 Determination of Exactly Synchronizable States by p-Groups

⎧ ⎪ ⎨e1 = α1 fr + α2 gs + α3 h t , e2 = β1 fr + β2 gs + β3 h t , ⎪ ⎩ e3 = γ1 fr + γ2 gs + γ3 h t .

(15.59)

Then, noting (15.52)–(15.53), it follows from (15.49) that

tT :

⎧ ⎪ ⎨φr = α1 u 1 + β1 u 2 + γ1 u 3 , ψs = α2 u 1 + β2 u 2 + γ2 u 3 , ⎪ ⎩ θt = α3 u 1 + β3 u 2 + γ3 u 3 .

(15.60)

Since e1 , e2 , e3 are linearly independent, the linear system (15.59) is invertible. Then, the corresponding exactly synchronizable state by 3-groups u = (u 1 , u 2 , u 3 )T can be uniquely determined by solving the linear system (15.60). In particular, when r = s = t = 1, the invariant subspace Span{ξ1 , η1 , ζ1 } of A T is bi-orthonormal to Ker(C3 ) = Span{e1 , e2 , e3 }. By Theorem 15.2, there exists a boundary control matrix D ∈ D N −3 , such that the exactly synchronizable state by 3-groups u = (u 1 , u 2 , u 3 )T is independent of applied boundary controls. (ii) A admits two eigenvectors fr and gs contained in Span{e1 , e2 , e3 }, corresponding to eigenvalues λ and μ, respectively. Let f 1 , f 2 , · · · , fr and g1 , g2 , · · · , gs be the Jordan chains corresponding to these eigenvectors of A: 

A f i = λ f i + f i+1 , 1  i  r, fr +1 = 0, Ag j = μg j + g j+1 , 1  j  s, gs+1 = 0

(15.61)

and let ξ1 , ξ2 , · · · , ξr and η1 , η2 , · · · , ηs be the Jordan chains corresponding to the related eigenvectors of A T : 

A T ξi = λξi + ξi−1 , 1  i  r, ξ0 = 0, A T η j = μη j + η j−1 , 1  j  s, η0 = 0,

(15.62)

such that ( f i , ξl ) = δil , (g j , ηm ) = δ jm

(15.63)

( f i , ηm ) = (g j , ξl ) = 0

(15.64)

and

for all i, l = 1, · · · , r and j, m = 1, · · · , s. Taking the inner product with ξi and η j on both sides of problem (II) and (II0), respectively, and denoting

15.4 Determination of Exactly Synchronizable States by 3-Groups

φi = (U, ξi ), ψ j = (U, η j ), i = 1, · · · , r ; j = 1, · · · , s,

197

(15.65)

we get again the reduced problems (15.55)–(15.56). Since Span{e1 , e2 , e3 } is A-marked, either fr or gs lies in Span{e1 , e2 , e3 }. For fixing idea, we assume that fr −1 ∈ Ker{e1 , e2 , e3 }. Then we have ⎧ ⎪ ⎨e1 = α1 fr + α2 fr −1 + α3 gs , e2 = β1 fr + β2 fr −1 + β3 gs , ⎪ ⎩ e3 = γ1 fr + γ2 fr −1 + γ3 gs .

(15.66)

Then, noting (15.63)–(15.64), it follows from (15.49) that

tT :

⎧ ⎪ ⎨φr = α1 u 1 + β1 u 2 + γ1 u 3 , φr −1 = α2 u 1 + β2 u 2 + γ2 u 3 , ⎪ ⎩ ψs = α3 u 1 + β3 u 2 + γ3 u 3 .

(15.67)

Since e1 , e2 , e3 are linearly independent, the linear system (15.66) is invertible, then the corresponding exactly synchronizable state by 3-groups u = (u 1 , u 2 , u 3 )T can be determined by solving the linear system (15.67). In particular, when r = 2, s = 1, the invariant subspace Span{ξ1 , ξ2 , η1 } of A T is bi-orthonormal to Span{e1 , e2 , e3 }. By Theorem 15.2, there exists a boundary control matrix D ∈ D N −3 , such that the corresponding exactly synchronizable state by 3groups u = (u 1 , u 2 , u 3 )T is independent of applied boundary controls. (iii) A admits only one eigenvector fr contained in Span{e1 , e2 , e3 }, corresponding to the eigenvalue λ. Let f 1 , f 2 , · · · , fr be the Jordan chain corresponding to this eigenvector of A: A f i = λ f i + f i+1 , 1  i  r,

fr +1 = 0,

(15.68)

and let ξ1 , ξ2 , · · · , ξr be the Jordan chain corresponding to the related eigenvector of A T : (15.69) A T ξi = λξi + ξi−1 , 1  i  r, ξ0 = 0, such that ( f i , ξl ) = δil , i, l = 1, · · · , r.

(15.70)

Taking the inner product with ξi on both sides of problem (II) and (II0), and denoting φi = (U, ξi ), i = 1, · · · r, we get again the reduced problem (15.55).

(15.71)

198

15 Determination of Exactly Synchronizable States by p-Groups

Since Span{e1 , e2 , e3 } is A-marked, fr −1 and fr −2 are necessarily contained in Span{e1 , e2 , e3 }. We have ⎧ ⎪ ⎨e1 = α1 fr + α2 fr −1 + α3 fr −2 , e2 = β1 fr + β2 fr −1 + β3 fr −2 , ⎪ ⎩ e3 = γ1 fr + γ2 fr −1 + γ3 fr −2 .

(15.72)

Then, noting (15.70), it follows from (15.49) that

tT :

⎧ ⎪ ⎨φr = α1 u 1 + β1 u 2 + γ1 u 3 , φr −1 = α2 u 1 + β2 u 2 + γ2 u 3 , ⎪ ⎩ φr −2 = α3 u 1 + β3 u 2 + γ3 u 3 .

(15.73)

Since e1 , e2 , e3 are linearly independent, the linear system (15.72) is invertible, then the corresponding exactly synchronizable state by 3-groups u = (u 1 , u 2 , u 3 )T can be uniquely determined by solving the linear system (15.73). In particular, when r = 3, the invariant subspace {ξ1 , ξ2 , ξ3 } of A T is bi-orthonormal to Span{e1 , e2 , e3 }. By Theorem 15.2, there exists a boundary control matrix D ∈ D N −3 , such that the exactly synchronizable state by 3-groups u = (u 1 , u 2 , u 3 )T is independent of applied boundary controls.

Part IV

Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls: Approximate Boundary Synchronization In this part we will introduce the concept of approximate boundary null controllability, approximate boundary synchronization and approximate boundary synchronization by groups, and establish the corresponding theory for system (II) of wave equations with Neumann boundary controls in the case with fewer boundary controls. Moreover, we will show that Kalman’s criterion of various kinds will play an important role in the discussion.

Chapter 16

Approximate Boundary Null Controllability

In this chapter, we will define the approximate boundary null controllability for system (II) and the D-observability for the adjoint problem, and show that these two concepts are equivalent to each other. Moreover, the corresponding Kalman’s criterion is introduced and studied.

16.1 Definitions For the coupled system (II) of wave equations with Neumann boundary controls, we still use the notations given in Chaps. 12 and 13. Definition 16.1 Let s > 21 . System (II) is approximately null controllable at the 0 , U 1 ) ∈ (H1−s ) N × (H−s ) N , there exists a time T > 0 if for any given initial data (U M sequence {Hn } of boundary controls in L with compact support in [0, T ], such that the sequence {Un } of solutions to the corresponding problem (II) and (II0) satisfies   Un (T ), Un (T ) −→ 0 in (H1−s ) N × (H−s ) N as n → +∞

(16.1)

or equivalently,   Un , Un −→ (0, 0) in (H1−s ) N × (H−s ) N as n → +∞

(16.2)

in the space Cloc ([T, +∞); (H1−s ) N ) × Cloc ([T, +∞); (H−s ) N ). Obviously, the exact boundary null controllability implies the approximate boundary null controllability. However, since we cannot get the convergence of the sequence {Hn } of boundary controls from Definition 16.1, generally speaking, the approximate boundary null controllability does not lead to the exact boundary null controllability. © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_16

201

202

Let

16 Approximate Boundary Null Controllability

 = (φ(1) , · · · , φ(N ) )T .

Consider the following adjoint problem: ⎧   −  + A T  = 0 ⎪ ⎪ ⎪ ⎨ = 0 ⎪ ∂ν  = 0 ⎪ ⎪ ⎩ t = 0 :  = 0 ,   =  1

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in ,

(16.3)

where A T is the transpose of A. As in the case with Dirichlet boundary controls (cf. Sect. 8.2), we give the following. Definition 16.2 For (0 , 1 ) ∈ (Hs ) N × (Hs−1 ) N (s > 21 ), the adjoint problem (16.3) is D-observable on [0, T ], if D T  ≡ 0 on [0, T ] × 1 ⇒ (0 , 1 ) ≡ 0, then  ≡ 0.

(16.4)

16.2 Equivalence Between the Approximate Boundary Null Controllability and the D-Observability In order to find the equivalence between the approximate boundary null controllability of the original system (II) and the D-observability of the adjoint problem (16.3), let C be the set of all the initial states (V (0), V  (0)) of the following backward problem: ⎧  V − V + AV = 0 ⎪ ⎪ ⎪ ⎨V = 0 ⎪ ∂ν V = D H ⎪ ⎪ ⎩ V (T ) = 0, V  (T ) = 0

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in 

(16.5)

with all admissible boundary controls H ∈ L M . By Proposition 12.12, we have the following. Lemma 16.3 For any given T > 0 and any given (V (T ), V  (T )) ∈ (H1−s ) N × (H−s ) N with s > 21 , and for any given boundary function H in L M , the backward problem (16.5) (in which the null final condition is replaced by the given final data) admits a unique weak solution such that V ∈ C 0 ([0, T ]; (H1−s ) N ) ∩ C 1 ([0, T ]; (H−s ) N ).

(16.6)

16.2 Equivalence Between the Approximate Boundary Null …

203

Lemma 16.4 System (II) possesses the approximate boundary null controllability if and only if C = (H1−s ) N × (H−s ) N .

(16.7)

Proof Assume that C = (H1−s ) N × (H−s ) N . By the definition of C, for any given 1 ) ∈ (H1−s ) N × (H−s ) N , there exists a sequence {Hn } of boundary controls 0 , U (U M in L with compact support in [0, T ], such that the sequence {Vn } of solutions to the corresponding backward problem (16.5) satisfies 

 0 , U 1 ) in (H1−s ) N × (H−s ) N as n → +∞. Vn (0), Vn (0) → (U

(16.8)

Recall that R:

1 , H ) → (U, U  ) 0 , U (U

(16.9)

is the continuous linear mapping defined by (12.73). We have 1 , Hn ) 0 , U R(U 1 − Vn (0), 0) + R(Vn (0), Vn (0), Hn ). 0 − Vn (0), U =R(U

(16.10)

On the other hand, by the definition of Vn , we have R(Vn (0), Vn (0), Hn )(T ) = 0.

(16.11)

Therefore 1 , Hn )(T ) = R(U 0 − Vn (0), U 1 − Vn (0), 0)(T ). 0 , U R(U

(16.12)

By Proposition 12.12 and noting (16.8), we then get 1 , Hn )(T ) (H1−s ) N ×(H−s ) N 0 , U R(U 1 − Vn (0)) (H1−s ) N ×(H−s ) N → 0 0 − Vn (0), U c (U

(16.13)

as n → +∞. Here and hereafter, c always denotes a positive constant. Thus, system (II) is approximately null controllable. Inversely, assume that system (II) is approximately null controllable. For any given 1 ) ∈ (H1−s ) N × (H−s ) N , there exists a sequence {Hn } of boundary controls 0 , U (U M in L with compact support in [0, T ], such that the sequence {Un } of solutions to the corresponding problem (II) and (II0) satisfies   0 , U 1 , Hn )(T ) → (0, 0) Un (T ), Un (T ) = R(U

(16.14)

204

16 Approximate Boundary Null Controllability

in (H1−s ) N × (H−s ) N as n → +∞. Taking such Hn as the boundary control, we solve the backward problem (16.5) and get the corresponding solution Vn . By the linearity of the mapping R, we have 1 , Hn ) − R(Vn (0), Vn (0), Hn ) 0 , U R(U 1 − Vn (0), 0). 0 − Vn (0), U =R(U

(16.15)

By Lemma 16.3 and noting (16.14), we have 1 − Vn (0), 0)(0) (H1−s ) N ×(H−s ) N 0 − Vn (0), U R(U

(16.16)

c (Un (T ) − Vn (T ), Un (T ) − Vn (T ) (H1−s ) N ×(H−s ) N =c (Un (T ), Un (T )) (H1−s ) N ×(H−s ) N → 0 as n → +∞. Noting (16.15), we then get 1 ) − (Vn (0), Vn (0)) (H1−s ) N ×(H−s ) N 0 , U (U 1 , Hn )(0) − R(Vn (0), Vn (0), Hn )(0) (H1−s ) N ×(H−s ) N 0 , U = R(U

(16.17)

0 − Vn (0), U 1 − Vn (0), 0)(0) (H1−s ) N ×(H−s ) N → 0 c R(U as n → +∞, which shows that C = (H1−s ) N × (H−s ) N .



Theorem 16.5 System (II) is approximately null controllable at the time T > 0 if and only if the adjoint problem (16.3) is D-observable on [0, T ]. Proof Assume that system (II) is not approximately null controllable at the time T > 0. By Lemma 16.4, there is a nontrivial vector (−1 , 0 ) ∈ C ⊥ . Here and hereafter, the orthogonal complement space is always defined in the sense of duality. Thus, (−1 , 0 ) ∈ (Hs−1 ) N × (Hs ) N . Taking (0 , 1 ) as the initial data, we solve the adjoint problem (16.3) and get the solution  ≡ 0. Multiplying  on both sides of the backward problem (16.5) and integrating by parts, we get V (0), 1 (H1−s ) N ;(Hs−1 ) N − V  (0), 0 (H−s ) N ;(Hs ) N T (D H, )ddt. = 0

(16.18)

1

The right-hand side of (16.18) is meaningful due to N  N  1  ∈ C 0 ([0, T ]; Hs ) ⊂ L 2 (0, T ; L 2 (1 )) , s > . 2

(16.19)

Noticing (V (0), V  (0)) ∈ C and (−1 , 0 ) ∈ C ⊥ , it is easy to see from (16.18) that for any given H in L M , we have

16.2 Equivalence Between the Approximate Boundary Null …



T 0

205

1

(D H, )ddt = 0.

Then, it follows that D T  ≡ 0 on [0, T ] × 1 .

(16.20)

But  ≡ 0, which implies that the adjoint problem (16.3) is not D-observable on [0, T ]. Inversely, assume that the adjoint problem (16.3) is not D-observable on [0, T ], then there exists a nontrivial initial data (0 , 1 ) ∈ (Hs ) N × (Hs−1 ) N , such that the solution  to the corresponding adjoint problem (16.3) satisfies (16.20). For any 1 ) ∈ C, there exists a sequence {Hn } of boundary controls in L M , such 0 , U given (U that the solution Vn to the corresponding backward problem (16.5) satisfies 0 , U 1 ) (Vn (0), Vn (0)) → (U

in (H1−s ) N × (H−s ) N

(16.21)

as n → +∞. Similarly to (16.18), multiplying  on both sides of the backward problem (16.5) and noting (16.20), we get Vn (0), 1 (H1−s ) N ;(Hs−1 ) N − Vn (0), 0 (H−s ) N ;(Hs ) N = 0.

(16.22)

Then, taking n → +∞ and noting (16.21), it follows from (16.22) that 1 ), (−1 , 0 ) (H1−s ) N ×(H−s ) N ;(Hs−1 ) N ×(Hs ) N = 0 0 , U (U

(16.23)

0 , U 1 ) ∈ C, which indicates that (−1 , 0 ) ∈ C ⊥ , thus C = (H1−s ) N × for all (U  (H−s ) N . 1 ) ∈ (H1 ) N × (H0 ) N , system (II) 0 , U Theorem 16.6 If for any given initial data (U is approximately null controllable for some s (> 21 ), then for any given initial data 0 , U 1 ) ∈ (H1−s ) N × (H−s ) N , system (II) possesses the same approximate bound(U ary null controllability, too. 1 ) ∈ (H1−s ) N × (H−s ) N (s > 1 ), by the den0 , U Proof For any given initial data (U 2 N N N 0n , U 1n )}n∈N sity of (H1 ) × (H0 ) in (H1−s ) × (H−s ) N , we can find a sequence {(U N N in (H1 ) × (H0 ) , satisfying 1n ) → (U 0 , U 1 ) in (H1−s ) N × (H−s ) N 0n , U (U

(16.24)

as n → +∞. By the assumption, for any fixed n  1, there exists a sequence {Hkn }k∈N of boundary controls in L M with compact support in [0, T ], such that the sequence of solutions {Ukn } to the corresponding problem (II) and (II0) satisfies (Ukn (T ), (Ukn ) (T )) → (0, 0)

in

(H1−s ) N × (H−s ) N

(16.25)

206

16 Approximate Boundary Null Controllability

as k → +∞. For any given n  1, let kn be an integer such that 1n , Hkn )(T ) (H1−s ) N ×(H−s ) N 0n , U R(U n

(16.26)

= (Uknn (T ), (Uknn ) (T )) (H1−s ) N ×(H−s ) N 

1 . 2n

Thus, we get a sequence {kn } with kn → +∞ as n → +∞. For the sequence {Hknn } of boundary controls in L M , we have 1n , Hkn )(T ) → 0 in (H1−s ) N × (H−s ) N 0n , U R(U n

(16.27)

as n → +∞. Therefore, by the linearity of R, the combination of (16.24) and (16.27) gives 1 , Hkn ) 0 , U R(U n n   1n , 0) + R(U 0n , U 1n , Hkn ) → (0, 0)  =R(U0 − U0 , U1 − U

(16.28)

n

in (H1−s ) N × (H−s ) N as n → +∞, which indicates that the sequence {Hknn } of boundary controls realizes the approximate boundary null controllability for any 1 ) ∈ (H1−s ) N × (H−s ) N . 0 , U  given initial data (U Remark 16.7 Since (H1 ) N × (H0 ) N ⊂ (H1−s ) N × (H−s ) N (s > 21 ) if system (II) 0 , U 1 ) ∈ (H1−s ) N × is approximately null controllable for any given initial data (U 1 N (H−s ) (s > 2 ), then it possesses the same approximate boundary null controllabil0 , U 1 ) ∈ (H1 ) N × (H0 ) N , too. ity for any given initial data (U Remark 16.8 Theorem 16.6 and Remark 16.7 indicate that for system (II), the approximate boundary null controllability for the initial data in (H1−s ) N × (H−s ) N (s > 21 ) is equivalent to the approximate boundary null controllability for the initial data in (H1 ) N × (H0 ) N with the same convergence space (H1−s ) N × (H−s ) N . Remark 16.9 In a similar way, we can prove that if for any given initial data 1 ) ∈ (H1−s ) N × (H−s ) N (s > 1 ), system (II) is approximately null control0 , U (U 2 0 , U 1 ) ∈ (H1−s  ) N × (H−s  ) N (s  > s), system lable, then for any given initial data (U (II) is still approximately null controllable. Corollary 16.10 If M = N , then, no matter whether the multiplier geometrical condition (12.3) is satisfied or not, system (II) is always approximately null controllable. Proof Since M = N , the observations given by (16.4) become  ≡ 0 on [0, T ] × 1 .

(16.29)

By Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]), we then get the Dobservability of the adjoint problem (16.3). Hence, by Theorem 16.5, we get the approximate boundary null controllability of system (II). 

16.3 Kalman’s Criterion. Total (Direct and Indirect) Controls

207

16.3 Kalman’s Criterion. Total (Direct and Indirect) Controls By Theorem 16.5, similarly to Theorem 8.9, we can give the following necessary condition for the approximate boundary null controllability. Theorem 16.11 If system (II) is approximately null controllable at the time T > 0, then we have necessarily the following Kalman’s criterion: rank(D, AD, · · · , A N −1 D) = N .

(16.30)

By Theorem 16.11, as in the case with Dirichlet boundary controls (cf. Sect. 8.3), for Neumann boundary controls, in Part IV, we should consider not only the number of direct boundary controls acting on 1 , which is equal to the rank of D, but also the number of total controls, given by the rank of the enlarged matrix (D, AD, · · · , A N −1 D), composed of the coupling matrix A and the boundary control matrix D, which should be equal to N in the case of approximate boundary null controllability. Therefore, the number of indirect controls should be equal to rank(D, AD, · · · , A N −1 D) – rank(D) = N − M. Generally speaking, similarly to the approximate boundary null controllability for a coupled system of wave equations with Dirichlet boundary controls, Kalman’s criterion is not sufficient in general. The reason is that Kalman’s criterion does not depend on T , then if it is sufficient, the approximate boundary null controllability of the original system (II), or the D-observability of the adjoint problem (16.3) could be immediately realized; however, this is impossible since the wave propagates with a finite speed. First of all, we will give an example to show the insufficiency of Kalman’s criterion. Theorem 16.12 Let μ2n and en be defined by ⎧ ⎨ −en = μ2n en in , on 0 , en = 0 ⎩ on 1 . ∂ν en = 0

(16.31)

 = {(m, n) : μn = μm , em = en on 1 }

(16.32)

Assume that the set

is not empty. For any given (m, n) ∈ , setting =

μ2m − μ2n , 2

(16.33)

208

16 Approximate Boundary Null Controllability

the adjoint problem ⎧  φ − φ + ψ = 0 ⎪ ⎪ ⎪ ⎨ψ  − ψ + φ = 0 ⎪ φ=ψ=0 ⎪ ⎪ ⎩ ∂ν φ = ∂ ν ψ = 0

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1

(16.34)

admits a nontrivial solution (φ, ψ) ≡ (0, 0) with the observation on the infinite time interval φ ≡ 0 on [0 + ∞) × 1 ,

(16.35)

hence the corresponding D-observability of the adjoint problem (16.34) fails Proof Let φ = (en − em ), ψ = (en + em ), λ2 =

μ2m + μ2n . 2

(16.36)

It is easy to check that (φ, ψ) satisfies the following system: ⎧ 2 λ φ + φ − ψ = 0 ⎪ ⎪ ⎪ ⎨λ2 ψ + ψ − φ = 0 ⎪ φ=ψ=0 ⎪ ⎪ ⎩ ∂ν φ = ∂ ν ψ = 0

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 .

(16.37)

Moreover, noting the definition (16.32) of , we have φ = 0 on 1 .

(16.38)

φλ = eiλt φ, ψλ = eiλt ψ.

(16.39)

Then, let

It is easy to see that (φλ , ψλ ) is a nontrivial solution to the adjoint system (16.34), which satisfies condition (16.35).  In order to illustrate the validity of the assumptions given in Theorem 16.12, we may examine the following situations, in which the set  is indeed not empty. (1)  = (0, π), 1 = {π}. In this case, we have 1 1 μn = n + , en = (−1)n sin(n + )x, en (π) = em (π) = 1. 2 2 Thus, (m, n) ∈  for all m = n.

(16.40)

16.3 Kalman’s Criterion. Total (Direct and Indirect) Controls

209

(2)  = (0, π) × (0, π), 1 = {π} × [0, π]. Let

μm,n =

1 1 (m + )2 + n 2 , em,n = (−1)m sin(m + )x sin ny. 2 2

(16.41)

We have em,n (π, y) = em  ,n (π, y) = sin ny, 0  y  π.

(16.42)

Thus, ({m, n}, {m  , n}) ∈  for all m = m  and n  1. Remark 16.13 Theorem 16.12 implies that Kalman’s criterion (16.30) is not sufficient in general. As a matter of fact, for the adjoint system (16.34) which satisfies the condition of observation (16.35), we have N = 2,  A=

0 , 0

D=

  1 10 , (D, AD) = , 0 0

(16.43)

and the corresponding Kalman’s criterion (16.30) is satisfied. Theorem 16.12 shows that Kalman’s criterion cannot guarantee the D-observability of the adjoint system (16.34) even the observation is given on the infinite time interval [0, +∞).

16.4 Sufficiency of Kalman’s Criterion for T > 0 Large Enough in the One-Space-Dimensional Case Similarly to the situation with Dirichlet boundary controls in the one-spacedimensional case, we will show that Kalman’s criterion (16.30) is also sufficient in certain cases for the approximate boundary null controllability of the original system. Now consider the following one-space-dimensional adjoint problem: ⎧   −  + A T  = 0, ⎪ ⎪ ⎪ ⎨(t, 0) = 0, ⎪ ∂ν (t, π) = 0, ⎪ ⎪ ⎩ t = 0 :  =  0 ,   = 1 ,

t > 0, 0 < x < π, t > 0, t > 0, 0 0 Large Enough …



 

=

μl m

(l)

αn(l,μ) eiβn t E n(l,μ) .

211

(16.54)

n∈Z l=1 μ=1

In particular, we have =

μl m (l,μ)

αn (l) n∈Z l=1 μ=1 iβn

(l)

eiβn t en w (l,μ) ,

(16.55)

and the condition of D-observation (16.45) leads to m

DT

μl (l,μ)  α n

μ=1

n∈Z l=1

iβn(l)

 (l) w (l,μ) eiβn t = 0 on [0, T ].

(16.56)

Now let us examine the sequence {βn(l) }1lm;n∈Z defined by (16.51): (1) (m) · · · β−1 < · · · < β−1 < β0(1) < · · · < β0(m) < β1(1) < · · · < β1(m) < · · · . (16.57)

First, for || > 0 small enough, the sequence {βn(l) }1lm;n∈Z is strictly increasing. On the other hand, for || > 0 small enough and for |n| > 0 large enough, similarly to (8.106) and (8.107), we check easily that (l) − βn(l) = O(1) βn+1

and

     βn(l+1) − βn(l) = O   . n

(16.58)

(16.59)

Thus, the sequence {βn(l) }1lm;n∈Z satisfies all the assumptions given in Theorem 8.22 (in which s = 1). Moreover, by definition given in (8.93), a computation shows (l) D + = m. Then the sequence {eiβn t }1lm;n∈Z is ω-linearly independent in L 2 (0, T ), provided that T > 2mπ. Theorem 16.14 Assume that A and D satisfy Kalman’s criterion (16.30). Assume furthermore that A T is diagonalizable with (16.46)–(16.47). Then the adjoint problem (16.44) is D-observable for || > 0 small enough, provided that T > 2mπ. (l)

Proof Since the sequence {eiβn t }1lm;n∈Z is ω-linearly independent in L 2 (0, T ) as T > 2mπ, it follows from (16.56) that D

T

μl (l,μ)  α n

μ=1

iβn(l)

 w (l,μ) = 0, 1  l  m, n ∈ Z.

(16.60)

212

16 Approximate Boundary Null Controllability

Noting Kalman’s criterion (16.30), by Proposition 2.12 (ii) with d = 0, Ker(D T ) does not contain any nontrivial invariant subspace of A T , then it follows that μl (l,μ)

αn μ=1

iβn(l)

w (l,μ) = 0, 1  l  m, n ∈ Z.

(16.61)

Then αn(l,μ) = 0, 1  μ  μl , 1  l  m, n ∈ Z,

(16.62)

hence,  ≡ 0, namely, the adjoint problem (16.44) is D-observable.



Moreover, similarly to Theorem 8.25, we have Theorem 16.15 Under the assumptions of Theorem 16.14, assume furthermore that A T possesses N distinct real eigenvalues: δ1 < δ2 < · · · < δ N .

(16.63)

Then the adjoint problem (16.44) is D-observable for || > 0 small enough, provided that T > 2π(N − M + 1), where M = rank(D).

16.5 Unique Continuation for a Cascade System of Two Wave Equations Consider the following problem for the cascade system of two wave equations: ⎧  ⎪ ⎪φ − φ = 0 ⎪ ⎪ ⎪ ⎪ψ  − ψ + φ = 0 ⎨ φ=ψ=0 ⎪ ⎪ ⎪ ∂ν φ = ∂ ν ψ = 0 ⎪ ⎪ ⎪ ⎩t = 0 : (φ, ψ, φ , ψ  ) = (φ , ψ , φ , ψ ) 0 0 1 1

in (0, T ) × , in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in 

(16.64)

with the Dirichlet observation aφ + bψ = 0 on [0, T ] × 1 , where a and b are constants. Let A=

  01 a and D = . 00 b

(16.65)

(16.66)

16.5 Unique Continuation for a Cascade System of Two Wave Equations

213

It is easy to see that Kalman’s criterion (16.30) holds if and only if b = 0. Moreover, we have (cf. [2]) Theorem 16.16 Let b = 0. If the solution (φ, ψ) to adjoint problem (16.64) with the initial data (φ0 , ψ0 , φ1 , ψ1 ) ∈ H10 () × H10 () × L 2 () × L 2 () satisfies the Dirichlet observation (16.65), then φ ≡ ψ ≡ 0, provided that T > 0 is large enough. Theorem 16.16 can be generalized to the following system of n blocks of cascade systems of two wave equations: ⎧  φ − φ j = 0 ⎪ ⎪ ⎪ j ⎨ ψ j − ψ j + φ j = 0 ⎪ φj = ψj = 0 ⎪ ⎪ ⎩ ∂ν φ j = ∂ν ψ j = 0

in (0, T ) × , in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1

(16.67)

for j = 1, . . . , n with the Dirichlet observations n

(a ji φ j + b ji ψ j ) = 0 on [0, T ] × 1 , i = 1, · · · , n.

(16.68)

j=1

Let N = 2n and

and let

and

 = (φ1 , ψ1 , · · · φn , ψn )T , ⎞ ⎛ 01 ⎟ ⎜ 00 ⎟ ⎜ ⎟ ⎜ . .. A=⎜ ⎟ ⎜  ⎟ ⎝ 01 ⎠ 00 ⎛

a11 ⎜b11 ⎜ ⎜ D = ⎜ ... ⎜ ⎝an1 bn1

⎞ a12 · · · a1n b12 · · · b1n ⎟ ⎟ .. .. ⎟ . . ··· . ⎟ ⎟ an2 · · · ann ⎠ bn2 · · · bnn

(16.69)

(16.70)

(16.71)

Setting (16.67) and (16.68) into the forms of (16.44) and (16.45), a straightforward computation shows that Kalman’s criterion (16.30) holds if and only if the matrix B = (bi j ) is invertible. Moreover, we have the following Corollary 16.17 Assume that the matrix B = (bi j ) is invertible. If the solution {(φi , ψi )} to system (16.67) with (φi0 , ψi0 ) ∈ H10 × L 2 () for i = 1, . . . , n satisfies the observations (16.68), then φi ≡ ψi ≡ 0 for i = 1, . . . , n, provided that T > 0 is large enough.

214

16 Approximate Boundary Null Controllability

Proof For i = 1, · · · , n, multiplying the first j-th equation of (16.67) by a ji and the second j-th equation of (16.67) by b ji , respectively, then summing up for j from 1 to n, we get N N N

 (a ji φ j + b ji ψ j ) − (a ji φ j + b ji ψ j ) + b ji φ j = 0 j=1

j=1

(16.72)

j=1

for i = 1, · · · , n. Let ui =

N

b ji φ j , vi =

j=1

N

(a ji φ j + b ji ψ j ), i = 1, · · · , n.

(16.73)

j=1

It follows from (16.72) that for each i = 1, · · · , n, we have ⎧  u i − u i = 0 ⎪ ⎪ ⎪ ⎨v  − v + u = 0 i i i ⎪ = v = 0 u i i ⎪ ⎪ ⎩ ∂ν u i = ∂ν vi = 0

in (0, T ) × , in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1

(16.74)

with the Dirichlet observations vi ≡ 0 on [0, T ] × 1 .

(16.75)

Applying Theorem 16.16 to each sub-system (16.74) with (16.75), it follows that u i ≡ vi ≡ 0 in [0, T ] × , i = 1, · · · , n.

(16.76)

Finally, since the matrix B is invertible, it follows from (16.76) that φi ≡ ψi ≡ 0 in [0, T ] × , i = 1, · · · , n. The proof is thus complete.

(16.77) 

Chapter 17

Approximate Boundary Synchronization

The approximate boundary synchronization is defined and studied in this chapter for system (II) with Neumann boundary controls.

17.1 Definition Definition 17.1 Let s > 21 . System (II) is approximately synchronizable at the 0 , U 1 ) ∈ (H1−s ) N × (H−s ) N , there exists time T > 0 if for any given initial data (U M a sequence {Hn } of boundary controls in L with compact support in [0, T ], such that the sequence {Un } of solutions to problem (II) and (II0) satisfies (l) u (k) n − u n → 0 as n → +∞

(17.1)

for 1  k, l  N in the space 0 1 ([T, +∞); H1−s ) ∩ Cloc ([T, +∞); H−s ). Cloc

(17.2)

Let C1 be the synchronization matrix of order (N − 1) × N , defined by ⎞ 1 −1 ⎟ ⎜ 1 −1 ⎟ ⎜ C1 = ⎜ ⎟. . . .. .. ⎠ ⎝ 1 −1 ⎛

(17.3)

C1 is a full row-rank matrix, and Ker(C1 ) = Span{e1 } © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_17

(17.4) 215

216

17 Approximate Boundary Synchronization

with e1 = (1, · · · , 1)T .

(17.5)

Obviously, the approximate boundary synchronization (17.1) can be equivalently rewritten as (17.6) C1 Un → 0 as n → +∞ in the space 0 1 ([T, +∞); (H1−s ) N −1 ) ∩ Cloc ([T, +∞); (H−s ) N −1 ). Cloc

(17.7)

17.2 Condition of C1 -Compatibility Similarly to Theorem 9.2, we have Theorem 17.2 Assume that system (II) is approximately synchronizable at the time T > 0, but not approximately null controllable. Then the coupling matrix A = (ai j ) should satisfy the following row-sum condition: N

ai j := a (i = 1, · · · , N ),

(17.8)

j=1

where a is a constant independent of i = 1, · · · , N . This condition is called the condition of C1 -compatibility. Remark 17.3 The condition of C1 -compatibility (17.8) for the approximate boundary synchronization is just the same as that given in (13.6) for the exact boundary synchronization. In particular, Lemma 13.3 remains valid in the present situation.

17.3 Fundamental Properties Under the condition of C1 -compatibility (17.8), setting W1 = (w (1) , · · · , w (N −1) )T = C1 U

(17.9)

and noting (13.11), the original problem (II) and (II0) for U can be transformed into following reduced system for W1 : ⎧ ⎨ W1 − W1 + A1 W1 = 0 in (0, +∞) × , on (0, +∞) × 0 , W =0 ⎩ 1 on (0, +∞) × 1 ∂ν W 1 = C 1 D H

(17.10)

17.3 Fundamental Properties

217

with the initial data t =0:

0 , W1 = C1 U 1 in . W1 = C 1 U

(17.11)

Accordingly, let 1 = (ψ (1) , · · · , ψ (N −1) )T .

(17.12)

Consider the following adjoint problem of the reduced system (17.10): ⎧ T 1 − 1 + A1 1 = 0 ⎪ ⎪ ⎪ ⎨ 1 = 0 ⎪ ∂ν 1 = 0 ⎪ ⎪ ⎩ 0 , 1 =  1 t = 0 : 1 = 

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in ,

(17.13)

which is called the reduced adjoint problem of system (II). From Definition 16.1 and Definition 17.1, we immediately get Lemma 17.4 Assume that the coupling matrix A satisfies the condition of C1 -compatibility (17.8). Then system (II) is approximately synchronizable at the time T > 0 if and only if the reduced system given by (17.10) is approximately null controllable at the time T > 0, or equivalently if and only if the reduced adjoint problem (17.13) is C1 D-observable on the time interval [0, T ] (cf. Definition 16.2). Corollary 17.5 Assume that the condition of C1 -compatibility (17.8) holds. If rank(C1 D) = N − 1, then system (II) is always approximately synchronizable. Proof Since (C1 D)T is an invertible matrix of order (N − 1), the condition of observation similarly given in Definition 16.2 becomes 1 ≡ 0 on [0, T ] × 1 .

(17.14)

Thus, by means of Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]), we get the C1 D-observability of the reduced adjoint problem (17.13) on [0, T ], then by Lemma 17.4, the approximate boundary synchronization of system (II).  Similarly to Lemma 9.5, we have Lemma 17.6 Under the condition of C1 -compatibility (17.8), if system (II) is approximately synchronizable at T > 0, then we have the following criterion of Kalman’s type: rank(C1 D, C1 AD, · · · , C1 A N −1 D) = N − 1.

(17.15)

218

17 Approximate Boundary Synchronization

17.4 Properties Related to the Number of Total Controls Since the rank of the enlarged matrix (D, AD, · · · , A N −1 D) measures the number of total (direct and indirect) controls, we now determine the minimal number of total controls, which is necessary to the approximate boundary synchronization of system (II), no matter whether the condition of C1 -compatibility (17.8) is satisfied or not. Similarly to Theorem 9.6, we have Theorem 17.7 Assume that system (II) is approximately synchronizable under the action of a boundary control matrix D. Then we necessarily have rank(D, AD, · · · , A N −1 D)  N − 1.

(17.16)

In other words, at least (N − 1) total controls are needed in order to realize the approximate boundary synchronization of system (II). When system (II) is approximately synchronizable under the minimal rank condition (17.17) rank(D, AD, · · · , A N −1 D) = N − 1, the coupling matrix A should possess some fundamental properties related to the synchronization matrix C1 . Similarly to Theorem 9.7, we have Theorem 17.8 Assume that system (II) is approximately synchronizable under the minimal rank condition (17.17). Then we have the following assertions: (i) The coupling matrix A satisfies the condition of C1 -compatibility (17.8). (ii) There exists a scalar function u as the approximately synchronizable state, such that (17.18) u (k) n → u as n → +∞ for all 1  k  n in the space 0 1 ([T, +∞); H1−s ) ∩ Cloc ([T, +∞); H−s ) Cloc

(17.19)

with s > 1/2. Moreover, the approximately synchronizable state u is independent of the sequence {Hn } of applied boundary controls. (iii) The transpose A T of the coupling matrix A admits an eigenvector E 1 such that (E 1 , e1 ) = 1, where e1 = (1, · · · , 1)T is the eigenvector of A, associated with the eigenvalue a given by (17.8). Remark 17.9 Under the hypothesis of Theorem 17.8, system (II) is approximately synchronizable in the pinning sense, while that originally given by Definition 17.1 is in the consensus sense. On the other hand, similarly to Theorem 9.12, we have

17.4 Properties Related to the Number of Total Controls

219

Theorem 17.10 Let A satisfy the condition of C1 -compatibility (17.8). Assume that A T admits an eigenvector E 1 such that (E 1 , e1 ) = 1 with e1 = (1, · · · , 1)T . Then there exists a boundary control matrix D satisfying the minimal rank condition (17.17), which realizes the approximate boundary synchronization of system (II). Moreover, the approximately synchronizable state u is independent of applied boundary controls.

17.5 An Example In this subsection, we will examine the approximate boundary synchronization for a coupled system of three wave equations, the reduced adjoint system of which is given by (16.64). For this purpose, we first look for all the matrices A of order 3, such that C1 A = A¯ 1 C1 . More precisely, let N = 3,

    01 1 −1 0 , C1 = . A¯ 1 = 00 0 1 −1

(17.20)

For getting A, we solve the linear system:   0 1 −1 C1 A = A¯ 1 C1 = . 00 0 Noting that the matrix

⎛ ⎞ 0 1 −1 A0 = ⎝0 0 0 ⎠ 00 0

(17.21)

(17.22)

satisfies (17.21), it is easy to see that ⎛ ⎞ ⎛ ⎞ αβγ α β+1 γ−1 γ ⎠, A = A0 + ⎝α β γ ⎠ = ⎝α β αβγ α β γ

(17.23)

where α, β, δ are arbitrarily given real numbers. Par construction, A satisfies the condition of C1 -compatibility (17.8) with the row-sum a = α + β + δ. Proposition 17.11 In the case α + β + γ = 0

(17.24)

α + β + γ = 0 and α = 0,

(17.25)

or

220

17 Approximate Boundary Synchronization

λ = α + β + γ is an eigenvalue of A associated with the eigenvector e1 = (1, 1, 1)T , and the corresponding eigenvector E 1 of A T can be chosen such that (E 1 , e1 ) = 1. While, in the case α + β + γ = 0 but α = 0, (17.26) λ = 0 is the only eigenvalue of A, and all the corresponding eigenvectors E 1 of A T necessarily satisfy (E 1 , e1 ) = 0. Proof First, a straightforward computation gives det (λI − A) = λ3 − (α + β + γ)λ2 .

(17.27)

(i) If α + β + γ = 0, then λ = α + β + γ is a simple eigenvalue of A with the eigenvector e1 . So, A T admits an eigenvector E 1 such that (E 1 , e1 ) = 1. More precisely, the vector E 1 is given by ⎛ ⎞ aα 1 ⎝ E 1 = 2 aβ + α⎠ with a = α + β + γ. a aγ − α

(17.28)

(ii) If α + β + γ = 0 and α = 0, then λ = 0 is the eigenvalue of A with multiplicity 3 and dim Ker (A) = 2. Moreover, the eigenvector E 1 of A T given by ⎛ ⎞ −2β 1⎝ β + 1⎠ E1 = 2 β+1

(17.29)

satisfies (E 1 , e1 ) = 1. (iii) If α + β + γ = 0 and α = 0, then λ = 0 is also the eigenvalue of A with multiplicity 3 and dim Ker (A) = 1. Then, the eigenvector E 1 of A T given by ⎛

⎞ 0 E1 = ⎝ 1 ⎠ −1

(17.30)

necessarily satisfies (E 1 , e1 ) = 0. The proof is complete.



Now let us consider the corresponding problem (II) and (II0) with the boundary control matrix D = (d1 , d2 , d3 )T and the coupling matrix A given by (17.23): ⎧  ⎪ ⎪ ⎪u − u + αu + (β + 1)v + (γ − 1)w = 0 ⎪  ⎪ ⎪ ⎨v − v + αu + βv + γw = 0 w  − w + αu + βv + γw = 0 ⎪ ⎪ ⎪u = v = w = 0 ⎪ ⎪ ⎪ ⎩∂ u = d h, ∂ v = d h, ∂ w = d h ν 1 ν 2 ν 3

in (0, +∞) × , in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1

(17.31)

17.5 An Example

221

with the initial data t =0:

(u, v, w) = (u 0 , v0 , w0 ), (u  , v  , w  ) = (u 1 , v1 , w1 ) in .

(17.32)

Since A satisfies the condition of C1 -compatibility, the reduced system (17.10) with A¯ 1 given by (17.20) is written as ⎧  y − y + z = 0 ⎪ ⎪ ⎪ ⎨z  − z = 0 ⎪ y=z=0 ⎪ ⎪ ⎩ ∂ν y = (d1 − d2 )h, ∂ν z = (d2 − d3 )h

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 ,

(17.33)

and the corresponding reduced adjoint system (17.13) becomes ⎧  ψ1 − ψ1 = 0 ⎪ ⎪ ⎪ ⎨ψ  − ψ + ψ = 0 2 1 2 ⎪ ψ1 = ψ2 = 0 ⎪ ⎪ ⎩ ∂ν ψ 1 = ∂ν ψ 2 = 0

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 .

(17.34)

Moreover, the C1 D-observation becomes (d1 − d2 )ψ1 + (d2 − d3 )ψ2 = 0 on [0, T ] × 1 .

(17.35)

Theorem 17.12 Let  satisfy the usual multiplier geometrical condition. Then, there exists a boundary control matrix D = (d1 , d2 , d3 )T such that system (17.31) is approximately synchronizable. Moreover, the approximately synchronizable state u is independent of applied boundary controls in two cases (17.24) and (17.25). Proof By Lemma 17.4, the approximate boundary synchronization of system (17.31) is equivalent to the C1 D-observability of the reduced adjoint system (17.34). By Theorem 16.16, under the multiplier geometrical condition, the reduced adjoint system (17.34), together with the C1 D-observation (17.35), has only the trivial solution ψ1 ≡ ψ2 ≡ 0, provided that d2 − d3 = 0 and that T > 0 is large enough. We get thus the approximate boundary synchronization of system (17.31). More precisely, in case (17.24), we can choose ⎧ γ 1 ⎪ ⎨d1 = σ − α , d2 = 0, d3 = 1, d1 = 0, d2 = 1, d3 = − βγ , ⎪ ⎩ d1 = 0, d2 = − βγ , d3 = 1,

if α = 0, if α = 0 and γ = 0, if α = 0 and β = 0,

(17.36)

while, in case (17.25), we can choose d1 = 0, d2 = −1, d3 = 1,

(17.37)

222

17 Approximate Boundary Synchronization

such that E 1T D = 0 in both cases, where E 1 is given by (17.28) and (17.29), respectively. Then, applying E 1 to system (17.31), and setting φ = (E 1 , Un ) with Un = (u n , vn , wn )T , it follows that ⎧  φ − φ + aφ = 0 ⎪ ⎪ ⎪ ⎨φ = 0 ⎪ ∂ν φ = 0, ⎪ ⎪ ⎩ 0 ), φ = (E 1 , U 1 ) t = 0 : φ = (E 1 , U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in .

(17.38)

Since φ is independent of applied boundary controls, and (E 1 , e1 ) = 1, the convergence (17.39) Un → ue1 as n → +∞ in the space C 0 ([T, +∞); (H0 )3 ) ∩ C 1 ([T, +∞); (H−1 )3 )

(17.40)

will imply that tT :

φ = (E 1 , Un ) → (E 1 , e1 )u = u.

(17.41)

Therefore, the approximately synchronizable state u is indeed independent of applied boundary controls h.  Remark 17.13 In two cases (17.24) and (17.25), under the minimal rank condition rank(D, AD, AD 2 ) = 2,

(17.42)

the approximately synchronizable state u is independent of applied boundary controls. While, in case (17.26), A is similar to a Jordan block of order 3, we must have the rank condition: (17.43) rank(D, AD, AD 2 ) = 3, which is necessary for the approximate boundary null controllability of the nilpotent system (17.31). We hope, but we do not know up to now if it is also sufficient.

Chapter 18

Approximate Boundary Synchronization by p-Groups

The approximate boundary synchronization by p-groups is introduced and studied in this chapter for system (II) with Neumann boundary controls.

18.1 Definition When rank(D, AD, · · · , A N −1 D), the number of total controls, is further reduced, we can consider the approximate boundary synchronization by p-groups ( p  1). In the special case p = 1, it is just the approximate boundary synchronization considered in Chap. 17. Let p  1 be an integer and let 0 = n0 < n1 < n2 < · · · < n p = N

(18.1)

be integers such that n r − n r −1  2 for all 1  r  p. The approximate boundary synchronization by p-groups means that the components of U are divided into p groups: (u (1) , · · · , u (n 1 ) ), (u (n 1 +1) , · · · , u (n 2 ) ), · · · , (u (n p−1 +1) , · · · , u (n p ) ),

(18.2)

and each group possesses the corresponding approximate boundary synchronization, respectively. Definition 18.1 Let s > 21 . System (II) is approximately synchronizable by 0 , U 1 ) ∈ (H1−s ) N × (H−s ) N , there p-groups at T > 0 if for any given initial data (U M exists a sequence {Hn } of boundary controls in L with compact support in [0, T ], such that the corresponding sequence {Un } of solutions to problem (II) and (II0) satisfies © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_18

223

224

18 Approximate Boundary Synchronization by p-Groups (l) 0 1 u (k) n − u n → 0 in Cloc ([T, +∞); H1−s ) ∩ Cloc ([T, +∞); H−s )

(18.3)

as n → +∞ for all n r −1 + 1  k, l  n r and 1  r  p. Let ⎞ 1 −1 ⎟ ⎜ 1 −1 ⎟ ⎜ Sr = ⎜ ⎟, 1  r  p .. .. ⎠ ⎝ . . 1 −1 ⎛

(18.4)

be an (n r − n r −1 − 1) × (n r − n r −1 ) matrix with full row-rank, and ⎛ ⎜ ⎜ Cp = ⎜ ⎝

⎞

S1 S2

..

⎟ ⎟ ⎟ ⎠

.

(18.5)

Sp be the (N − p) × N matrix of synchronization by p-groups. Clearly, Ker(C p ) = Span{e1 , · · · , e p },

(18.6)

where (er )i =

1, n r −1 + 1  i  n r , 0, otherwise

(18.7)

for 1  r  p. Thus, the approximate boundary synchronization by p-groups (18.3) can be written as C p Un → 0 as n → +∞

(18.8)

0 1 ([T, +∞); (H1−s ) N − p ) ∩ Cloc ([T, +∞); (H−s ) N − p ). Cloc

(18.9)

in the space

18.2 Fundamental Properties Similarly to Lemma 13.3, the coupling matrix A satisfies the condition of C p -compatibility if Ker(C p ) is an invariant subspace of A: AKer(C p ) ⊆ Ker(C p ),

(18.10)

18.2 Fundamental Properties

225

or equivalently, there exists a unique matrix A p of order (N − p), such that C p A = A pC p.

(18.11)

A p is called the reduced matrix of A by C p . Under the condition of C p -compatibility (18.10), setting W p = C p U = (w (1) , · · · , w (N − p) )T

(18.12)

and noting (18.11), from problem (II) and (II0) for U we get the following reduced system for W p : ⎧  ⎨ W p − W p + A p W p = 0 in (0, +∞) × , on (0, +∞) × 0 , W =0 ⎩ p on (0, +∞) × 1 ∂ν W p = C p D H

(18.13)

with the initial data t =0:

0 , W p = C p U 1 in . W p = C pU

(18.14)

Accordingly, let  p = (ψ (1) , · · · , ψ (N − p) )T .

(18.15)

Consider the following reduced adjoint problem: ⎧ T  p −  p + A p  p = 0 ⎪ ⎪ ⎪ ⎨ p = 0 ⎪ ∂ ⎪ νp = 0 ⎪ ⎩  p0 ,  p =   p1 t = 0 : p = 

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in .

(18.16)

By Definitions 16.1 and 18.1, and using Theorem 16.5, we get Lemma 18.2 Assume that the coupling matrix A satisfies the condition of C p -compatibility (18.10). Then system (II) is approximately synchronizable by pgroups at the time T > 0 if and only if the reduced system (18.13) is approximately null controllable at the time T > 0, or equivalently, the reduced adjoint problem (18.16) is C p D-observable on the time interval [0, T ], namely, the partial observation (C p D)T ∂ν  p ≡ 0 on [0, T ] × 1  p0 =   p1 ≡ 0, then  p ≡ 0. implies 

(18.17)

226

18 Approximate Boundary Synchronization by p-Groups

Thus, by Theorem 16.11 and noting (18.11), it is easy to get Theorem 18.3 Under the condition of C p -compatibility (18.10), assume that system (II) is approximately synchronizable by p-groups, we necessarily have the following criterion of Kalman’s type: rank(C p D, C p AD, · · · , C p A N −1 D) = N − p.

(18.18)

18.3 Properties Related to the Number of Total Controls We now consider rank(D, AD, · · · , A N −1 D), the number of total (direct and indirect) controls, and determine the minimal number of total controls, which is necessary to the approximate boundary synchronization by p-groups for system (II), no matter whether the condition of C p -compatibility (18.10) is satisfied or not. Similarly to Theorem 10.4, we have Theorem 18.4 Assume that system (II) is approximately synchronizable by p-groups under the action of a boundary control matrix D. Then, we necessarily have rank(D, AD, · · · , A N −1 D)  N − p.

(18.19)

In other words, at least (N − p) total controls are needed in order to realize the approximate boundary synchronization by p-groups for system (II). Based on Theorem 18.4, we consider the approximate boundary synchronization by p-groups for system (II) under the minimal rank condition rank(D, AD, · · · , A N −1 D) = N − p.

(18.20)

In this case, the coupling matrix A should possess some fundamental properties related to C p , the matrix of synchronization by p-groups. Similarly to Theorem 10.5, we have Theorem 18.5 Assume that system (II) is approximately synchronizable by p-groups under the minimal rank condition (18.20). Then we have the following assertions: (i) The coupling matrix A satisfies the condition of C p -compatibility (18.10). (ii) There exists some linearly independent scalar functions u 1 , · · · , u p such that u (k) n → u r as n → +∞

(18.21)

for all nr −1 + 1  k  n r and 1  r  p in the space 0 1 ([T, +∞); H1−s ) ∩ Cloc ([T, +∞); H−s ) Cloc

(18.22)

18.3 Properties Related to the Number of Total Controls

227

with s > 1/2. Moreover, the approximately synchronizable state by p-groups u = (u 1 , · · · , u p )T is independent of applied boundary controls Hn . (iii) A T admits an invariant subspace Span{E 1 , · · · , E p }, which is contained in Ker(D T ) and bi-orthonormal to Ker(C p )=Span{e1 , · · · , e p }. Remark 18.6 Under hypothesis (18.20), system (II) is approximately synchronizable by p-groups in the pinning sense, while that originally given by Definition 18.1 is in the consensus sense. Remark 18.7 Since the invariant subspace Span{E 1 , · · · , E p } of A T is bi-orthonormal to the invariant subspace Span{e1 , · · · , e p } of A, by Proposition 2.8, the invariant subspace Span{E 1 , · · · , E p }⊥ of A is a supplement of Span{e1 , · · · , e p } Moreover, similarly to Theorem 10.10, we have Theorem 18.8 Let A satisfy the condition of C p -compatibility (18.10). Assume that Ker(C p ) admits a supplement which is also invariant for A. Then there exists a boundary control matrix D satisfying the minimal rank condition (18.20), which realizes the approximate boundary synchronization by p-groups for system (II) in the pinning sense.

Part V

Synchronization for a Coupled System of Wave Equations with Coupled Robin Boundary Controls: Exact Boundary Synchronization We consider the following coupled system of wave equations with coupled Robin boundary controls: ⎧  ⎨ U − U + AU = 0 U =0 ⎩ ∂ν U + BU = D H

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1

(III)

with the initial condition t =0:

1 in , 0 , U  = U U =U

(III0)

where  ⊂ Rn is a bounded domain with smooth boundary  = 1 ∪  0 such that 2  1 ∩  0 = ∅ and mes(1 ) > 0; “” stands for the time derivative;  = nk=1 ∂∂x 2 is  T T k  the Laplacian operator; U = u (1) , · · · , u (N ) and H = h (1) , · · · , h (M) (M  N ) denote the state variables and the boundary controls, respectively; the internal coupling matrix A = (ai j ) and the boundary coupling matrix B are of order N, and D as the boundary control matrix is a full column-rank matrix of order N × M, all with constant elements. The exact boundary synchronization and the exact boundary synchronization by groups for system (III) will be discussed in this part, while, the approximate boundary synchronization and the approximate boundary synchronization by groups for system (III) will be considered in the next part (Part VI).

Chapter 19

Preliminaries on Problem (III) and (III0)

In order to consider the exact boundary controllability and the exact boundary synchronization of system (III), we first give some necessary results on problem (III) and (III0) in this chapter.

In order to consider the exact boundary controllability and the exact boundary synchronization of system (III), we first give some necessary results on problem (III) and (III0) in this chapter.

19.1 Regularity of Solutions with Neumann Boundary Conditions Similarly to the problem of wave equations with Neumann boundary conditions, a problem with Robin boundary conditions no longer enjoys the hidden regularity as in the case with Dirichlet boundary conditions. As a result, the solution to problem (III) and (III0) with coupled Robin boundary conditions is not smooth enough in general for the proof of the non-exact boundary controllability of the system. In order to overcome this difficulty, we should deeply study the regularity of solutions to wave equations with Neumann boundary conditions. For this purpose, consider the following second-order hyperbolic problem on a bounded domain  ⊂ R n (n  2) with boundary : ⎧  ⎪ in (0, T ) ×  = Q, ⎨ y + A(x, ∂)y = f ∂y = g on (0, T ) ×  = , ∂ν ⎪ ⎩ A  t = 0 : y = y0 , y = y1 in ,

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_19

(19.1)

231

232

19 Preliminaries on Problem (III) and (III0)

where n 

 ∂2 ∂ ai j (x) + bi (x) + c0 (x), A(x, ∂) = − ∂x ∂x ∂x i j i i, j=1 i=1 n

(19.2)

in which ai j (x) with ai j (x) = a ji (x), b j (x) and c0 (x) (i, j = 1, · · · , n) are smooth real coefficients, and the principal part of A(x, ∂) is supposed to be uniformly strong elliptic in : n n   ai j (x)ηi η j  c η 2j (19.3) i, j=1

j=1

for all x ∈  and for all η = (η1 , · · · , ηn ) ∈ Rn , where c > 0 is a positive constant; ∂y is the outward normal derivative associated with A: moreover, ∂ν A  ∂y ∂y = ai j (x) νj, ∂ν A ∂x i i=1 j=1 N

N

(19.4)

ν = (ν1 , · · · , νn )T being the unit outward normal vector on the boundary . Define the operator A by  A = A(x, ∂), D(A) = y ∈ H 2 () :

 ∂y = 0 on  . ∂ν A

(19.5)

We know that if −A is closed, and if −A is maximal dissipative, namely, if −A is dissipative: Re(Au, u)  0, ∀u ∈ D(A), and it does not possess any proper dissipative extension, then for any given α with 0 < α < 1, the fractional power Aα can be defined in a natural way (cf. [6, 24, 70]), for example, we have Aα u =

sin πα π



∞

λα−1 A(λ + A)−1 u dλ, u ∈ D(A).

(19.6)

0

We can verify that −A given by (19.5) is closed, and there exists a constant c > 0 so large that −(cI + A) is maximal dissipative, thus we can define the fractional powers of (cI + A). Since a suitable translation of the operator does not change the regularity of the solution in [0, T ] with T < +∞, for any α with 0 < α < 1, the fractional powers Aα can be well defined, moreover, we have y D(Aα ) = Aα y L 2 () .

(19.7)

In [30] (cf. also [29, 31]), Lasiecka and Triggiani got the optimal regularity for the solution to problem (19.1) by means of the theory of cosine operator. In particular, more regularity results can be obtained when the domain is a parallelepiped. For conciseness and clarity, we list only those results which are needed in what follows.

19.1 Regularity of Solutions with Neumann Boundary Conditions

233

Let  > 0 be an arbitrarily given small number. Here and hereafter, we always assume that α and β are given, respectively, as follows: ⎧ α = 3/5 − , β = 3/5,  is a smooth bounded domain ⎪ ⎪ ⎪ ⎨ and A(x, ∂) is defined by (19.2), ⎪ α = β = 2/3,  is a sphere and A(x, ∂) = −, ⎪ ⎪ ⎩ α = β = 3/4 − ,  is a parallelepiped and A(x, ∂) = −.

(19.8)

Lemma 19.1 Assume that y0 ≡ y1 ≡ 0 and f ≡ 0. For any given g ∈ L 2 (0, T ; L 2 ()), problem (19.1) admits a unique solution y such that (y, y  ) ∈ C 0 ([0, T ]; H α () × H α−1 ())

(19.9)

and y| ∈ H 2α−1 () = L 2 (0, T ; H 2α−1 ()) ∩ H 2α−1 (0, T ; L 2 ()),

(19.10)

where H α () denotes the usual Sobolev space of order α and  = (0, T ) × . Lemma 19.2 Assume that y0 ≡y1 ≡ 0 and g ≡ 0. For any given f ∈L 2 (0, T ; L 2 ()), the unique solution y to problem (19.1) satisfies

and

(y, y  ) ∈ C 0 ([0, T ]; H 1 () × L 2 ())

(19.11)

y| ∈ H β ().

(19.12)

Lemma 19.3 Assume that f ≡ 0 and g ≡ 0. (1) If (y0 , y1 ) ∈ H 1 () × L 2 (), then problem (19.1) admits a unique solution y such that (19.13) (y, y  ) ∈ C 0 ([0, T ]; H 1 () × L 2 ()) and

y| ∈ H β ().

(19.14)

(2) If (y0 , y1 ) ∈ L 2 () × (H 1 ()) , where (H 1 ()) denotes the dual space of H () with respect to L 2 (), then problem (19.1) admits a unique solution y such that (19.15) (y, y  ) ∈ C 0 ([0, T ]; L 2 () × (H 1 ()) ) 1

and

y| ∈ H α−1 ().

(19.16)

234

19 Preliminaries on Problem (III) and (III0)

Remark 19.4 The regularities given in Lemmas 19.1, 19.2, and 19.3 remain true when we replace the Neumann boundary condition on the whole boundary  by the homogeneous Dirichlet boundary condition on 0 and the Neumann boundary condition on 1 with 0 ∪ 1 =  and  0 ∩  1 = ∅. Remark 19.5 In the results mentioned above, the mappings from the given data to the solution are all continuous with respect to the corresponding topologies.

19.2 Well-Posedness of a Coupled System of Wave Equations with Coupled Robin Boundary Conditions We now prove the well-posedness of problem (III) and (III0). Throughout this part and the next part, when mes(0 ) > 0, we define H0 = L 2 (), H1 = H10 (),

(19.17)

in which H10 () is the subspace of H 1 (), composed of all the functions with the null trace on 0 , while, when mes(0 ) = 0, instead of (19.17), we define

 H0 = u : u ∈ L 2 (), ud x = 0 , H1 = H 1 () ∩ H0 . (19.18) 

For simplifying the statement, we often assume that mes(0 ) = 0; however, all the conclusions are still valid when mes(0 ) = 0. Let  = (φ(1) , · · · , φ(N ) )T . We first consider the following adjoint system: ⎧  T ⎪ ⎨ −  + A  = 0 in (0, +∞) × , =0 on (0, +∞) × 0 , ⎪ ⎩ T on (0, +∞) × 1 ∂ν  + B  = 0 with the initial data t =0:

1 in , 0 ,  =  =

(19.19)

(19.20)

where A T and B T denote the transposes of A and B, respectively. Theorem 19.6 Assume that B is similar to a real symmetric matrix. Then for any 0 ,  1 ) ∈ (H1 ) N × (H0 ) N , the adjoint problem (19.19), (19.20) admits a given ( unique weak solution 0 ([0, +∞); (H1 ) N × (H0 ) N ) (,  ) ∈ Cloc

in the sense of C0 -semigroup, where H1 and H0 are defined by (19.17).

(19.21)

19.2 Well-Posedness of a Coupled System of Wave Equations …

235

Proof Without loss of generality, we assume that B is a real symmetric matrix. We first formulate system (19.19) into the following variational form: 

)d x + ( , 



d x +

∇, ∇ 

1

)d + (, B 



)d x = 0 (, A

(19.22) ∈ (H1 ) N , where (·, ·) denotes the inner product of R N , for any given test function  while ·, · denotes the inner product of M N ×N (R). Recalling the following interpolation inequality [66]: 1

|φ|2 d  cφ H 1 () φ L 2 () , ∀φ ∈ H 1 (),

we have

1

(, B)d  B

1

||2 d  cB(H1 ) N (H0 ) N ,

then it is easy to see that



∇, ∇d x +

1

(, B)d + λ2(H0 ) N  c 2(H1 ) N

for some suitable constants λ > 0 and c > 0. Therefore, the bilinear symmetric form 

d x +

∇, ∇ 

1

)d (, B 

is coercive in (H1 ) N × (H0 ) N . Moreover, the nonsymmetric part in (19.22) satisfies 

)d x  A(H0 ) N  (H0 ) N . (, A

By Theorem 1.1 of Chap. 8 in [60], the variational problem (19.22) with the initial data (19.20) admits a unique solution  with the smoothness (19.21). The proof is complete.  Remark 19.7 From now on, in order to guarantee the well-posedness of problem (III) and (III0), we always assume that B is similar to a real symmetric matrix. Definition 19.8 U is a weak solution to problem (III) and (III0) if 0 1 U ∈ Cloc ([0, +∞); (H0 ) N ) ∩ Cloc ([0, +∞); (H−1 ) N ),

(19.23)

0 ,  1 ) ∈ (H1 ) N × where H−1 denotes the dual space of H1 , such that for any given ( N (H0 ) and for all given t  0, we have

236

19 Preliminaries on Problem (III) and (III0)

(U  (t), −U (t)), ((t),  (t)) t (D H (τ ), (τ ))d xdt, =

(U1 , −U0 ),(0 , 1 ) +

(19.24)

1

0

in which (t) is the solution to the adjoint problem (19.19) and (19.20), and

·, · denotes the duality between the spaces (H−1 ) N × (H0 ) N and (H1 ) N × (H0 ) N . Theorem 19.9 Assume that B is similar to a real symmetric matrix. For any given 2 0 , U 1 ) ∈ (H0 ) N × (H−1 ) N , problem (III) and H ∈ L loc (0, +∞; (L 2 (1 )) M ) and (U (III0) admits a unique weak solution U . Moreover, the mapping 1 , H ) → (U, U  ) 0 , U (U

(19.25)

is continuous with respect to the corresponding topologies. Proof Let  be the solution to the adjoint problem (19.19) and (19.20). Define a linear functional as follows: 0 ,  1 ) L t ( 0 ), ( 1 , −U 0 ,  1 ) + =

(U

t 0

1

(19.26) (D H (τ ), (τ ))d xdt.

Clearly, L t is bounded in (H1 ) N × (H0 ) N . Let St be the semigroup in (H1 ) N × (H0 ) N , corresponding to the adjoint problem (19.19), (19.20). L t ◦ St−1 is bounded in (H1 ) N × (H0 ) N . Then, by Riesz–Fréchet representation theorem, for any given 1 ) ∈ (H1 ) N ×(H0 ) N , there exists a unique (U  (t), −U (t))∈(H−1 ) N × (H0 ) N , 0 ,  ( such that L t ◦ St−1 ((t),  (t)) =

(U  (t), −U (t)), ((t),  (t)).

(19.27)

By 0 ,  1 ) L t ◦ St−1 ((t),  (t)) = L t (

(19.28)

0 ,  1 ) ∈ (H1 ) N × (H0 ) N , (19.24) holds, then (U, U  ) is the unique for any given ( weak solution to problem (III) and (III0). Moreover, we have (U  (t), −U (t))(H−1 ) N ×(H0 ) N = L t ◦ St−1  1 )(H0 ) N ×(H−1 ) N + H  L 2 (0,T ;(L 2 (1 )) M ) ) 0 , U  c((U

(19.29)

for all t ∈ [0, T ]. At last, by a classic argument of density, we obtain the regularity desired by (19.23). 

19.3 Regularity of Solutions to Problem (III) and (III0)

237

19.3 Regularity of Solutions to Problem (III) and (III0) We have Theorem 19.10 Assume that B is similar to a real symmetric matrix. For any 0 , U 1 ) ∈ (H1 ) N × (H0 ) N , the given H ∈ L 2 (0, T ; (L 2 (1 )) M ) and any given (U weak solution U to problem (III) and (III0) satisfies (U, U  ) ∈ C 0 ([0, T ]; (H α ()) N × (H α−1 ()) N )

(19.30)

U |1 ∈ (H 2α−1 (1 )) N ,

(19.31)

and where 1 = (0, T ) × 1 and α is defined by (19.8). Moreover, the linear mapping 1 , H ) → (U, U  ) 0 , U (U is continuous with respect to the corresponding topologies. Proof Noting (19.13) and (19.14) in Lemma 19.3, it is easy to see that we need only 0 ≡ U 1 ≡ 0. Assume that  is sufficiently smooth, for to consider the case that U 3 example, with C boundary, there exists a function h ∈ C 2 (), such that ∇h = ν on 1 ,

(19.32)

where ν is the unit outward normal vector on the boundary 1 (cf. [62]). Let λ be an eigenvalue of B T and let e be the corresponding eigenvector: B T e = λe. Defining φ = (e, U ), we have

Let

⎧  φ − φ = −(e, AU ) ⎪ ⎪ ⎨ φ=0 ∂ν φ + λφ = (e, D H ) ⎪ ⎪ ⎩ t = 0 : φ = 0, φ = 0 ψ = eλh φ.

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in .

(19.33)

(19.34)

(19.35)

Problem (III) and (III0) can be rewritten to the following problem with Neumann boundary conditions:

238

19 Preliminaries on Problem (III) and (III0)

⎧  λh ⎪ ⎪ ψ − ψ + b(ψ) = −e (e, AU ) ⎨ ψ=0 ∂ν ψ = eλh (e, D H ) ⎪ ⎪ ⎩ t = 0 : ψ = 0, ψ  = 0

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in ,

(19.36)

where b(ψ) = 2λ∇h · ∇ψ + λ(h − λ|∇h|2 )ψ is a first-order linear form of ψ with smooth coefficients. By Theorem 19.9, U ∈ C 0 ([0, T ]; (H0 ) N ). By (19.11) in Lemma 19.2 and Remark 19.4, the solution ψ to the following problem with homogeneous Neumann boundary conditions: ⎧  λh ⎪ ⎪ ψ − ψ + b(ψ) = −e (e, AU ) ⎨ ψ=0 ∂ν ψ = 0 ⎪ ⎪ ⎩ t = 0 : ψ = 0, ψ  = 0 satisfies

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in 

(ψ, ψ  ) ∈ C 0 ([0, T ]; H 1 () × L 2 ()).

(19.37)

(19.38)

Next, we consider the following problem with inhomogeneous Neumann boundary conditions but without internal force terms: ⎧  ψ − ψ + b(ψ) = 0 ⎪ ⎪ ⎨ ψ=0 ∂ ⎪ ν ψ = eλh (e, D H ) ⎪ ⎩ t = 0 : ψ = 0, ψ  = 0

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in .

(19.39)

By (19.9) and (19.10) in Lemma 19.1 and Remark 19.4, we have (ψ, ψ  ) ∈ C 0 ([0, T ]; H α () × H α−1 ())

(19.40)

ψ|1 ∈ H 2α−1 (1 ),

(19.41)

and where α is given by (19.8). Since this regularity result holds for all the eigenvectors of B T , and all the eigenvectors of B T constitute a set of basis in R N , we get the desired (19.30) and (19.31). 

Chapter 20

Exact Boundary Controllability and Non-exact Boundary Controllability

In this chapter, we will study the exact boundary controllability and the non-exact boundary controllability for the coupled system (III) of wave equations with coupled Robin boundary controls. In this chapter, we will study the exact boundary controllability and the non-exact boundary controllability for the coupled system (III) of wave equations with coupled Robin boundary controls. We will prove that when the number of boundary controls is equal to N , the number of state variables, system (III) is exactly controllable for 1 ) ∈ (H1 ) N × (H0 ) N , while, when M < N and  is a 0 , U any given initial data (U parallelepiped, system (III) is not exactly boundary controllable in (H1 ) N × (H0 ) N .

20.1 Exact Boundary Controllability Definition 20.1 System (III) is exactly null controllable in the space (H1 ) N × 1 , U 0 ) ∈ (H0 ) N , if there exists a positive constant T > 0, such that for any given (U N N 2 2 M (H1 ) × (H0 ) , there exists a boundary control H ∈ L (0, T ; (L (1 )) ), such that problem (III) and (III0) admits a unique weak solution U satisfying the final condition (20.1) t = T : U = U  = 0. Theorem 20.2 Assume that M = N . Assume furthermore that B is similar to a real symmetric matrix. Then there exists a time T > 0, such that for any given initial data 1 ) ∈ (H1 ) N ×(H0 ) N , there exists a boundary control H ∈L 2 (0, T ; (L 2 (1 )) N ), 0 , U (U such that system (III) is exactly null controllable at the time T and the boundary control continuously depends on the initial data: 0 , U 1 )(H1 ) N ×(H0 ) N , H  L 2 (0,T ;(L 2 (1 )) N )  c(U

(20.2)

where c > 0 is a positive constant. © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_20

239

240

20 Exact Boundary Controllability and Non-exact Boundary Controllability

Proof We first consider the corresponding problem (II) and (II0). By Theorem 12.14, 1 ) ∈ (H1 ) N × (H0 ) N , there exists a boundary con0 , U for any given initial data (U 2 2  ∈ L loc (0, +∞; (L (1 )) N ) with compact support in [0, T ], such that system trol H (II) with Neumann boundary controls is exactly controllable at the time T , and the  continuously depends on the initial data: boundary control H 0 , U 1 )(H1 ) N ×(H0 ) N ,  L 2 (0,T ;(L 2 (1 )) N )  c1 (U H

(20.3)

where c1 > 0 is a positive constant. Noting that M = N , D is invertible and the boundary condition in system (II)  on (0, T ) × 1 ∂ν U = D H

(20.4)

 + D −1 BU ) := D H on (0, T ) × 1 . ∂ν U + BU = D( H

(20.5)

can be rewritten as

Thus, problem (II) and (II0) with (20.4) can be equivalently regarded as problem (III) and (III0) with (20.5). In other words, the boundary control H is given by  + D −1 BU on (0, T ) × 1 , H=H

(20.6)

where U is the solution to problem (II) and (II0) with (20.4), realizes the exact boundary controllability of system (III). It remains to check that H given in (20.6) belongs to the control space L 2 (0, T ; 2 (L (1 )) N ) with continuous dependence (20.2). By the regularity result given in Theorem 19.10 (in which we take B = 0), the trace U |1 ∈ (H 2α−1 (1 )) N , where α is defined by (19.8). Since 2α − 1 > 0, we have H ∈ L 2 (0, T ; (L 2 (1 )) N ). Moreover, still by Theorem 19.9, we have   1 )(H1 ) N ×(H0 ) N +  H  L 2 (0,T ;(L 2 (1 )) N ) , 0 , U U (L 2 (1 )) N  c2 (U

(20.7)

where c2 > 0 is another positive constant. Noting (20.6), (20.2) follows from (20.3) and (20.7). The proof is complete. 

20.2 Non-exact Boundary Controllability Differently from the cases with Neumann boundary controls, the non-exact boundary controllability for a coupled system with coupled Robin boundary controls in a general domain is still an open problem. Fortunately, for some special domain, the solutions to problem (III) and (III0) may possess higher regularity. In particular, when  is a parallelepiped, the optimal regularity of trace U |1 almost reaches

20.2 Non-exact Boundary Controllability

241

H 1/2 (1 ). This benefits a lot in the proof of the non-exact boundary controllability with fewer boundary controls. Lemma 20.3 Let L be a compact linear mapping from L 2 () to L 2 (0, T ; L 2 ()), and let R be a compact linear mapping from L 2 () to L 2 (0, T ; H 1−α (1 )), where α is defined by (19.8). Then we cannot find a T > 0, such that for any given θ ∈ L 2 (), the solution to the following problem: ⎧  w − w = Lθ ⎪ ⎪ ⎨ w=0 ∂ ⎪ ν w = Rθ ⎪ ⎩ t = 0 : w = 0, w = θ satisfies the final condition

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in 

w(T ) = w (T ) = 0.

(20.8)

(20.9)

Proof For any given θ ∈ L 2 (), let φ be the solution to the following problem: ⎧  φ − φ = 0 ⎪ ⎪ ⎨ φ=0 ∂ ⎪ νφ = 0 ⎪ ⎩ t = 0 : φ = θ, φ = 0

in (0, T ) × , on (0, T ) × 0 , on (0, T ) × 1 , in .

(20.10)

By (19.13) and (19.14) in Lemma 19.3, we have φ L 2 (0,T ;L 2 ())  cθ L 2 ()

(20.11)

φ L 2 (0,T ;H α−1 (1 ))  cθ L 2 () .

(20.12)

and Noting (20.9) and taking the inner product with φ on both sides of (20.8) and integrating by parts, it is easy to get θ2L 2 () =

0

T





T

Lθφd x + 0

1

Rθφd.

(20.13)

Noting (20.11)–(20.12), we then have θ L 2 ()  c(Lθ L 2 (0,T ;L 2 ()) + Rθ L 2 (0,T ;H 1−α ()) ) for all θ ∈ L 2 (), which contradicts the compactness of L and R.

(20.14) 

Remark 20.4 When  is a parallelepiped with 0 = ∅, by Fourier analysis, the solution to the adjoint problem (19.19) and (19.20) is sufficiently smooth. Therefore, Lemma 20.3 is still valid.

242

20 Exact Boundary Controllability and Non-exact Boundary Controllability

Theorem 20.5 Assume that rank(D) = M < N and that B is similar to a real symmetric matrix. Assume furthermore that  ⊂ Rn is a parallelepiped with 0 = ∅. Then, no matter how large T > 0 is, system (III) is not exactly null controllable for 1 ) ∈ (H1 ) N × (H0 ) N . 0 , U any given initial data (U Proof Since M < N , there exists an e ∈ R N , such that D T e = 0. Take the special initial data (20.15) t = 0 : U = 0, U  = eθ for system (III). If the system is exactly controllable for any given θ ∈ L 2 () at the time T > 0, then there exists a boundary control H ∈ L 2 (0, T ; (L 2 ()) M ), such that the corresponding solution satisfies U (T ) = U  (T ) = 0.

(20.16)

Let w = (e, U ), Lθ = −(e, AU ), Rθ = −(e, BU )| .

(20.17)

Noting that D T e = 0, we see that w satisfies problem (20.8) and the final condition (20.9). By Lemma 20.3, in order to show Theorem 20.5, it suffices to show that the linear mapping L is compact from L 2 () to L 2 (0, T ; L 2 ()), and R is compact from L 2 () to H 1−α (). Assume that system (III) with special initial data (20.15) is exactly null controllable. The linear mapping θ → H is continuous from L 2 () to L 2 (0, T ; (L 2 ()) M ). By Theorem 19.10, (θ, H ) → (U, U  ) is a continuous mapping from L 2 () × L 2 (0, T ; (L 2 ()) M ) to C 0 ([0, T ]; (H α ()) N ) ∩ C 1 ([0, T ]; (H α−1 ()) N ). Besides, by Lions’ compact embedding theorem (cf. Theorem 5.1 in [63]), the embedding L 2 (0, T ; (H α ()) N ) ∩ H 1 (0, T ; (H α−1 ()) N )} ⊂ L 2 (0, T ; (L 2 ()) N ) is compact, hence the linear mapping L is compact from L 2 () to L 2 (0, T ; L 2 ()). On the other hand, by (19.31), H → U | is a continuous mapping from L 2 (0, T ; 2 (L ()) M ) to (H 2α−1 ()) N , then R : θ → −(e, BU )| is a continuous mapping from L 2 () into H 2α−1 (1 ). When  is a parallelepiped, α = 3/4 − , then 2α − 1 > 1 − α, hence H 2α−1 () → H 1−α () is a compact embedding; therefore, R is  a compact mapping from L 2 () to H 1−α (). The proof is complete. Remark 20.6 We obtain the non-exact boundary controllability for system (III) with coupled Robin boundary controls in a parallelepiped  when there is a lack of boundary controls. The main idea is to use the compact perturbation theory which has a higher requirement on the regularity of the solution to the original problem with coupled Robin boundary condition. How to generalize this result to the general domain is still an open problem.

Chapter 21

Exact Boundary Synchronization

Based on the results of the exact boundary controllability and the non-exact boundary controllability, we study the exact boundary synchronization for system (III) with coupled Robin boundary controls.

21.1 Exact Boundary Synchronization Based on the results of the exact boundary controllability and the non-exact boundary controllability, we continue to study the exact boundary synchronization for system (III) under coupled Robin boundary controls. Let ⎞ 1 −1 ⎟ ⎜ 1 −1 ⎟ ⎜ C1 = ⎜ ⎟ .. .. ⎠ ⎝ . . 1 −1 (N −1)×N ⎛

(21.1)

be the corresponding full row-rank matrix of synchronization. We have Ker(C1 ) = Span{e1 },

(21.2)

where e1 = (1, · · · , 1)T . Definition 21.1 System (III) is exactly synchronizable at the time T > 0 if for any 0 , U 1 ) ∈ (H1 ) N × (H0 ) N , there exists a boundary control H ∈ given initial data (U 2 2 (L loc (0, +∞; L (1 ))) M with compact support in [0, T ], such that the corresponding solution U = U (t, x) to the mixed problem (III) and (III0) satisfies tT :

C1 U ≡ 0 in .

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_21

(21.3) 243

244

21 Exact Boundary Synchronization

Theorem 21.2 Let  ⊂ Rn be a parallelepiped. Assume that the coupled system (III) of wave equations with coupled Robin boundary controls is exactly boundary synchronizable. Then we have rank(C1 D) = N − 1.

(21.4)

Proof If K er (D T ) ∩ I m(C1T ) = {0}, then, by Proposition 2.11, we get rank(C1 D) = rank(D T C1T ) = rank(C1T ) = N − 1.

(21.5)

Otherwise, if K er (D T ) ∩ I m(C1T ) = {0}, then there exists a vector E = 0, such that D T C1T E = 0.

(21.6)

Let w = (E, C1 U ), Lθ = −(E, C1 AU ), Rθ = −(E, C1 BU ).

(21.7)

1 ) = (0, θe1 ), where θ ∈ L 2 (), we can still get 0 , U For any given initial data (U problem (20.8) for w. By the exact boundary synchronization of system (III), for this initial data, there exists a boundary control H ∈ L 2 (0, T, (L 2 (1 )) M ), such that (21.3) holds, then the corresponding solution w satisfies (20.9). Similarly to the proof of Theorem 20.5, when  is a parallelepiped, we can prove that L is a compact linear mapping from L 2 () to L 2 (0, T ; L 2 ()), and R is a compact linear mapping from L 2 () to L 2 (0, T ; H 1−α (1 )). It contradicts Lemma 20.3. The proof is complete. 

21.2 Conditions of C1 -Compatibility Remark 21.3 If rank(D) = N , by Theorem 20.2, system (III) is exactly null controllable, then exactly synchronizable. In order to exclude this trivial situation, in what follows, we always assume that rank(D) = N − 1. Theorem 21.4 Let  ⊂ Rn be a parallelepiped. Assume that rank(D) = N − 1 and that B is similar to a real symmetric matrix. If the coupled system (III) of wave equations with coupled Robin boundary controls is exactly boundary synchronizable, then we have the following conditions of C1 -compatibility: AK er (C1 ) ⊆ K er (C1 ) and B K er (C1 ) ⊆ K er (C1 ).

(21.8)

Proof Let e1 = (1, · · · , 1)T . Then, we have tT :

U = ue1 ,

where u is the corresponding exactly synchronizable state.

(21.9)

21.2 Conditions of C1 -Compatibility

245

Taking the inner product with C1 on both sides of (III), we get tT :

C1 Ae1 u = 0 in 

and

C1 Be1 u = 0 on 1 .

(21.10)

By Theorem 20.5, system (III) is not exactly boundary controllable when rank(D) = 1 ) such that 0 , U N − 1, hence there exists at least an initial data (U tT :

u ≡ 0 in .

(21.11)

Therefore, we get C1 Ae1 = 0. Noting that Ker(C1 ) = Span{e1 }, we have AKer(C1 ) ⊆ Ker(C1 ).

(21.12)

We next prove C1 Be1 = 0. Otherwise, by the second formula in (21.10), we get u|1 ≡ 0. By (21.9), it is easy to see that as t  T , system (III) becomes tT :

⎧ ⎨ u − u + au = 0 in (0, T ) × , u=0 on (0, T ) × 0 , ⎩ on (0, T ) × 1 , ∂ν u = u = 0

(21.13)

where a is defined by (17.8). Therefore, by Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]), we have u ≡ 0 as t  T . It contradicts the non-exact boundary controllability of system (III). The proof is then complete.  Theorem 21.5 Assume that rank(D) = N − 1 and that B is similar to a real symmetric matrix. Assume furthermore that both A and B satisfy the conditions of C1 -compatibility (21.8). Then there exists a boundary control matrix D with rank(C1 D) = N − 1, such that system (III) is exactly synchronizable, and the boundary control H possesses the following continuous dependence: 0 , U 1 ) (H1 ) N −1 ×(H0 ) N −1 .

H L 2 (0,T,(L 2 (1 )) N −1 )  c C1 (U

(21.14)

Proof Since both A and B satisfy the conditions of C1 -compatibility (21.8), by Proposition 2.15, there exist matrices A1 and B 1 of order (N − 1), such that C 1 A = A1 C 1

and

C1 B = B 1 C1 .

(21.15)

W = C1U

and

D 1 = C1 D.

(21.16)

Let

W satisfies

⎧ ⎨ W − W + A1 W = 0 in (0, T ) × , W =0 on (0, T ) × 0 , ⎩ ∂ν W + B 1 W = D 1 H on (0, T ) × 1

(21.17)

246

21 Exact Boundary Synchronization

with the initial data t =0:

0 , W = C1 U 1 in . W = C1U

(21.18)

0 , U 1 ) ∈ Noting that C1 is a surjection from R N to R N −1 , any given initial data (U 0 , C1 U 1 ) of the reduced (H1 ) N × (H0 ) N corresponds to a unique initial data (C1 U system (21.17). Then, it follows that the exact boundary synchronization for system (III) is equivalent to the exact boundary null controllability for the reduced system (21.17), and then the boundary control H , which realizes the exact boundary null controllability for the reduced system (21.17), is just the boundary control which realizes the exact boundary synchronization for system (III). By Proposition 2.21, when B is similar to a real symmetric matrix, its reduced matrix B 1 is also similar to a real symmetric matrix, which guarantees the wellposedness of the reduced problem (21.17) and (21.18). Defining the boundary control matrix D by D = C1T , D 1 = C1 D = C1 C1T

(21.19)

is an invertible matrix of order (N − 1), by Theorem 20.2, the reduced system (21.17) is exactly boundary controllable, then system (III) is exactly synchronizable. Moreover, by (20.2), we get (21.14). 

Chapter 22

Determination of Exactly Synchronizable States

When system (III) possesses the exact boundary synchronization, the corresponding exactly synchronizable states will be studied in this chapter. Although under certain conditions, the exact boundary synchronization can be realized by fewer boundary controls; however, exactly synchronizable states are a priori unknown, which depend not only on the given initial data, but also on applied boundary controls in general. In this section, we will discuss the determination and the estimate of exactly synchronizable states.

22.1 Determination of Exactly Synchronizable States Theorem 22.1 Assume that both A and B satisfy the conditions of C1 -compatibility (21.8) and that B is similar to a real symmetric matrix. Assume furthermore that A T and B T have a common eigenvector E 1 ∈ R N , such that (E 1 , e1 ) = 1, where e1 = (1, · · · , 1)T . Then there exists a boundary control matrix D with rank(D) = N − 1, such that system (III) is exactly synchronizable, and the exactly synchronizable state is independent of applied boundary controls. Proof Taking the boundary control matrix D such that Ker(D T ) = Span{E 1 },

(22.1)

it is evident that rank(D) = N − 1. Noticing that (E 1 , e1 ) = 1, we have Ker(C1 ) ∩ Im(D) = Span(e1 ) ∩ {Span(E 1 )}⊥ = {0}. © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_22

(22.2) 247

248

22 Determination of Exactly Synchronizable States

Thus, by Proposition 2.15, we have rank(C1 D) = rank(D) = N − 1.

(22.3)

Then, by Theorem 21.5, system (III) is exactly synchronizable. Next, we prove that the exactly synchronizable state is independent of boundary controls which realize the exact boundary synchronization. Noting that E 1 is a common eigenvector of A T and B T , and B is similar to a real symmetric matrix, there exist λ ∈ C and μ ∈ R, such that A T E 1 = λE 1 and B T E 1 = μE 1 .

(22.4)

Noting (22.1), we have D T E 1 = 0, then φ = (E 1 , U ) satisfies ⎧  φ − φ + λφ = 0 ⎪ ⎪ ⎨ φ=0 ∂ ⎪ ν φ + μφ = 0 ⎪ ⎩ 0 ), φ = (E 1 , U 1 ) t = 0 : φ = (E 1 , U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in .

(22.5)

Obviously, the solution φ to this problem is independent of boundary controls H . On the other hand, noting that tT :

φ = (E 1 , U ) = (E 1 , e1 )u = u,

(22.6)

the exactly synchronizable state is determined by the solution to problem (22.5), and is independent of boundary controls H . The proof is complete.  Remark 22.2 By Proposition 6.12, a matrix D satisfying rank(C1 D) = rank(D) = N − 1 has the following form: D = C1T D1 + e1 D0 , where D1 is an invertible matrix of order (N − 1) and D0 is a matrix of order 1 × (N − 1). Furthermore, if (E 1 , e1 ) = 1 and D T E 1 = 0, then E 1T C1T D1 + D0 = 0, it follows that D = (I − e1 E 1T )C1T D1 . Theorem 22.3 Assume that both A and B satisfy the conditions of C1 -compatibility (21.8). Assume furthermore that B is similar to a real symmetric matrix. Assume finally that system (III) is exactly synchronizable. Let E 1 = 0 be a vector in R N , such that the projection φ = (E 1 , U ) is independent of boundary controls H which

22.1 Determination of Exactly Synchronizable States

249

realize the exact boundary synchronization. Then E 1 must be a common eigenvector of A T and B T , E 1 ∈ Ker(D T ) and (E 1 , e1 ) = 0. 0 , U 1 ) = (0, 0) in problem (III) and (III0), by Theorem 19.10, the Proof Taking (U linear mapping F : H → (U, U  ) is continuous from L 2 (0, T ; (L 2 (1 )) M ) to C 0 ([0, T ]; (H α ()) N × (H α−1 ()) N ),  be the Fréchet derivative of U in the direction where α is defined by (19.8). Let U : of H .  = F  (0) H U

(22.7)

 satisfies the same problem as U : By linearity, U ⎧   − U  + AU =0 U ⎪ ⎪ ⎪ ⎨ =0 U  + BU  = DH  ⎪ ∂ν U ⎪ ⎪ ⎩    t =0: U =U =0

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 in .

(22.8)

Since the projection φ = (E 1 , U ) is independent of boundary controls H which realize the exact boundary synchronization, we have ) ≡ 0, ∀ H  ∈ L 2 (0, T ; (L 2 (1 )) M ). (E 1 , U

(22.9)

/ Im(C1T ). Otherwise, there exists a vector R ∈ R N −1 , such We first prove that E 1 ∈ T that E 1 = C1 R, then ) = 0. ) = (E 1 , U (R, C1 U

(22.10)

 is the solution to the reduced system (21.17) with the zero initial Noting that C1 U data, by the exact boundary synchronization of system (III), the reduced system  at the time T (21.17) has the exact boundary controllability, then the value of C1 U can be arbitrarily chosen, and as a result, from (22.10), we get R = 0, / Im(C1T ). Hence, noting (21.2), we which contradicts the fact that E 1 = 0, then E 1 ∈ can choose E 1 such that (E 1 , e1 ) = 1. Thus, {E 1 , C1T } constitutes a basis in R N , and hence there exist a λ ∈ C and a vector Q ∈ R N −1 , such that (22.11) A T E 1 = λE 1 + C1T Q.

250

22 Determination of Exactly Synchronizable States

Then by (22.11), taking the inner product with E 1 on both sides of the equations in (22.8), and noting (22.9), we get , A T E 1 ) = (U , C1T Q) = (C1 U , Q). , E 1 ) = (U 0 = (AU

(22.12)

Thus, by the exact boundary controllability for the reduced system (21.17), we can get Q = 0, then it follows from (22.11) that A T E 1 = λE 1 . On the other hand, by (22.10), taking the inner product with E 1 on both sides of the boundary condition on 1 in (22.8), we get ) = (E 1 , D H ) on 1 . (E 1 , B U

(22.13)

By Theorem 19.10, we have ) H 2α−1 ((0,T )×1 ) (E 1 , D H ) H 2α−1 ((0,T )×1 )  c H  L 2 (0,T ;(L 2 (1 )) M ) , =(E 1 , B U

(22.14)

here and hereafter, c denotes a positive constant.  = D T E1v We claim that D T E 1 = 0, namely, E 1 ∈ Ker(D T ). Otherwise, taking H in (22.14), we get a contradiction: v H 2α−1 (1 )  cv L 2 (0,T ;L 2 (1 )) , ∀v ∈ H 2α−1 (1 ), because of 2α − 1 > 0. Thus, we have

) = 0 on 1 . (E 1 , B U

(22.15)

(22.16)

Similarly, since {E 1 , C1T } constitutes a basis in R N , there exist a μ ∈ R and a vector P ∈ R N −1 , such that (22.17) B T E 1 = μE 1 + C1T P. Substituting it into (22.16) and noting (22.9), we have ) + (C1T P, U ) = (P, C1 U ) = 0, (μE 1 , U

(22.18)

then by the exact boundary controllability for the reduced system (21.17), we get P = 0, and hence we have B T E 1 = μE 1 . Thus, E 1 must be a common eigenvector of A T and B T . The proof is complete. 

22.2 Estimation of Exactly Synchronizable States

251

22.2 Estimation of Exactly Synchronizable States Theorem 22.4 Assume that both A and B satisfy the conditions of C1 -compatibility (21.8), namely, there exist real numbers a and b such that Ae1 = ae1 and Be1 = be1 with e1 = (1, · · · , 1)T .

(22.19)

Assume furthermore that B is similar to a real symmetric matrix. Then there exists a boundary control matrix D such that system (III) is exactly synchronizable, and each exactly synchronizable state u satisfies the following estimate: (u, u  )(T ) − (φ, φ )(T ) H α+1 ()×H α () 0 , U 1 )(H1 ) N −1 ×(H0 ) N −1 , cC1 (U

(22.20)

where α is defined by (19.8), and φ is the solution to the following problem: ⎧  φ − φ + aφ = 0 ⎪ ⎪ ⎨ φ=0 ∂ ⎪ ν φ + bφ = 0 ⎪ ⎩ 1 , E 1 ) 0 , E 1 ), φ = (U t = 0 : φ = (U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in .

(22.21)

Proof We first show that there exists an eigenvector E 1 ∈ R N of B T , associated with the eigenvalue b, satisfying (E 1 , e1 ) = 1. Let P be a matrix such that P B P −1 is a real symmetric matrix. Setting E 1 = P T Pe1 ,

(22.22)

B T E 1 =P T (P B P −1 )T Pe1 = P T P B P −1 Pe1

(22.23)

we have

=P P Be1 = b P Pe1 = bE 1 T

T

and (E 1 , e1 ) = (P T Pe1 , e1 ) = Pe1 2 > 0.

(22.24)

In particular, we can choose P such that (E 1 , e1 ) = 1. Next, define the boundary control matrix D such that K er (D T ) = Span{E 1 }.

(22.25)

Noting that Ker(C1 ) = Span{e1 } and (E 1 , e1 ) = 1, we have Ker(C1 ) ∩ Im(D) = {e1 } ∩ {E 1 }⊥ = {0},

(22.26)

252

22 Determination of Exactly Synchronizable States

then, by Proposition 2.11, we have rank(C1 D) = rank(D) = N − 1.

(22.27)

Thus, by Theorem 21.5, system (III) is exactly synchronizable. Taking the inner product with E 1 on both sides of problem (III) and (III0) and denoting ψ = (E 1 , U ), we get ⎧  ψ − ψ + aψ = (a E 1 − A T E 1 , U ) ⎪ ⎪ ⎨ ψ=0 ∂ ⎪ ν ψ + bψ = 0 ⎪ ⎩ 1 , E 1 ) 0 , E 1 ), ψ  = (U t = 0 : ψ = (U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , in .

(22.28)

Since

we have

(a E 1 − A T E 1 , e1 ) = (E 1 , ae1 − Ae1 ) = 0,

(22.29)

a E 1 − A T E 1 ∈ {Span(e1 )}⊥ = Im(C1T ).

(22.30)

Therefore, there exists a vector R ∈ R N −1 , such that a E 1 − A T E 1 = C1T R.

(22.31)

Let ϕ = ψ − φ, where φ is the solution to problem (22.21). By (22.21) and (22.28), we have ⎧  ϕ − ϕ + aϕ = (R, C1 U ) in (0, +∞) × , ⎪ ⎪ ⎨ ϕ=0 on (0, +∞) × 0 , (22.32) ϕ + bϕ = 0 on (0, +∞) × 1 , ∂ ⎪ ν ⎪ ⎩  in . t =0: ϕ=ϕ =0 Then, by the classic semigroups theory (cf. [70]), we have (ϕ, ϕ )(T ) H α+1 ()×H α ()  c1 (R, C1 U ) L 2 (0,T ;H α ()) ,

(22.33)

here and hereafter ci for i = 1, 2, · · · denote different positive constants. Recall that W = C1 U is the solution to the reduced problem (21.17)–(21.18). Noting (20.2), it follows from Theorem 19.10 that C1 U  L 2 (0,T ;(H α ()) N −1 0 , U 1 )(H1 ) N −1 ×(H0 ) N −1 + D 1 H  L 2 (0,T ;(L 2 (1 )) N −1 ) ) c2 (C1 (U 0 , U 1 )(H1 ) N −1 ×(H0 ) N −1 . c3 C1 (U

(22.34)

22.2 Estimation of Exactly Synchronizable States

253

Thus, we get 0 , U 1 )(H1 ) N −1 ×(H0 ) N −1 . (ϕ, ϕ )(T ) H α+1 ()×H α ()  c4 C1 (U

(22.35)

On the other hand, we have tT :

ψ = (E 1 , U ) = (E 1 , e1 )u = u,

thus ϕ = u − φ for t  T . Therefore, by (22.35) we get (22.20).

(22.36) 

Chapter 23

Exact Boundary Synchronization by p-Groups

The exact boundary synchronization by groups will be considered in this chapter for system (III) with further lack of coupled Robin boundary controls.

23.1 Definition When there is a further lack of boundary controls, similarly to the case with Dirichlet or Neumann boundary controls, we consider the exact boundary synchronization by p-groups for system (III). Let the components of U be divided into p groups: (u (1) , · · · , u (n 1 ) ), (u (n 1 +1) , · · · , u (n 2 ) ), · · · , (u (n p−1 +1) , · · · , u (n p ) ),

(23.1)

where 0 = n 0 < n 1 < n 2 < · · · < n p = N are integers such that n r − n r −1  2 for all r = 1, · · · , p. Definition 23.1 System (III) is exactly synchronizable by p-groups at the time 0 , U 1 ) ∈ (H1 ) N × (H0 ) N , there exists a boundT > 0 if for any given initial data (U 2 2 ary control H ∈ L loc (0, +∞; (L (1 )) M ) with compact support in [0, T ], such that the corresponding solution U = U (t, x) to problem (III) and (III0) satisfies tT :

u (i) = u r , n r −1 + 1  i  n r , 1  r  p,

(23.2)

where u = (u 1 , · · · , u p )T , being unknown a priori, is called the corresponding exactly synchronizable state by p-groups.

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_23

255

256

23 Exact Boundary Synchronization by p-Groups

Let Sr be an (n r − n r −1 − 1) × (n r − n r −1 ) full row-rank matrix: ⎞ 1 −1 ⎟ ⎜ 1 −1 ⎟ ⎜ Sr = ⎜ ⎟ , 1  r  p, . . .. .. ⎠ ⎝ 1 −1 ⎛

(23.3)

and let C p be the following (N − p) × N matrix of synchronization by p-groups: ⎛ ⎜ ⎜ Cp = ⎜ ⎝

⎞

S1 S2

..

⎟ ⎟ ⎟. ⎠

.

(23.4)

Sp Evidently, we have Ker(C p ) = Span{e1 , · · · , e p },

(23.5)

where for 1  r  p, (er )i =

1, n r −1 + 1  i  n r , 0, others.

Thus, the exact boundary synchronization by p-groups (23.2) can be equivalently written as tT :

C p U ≡ 0,

(23.6)

or tT :

U=

p

u r er .

(23.7)

r =1

23.2 Exact Boundary Synchronization by p-Groups Theorem 23.2 Let  ⊂ Rn be a parallelepiped. Assume that system (III) is exactly synchronizable by p-groups. Then, we have rank(C p D) = N − p.

(23.8)

23.2 Exact Boundary Synchronization by p-Groups

257

In particular, we have rank(D)  N − p.

(23.9)

Proof If K er (D T ) ∩ I m(C Tp ) = {0}, by Proposition 2.11, we have rank(C p D) = rank(D T C Tp ) = rank(C Tp ) = N − p.

(23.10)

Next, we prove that it is impossible to have K er (D T ) ∩ I m(C Tp ) = {0}. Otherwise, there exists a vector E = 0, such that D T C Tp E = 0.

(23.11)

Let w = (E, C p U ), Lθ = −(E, C p AU ), Rθ = −(E, C p BU ).

(23.12)

We get again problem (20.8) for w. Besides, the exact boundary synchronization by p-groups for system (III) indicates that the final condition (20.9) holds. Similarly to the proof of Theorem 21.2, we get a contradiction to Lemma 20.3.  Theorem 23.3 Let C p be the (N − p) × N matrix of synchronization by p-groups defined by (23.3)–(23.4). Assume that both A and B satisfy the following conditions of C p -compatibility: AK er (C p ) ⊆ K er (C p ), B K er (C p ) ⊆ K er (C p ).

(23.13)

Then there exists a boundary control matrix D satisfying rank(D) = rank(C p D) = N − p,

(23.14)

such that system (III) is exactly synchronizable by p-groups, and the corresponding boundary control H possesses the following continuous dependence: 0 , U 1 ) (H1 ) N − p ×(H0 ) N − p , H L 2 (0,T,(L 2 (1 )) N − p )  c C p (U

(23.15)

where c > 0 is a positive constant. Proof Since both A and B satisfy the conditions of C p -compatibility (23.13), by Proposition 2.15, there exist matrices A p and B p of order (N − p), such that C p A = A pC p, C p B = B pC p.

(23.16)

Let W = C p U,

D p = C p D.

(23.17)

258

We have

23 Exact Boundary Synchronization by p-Groups

⎧

⎨ W − W + A p W = 0 in (0, +∞) × , W =0 on (0, +∞) × 0 , ⎩ ∂ν W + B p W = D p H on (0, +∞) × 1

(23.18)

with the initial data: t =0:

0 , W = C p U 1 in . W = C pU

(23.19)

Noting that C p is a surjection from R N to R N − p , the exact boundary synchronization by p-groups for system (III) is equivalent to the exact boundary null controllability for the reduced system (23.18), and the boundary control H , which realizes the exact boundary null controllability for the reduced system (23.18), must be the boundary control which realizes the exact boundary synchronization by p-groups for system (III). Let D be defined by Ker(D T ) = Span{e1 , · · · , e p } = Ker(C p ).

(23.20)

We have rank(D) = N − P, and Ker(C p ) ∩ Im(D) = Ker(C p ) ∩ {Ker(C p )}⊥ = {0}.

(23.21)

By Proposition 2.11, we get rank(C p D) = rank(D) = N − p, thus D p is an invertible matrix of order (N − p). By Theorem 20.2, the reduced system (23.18) is exactly null controllable, then system (III) is exactly synchronizable by p-groups. By (20.2), we get (23.15). 

Chapter 24

Necessity of the Conditions of C p -Compatibility

In this chapter, we will discuss the necessity of the conditions of C p -compatibility for system (III) with coupled Robin boundary controls. This problem is closely related to the number of applied boundary controls.

In this chapter, we will discuss the necessity of the conditions of C p -compatibility. This problem is closely related to the number of applied boundary controls.

24.1 Condition of C p -Compatibility for the Internal Coupling Matrix We first study the condition of C p -compatibility for the coupling matrix A. Theorem 24.1 Let  ⊂ Rn be a parallelepiped. Assume that M = rank(D) = N − p. If system (III) is exactly synchronizable by p-groups, then the coupling matrix A = (ai j ) should satisfy the following condition of C p -compatibility: AKer(C p ) ⊆ Ker(C p ).

(24.1)

C p Aer = 0, 1  r  p.

(24.2)

Proof It suffices to prove that

By (23.7), taking the inner product with C p on both sides of the equations in system (III), we get p  C p Aer u r = 0 in . (24.3) tT : r =1

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_24

259

260

24 Necessity of the Conditions of C p -Compatibility

If (24.2) does not hold, then there exist constant coefficients αr (1  r  p), not all equal to zero, such that p  αr u r = 0 in . (24.4) r =1

Let c p+1 =

p  αr erT . er 2 r =1

(24.5)

Noting (er , es ) = er 2 δr s , we have tT :

c p+1 U =

p 

αr u r = 0 in .

(24.6)

r =1

Let p−1 = C



 Cp . c p+1

(24.7)

p−1 ) = / Im(C Tp ), then rank(C Noting (23.5) and (24.5), it is easy to see that c Tp+1 ∈ T T  N − p + 1. Since rank(D) = N − p, we have Ker(D ) ∩ Im(C p−1 ) = {0}, then there exists a vector E = 0, such that Tp−1 E = 0. DT C

(24.8)

Let p−1 AU ), Rθ = −(E, C p−1 BU ). p−1 U ), Lθ = −(E, C w = (E, C We get again problem (20.8) for w. Noting (23.6) and (24.6), we have p−1 U ) = (E, t = T : w(T ) = (E, C



 Cp U ) = 0. c p+1

(24.9)

Similarly, we have w  (T ) = 0, then (20.9) holds. Noting that  ⊂ Rn is a parallelepiped, similarly to the proof of Theorem 21.2, we can get a conclusion that contradicts Lemma 20.3.  Remark 24.2 The condition of C p -compatibility (24.1) is equivalent to the fact that there exist constants αr s (1  r, s  p) such that Aer =

p  s=1

αsr es 1  r  p,

(24.10)

24.1 Condition of C p -Compatibility for the Internal Coupling Matrix

261

or A satisfies the following row-sum condition by blocks: ns 

ai j = αr s , n r −1 + 1  i  n r , 1  r, s  p.

(24.11)

j=n s−1 +1

In particular, when αsr = 0, 1  r, s  p,

(24.12)

we say that A satisfies the zero-sum condition by blocks. In this case, we have Aer = 0, 1  r  p.

(24.13)

24.2 Condition of C p -Compatibility for the Boundary Coupling Matrix Comparing with the internal coupling matrix A, the study on the necessity of the condition of C p -compatibility for the boundary coupling matrix B is more complicated. It concerns the regularity of solution to the problem with coupled Robin boundary conditions. Let (i)

εi = (0, · · · , 1 , · · · , 0)T , 1  i  N

(24.14)

be a set of classical orthogonal basis in R N , and let Vr = Span{εnr −1 +1 , · · · , εnr }, 1  r  p.

(24.15)

er ∈ Vr , 1  r  p.

(24.16)

Obviously, we have

In what follows, we will discuss the necessity of the condition of C p -compatibility for the boundary coupling matrix B under the assumption that Aer ∈ Vr and Ber ∈ Vr (1  r  p). Theorem 24.3 Let  ⊂ Rn be a parallelepiped. Assume that M = rank(D) = N − p and (24.17) Aer ∈ Vr , Ber ∈ Vr (1  r  p).

262

24 Necessity of the Conditions of C p -Compatibility

If system (III) is exactly synchronizable by p-groups, then the boundary coupling matrix B should satisfy the following condition of C p -compatibility: B K er (C p ) ⊆ K er (C p ).

(24.18)

Proof Noting (23.7), we have ⎧ p  ⎪ ⎪ ⎪ (u r er − u r er + u r Aer ) = 0 in (T, +∞) × , ⎪ ⎨ r =1

p  ⎪ ⎪ ⎪ (∂ν u r er + u r Ber ) = 0 ⎪ ⎩

(24.19) on (T, +∞) × 1 .

r =1

Noting (24.16)–(24.17) and the fact that subspaces Vr (1  r  p) are orthogonal to each other, for 1  r  p, we have

u r er − u r er + u r Aer = 0 in (T, +∞) × , ∂ν u r er + u r Ber = 0

on (T, +∞) × 1 .

(24.20)

Taking the inner product with C p on both sides of the boundary condition on 1 in (24.20), and noting (23.5), we get u r C p Ber ≡ 0 on (T, +∞) × 1 , 1  r  p.

(24.21)

We claim that C p Ber = 0 (r = 1, · · · , p), which just means that B satisfies the condition of C p -compatibility (24.18). Otherwise, there exists an r¯ (1  r¯  p) such that C p Ber¯ = 0, and consequently, we have u r¯ ≡ 0 on (T, +∞) × 1 .

(24.22)

Then, it follows from the boundary condition in system (24.20) that ∂ν u r¯ ≡ 0 on (T, +∞) × 1 .

(24.23)

Hence, applying Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]) to (24.20) yields that (24.24) u r¯ ≡ 0 in (T, +∞) × , then it is easy to check that tT :

erT¯ U ≡ 0 in .

(24.25)

24.2 Condition of C p -Compatibility for the Boundary Coupling Matrix

  p−1 = CTp . C er¯

Let

263

(24.26)

p−1 ) = N − p + 1. On the other Since erT¯ ∈ / Im(C Tp ), it is easy to show that rank(C T Tp−1 ) = {0}, then there hand, since rank(Ker(D )) = p, we have Ker(D T ) ∩ Im(C exists a vector E = 0, such that Tp−1 E = 0. DT C

(24.27)

Let p−1 AU ), Rθ = −(E, C p−1 BU ). p−1 U ), Lθ = −(E, C w = (E, C

(24.28)

We get again problem (20.8) for w, and (20.9) holds. Therefore, noting that  ⊂ Rn is a parallelepiped, similarly to the proof of Theorem 21.2, we can get a conclusion that contradicts Lemma 20.3.  Remark 24.4 Noting that condition (24.17) obviously holds for p = 1, so, the conditions of C1 -compatibility (21.8) are always satisfied for both A and B in the case p = 1. In the next result, we will remove the restricted conditions (24.17) in the case p = 2, and further study the necessity of the condition of C2 -compatibility for B. Theorem 24.5 Let  ⊂ Rn be a parallelepiped and let the matrix B be similar to a real symmetric matrix. Assume that the coupling matrix A satisfies the zerosum condition by blocks (24.13). Assume furthermore that system (III) is exactly synchronizable by 2-groups with rank(D) = N − 2. Then the coupling matrix B necessarily satisfies the condition of C2 -compatibility: B K er (C2 ) ⊆ K er (C2 ).

(24.29)

Proof By the exact boundary synchronization by 2-groups, we have tT :

U = e1 u 1 + e2 u 2 in .

(24.30)

Noting (24.13), as t  T , we have AU = Ae1 u 1 + Ae2 u 2 = 0, then it is easy to see that ⎧  ⎨ U − U = 0 in (T, +∞) × , U =0 on (T, +∞) × 0 , ⎩ ∂ν U + BU = 0 on (T, +∞) × 1 .

(24.31)

264

24 Necessity of the Conditions of C p -Compatibility

Let P be a matrix such that  B = P B P −1 is a real symmetric matrix. Denote u = (u 1 , u 2 )T .

(24.32)

Taking the inner product on both sides of (24.31) with P T Pei for i = 1, 2, we get ⎧  ⎨ Lu − Lu = 0 in (T, +∞) × , Lu = 0 on (T, +∞) × 0 , ⎩ L∂ν u + u = 0 on (T, +∞) × 1 ,

(24.33)

where the matrices L and  are given by B Pei , Pe j ), 1  i, j  2. L = (Pei , Pe j ) and  = ( 

(24.34)

Clearly, L is a symmetric and positive definite matrix and  is a symmetric matrix. 1 Taking the inner product on both sides of (24.33) with L − 2 and denoting w = 1 L 2 u, we get ⎧  ⎨ w − w = 0 in (T, +∞) × , w=0 on (T, +∞) × 0 , (24.35) ⎩ w = 0 on (T, +∞) × 1 , ∂ν w +   = L − 2 L − 2 is also a symmetric matrix. where  On the other hand, taking the inner product with C2 on both sides of the boundary condition on 1 in system (III) and noting (24.30), we get 1

1

tT :

C2 Be1 u 1 + C2 Be2 u 2 ≡ 0 on 1 .

(24.36)

We claim that C2 Be1 = C2 Be2 = 0, namely, B satisfies the condition of C2 compatibility (24.29). Otherwise, without loss of generality, we may assume that C2 Be1 = 0. Then it follows from (24.36) that there exists a nonzero vector D2 ∈ R2 , such that tT : Denoting

D2T u ≡ 0 on 1 .

(24.37)

T = D2T L − 21 , D

we then have tT :

T w = 0 on 1 , D

(24.38)

1

in which w = L 2 u. By Theorem 28.5, the following Kalman’s criterion    =2  D) rank( D,

(24.39)

24.2 Condition of C p -Compatibility for the Boundary Coupling Matrix

265

is sufficient for the unique continuation of system (24.35) under the observation (24.38) on the infinite horizon [T, +∞). Since rank(D) = N − 2, by Theorem 20.5, system (III) is not exactly null controllable. So, the rank condition (24.39) does not hold. Thus, by Proposition 2.12 (i), there exists a vector E = 0 in R2 , such that T E = 0. T E =   E = μE and D 

(24.40)

Noting (24.38) and the second formula of (24.40), we have both E and w|1 ∈  Since dim Ker( D)  = 1, there exists a constant α such that w = αE on 1 . Ker( D). Therefore, noting the first formula of (24.40), we have w =  αE = μαE = μw on 1 .  Thus, (24.35) can be rewritten as ⎧  ⎨ w − w = 0 in (T, +∞) × , w=0 on (T, +∞) × 0 , ⎩ ∂ν w + μw = 0 on (T, +∞) × 1 .

(24.41)

(24.42)

T w. Noting (24.38), it follows from (24.42) that Let z = D ⎧  ⎨ z − z = 0 in (T, +∞) × , z=0 on (T, +∞) × 0 , ⎩ ∂ν z = z = 0 on (T, +∞) × 1 .

(24.43)

Then, by Holmgren’s uniqueness theorem, we have tT :

T w = D2T u ≡ 0 in . z=D

(24.44)

Let D2T = (α1 , α2 ). Define the following row vector: c3 =

α1 e1T α2 e2T + . 2 e1  e2 2

(24.45)

Noting (e1 , e2 ) = 0 and (24.44), we have tT : Let

c3 U = α1 u 1 + α2 u 2 = D2T u ≡ 0 in .   1 = C2 . C c3

(24.46)

(24.47)

266

24 Necessity of the Conditions of C p -Compatibility

1 ) = N − 1, and Ker(D T ) ∩ Since c3T ∈ / Im(C2T ), it is easy to see that rank(C T  = 0, such that 1 ) = {0}, thus there exists a vector E Im(C 1T E  = 0. DT C

(24.48)

Let  C 1 AU ), Rθ = −( E,  C 1 BU ).  C 1 U ), Lθ = −( E, u = ( E,

(24.49)

We get again problem (20.8) for w, satisfying (20.9). Noting that  is a parallelepiped, similarly to the proof of Theorem 21.2, we get a contradiction to Lemma 20.3. 

Chapter 25

Determination of Exactly Synchronizable States by p-Groups

When system (III) possesses the exact boundary synchronization by p-groups, the corresponding exactly synchronizable states by p-groups will be studied in this chapter.

In general, exactly synchronizable states by p-groups depend not only on initial data, but also on applied boundary controls. However, when the coupling matrices A and B satisfy certain algebraic conditions, the exactly synchronizable state by p-groups can be independent of applied boundary controls. In this section, we first discuss the case when the exactly synchronizable state by p-groups is independent of applied boundary controls, then we present the estimate on each exactly synchronizable state by p-groups in general situation.

25.1 Determination of Exactly Synchronizable States by p-Groups Theorem 25.1 Assume that both A and B satisfy the conditions of C p -compatibility (23.13) and that B is similar to a real symmetric matrix. Assume furthermore that A T and B T possess a common invariant subspace V , bi-orthonormal to Ker(C p ). Then there exists a boundary control matrix D with rank(D) = rank(C p D) = N − p, such that system (III) is exactly synchronizable by p-groups, and the exactly synchronizable state by p-groups u = (u 1 , · · · , u p )T is independent of applied boundary controls. Proof Define the boundary control matrix D by Ker(D T ) = V.

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_25

(25.1)

267

268

25 Determination of Exactly Synchronizable States by p-Groups

Since V is bi-orthonormal to Ker(C p ), by Proposition 2.5, we have Ker(C p ) ∩ Im(D) = Ker(C p ) ∩ V ⊥ = {0},

(25.2)

then, by Proposition 2.11, we have rank(C p D) = rank(D) = N − p.

(25.3)

Thus, by Theorem 23.3, system (III) is exactly synchronizable by p-groups. By (25.2), noting Ker(C p ) = Span{e1 , · · · , e p }, we may write V = Span{E 1 , · · · , E p } with (er , E s ) = δr s (r, s = 1, · · · , p).

(25.4)

Since V is a common invariant subspace of A T and B T , there exist constants αr s and βr s (r, s = 1, · · · , p) such that A T Er =

p 

αr s E s and B T Er =

s=1

For r = 1, · · · , p, let

p 

βr s E s .

(25.5)

s=1

φr = (Er , U ).

(25.6)

By system (III) and noting (25.1), for r = 1, · · · , p, we have ⎧ p  ⎪ ⎪  ⎪ φ − φ + αr s φs = 0 in (0, +∞) × , ⎪ r r ⎪ ⎪ ⎪ s=1 ⎪ ⎨ φr = 0 on (0, +∞) × 0 , p  ⎪ ⎪ ⎪ ∂ν φr + βr s φs = 0 on (0, +∞) × 1 , ⎪ ⎪ ⎪ ⎪ s=1 ⎪ ⎩ 0 ), φr = (Er , U 1 ) in . t = 0 : φr = (Er , U

(25.7)

On the other hand, for r = 1, · · · , p, we have tT :

φr = (Er , U ) =

p p   (Er , es )u s = δr s u s = u r . s=1

(25.8)

s=1

Thus, the exactly synchronizable state by p-groups u = (u 1 , · · · , u p )T is entirely determined by the solution to problem (25.7), which is independent of applied boundary controls H .  The following result gives the counterpart of Theorem 25.1. Theorem 25.2 Assume that both A and B satisfy the conditions of C p -compatibility (23.13) and that B is similar to a real symmetric matrix. Assume furthermore that

25.1 Determination of Exactly Synchronizable States by p-Groups

269

system (III) is exactly synchronizable by p-groups. Let V = Span{E 1 , · · · , E p } be a subspace of dimension p. If the projection functions φr = (Er , U ), r = 1, · · · , p

(25.9)

are independent of applied boundary controls H which realize the exact boundary synchronization by p-groups, then V is a common invariant subspace of A T and B T , and bi-orthonormal to Ker(C p ). 0 , U 1 ) = (0, 0). By Theorem 19.10, the linear mapping Proof Let (U F:

H → (U, U  )

is continuous from L 2 (0, T ; (L 2 (1 )) M ) to C 0 ([0, T ]; (H α ()) N × (H α−1 ()) N ), where α is defined by (19.8). ∈ Let F  denote the Fréchet derivative of the application F. For any given H 2 L (0, T ; (L 2 (1 )) M ), we define .  = F  (0) H U

(25.10)

 satisfies a system similar to that of U : By linearity, U ⎧   − U  + AU =0 U ⎪ ⎪ ⎪ ⎨U =0  + BU  = DH  ⎪ ∂ν U ⎪ ⎪ ⎩ =U  = 0 t =0: U

in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 in .

(25.11)

Since the projection functions φr = (Er , U ) (r = 1, · · · , p) are independent of applied boundary controls H , we have ) ≡ 0, ∀ H  ∈ L 2 (0, T ; (L 2 (1 )) M ), r = 1, · · · , p. (Er , U

(25.12)

/ Im(C Tp ) for r = 1, · · · , p. Otherwise, there exist an r¯ First, we prove that Er ∈ N−p and a vector Rr¯ ∈ R , such that Er¯ = C Tp Rr¯ , then we have ) = (Rr¯ , C p U ), ∀ H  ∈ L 2 (0, T ; (L 2 (1 )) M ). 0 = (Er¯ , U

(25.13)

 is the solution to the corresponding reduced problem (23.18) and (23.19), Since C p U noting the equivalence between the exact boundary synchronization by p-groups for the original system and the exact boundary controllability for the reduced system, from the exact boundary synchronization by p-groups for system (III), we know that  at the time the reduced system (23.18) is exactly controllable, then the value of C p U T can be chosen arbitrarily, thus we get Rr¯ = 0, which contradicts Er¯ = 0. Then, / Im(C Tp ) (r = 1, · · · , p). Thus, V ∩ {Ker(C p )}⊥ = V ∩ Im(C Tp ) = we have Er ∈

270

25 Determination of Exactly Synchronizable States by p-Groups

{0}. Hence, by Propositions 2.4 and 2.5, V is bi-orthonormal to Ker(C p ), and then (V, C Tp ) constitutes a set of basis in R N . Therefore, there exist constant coefficients αr s (r, s = 1, · · · , p) and vectors Pr ∈ R N − p (r = 1, · · · , p), such that A T Er =

p 

αr s E s + C Tp Pr , r = 1, · · · , p.

(25.14)

s=1

Taking the inner product with Er on both sides of the equations in (25.11) and noting (25.12), we get , Er ) = (U , A T Er ) = (U , C Tp Pr ) = (C p U , Pr ) 0 = (AU

(25.15)

for r = 1, · · · , p. Similarly, by the exact boundary controllability for the reduced system (23.18), we get Pr = 0 (r = 1, · · · , p), thus we have A T Er =

p 

αr s E s , r = 1, · · · , p,

s=1

which means that V is an invariant subspace of A T . On the other hand, noting (25.12) and taking the inner product with Er on both sides of the boundary condition on 1 in (25.11), we get ) = (Er , D H ) on 1 , r = 1, · · · , p. (Er , B U

(25.16)

By Theorem 19.10, for r = 1, · · · , p, we have ) H 2α−1 (1 ) (Er , D H ) H 2α−1 (1 )  c H  L 2 (0,T ;(L 2 (1 )) M ) . =(Er , B U

(25.17)

We claim that D T Er = 0 for r = 1, · · · , p. Otherwise, for r = 1, · · · , p, setting  = D T Er v, it follows from (25.17) that H v H 2α−1 (1 )  cv L 2 (0,T ;L 2 (1 )) .

(25.18)

Since 2α − 1 > 0, it contradicts the compactness of H 2α−1 (1 ) → L 2 (1 ). Thus, by (25.16) we have ) = 0 on (0, T ) × 1 , r = 1, · · · , p. (Er , B U

(25.19)

Similarly, there exist constants βr s (r, s = 1, · · · , p) and vectors Q r ∈ R N − p (r = 1, · · · , p), such that B T Er =

p  s=1

βr s E s + C Tp Q r , r = 1, · · · , p.

(25.20)

25.1 Determination of Exactly Synchronizable States by p-Groups

271

Substituting it into (25.19) and noting (25.12), we have p 

) + (C Tp Q r , U ) = (Q r , C p U ) = 0, r = 1, · · · , p. βr s (E s , U

(25.21)

s=1

Similarly, by the exact boundary controllability for the reduced system (23.18), we get Q r = 0 (r = 1, · · · , p), then we have B Er = T

p 

βr s E s , r = 1, · · · , p,

(25.22)

s=1

which indicates that V is also an invariant subspace of B T . The proof is complete. 

25.2 Estimation of Exactly Synchronizable States by p-Groups When A and B do not satisfy all the conditions mentioned in Theorem 25.1, exactly synchronizable states by p-groups may depend on applied boundary controls. We have the following Theorem 25.3 Assume that both A and B satisfy the conditions of C p -compatibility (23.13). Then there exists a boundary control matrix D such that system (III) is exactly synchronizable by p-groups, and each exactly synchronizable state by pgroups u = (u 1 , · · · , u p )T satisfies the following estimate: (u, u  )(T ) − (φ, φ )(T )(H α+1 ()) p ×(H α ()) p 0 , U 1 )(H1 ) N − p ×(H0 ) N − p , cC p (U

(25.23)

where α is defined by (19.8), c is a positive constant, and φ = (φ1 , · · · , φ p )T is the solution to the following problem (1  r  p): ⎧ p  ⎪ ⎪  ⎪ φ − φ + αr s φs = 0 in (0, +∞) × , ⎪ r r ⎪ ⎪ ⎪ s=1 ⎪ ⎨ φr = 0 on (0, +∞) × 0 , p  ⎪ ⎪ ⎪ ∂ν φr + βr s φs = 0 on (0, +∞) × 1 , ⎪ ⎪ ⎪ ⎪ s=1 ⎪ ⎩ 0 ), φr = (Er , U 1 ) in , t = 0 : φr = (Er , U

(25.24)

272

25 Determination of Exactly Synchronizable States by p-Groups

in which Aer =

p 

αsr es and Ber =

s=1

p 

βsr es , r = 1, · · · , p.

(25.25)

s=1

Proof We first show that there exists a subspace V which is invariant for B T and bi-orthonormal to Ker(C p ). Let B = P −1 P, where P is an invertible matrix, and  be a symmetric matrix. Let V = Span(E 1 , · · · , E p } in which Er = P T Per , r = 1, · · · , p.

(25.26)

Noting (23.5) and the fact that Ker(C p ) is an invariant subspace of B, we get B T Er = P T P Ber ⊆ P T PKer(C p ) ⊆ V, r = 1, · · · , p,

(25.27)

then V is invariant for B T . We next show that V ⊥ ∩ K er (C p ) = {0}. Then, noting that dim(V ) = dim (C p ) = p, by Propositions 2.4 and 2.5, V is bi-orthonormal to K er (C p ). For this purpose, let a1 , · · · , a p be coefficients such that p 

ar er ∈ V ⊥ .

(25.28)

r =1

Then (

p  r =1

ar er , E s ) = (

p 

ar Per , Pes ) = 0, s = 1, · · · , p.

(25.29)

r =1

It follows that (

p 

ar Per ,

r =1

p 

as Pes ) = 0,

(25.30)

s=1

then a1 = · · · = a p = 0, namely, V ⊥ ∩ K er (C p ) = {0}. Denoting Ber =

p 

βsr es , r = 1, · · · , p,

(25.31)

s=1

a direct calculation yields that B T Er =

p  s=1

βr s E s , r = 1, · · · , p.

(25.32)

25.2 Estimation of Exactly Synchronizable States by p-Groups

273

Define the boundary control matrix D by K er (D T ) = V.

(25.33)

Noting (23.5), we have Ker(C p ) ∩ Im(D)

(25.34) ⊥

=Ker(C p ) ∩ {Ker(D )} = Ker(C p ) ∩ V T

⊥

= {0},

then, by Proposition 2.11, we have rank(C p D) = rank(D) = N − p.

(25.35)

Therefore, by Theorem 23.3, system (III) is exactly synchronizable by p-groups. Denoting ψr = (Er , U )(r = 1, · · · , p), we have (Er , AU ) = (A T Er , U )  p

=(

αr s E s + A T Er −

s=1 p

=





αr s E s , U )

s=1

αr s (E s , U ) + (A T Er −

s=1 p

=

(25.36) p

p 

αr s E s , U )

s=1



αr s ψs + (A Er − T

s=1

p 

αr s E s , U ).

s=1

By the assumption that V is bi-orthonormal to Ker(C p ), without loss of generality, we may assume that (Er , es ) = δr s (r, s = 1, · · · , p).

(25.37)

Then, for any given t = 1, · · · , p, by the first formula of (25.25), we get (A T Er −

p 

αr s E s , et ) = (Er , Aet ) −

s=1

=

p 

p 

αr s (E s , et )

s=1

αst (Er , es ) − αr t = αr t − αr t = 0,

s=1

hence A T Er −

p  s=1

αr s E s ∈ {K er (C p )}⊥ = Im(C Tp ), r = 1, · · · , p.

(25.38)

274

25 Determination of Exactly Synchronizable States by p-Groups

Thus, there exist Rr ∈ R N − p (r = 1, · · · , p) such that A T Er −

p 

αr s E s = C Tp Rr , r = 1, · · · , p.

(25.39)

s=1

Taking the inner product on both sides of problem (III) and (III0) with Er , and noting (25.32)–(25.33), for r = 1, · · · , p, we have ⎧ p  ⎪ ⎪  ⎪ ψr − ψr + αr s ψs = −(Rr , C p U ) in (0, +∞) × , ⎪ ⎪ ⎪ ⎪ s=1 ⎪ ⎨ ψr = 0 on (0, +∞) × 0 , p  ⎪ ⎪ ⎪ ∂ν ψr + βr s ψs = 0 on (0, +∞) × 1 , ⎪ ⎪ ⎪ ⎪ s=1 ⎪ ⎩ 0 ), ψr = (Er , U 1 ) in . t = 0 : ψr = (Er , U

(25.40)

Similarly to the proof of Theorem 22.4, we get (ψ, ψ  )(T ) − (φ, φ )(T )(H α+1 ()) p ×(H α ()) p 0 , U 1 )(H 1 ()) N − p ×(L 2 ()) N − p . cC p (U

(25.41)

On the other hand, noting (25.37), it is easy to see that tT :

ψr = (Er , U ) =

p  (Er , es )u s = u r , r = 1, · · · , p.

(25.42)

s=1

Substituting it into (25.41), we get (25.23).



Part VI

Synchronization for a Coupled System of Wave Equations with Coupled Robin Boundary Controls: Approximate Boundary Synchronization In this part, the approximate boundary synchronization and the approximate boundary synchronization by groups will be discussed so that they can be realized by means of substantially fewer number of boundary controls than the number of state variables for system (III) with coupled Robin boundary controls under suitable hypotheses.

Chapter 26

Some Algebraic Lemmas

In order to study the approximate boundary synchronization for system (III) with coupled Robin boundary controls, some algebraic lemmas are given in this chapter.

Let A be a matrix of order N and D a full column-rank matrix of order N × M with M  N . By Proposition 2.12, we know the following Kalman’s criterion: rank(D, AD, · · · , A N −1 D)  N − d

(26.1)

holds if and only if the dimension of any given subspace, contained in Ker(D T ) and invariant for A T , does not exceed d. In particular, the equality holds if and only if the dimension of the largest subspace, contained in Ker(D T ) and invariant for A T , is exactly equal to d. Let A and B be two matrices of order N and D a full column-rank matrix of order N × M with M  N . For any given nonnegative integers p, q, · · · , r, s  0, we define a matrix of order N × M by R( p,q,··· ,r,s) = A p B q · · · Ar B s D.

(26.2)

We construct an enlarged matrix R = (R( p,q,··· ,r,s) , R( p ,q  ,··· ,r  ,s  ) , · · · )

(26.3)

by the matrices R( p,q,··· ,r,s) for all possible ( p, q, · · · , r, s), which, by Cayley– Hamilton’s Theorem, essentially constitute a finite set M with dim(M)  M N . Lemma 26.1 Ker(RT ) is the largest subspace of all the subspaces which are contained in Ker(D T ) and invariant for A T and B T .

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_26

277

278

26 Some Algebraic Lemmas

Proof First, noting that Im(D) ⊆ Im(R), we have Ker(RT ) ⊆ Ker(D T ). We now show that Ker(RT ) is invariant for A T and B T . Let x ∈ Ker(RT ). We have D T (B T )s (A T )r · · · (B T )q (A T ) p x = 0

(26.4)

for any given integers p, q, · · · , r, s  0. Then, it follows that A T x ∈ Ker(RT ), namely, Ker(RT ) is invariant for A T . Similarly, Ker(RT ) is invariant for B T . Thus, the subspace Ker(RT ) is contained in Ker(D T ) and invariant for both A T and B T . Now let V be another subspace, contained in Ker(D T ) and invariant for A T and T B . For any given y ∈ V , we have A T y ∈ V,

B T y ∈ V and D T y = 0.

(26.5)

Then, it is easy to see that (B T )s (A T )r · · · (B T )q (A T ) p y ∈ V

(26.6)

for any given integers p, q, · · · , r, s  0. Thus, by the first formula of (26.5), we have (26.7) D T (B T )s (A T )r · · · (B T )q (A T ) p y = 0 for any given integers p, q, · · · , r, s  0, namely, we have V ⊆ Ker(RT ).

(26.8) 

The proof is then complete.

By the rank-nullity theorem, we have rank(R) + dim Ker(RT ) = N . The following lemma is a dual version of Lemma 26.1. Lemma 26.2 Let d  0 be an integer. Then (i) the rank condition rank(R)  N − d

(26.9)

holds true if and only if the dimension of any given subspace, contained in Ker(D T ) and invariant for A T and B T , does not exceed d; (ii) the rank condition rank(R) = N − d (26.10) holds true if and only if the dimension of the largest subspace, contained in Ker(D T ) and invariant for A T and B T , is exactly equal to d. Proof (i) Let V be a subspace which is contained in Ker(D T ) and invariant for A T and B T . By Lemma 26.1, we have V ⊆ K er (RT ).

(26.11)

26 Some Algebraic Lemmas

279

Assume that (26.9) holds, it follows from (26.11) that N − d  rank(R) = N − dim Ker(RT )  N − dim(V ),

(26.12)

namely, dim(V )  d.

(26.13)

Conversely, assume that (26.13) holds for any given subspace V which is contained in Ker(D T ) and invariant for A T and B T . In particular, by Lemma 26.1, we have dim Ker(RT )  d. Then, it follows that rank(R) = N − dim Ker(RT )  N − d.

(26.14)

(ii) Noting that (26.10) can be written as rank(R)  N − d

(26.15)

rank(R)  N − d.

(26.16)

and By (i), the rank condition (26.15) means that dim(V )  d for any given invariant subspace V of A T and B T , contained in Ker(D T ). We claim that there exists a subspace V0 , which is contained in Ker(D T ) and invariant for A T and B T , such that dim(V0 ) = d. Otherwise, all the subspaces of this kind have a dimension fewer than or equal to (d − 1). By (i), we get rank(R)  N − d + 1, which contradicts (26.16). It proves (ii). The proof is then complete.

(26.17) 

Remark 26.3 In the special case that B = I , we can write R = (D, AD, · · · , A N −1 D).

(26.18)

Then, by Lemma 26.2, we find again that Kalman’s criterion (26.1) holds if and only if the dimension of any given subspace, contained in Ker(D T ) and invariant for A T , does not exceed d. In particular, the equality holds if and only if the dimension of the largest subspace, contained in Ker(D T ) and invariant for A T , is exactly equal to d.

Chapter 27

Approximate Boundary Null Controllability

In this chapter, we will define the approximate boundary null controllability for system (III) and the D-observability for the adjoint problem, and show that these two concepts are equivalent to each other. Definition 27.1 For (0 , 1 ) ∈ (H1 ) N × (H0 ) N , the adjoint system (19.19) is Dobservable on a finite interval [0, T ] if the observation D T  ≡ 0 on [0, T ] × 1

(27.1)

implies that 0 = 1 ≡ 0, then  ≡ 0. Proposition 27.2 If the adjoint system (19.19) is D-observable, then we necessarily have rank(R) = N . In particular, if M = N , namely, D is invertible, then system (19.19) is D-observable. Proof Otherwise, we have dim Ker(RT ) = d  1. Let Ker(RT ) = Span{E 1 , · · · , E d }. By Lemma 26.1, Ker(RT ) is contained in Ker(D T ) and invariant for A T and B T , namely, we have (27.2) D T Er = 0 for 1  r  d and there exist coefficients αsr and βsr such that d d   αr s E s and B T Er = βr s E s A T Er = s=1

for 1  r  d.

(27.3)

s=1

In what follows, we restrict system (19.19) on the subspace Ker(RT ) and look for a solution of the form d  = φs E s , (27.4) s=1

which, because of (27.2), obviously satisfies the D-observation (27.1). © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_27

281

282

27 Approximate Boundary Null Controllability

Inserting the function (27.4) into system (19.19) and noting (27.3), it is easy to see that for 1  r  d, we have ⎧ d  ⎪ ⎨φr − φr + s=1 αsr φs = 0 in (0, +∞) × , on (0, +∞) × 0 , φr = 0 ⎪  ⎩ on (0, +∞) × 1 . ∂ν φr + ds=1 βsr φs = 0

(27.5)

For any nontrivial initial data t =0:

φr = φ0r , φr = φ1r (1  r  d),

(27.6)

we have  ≡ 0. This contradicts the D-observability of system (19.19). Moreover, when D is invertible, the D-observation (27.1) implies that ∂ν  ≡  ≡ 0 on (0, T ) × 1 .

(27.7)

Then, Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]) implies well  ≡ 0, provided that T > 0 is large enough.  Definition 27.3 System (III) is approximately null controllable at the time T > 0 1 ) ∈ (H0 ) N × (H−1 ) N , there exists a sequence {Hn } 0 , U if for any given initial data (U M of boundary controls in L with compact support in [0, T ], such that the sequence {Un } of solutions to problem (III) and (III0) satisfies 0 1 u (k) n −→ 0 in Cloc ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 )

(27.8)

for all 1  k  N as n → +∞. By a similar argument as in Chaps. 8 and 16, we can prove the following Proposition 27.4 System (III) is approximately null controllable at the time T > 0 if and only if its adjoint system (19.19) is D-observable on the interval [0, T ]. Corollary 27.5 If system (III) is approximately null controllable, then we necessarily have rank(R) = N . In particular, as M = N , namely, D is invertible, system (III) is approximately null controllable. Proof This corollary follows immediately from Propositions 27.2 and 27.4. However, here we prefer to give a direct proof from the point of view of control. Suppose that dim Ker(RT ) = d  1. Let Ker(RT ) = Span{E 1 , · · · , E d }. By Lemma 26.1, Ker(RT ) is contained in Ker(D T ) and invariant for both A T and B T , then we still have (27.2) and (27.3). Applying Er to problem (III) and (III0) and setting u r = (Er , U ) for 1  r  d, it follows that for 1  r  d, we have ⎧  d ⎨ u r − u r + s=1 αr s u s = 0 in (0, +∞) × , on (0, +∞) × 0 , ur = 0  ⎩ ∂ν u r + ds=1 βr s u s = 0 on (0, +∞) × 1

(27.9)

27 Approximate Boundary Null Controllability

283

with the initial condition t =0:

0 ), u r = (Er , U 1 ) in . u r = (Er , U

(27.10)

Thus, the projections u 1 , · · · , u d of U on the subspace Ker(RT ) are independent of applied boundary controls H , therefore, uncontrollable. This contradicts the approximate boundary null controllability of system (III). The proof is then complete. 

Chapter 28

Unique Continuation for Robin Problems

We consider the unique continuation for Robin problems in this chapter.

28.1 General Consideration By Proposition 27.2, rank(R) = N is a necessary condition for the D-observability of the adjoint system (19.19). Proposition 28.1 Let  μ = sup dim Ker α,β∈C

 A T − αI . BT − β I

(28.1)

Assume that Ker(RT ) = {0}.

(28.2)

Then we have the following lower bound estimate: rank(D)  μ.

(28.3)

Proof Let α, β ∈ C, such that  V = Ker

A T − αI BT − β I

 (28.4)

is of dimension μ. It is easy to see that any given subspace W of V is still invariant for A T and B T , then by Lemma 26.1, condition (28.2) implies that Ker(D T ) ∩ V = {0}. Then, it follows that © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_28

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28 Unique Continuation for Robin Problems

dim Ker(D T ) + dim (V )  N ,

(28.5)

μ = dim (V )  N − dim Ker(D T ) = rank(D).

(28.6)

namely, 

The proof is complete.

In general, the condition dim Ker(RT ) = 0 does not imply rank(D) = N , so, the D-observation (27.1) does not imply  = 0 on (0, T ) × 1 .

(28.7)

Therefore, the unique continuation for the adjoint system (19.19) with D-observation (27.1) is not a standard type of Holmgren’s uniqueness theorem (cf. [18, 62, 75]). Let us consider the following adjoint system (19.19) with  = (φ, ψ)T : ⎧ ⎪ φ − φ + aφ + bψ = 0 ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ψ − ψ + cφ + dψ = 0 φ=ψ=0 ⎪ ⎪ ⎪ ∂ ⎪ ν φ + αφ = 0 ⎪ ⎪ ⎩∂ ψ + βψ = 0 ν

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , on (0, +∞) × 1

(28.8)

with the observation of rank one: d1 φ + d2 ψ = 0 on [0, T ] × 1 ,

(28.9)

where a, b, c, d; α, β; and d1 , d2 are constants. In system (28.8), since the boundary coupling matrix B is always assumed to be similar to a real symmetric matrix, without loss of generality, we suppose that B is a diagonal matrix. Proposition 28.2 The following results can be easily checked. (a) Assume that A T and B T admit only one common eigenvector E. Then Ker(RT ) = {0} if and only if (E, D) = 0. (b) Assume that A T and B T admit only two common eigenvectors E 1 and E 2 . Then Ker(RT ) = {0} if and only if (E i , D) = 0 for i = 1, 2. (c) Assume that A T and B T have no common eigenvector. Then Ker(RT ) = {0} if and only if D = 0. The above conditions are only necessary for the unique continuation. We only know a few results about their sufficiency as follows.

28.2 Examples in Higher Dimensional Cases

287

28.2 Examples in Higher Dimensional Cases The first example is a cascade system with Neumann boundary conditions: ⎧  φ − φ = 0 ⎪ ⎪ ⎪ ⎨ψ  − ψ + φ = 0 ⎪ φ=ψ=0 ⎪ ⎪ ⎩ ∂ν φ = ∂ν ψ = 0

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1

with the initial data in H10 () × H10 () × L 2 () × L 2 (). Let   00 AT = 10

(28.10)

(28.11)

denote the corresponding coupling matrix of system (28.10). Since (0, 1)T is the only eigenvector of A T , by Proposition 28.2, Ker(R) = {0} holds true if and only if d2 = 0. The following result shows that it is also sufficient for the unique continuation. Theorem 28.3 ([3]) Assume that d2 = 0. Let (φ, ψ) be a solution to system (28.10). Then the partial observation (28.9) implies that φ ≡ ψ ≡ 0, provided that T > 0 is large enough. Remark 28.4 Let us consider the following slightly modified system: ⎧ ⎪ φ − φ = 0 ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ψ − ψ + φ = 0 φ=ψ=0 ⎪ ⎪ ⎪ ∂ν φ + αφ = 0 ⎪ ⎪ ⎪ ⎩∂ ψ + βψ = 0 ν

in (0, +∞) × , in (0, +∞) × , on (0, +∞) × 0 , on (0, +∞) × 1 , on (0, +∞) × 1 .

(28.12)

Since (0, 1)T is the only common eigenvector of A T and B T , by Proposition 28.2, Ker(RT ) = {0} if and only if d2 = 0. Unfortunately, the multiplier approach used in [3] is quite technically delicate, we don’t know up to now if it can be adapted to get the unique continuation for system (28.12) with the partial observation (28.9). The second example is a system of decoupled wave equations with Robin boundary conditions: ⎧  φ − φ = 0 in (0, +∞) × , ⎪ ⎪ ⎪ ⎪ ⎨ ψ  − ψ = 0 in (0, +∞) × , φ=ψ=0 on (0, +∞) × 0 , (28.13) ⎪ ⎪ ⎪ ∂ν φ + αφ = 0 on (0, +∞) × 1 , ⎪ ⎩ ∂ν ψ + βψ = 0 on (0, +∞) × 1 .

288

28 Unique Continuation for Robin Problems

If α = β, then dim Ker(RT ) > 0 for any given D = (d1 , d2 )T . So, we assume that α = β, then (1, 0)T and (0, 1)T are the only common eigenvectors of A T and B T . By Proposition 28.2, Ker(RT ) = {0} holds if and only if d1 d2 = 0. Moreover, we replace the observation on the finite horizon (28.9) by the observation on the infinite horizon: (28.14) d1 φ + d2 ψ = 0 on (0, +∞) × 1 . The following result confirms that Ker(RT ) = {0} or equivalently, rank(R) = 2 is sufficient for the unique continuation of system (28.13) with the observation on the infinite horizon (28.14). Theorem 28.5 Assume that α > 0, β > 0, α = β, and d1 d2 = 0. Let (φ, ψ) be a solution to system (28.13). Then the partial observation on the infinite horizon (28.14) implies that φ ≡ ψ ≡ 0. Proof We first recall Green’s formula 





uvd x = −

 

∇u · ∇vd x +



∂ν uvd

(28.15)

and Rellich’s identity (cf. [62]):  u(m · ∇u)d x = (n − 2) |∇u|2 d x    +2 ∂ν u(m · ∇u)d − (m, ν)|∇u|2 d, 

2



(28.16)



where m = x − x0 and ν stands for the unit outward normal vector on the boundary . The energy of system (28.13) can be defined as  1 (|φt |2 + |∇φ|2 + |ψt |2 + |∇ψ|2 )d x E(t) = 2   1 + (α|φ|2 + β|ψ|2 )d. 2 1

(28.17)

It is easy to check that E  (t) = 0 which yields the conservation of energy E(t) = E(0), ∀ t  0.

(28.18)

Now, taking the inner product with 2m · ∇u on both sides of the first equation in (28.13) and integrating by parts, we get

28.2 Examples in Higher Dimensional Cases

289



T φt (m · ∇φ)d x 0    T φt m · ∇φt d xdt + = 

0

(28.19) T



0



φ(m · ∇φ)d xdt.

Then, using Green’s formula in the first term and Rellich’s identity in the second term on the right-hand side, it follows that

T  T  φt (m · ∇φ)d x = (m, ν)|φt |2 ddt 0 0    T  T |φt |2 d xdt + (n − 2) |∇φ|2 d xdt −n 0 0     T ∂ν φ(m · ∇φ)ddt − (m, ν)|∇φ|2 ddt. +2 

2



0

(28.20)



Noting the conservation of energy (28.18) and using Cauchy–Schwarz inequality on the left-hand side, it follows that 

T





T



|φt | d xdt + (2 − n) |∇φ|2 d xdt 0   T (m, ν)|φt |2 ddt cE(0) + 0   T (2∂ν φ(m · ∇φ) − (m, ν)|∇φ|2 )d, + 2

n

0

(28.21)



0



here and hereafter, c denotes a positive constant independent of T . Recall the usual multiplier geometrical condition: (m, ν)  0, ∀x ∈ 0 ; (m, ν)  δ, ∀x ∈ 1 ,

(28.22)

where δ > 0 is a positive constant. Then on 1 , we have 2∂ν φ(m · ∇φ) − (m, ν)|∇φ|2 2 m ∞ |∂ν φ| · |∇φ| − δ|∇φ| 

(28.23)

2

m 2∞

m 2∞ 2 2 |∂ν φ|2 = α |φ| . δ δ

While, noting that φ = 0 on 0 , we have ∇φ = ∂ν φν,

(28.24)

290

28 Unique Continuation for Robin Problems

then 2∂ν φ(m · ∇φ) − (m, ν)|∇φ|2 = (m, ν)|∂ν φ|2  0.

(28.25)

Substituting (28.23) and (28.25) into (28.21), it is easy to get 

T







T

|φt | d xdt + (2 − n) |∇φ|2 d xdt (28.26) 0   T  

m 2∞ α2 T (m, ν)|φt |2 ddt + |φ|2 ddt cE(0) + δ 0 0 1 1  T (|φ|2 + |φt |2 )ddt. cE(0) + c 2

n

0



1

0

Next, taking the inner product with u on both sides of the first equation of system (28.13) and integrating by parts, we have

T  T  φt φd x = |φt |2 d xdt 0 0    T  T φ∂ν φddt − |∇φ|2 d xdt. + 

1

0

0

(28.27)



Using the conservation of energy (28.18), it follows that 

T

− 0



 

|φt |2 d xdt + 0

T

 

|∇φ|2 d xdt  cE(0).

(28.28)

By (28.26) +(n − 1)×(28.28), we get 

T

 

0

 c

(|φt |2 + |∇φ|2 )d xdt

(28.29)



T

1

0

(|φ|2 + |φt |2 )ddt + cE(0).

Similarly, we have 

T

 

0



T

c 0

(|ψt |2 + |∇ψ|2 )d xdt



1

(|ψ|2 + |ψt |2 )ddt + cE(0).

(28.30)

28.2 Examples in Higher Dimensional Cases

291

Combing (28.29) and (28.30), we have 

T

 E(t)dt  c

0



T

1

0

(|φ|2 + |ψ|2 + |φt |2 + |ψt |2 )ddt + cE(0).

(28.31)

Finally, taking the inner product with ψ on both sides of the first equation of (28.13) and integrating by parts, we get

T  T  φt ψd x = φt ψt d xdt 0 0    T  T φψddt − ∇φ · ∇ψd xdt. −α 

1

0

0

(28.32)



Similarly, taking the inner product with φ on both sides of the second equation of (28.13) and integrating by parts, we get

T  T  ψt φd x = φt ψt d xdt 0   0  T  T φψddt − ∇φ · ∇ψd xdt. −β 

1

0

0

(28.33)



Thus, we get  

(φt ψ − ψt φ)d x

T 0



T

= (α − β) 0

 1

φψddt.

(28.34)

By the boundary observation (28.14), we have ψ = − dd21 φ on 1 , then, by Cauchy– Schwarz’ inequality and noting (28.18), it comes from (28.34) that 

T

 1

0

(|φ|2 + |ψ|2 )ddt  cE(0).

(28.35)

Because of the linearity, we also have  0

T

 1

 (|φt |2 + |ψt |2 )ddt  c E(0),

(28.36)

where   = 1 (|φtt |2 + |∇φt |2 + |ψtt |2 + |∇ψt |2 )d x E(t) 2   1 + (α|φt |2 + β|ψt |2 )d. 2 1

(28.37)

292

28 Unique Continuation for Robin Problems

Noting the conservation of energy (28.18), and substituting (28.35)–(28.36) into (28.31), we get  T E(0)  c(E(0) + E(0)). (28.38) Taking T → +∞, we have E(0) = 0, then by (28.18), we get E(t) ≡ 0 for all t  0. The proof is complete. 

28.3 One-Dimensional Case In the one-space-dimensional case, Theorem 28.5 can be further improved to the observation of finite horizon. Theorem 28.6 Assume that α > 0, β > 0, α = β, and d1 d2 = 0. Then the following one-dimensional system ⎧  φ − φx x = 0 0 < x < 1, ⎪ ⎪ ⎪ ⎪ 0 < x < 1, ⎨ ψ  − ψx x = 0 φ(t, 0) = ψ(t, 0) = 0, ⎪ ⎪ φx (t, 1) + αφ(t, 1) = 0, ⎪ ⎪ ⎩ ψx (t, 1) + βψ(t, 1) = 0

(28.39)

with the observation of finite horizon d1 φ(t, 1) + d2 ψ(t, 1) = 0 ∀t ∈ [0, T ]

(28.40)

only has the trivial solution, provided that T > 0 is large enough. Proof Here we only give a sketch of the proof which is similar to that in Theorem 8.24. Consider the following eigenvalue problem: ⎧ 2 λ u + u x x = 0, 0 < x < 1, ⎪ ⎪ ⎪ 2 ⎪ 0 < x < 1, ⎨ λ v + vx x = 0, u(0) = v(0) = 0, ⎪ ⎪ u x (1) + αu(1) = 0, ⎪ ⎪ ⎩ vx (1) + βv(1) = 0.

(28.41)

u = sin λx, v = sin λx.

(28.42)

Let

By the last two formulas in (28.41), we have λ cos λ + α sin λ = 0, λ cos λ + β sin λ = 0.

(28.43)

28.3 One-Dimensional Case

293

Rewrite the first formula of (28.43) as tan λ +

λ = 0. α

(28.44)

Noting (28.44), we have 1 − tan2 λ α 2 − λ2 = 1 + tan2 λ α 2 + λ2

(28.45)

αλ tan λ =− 2 . 2 1 + tan λ α + λ2

(28.46)

cos2 λ − sin2 λ = and sin λ cos λ = Then, by

e2iλ = cos 2λ + i sin 2λ = cos2 λ − sin2 λ + 2i sin λ cos λ, we have e2iλ =

α 2 − λ2 2αλi λ + αi . − 2 =− 2 2 2 α +λ α +λ λ − αi

(28.47)

(28.48)

The asymptotic expansion of e2iλ at λ = ∞ gives e2iλ = −1 −

O(1) 2αi + 2 . λ λ

(28.49)

Taking the logarithm on both sides, we get 2αi O(1) 2iλ = (2n + 1)πi + ln 1 + + 2 λ λ O(1) 2αi + 2 i, = (2n + 1)πi + λ λ then

2α O(1) 1 + 2 . λ = λαn := (n + )π + 2 λ λ

(28.50)

Noting λαn ∼ nπ, we get α O(1) 1 + 2 . λαn = (n + )π + 2 nπ n

(28.51)

Similarly, by the second formula of (28.43), we have 1 β O(1) λ = λβn := (n + )π + + 2 . 2 nπ n

(28.52)

294

28 Unique Continuation for Robin Problems

Noting α = β, it follows from (28.44) that λαm = λβn , ∀m, n ∈ Z.

(28.53)

Besides, by the monotonicity of the function λ → tan λ + αλ , we have λαm = λαn , λβm = λβn , ∀m, n ∈ Z, m = n.

(28.54) β

Without loss of generality, assuming that α > β > 0, we can arrange {λαn } ∪ {λn } into an increasing sequence β

β

· · · < λα−n < λ−n < · · · < λ1 < λα1 < · · · < λβn < λαn < · · · .

(28.55)

On the other hand, noting a > β and using the expressions (28.51)–(28.52), there exists a positive constant γ > 0, such that β

λαn+1 − λαn  2γ, λn+1 − λβn  2γ

(28.56)

1  λαn − λβn  γ |n|

(28.57)

and

for all n ∈ Z with |n| large enough. β So, the sequence {λαn } ∪ {λn } satisfies the conditions (8.87), (8.91), and (8.92) in Theorem 8.22 with c = 1, m = 2, and s = 1. Moreover, the upper density D + β defined in (8.93) for the sequence {λαn } ∪ {λn } is equal to 2. β Finally, we easily check that the system of eigenvectors {E nα , E n }n∈Z with  E nα

=

sin λαn x λαn sin λαn x



 ,

E nβ

=

β

sin λn x β λn β sin λn x

 , n∈Z

(28.58)

forms a Hilbert basis in (H 1 ()) N × (L 2 ()) N . Then any given solution to system (28.39) can be represented by       β ψ φ α iλαn t α c e E , cnβ eiλn t E nβ . = = n n ψ φ n∈Z

(28.59)

n∈Z

By the boundary observation (28.40), we have  n∈Z

α

d1 cnα eiλn t

β sin λn sin λαn  + d2 cnβ eiλn t β = 0. λαn λn

β

n∈Z

(28.60)

28.3 One-Dimensional Case

295 α

β

Then, all the conditions of Theorem 8.22 are verified, so the sequence {eiλn t , eiλn t }n∈Z is ω-linearly independent in L 2 (0, T ), provided that T > 4π. It follows that d1 cnα namely,

β

sin λαn sin λn = 0, d2 cnβ β = 0, ∀ n ∈ Z, λαn λn cnα = 0, cnβ = 0, ∀n ∈ Z,

hence φ ≡ ψ ≡ 0. The proof is complete.

(28.61)

(28.62) 

Chapter 29

Approximate Boundary Synchronization

The approximate boundary synchronization is defined and studied in this chapter for system (III) with coupled Robin boundary controls.

29.1 Definitions Definition 29.1 System (III) is approximately synchronizable at the time T > 0 1 ) ∈ (H0 ) N × (H−1 ) N , there exists a sequence 0 , U if for any given initial data (U M {Hn } of boundary controls in L with compact support in [0, T ], such that the corresponding sequence {Un } of solutions to problem (III) and (III0) satisfies (l) 0 1 u (k) n − u n → 0 in Cloc ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 )

(29.1)

for all k, l with 1  k, l  N as n → +∞. Define the synchronization matrix of order (N − 1) × N by ⎞ 1 −1 ⎟ ⎜ 1 −1 ⎟ ⎜ C1 = ⎜ ⎟. .. .. ⎠ ⎝ . . 1 −1

(29.2)

Ker(C1 ) = Span{e1 } with e1 = (1, · · · , 1)T .

(29.3)

⎛

Clearly,

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_29

297

298

29 Approximate Boundary Synchronization

Then, the approximate boundary synchronization (29.1) can be equivalently rewritten as (29.4) C1 Un → 0 as n → +∞ in the space 0 1 ([T, +∞); (H0 ) N −1 ) ∩ Cloc ([T, +∞); (H−1 ) N −1 ). Cloc

(29.5)

Definition 29.2 The matrix A satisfies the condition of C1 -compatibility if there exists a unique matrix A1 of order (N − 1), such that C 1 A = A1 C 1 .

(29.6)

The matrix A1 is called the reduced matrix of A by C1 . Remark 29.3 The condition of C1 -compatibility (29.6) is equivalent to AKer(C1 ) ⊆ Ker(C1 ).

(29.7)

Then, noting (29.3), the vector e1 = (1, · · · , 1)T is an eigenvector of A, corresponding to the eigenvalue a given by a=

N

ai j , i = 1, · · · , N ,

(29.8)

j=1

where Nj=1 ai j is independent of i = 1, · · · , N . Equation (29.8) is called the rowsum condition, which is also equivalent to the condition of C1 -compatibility (29.6) or (29.7). Similarly, the matrix B satisfies the condition of C1 -compatibility if there exists a unique matrix B 1 of order (N − 1), such that C1 B = B 1 C1 ,

(29.9)

BKer(C1 ) ⊆ Ker(C1 ).

(29.10)

which is equivalent to the fact that

Moreover, the vector e = (1, · · · , 1)T is also an eigenvector of B, corresponding to the eigenvalue b given by b=

N

bi j , i = 1, · · · , N ,

j=1

where the row-sum

N j=1

bi j is independent of i = 1, · · · , N .

(29.11)

29.2 Fundamental Properties

299

29.2 Fundamental Properties Theorem 29.4 Assume that system (III) is approximately synchronizable. Then we necessarily have rank(R)  N − 1. Proof Otherwise, let Ker(RT ) = Span{E 1 , · · · , E d } with d > 1. Noting that dim Im(C1T ) + dim Ker(RT ) = N − 1 + d > N ,

(29.12)

there exists a unit vector E ∈ I m(C1T ) ∩ Ker(RT ). Let E = C1T x with x ∈ R N −1 . The approximate boundary synchronization (29.4) implies that (E, Un ) = (x, C1 Un ) → 0

(29.13)

as n → +∞ in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

On the other hand, since E ∈ Ker(RT ), we have E=

d

αr Er ,

(29.14)

r =1

where the coefficients α1 , · · · , αd are not all zero. By Lemma 26.1, Ker(RT ) is contained in Ker(D T ) and invariant for both A T and B T ; therefore, we still have (27.2) and (27.3). Thus, applying Er to problem (III) and (III0) and setting u r = (Er , Un ) for 1  r  d, we find again problem (27.9)–(27.10) with homogeneous boundary conditions. Noting that problem (27.9)–(27.10) is independent of n, it follows from (29.13) and (29.14) that d

αr u r (T ) ≡

r =1

d

αr u r (T ) ≡ 0.

(29.15)

r =1

Then, by well-posedness, it is easy to see that d

0 ) ≡ αr (Er , U

r =1

d

1 ) ≡ 0 αr (Er , U

(29.16)

r =1

1 ) ∈ (H0 ) N × (H−1 ) N . This yields 0 , U for any given initial data (U d r =1

αr Er = 0.

(29.17)

300

29 Approximate Boundary Synchronization

Because of the linear independence of the vectors E 1 , · · · , E d , we get a contradiction  α1 = · · · = αd = 0. Theorem 29.5 Assume that system (III) is approximately synchronizable under the minimal rank(R) = N − 1. Then, we have the following assertions: (i) There exists a vector E 1 ∈Ker(RT ), such that (E 1 , e1 )=1 with e1 =(1, · · · , 1)T . 1 ) ∈ (H0 ) N × (H−1 ) N , there exists a unique 0 , U (ii) For any given initial data (U scalar function u such that 0 1 u (k) n → u in Cloc ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 )

(29.18)

for all 1  k  N as n → +∞. (iii) The matrices A and B satisfy the conditions of C1 -compatibility (29.6) and (29.9), respectively. Proof (i) Noting that dim Ker(RT ) = 1, by Lemma 26.1, there exists a nonzero vector E 1 ∈ Ker(RT ), such that D T E 1 = 0,

A T E 1 = αE 1 and B T E 1 = β E 1 .

(29.19)

/ Im(C1T ). Otherwise, applying E 1 to problem (III) and (III0) with We claim that E 1 ∈ U = Un and H = Hn , and setting u = (E 1 , Un ), it follows that ⎧  ⎨ u − u + αu = 0 in (0, +∞) × , u=0 on (0, +∞) × 0 , ⎩ on (0, +∞) × 1 ∂ν u + βu = 0

(29.20)

with the following initial data: t =0:

0 ), u  = (E 1 , U 1 ) in . u = (E 1 , U

(29.21)

Suppose that E 1 ∈ Im(C1T ), there exists a vector x ∈ R N −1 , such that E 1 = C1T x. Then, the approximate boundary synchronization (29.4) implies (u(T ), u  (T )) = ((x, C1 Un (T )), (x, C1 Un (T ))) → (0, 0)

(29.22)

in the space H0 × H−1 as n → +∞. Then, since u is independent of n, we have u(T ) ≡ u  (T ) ≡ 0.

(29.23)

Thus, because of the well-posedness of problem (29.20)–(29.21), it follows that 0 ) = (E 1 , U 1 ) = 0 (E 1 , U

(29.24)

1 ) ∈ (H0 ) N × (H−1 ) N . This yields a contradiction 0 , U for any given initial data (U E 1 = 0.

29.2 Fundamental Properties

301

Since E 1 ∈ / Im(C1T ), noting that Im(C1T ) = (Span{e1 })⊥ , we have (E 1 , e1 ) = 0. Without loss of generality, we can take E 1 such  that (E 1 , e1 ) = 1. C 1 is invertible. Moreover, we have (ii) Since E 1 ∈ / I m(C1T ), the matrix E 1T 

   C1 0 e = . E 1T 1 1

(29.25)

       0 0 C1 C 1 Un → = u U = n E 1T (E 1 , Un ) u 1

(29.26)

Noting (29.4), we have 

as n → +∞ in the space 0 1 Cloc ([T, +∞); (H0 ) N ) ∩ Cloc ([T, +∞); (H−1 ) N ).

(29.27)

Then, noting (29.25), it follows that  Un =

C1 E 1T

−1 

C 1 Un E 1T Un





C1 →u E 1T

−1   0 = ue1 1

(29.28)

in the space (29.27), namely, (29.18) holds. (iii) Applying C1 to system (III) with U = Un and H = Hn , and passing to the limit as n → +∞, it follows from (29.4) and (29.28) that C1 Ae1 u = 0 in [T, +∞) × 

(29.29)

C1 Be1 u = 0 on [T, +∞) × 1 .

(29.30)

and 1 ), we have 0 , U We claim that at least for an initial data (U u ≡ 0 on [T, +∞) × 1 .

(29.31)

Otherwise, it follows from system (29.20) that ∂ν u ≡ u ≡ 0 on [T, +∞) × 1 ,

(29.32)

then, by Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]), we get u ≡ 0 1 ), namely, system (III) is approximately null control0 , U for all the initial data (U lable under the condition dim Ker(RT ) = 1. This contradicts Corollary 27.5. Then, it follows from (29.29) and (29.30) that C1 Ae = 0 and C1 Be = 0, which give the conditions of C1 -compatibility for A and B, respectively. The proof is complete. 

302

29 Approximate Boundary Synchronization

Remark 29.6 Theorem 29.5 (ii) indicates that under the minimal rank condition rank(D) = N − 1, the approximate boundary synchronization in the consensus sense (29.1) is actually that in the pinning sense (29.18) with the approximately synchronizable state u. Assume that both A and B satisfy the corresponding conditions of C1 compatibility, namely, there exist two matrices A1 and B 1 such that C1 A = A1 C1 and C1 B = B 1 C1 , respectively. Setting W = C1 U in problem (III) and (III0), we get the following reduced system: ⎧  ⎨ W − W + A1 W = 0 in (0, +∞) × , W =0 on (0, +∞) × 0 , ⎩ ∂ν W + B 1 W = C1 D H on (0, +∞) × 1

(29.33)

with the initial condition t =0:

0 , W  = C1 U 1 in . W = C1U

(29.34)

Since B is similar to a real symmetric matrix, by Proposition 2.21, so is its reduced matrix B 1 . Then, by Theorem 19.9, the reduced problem (29.33)–(29.34) is wellposed in the space (H0 ) N −1 × (H−1 ) N −1 . Accordingly, consider the reduced adjoint system ⎧ T  ⎪ ⎨ −  + A1  = 0 in (0, T ) × , =0 on (0, T ) × 0 , ⎪ T ⎩ ∂ν  + B 1  = 0 on (0, T ) × 1

(29.35)

with the C1 D-observation (C1 D)T  ≡ 0 on (0, T ) × 1 .

(29.36)

Obviously, we have Proposition 29.7 Under the conditions of C1 -compatibility for A and B, system (III) is approximately synchronizable if and only if the reduced system (29.33) is approximately null controllable, or equivalently if and only if the reduced adjoint system (29.35) is C1 D-observable. Theorem 29.8 Assume that A and B satisfy the conditions of C1 -compatibility (29.6) and (29.9), respectively. Assume furthermore that A T and B T admit a common eigenvector E 1 such that (E 1 , e1 ) = 1 with e1 = (1, · · · , 1)T . Let D be defined by Im(D) = (Span{E 1 })⊥ .

(29.37)

Then system (III) is approximate synchronizable. Moreover, we have rank(R) = N − 1.

29.2 Fundamental Properties

303

Proof Since (E 1 , e1 ) = 1, noting (29.37), we have e1 ∈ / Im(D) and Ker(C1 ) ∩ Im(D) = {0}. Therefore, by Proposition 2.11, we have rank(C1 D) = rank(D) = N − 1.

(29.38)

Thus, the reduced adjoint system (29.35) is C1 D-observable because of Holmgren’s uniqueness theorem. By Proposition 29.7, system (III) is approximate synchronizable. Noting (29.37), we have E 1 ∈ Ker(D T ). Moreover, since E 1 is a common eigenvector of A T and B T , we have E 1 ∈ Ker(RT ), hence dim Ker(RT )  1, namely, rank(R)  N − 1. On the other hand, since rank(R)  rank(D) = N − 1, we get rank(R) = N − 1. The proof is complete. 

Chapter 30

Approximate Boundary Synchronization by p-Groups

The approximate boundary synchronization by p-groups is introduced and studied in this chapter for system (III) with coupled Robin boundary controls.

30.1 Definitions Let p  1 be an integer and let 0 = n0 < n1 < n2 < · · · < n p = N

(30.1)

be integers such that n r − n r −1  2 for all 1  r  p. We rearrange the components of the state variable U into p-groups: (u (1) , · · · , u (n 1 ) ), (u (n 1 +1) , · · · , u (n 2 ) ), · · · , (u (n p−1 +1) , · · · , u (n p ) ).

(30.2)

Definition 30.1 System (III) is approximately synchronizable by p-groups at the 1 ) ∈ (H0 ) N × (H−1 ) N , there exists a 0 , U time T > 0 if for any given initial data (U M sequence {Hn } of boundary controls in L with compact support in [0, T ], such that the corresponding sequence {Un } of solutions to problem (III) and (III0) satisfies (l) 0 1 u (k) n − u n → 0 in Cloc ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 )

(30.3)

for n r −1 + 1  k, l  n r and 1  r  p as n → +∞.

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_30

305

306

30 Approximate Boundary Synchronization by p-Groups

Let Sr be the following (n r − n r −1 − 1) × (n r − n r −1 ) matrix: ⎛

1 ⎜0 ⎜ Sr = ⎜ . ⎝ ..

−1 1 .. .

0 −1 .. .

··· ··· .. .

0 0 .. .

⎞

⎟ ⎟ ⎟. ⎠ 0 0 · · · 1 −1

(30.4)

Let C p be the following (N − p) × N full row-rank matrix of synchronization by p-groups: ⎛ ⎞ S1 ⎜ S2 ⎟ ⎜ ⎟ . (30.5) Cp = ⎜ . .. ⎟ ⎝ ⎠ Sp For 1  r  p, setting

1, n r −1 + 1  i  n r , 0, otherwise,

(30.6)

Ker(C p ) = Span{e1 , e2 , · · · , e p }.

(30.7)

(er )i = it is clear that

Moreover, the approximate boundary synchronization by p-groups (30.3) can be equivalently rewritten as (30.8) C p Un → 0 as n → +∞ in the space 0 1 ([T, +∞); (H0 ) N − p ) ∩ Cloc ([T, +∞); (H−1 ) N − p ). Cloc

(30.9)

Definition 30.2 The matrix A satisfies the condition of C p -compatibility if there exists a unique matrix A p of order (N − p), such that C p A = A pC p.

(30.10)

The matrix A p is called the reduced matrix of A by C p . Remark 30.3 By Proposition 2.15, the condition of C p -compatibility (30.10) is equivalent to (30.11) AKer(C p ) ⊆ Ker(C p ). Moreover, the reduced matrix A p is given by A p = C p AC Tp (C p C Tp )−1 .

(30.12)

30.1 Definitions

307

Similarly, the matrix B satisfies the condition of C p -compatibility if there exists a unique matrix B p of order (N − p), such that C p B = B pC p,

(30.13)

BKer(C p ) ⊆ Ker(C p ).

(30.14)

which is equivalent to

Assume that A and B satisfy the conditions of C p -compatibility (30.10) and (30.13), respectively. Setting W = C p U in problem (III) and (III0), we get the following reduced system: ⎧  ⎨ W − W + A p W = 0 in (0, T ) × , W =0 on (0, T ) × 0 , ⎩ ∂ν W + B p W = C p D H on (0, T ) × 1

(30.15)

with the initial condition t =0:

0 , W  = C p U 1 in . W = C pU

(30.16)

Since B is similar to a real symmetric matrix, by Proposition 2.21, the reduced matrix B p is also similar to a real symmetric matrix. Then, by Theorem 19.19, the reduced problem (30.15)–(30.16) is well-posed in the space (H0 ) N − p × (H−1 ) N − p . Accordingly, consider the reduced adjoint system ⎧ T  ⎪ ⎨ −  + A p  = 0 in (0, +∞) × , =0 on (0, +∞) × 0 , ⎪ T ⎩ ∂ν  + B p  = 0 on (0, +∞) × 1

(30.17)

together with the C p D-observation (C p D)T  ≡ 0 on (0, T ) × 1 .

(30.18)

We have Proposition 30.4 Assume that A and B satisfy the conditions of C p -compatibility (30.10) and (30.13), respectively. Then system (III) is approximately synchronizable by p-groups if and only if the reduced system (30.15) is approximately null controllable, or equivalently if and only if the reduced adjoint system (30.17) is C p D-observable. Corollary 30.5 Under the conditions of C p -compatibility (30.10) and (30.13), if system (III) is approximately synchronizable by p-groups, we necessarily have the following rank condition: (30.19) rank(C p R) = N − p.

308

30 Approximate Boundary Synchronization by p-Groups

Proof Let R be the matrix defined by (26.2)–(26.3) corresponding to the reduced matrices A p , B p and D = C p D. Noting (30.10) and (30.13), we have r

s

r

s

A p B p D = A p B p C p D = C p Ar B s D,

(30.20)

R = C p R.

(30.21)

then

Under the assumption that system (III) is approximately synchronizable by p-groups, by Proposition 30.4, the reduced system (30.15) is approximately null controllable, then by Corollary 27.5, we have rank(R) = N − p, which together with (30.21) implies (30.19). 

30.2 Fundamental Properties Proposition 30.6 Assume that system (III) is approximately synchronizable by p-groups. Then, we necessarily have rank(R)  N − p. Proof Assume that dim Ker(RT )=d with d > p. Let Ker(RT )=Span{E 1 , · · · , E d }. Since dim Ker(RT ) + dim Im(C Tp ) = d + N − p > N , we have Ker(RT ) ∩ Im(C Tp ) = {0}. Hence, there exists a nonzero vector x ∈ R N −d and coefficients β1 , · · · , βd not all zero, such that d 

βr Er = C Tp x.

(30.22)

r =1

Moreover, by Lemma 26.1, we still have (27.2) and (27.3). Then, applying Er to problem (III) and (III0) with U = Un and H = Hn and setting u r = (Er , Un ) for 1  r  d, it follows that ⎧  d ⎨ u r − u r + s=1 αr s u s = 0 in (0, +∞) × , on (0, +∞) × 0 , ur = 0 d ⎩ on (0, +∞) × 1 ∂ν u r + s=1 βr s u s = 0

(30.23)

with the initial condition t =0:

0 ), u r = (Er , U 1 ) in . u r = (Er , U

(30.24)

30.2 Fundamental Properties

309

Noting (30.8), it follows from (30.22) that d 

βr u r = (x, C p Un ) → 0

(30.25)

r =1

as n → +∞ in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

Since the functions u 1 , · · · , u d are independent of n and of applied boundary controls Hn , it follows that d 

βr u r (T ) =

r =1

d 

βr u r (T ) = 0 in .

(30.26)

r =1

Then, it follows from the well-posedness of problem (30.23)–(30.24) that d 

0 ) = βr (Er , U

r =1

d 

1 ) = 0 βr (Er , U

(30.27)

r =1

1 ) ∈ (H0 ) N × (H−1 ) N . In particular, we get 0 , U for any given initial data (U d 

βr Er = 0,

(30.28)

r =1

then a contradiction: β1 = · · · = βd = 0, because of the linear independence of the  vectors E 1 , · · · , E d . The proof is achieved. Theorem 30.7 Let A and B satisfy the conditions of C p -compatibility (30.10) and (30.13), respectively. Assume that A T and B T admit a common invariant subspace V, which is bi-orthonormal to K er (C p ). Then, setting the boundary control matrix D by (30.29) Im(D) = V ⊥ , system (III) is approximately synchronizable by p-groups. Moreover, we have rank(R) = N − p. Proof Since V is bi-orthonormal to Ker(C p ), we have Ker(C p ) ∩ V ⊥ = Ker(C p ) ∩ Im(D) = {0},

(30.30)

therefore, by Proposition 2.11, we have rank(C p D) = rank(D) = N − p.

(30.31)

310

30 Approximate Boundary Synchronization by p-Groups

Thus, the C p D-observation (29.36) becomes the full observation  ≡ 0 on (0, T ) × 1 .

(30.32)

By Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]), the reduced adjoint system (30.17) is C p D-observable and the reduced system (30.15) is approximately null controllable. Then, by Proposition 30.4, the original system (III) is approximately synchronizable by p-groups. Noting that Ker(D T ) = V , by Lemma 26.1, it is easy to see that rank(R) = N − p. The proof is then complete.  Theorem 30.8 Assume that system (III) is approximately synchronizable by p-groups. Assume furthermore that rank(R) = N − p. Then, we have the following assertions: (i) Ker(RT ) is bi-orthonormal to Ker(C p ). 1 ) ∈ (H0 ) N × (H−1 ) N , there exist unique 0 , U (ii) For any given initial data (U scalar functions u 1 , u 2 , · · · , u p such that 0 1 −1 ()) u (k) n → u r in Cloc ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H

(30.33)

for n r −1 + 1  k  n r and 1  r  p as n → +∞. (iii) The coupling matrices A and B satisfy the conditions of C p -compatibility (30.10) and (30.13), respectively. Proof (i) We claim that Ker(RT ) ∩ Im(C Tp ) = {0}. Then, noting that Ker(RT ) and Ker(C p ) have the same dimension p and Ker(RT ) ∩ {Ker(C p )}⊥ = Ker(RT ) ∩ Im(C Tp ) = {0},

(30.34)

by Proposition 2.5, Ker(RT ) and Ker(C p ) are bi-orthonormal. Let Ker(RT ) = Span{E 1 , · · · , E p } and Ker(C P ) = Span{e1 , · · · , e p } such that (Er , es ) = δr s , r, s = 1, · · · , p.

(30.35)

Now we return to check that Ker(RT )∩Im(C Tp )={0}. If Ker(RT )∩Im(C Tp ) = {0}, then there exists a nonzero vector x ∈ R N − p and some coefficients β1 , · · · , β p not all zero, such that p  βr Er = C Tp x. (30.36) r =1

By Lemma 26.1, we still have (27.2) and (27.3) with d = p. For 1  r  p, applying Er to problem (III) and (III0) with U = Un and H = Hn , and setting u r = (Er , U ),

(30.37)

30.2 Fundamental Properties

311

it follows that ⎧  p ⎨ u r − u r + s=1 αr s u s = 0 in (0, +∞) × , on (0, +∞) × 0 , ur = 0  ⎩ p on (0, +∞) × 1 ∂ν u r + s=1 βr s u s = 0

(30.38)

with the initial condition t =0:

0 ), u r = (Er , U 1 ). u r = (Er , U

(30.39)

Noting (30.8), we have p 

βr u r = (x, C p Un ) → 0 as n → +∞

(30.40)

r =1

in the space 0 1 ([T, +∞); H0 ) ∩ Cloc ([T, +∞); H−1 ). Cloc

(30.41)

Since the functions u 1 , · · · , u p are independent of n and of applied boundary controls, we have p p   βr u r (T ) ≡ βr u r (T ) ≡ 0 in . (30.42) r =1

r =1

Then, it follows from the well-posedness of problem (30.38)–(30.39) that p 

0 ) = βr (Er , U

p 

r =1

1 ) = 0 in  βr (Er , U

(30.43)

r =1

1 ) ∈ (H0 ) N × (H−1 ) N . In particular, we get 0 , U for any given initial data (U p 

βr Er = 0,

(30.44)

r =1

then, a contradiction: β1 = · · · = β p = 0, because of the linear independence of the vectors E 1 , · · · , E p . (ii) Noting (30.8) and (30.37), we have ⎛ ⎞ ⎛ ⎞ ⎞ C p Un Cp 0 T T ⎜u1 ⎟ ⎜ E 1 Un ⎟ ⎜ E1 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ .. ⎟ Un = ⎜ .. ⎟ → ⎜ .. ⎟ ⎝.⎠ ⎝ . ⎠ ⎝ . ⎠ E Tp E Tp Un up ⎛

(30.45)

312

30 Approximate Boundary Synchronization by p-Groups

as n → +∞ in the space 0 1 ([T, +∞); (H0 ) N ) ∩ Cloc ([T, +∞); (H−1 ) N ), Cloc

(30.46)

where u , · · · , u p are given by (30.37). Since Ker(RT ) ∩ Im(C Tp ) = {0}, the matrix ⎛ ⎞1 Cp ⎜ E 1T ⎟ ⎜ ⎟ ⎜ .. ⎟ is invertible. Thus, it follows from (30.45) that there exists U such that ⎝ . ⎠ E Tp ⎛

⎞−1 ⎛ ⎞ Cp 0 ⎜ E 1T ⎟ ⎜ u 1 ⎟ ⎜ ⎟ ⎜ ⎟ Un → ⎜ .. ⎟ ⎜ . ⎟ =: U ⎝ . ⎠ ⎝ .. ⎠ E Tp

(30.47)

up

as n → +∞ in the space (30.46). Moreover, (30.8) implies that tT :

C p U ≡ 0 in .

Noting (30.7), (30.35) and (30.37), it follows that tT :

p p   (Er , U )er = u r er in . U= r =1

(30.48)

r =1

Noting (30.6), we get then (30.33). (iii) Applying C p to system (III) with U = Un and H = Hn , and passing to the limit as n → +∞, by (30.8), (30.47) and (30.48), it is easy to get that p 

C p Aer u r (T ) ≡ 0 in 

(30.49)

C p Ber u r (T ) ≡ 0 on 1 .

(30.50)

r =1

and

p  r =1

On the other hand, because of the time-invertibility, system (30.38) defines an isomorphism from (H1 ) p × (H0 ) p onto (H1 ) p × (H0 ) p for all t  0. Then, it follows from (30.49) and (30.50) that C p Aer = 0 and C p Ber = 0 for 1  r  p.

(30.51)

30.2 Fundamental Properties

313

We get thus the conditions of C p -compatibility for A and B, respectively. The proof is complete.  Remark 30.9 In general, the convergence (30.8) of the sequence {C p Un } does not imply the convergence of the sequence of solutions {Un }. In fact, we even don’t know if the sequence {Un } is bounded. However, under the rank condition rank(R) = N − p, the convergence (30.8) actually implies the convergence (30.33). Moreover, the functions u 1 , · · · , u p , called the approximately synchronizable state by p-groups, are independent of applied boundary controls. In this case, system (III) is approximately synchronizable by p-groups in the pinning sense, while that originally given by Definition 30.1 is in the consensus sense. Let D p be the set of all the boundary control matrices D which realize the approximate boundary synchronization by p-groups for system (III). In order to show the dependence on D, we prefer to write R D instead of R given in (26.3). Then, we may define the minimal rank as (30.52) N p = inf rank(R D ). D∈D p

Noting that rank(R D ) = N − dim Im(RTD ), because of Proposition 30.6, we have N p  N − p.

(30.53)

Np = N − p

(30.54)

Moreover, we have the following Corollary 30.10 The equality

holds if and only if the coupling matrices A and B satisfy the conditions of C p compatibility (30.10) and (30.13), respectively, and A T and B T possess a common invariant subspace, which is bi-orthonormal to Ker(C p ). Moreover, the approximate boundary synchronization by p-groups is in the pinning sense. Proof Assume that (30.54) holds. Then there exists a boundary control matrix D ∈ D p , such that dim Ker(RTD ) = p. By Theorem 30.8, the coupling matrices A and B satisfy the conditions of C p -compatibility (30.10) and (30.13), respectively, and Ker(RTD ) which, by Lemma 26.1, is bi-orthonormal to Ker(C p ), is invariant for both A T and B T . Moreover, the approximate boundary synchronization by p-groups is in the pinning sense. Conversely, let V be a subspace which is invariant for both A T and B T , and bi-orthonormal to Ker(C p ). Noting that A and B satisfy the conditions of C p compatibility (30.10) and (30.13), respectively, by Theorem 30.7, there exists a boundary control matrix D ∈ D p , such that dim Ker(RTD ) = p, which together with (30.53) implies (30.54).  Remark 30.11 When N p > N − p, the situation is more complicated. We don’t know if the conditions of C p -compatibility (30.10) and (30.13) are necessary, either if the approximate boundary synchronization by p-groups is in the pinning sense.

Chapter 31

Approximately Synchronizable States by p-Groups

When system (III) is approximately synchronizable by p-groups, the corresponding approximately synchronizable states by p-groups will be considered in this chapter. In Theorem 30.8, we have shown that if system (III) is approximately synchronizable by p-groups under the condition dim Ker(RT ) = p, then A and B satisfy the corresponding conditions of C p -compatibility, and Ker(RT ) is bi-orthonormal to Ker(C p ); moreover, the approximately synchronizable state by p-groups is independent of applied boundary controls. The following is the counterpart. Theorem 31.1 Let A and B satisfy the conditions of C p -compatibility (30.10) and (30.13), respectively. Assume that system (III) is approximately synchronizable by p-groups. If the projection of the solution U to problem (III) and (III0) on a subspace V of dimension p is independent of applied boundary controls, then V = Ker(RT ). Moreover, Ker(RT ) is bi-orthonormal to Ker(C p ). 0 = U 1 = 0, by Theorem 19.9, the linear map Proof Fixing U F : H →U 2 is continuous, then, infinitely differential from L loc (0, +∞; (L 2 (1 )) M ) to 0 1 N N Cloc ([0, +∞); (H0 ) ) ∩ Cloc ([0, +∞); (H−1 ) ). For any given boundary control 2  be the Fréchet derivative of U on H :  ∈ L loc (0, +∞; (L 2 (1 )) M ), let U H  = F  (0) H . U Then, it follows from problem (III) and (III0) that ⎧   + AU  = 0 in (0, +∞) × ,  − U U ⎪ ⎪ ⎨ U =0 on (0, +∞) × 0 ,  + BU  = DH  on (0, +∞) × 1 , ∂ U ⎪ ν ⎪ ⎩ =U  = 0 in . t =0: U © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_31

(31.1)

315

316

31 Approximately Synchronizable States by p-Groups

Let V = Span{E 1 , · · · , E p }. Then, the independence of the projection of U on the subspace V , with respect to the boundary controls, implies that ) ≡ 0 in (0, +∞) ×  (Er , U

for 1  r  p.

(31.2)

/ Im(C Tp ) for any given r with 1  r  p. Otherwise, there We first show that Er ∈ exists an r¯ with 1  r¯  p and a vector xr¯ ∈ R N − p , such that Er¯ = C Tp xr¯ . Then, it follows from (31.2) that ) = (xr¯ , C p U ). 0 = (Er¯ , U  is the solution to the reduced system (30.15) with H = H , which Since W = C p U is approximately controllable, we get thus xr¯ = 0, which contradicts Er¯ = 0. Thus, since dim Im(C Tp ) = N − p and dim(V ) = p, we have V ⊕ Im(C Tp ) = R N . Then, for any given r with 1  r  p, there exists a vector yr ∈ R N − p , such that A T Er =

p 

αr s E s + C Tp yr .

s=1

Noting (31.2) and applying Er to system (31.1), it follows that , A T Er ) = (U , C Tp yr ) = (C p U , Er ) = (U , yr ). 0 = (AU Once again, the approximate controllability of the reduced system (30.15) implies that yr = 0 for 1  r  p. Then, it follows that A T Er =

p 

αr s E s , 1  r  p.

s=1

So, the subspace V is invariant for A T . By the sharp regularity given in [30] (cf. also [29, 31]) on the problem of Neumann, we have improved the regularity of the solution to problem (III) and (III0). In fact, setting ⎧ ⎪ ⎨3/5 − ,  is a bounded smooth domain, α = 2/3, (31.3)  is a sphere, ⎪ ⎩ 3/4 − ,  is a parallelepiped, where  > 0 is a sufficiently small number, by Theorem 19.10, the trace 2α−1 ((0, +∞) × 1 )) N U |1 ∈ (Hloc

1 , H ). 0 , U with the corresponding continuous dependence with respect to (U

(31.4)

31 Approximately Synchronizable States by p-Groups

317

Next, noting (31.2) and applying Er (1  r  p) to the boundary condition on 1 in (31.1), we get ) = (Er , B U ). (D T Er , H 2α−1 2 ((0, +∞) × 1 ) is compactly embedded in L loc ((0, +∞) × Since 2α − 1 > 0, Hloc T 1 ), we get then D Er = 0 for all 1  r  p, namely,

V ⊆ Ker(D T ).

(31.5)

In particular, the subspace V is contained in Ker(D T ). Moreover, for 1  i  p, we have ) = 0 on (0, +∞) × 1 . (Er , B U

(31.6)

Now, let xr ∈ R N − p , such that B T Er =

p 

βr s E s + C Tp xr .

(31.7)

s=1

Noting (31.2) and inserting the expression (31.7) into (31.6), it follows that ) = 0 on (0, +∞) × 1 . (xr , C p U Once again, because of the approximate boundary controllability of the reduced system (30.15), we deduce that xr = 0 for 1  r  p. Then, we get B T Er =

p 

βr s E s , 1  r  p.

s=1

So, the subspace V is also invariant for B T . Finally, since dim(V ) = p, by Lemma 26.1 and Proposition 30.6, we have Ker(RT ) = V . Then, by assertion (i) of Theorem 30.8, Ker(RT ) is bi-orthonormal  to Ker(C p ). This achieves the proof. Let d be a column vector of D and contained in Ker(C p ). Then d will be canceled in the product matrix C p D, and therefore it cannot give any effect to the reduced system (30.15). However, the vectors in Ker(C p ) may play an important role for the approximate boundary controllability. More precisely, we have the following Theorem 31.2 Let A and B satisfy the conditions of C p -compatibility (30.10) and (30.13), respectively. Assume that system (III) is approximately synchronizable by p-groups under the action of a boundary control matrix D. Assume furthermore that e1 , · · · , e p ∈ Im(D),

(31.8)

318

31 Approximately Synchronizable States by p-Groups

where e1 , · · · , e p are given by (30.6). Then system (III) is actually approximately null controllable. Proof By Proposition 27.4, it is sufficient to show that the adjoint system (19.19) is D-observable. For 1  r  p, applying er to the adjoint system (19.19) and noting φr = (er , ), it follows that ⎧ p  ⎪ ⎨φr − φr + s=1 αsr φs = 0 in (0, +∞) × , on (0, +∞) × 0 , φr = 0 ⎪ p ⎩ on (0, +∞) × 1 , ∂ν φr + s=1 βsr φs = 0

(31.9)

where the constant coefficients αsr and βsr are given by Aer =

p  s=1

αsr es and Ber =

p 

βsr es , 1  r  p.

(31.10)

s=1

On the other hand, noting (31.8), the D-observation (27.1) implies that φr ≡ 0 on (0, T ) × 1

(31.11)

for 1  r  p. Then, by Holmgren’s uniqueness theorem (cf. Theorem 8.2 in [62]), we get (31.12) φr ≡ 0 in (0, +∞) ×  for 1  r  p. Thus,  ∈ Im(C Tp ), then we can write  = C Tp  and the adjoint system (19.19) becomes ⎧ T  T T T ⎪ ⎨C p  − C p  + A C p  = 0 in (0, +∞) × , on (0, +∞) × 0 , C Tp  = 0 ⎪ ⎩ T on (0, +∞) × 1 . C p ∂ν  + B T C Tp  = 0

(31.13)

Noting the conditions of C p -compatibility (30.10) and (30.13), it follows that ⎧ T T  ⎪ ⎨C p ( −  + A p ) = 0 in (0, +∞) × , on (0, +∞) × 0 C Tp  = 0 ⎪ T ⎩ T C p (∂ν  + B p ) = 0 on (0, +∞) × 1 .

(31.14)

Since the mapping C Tp is injective, we find again the reduced adjoint system (30.17). Accordingly, the D-observation (27.1) implies that D T  ≡ D T C Tp  ≡ 0 on [0, T ] × 1 .

(31.15)

31 Approximately Synchronizable States by p-Groups

319

Since system (III) is approximately synchronizable by p-groups under the action of the boundary control matrix D, by Proposition 30.4, the reduced adjoint system (30.17) for  is C p D-observable, therefore,  ≡ 0, then  ≡ 0. So, the adjoint system (19.19) is D-observable, then by Proposition 27.4, system (III) is approximately null controllable. 

Chapter 32

Closing Remarks

Some closing remarks including related literatures and prospects are given in this chapter.

32.1 Related Literatures The main contents of this monograph have been published or will be published in a series of papers written by the authors with their collaborators (cf. [3, 35–56]). For closing this monograph, let us comment on some related literatures. One motivation for studying the synchronization consists of establishing a weak exact boundary controllability in the case of fewer boundary controls. In order to realize the exact boundary controllability, because of its uniform character with respect to the state variables, the number of boundary controls should be equal to the degrees of freedom of the considered system. However, when the components of initial data are allowed to have different levels of energy, the exact boundary controllability by means of only one boundary control for a system of two wave equations was established in [65, 72], and for the cascade system of N wave equations in [2]. In [10], the authors established the controllability of two coupled wave equations on a compact manifold with only one local distributed control. Moreover, both the optimal time of controllability and the controllable spaces are given in the cases with the same or different wave speeds. The approximate boundary null controllability is more flexible with respect to the number of applied boundary controls. In [37, 53, 55], for a coupled system of wave equations with Dirichlet, Neumann or Robin boundary controls, some fundamental algebraic properties on the coupling matrices are used to characterize the unique continuation for the solution to the corresponding adjoint systems. Although these © Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8_32

321

322

32 Closing Remarks

criteria are only necessary in general, they open an important way to the research on the unique continuation for a system of hyperbolic partial differential equations. In contrast with hyperbolic systems, in [4] (also [5, 13] and the reference therein), it was shown that Kalman’s criterion is sufficient to the exact boundary null controllability for systems of parabolic equations. Recently, in [76], the authors have established the minimal time for a control problem related to the exact synchronization for a linear parabolic system. The average controllability proposed in [67, 86] gives another way to deal with the controllability with fewer controls. The observability inequality is particularly interesting for a trial on the decay rate of approximate controllability.

32.2 Prospects The study of exact and approximate boundary synchronizations for systems of PDEs has just begun, and there are still many problems worthy of further research. 1. We have discussed above a coupled system of wave equations with Dirichlet, Neumann, or coupled Robin boundary controls. In the case of coupled Robin boundary controls, since the solution has less trace regularity, we can only obtain a relatively complete result when the domain  is a parallelepiped. It is still an open problem to get a relevant result for more general domains. Moreover, besides the original coupling matrix A, there is one more boundary coupling matrix B, which brings more difficulties and opportunities to the study. 2. The above research is limited to the linear situation, and the method employed is also linear. However, the study of synchronization should be extended to the nonlinear situation. At present, in the one-space-dimensional case, for the coupled system of quasilinear wave equations with various boundary controls, the exact boundary synchronization has been achieved in the framework of classical solutions (cf. [20, 68]) and there is still larger developing space for more general research. 3. In addition to the study of exact boundary controllability of hyperbolic systems on the whole space domain (cf. [32, 62, 73]), in the recent years, due to the demand of applications, in the one-space-dimensional situation, the research on exact boundary controllability of hyperbolic systems at a node has been developed, which is called the exact boundary controllability of nodal profile (cf. [16, 33, 59]). Correspondingly, the investigation on the approximate boundary controllability of nodal profile as well as on the exact and approximate boundary synchronizations of nodal profile should also be carried further. 4. We can also probe into similar problems on a complicate domain formed by networks. The corresponding results in the one-space-dimensional case on the exact boundary controllability and on the exact boundary controllability of nodal profile can be found in [32, 59], etc., and the synchronization on a tree-like network for a coupled system of wave equations should be carried out. 5. Some essentially new results can be obtained if we study the phenomena of synchronization through coupling among individuals with possibly different motion

32.2 Prospects

323

laws (governing equations), whose nature is yet to be explored. The research on the existence of the exactly synchronizable state for a coupled system of wave equations with different wave speeds has been initiated. 6. The stability of the exactly synchronizable state or the approximately synchronizable state should be studied systematically. 7. It is worth to consider the generalized exact boundary synchronization (cf. [58] in the 1-D case and [77–80] in higher dimensional case) and the generalized approximate boundary synchronization. 8. The coupled system of first-order linear or quasilinear hyperbolic systems has wider implications and applications than the coupled system of wave equations, and to do the study of similar problems on it will be of great significance, though more difficult (cf. [68]). 9. To do similar research on the coupled system of other linear or nonlinear evolution equations (such as beam equations, plate equations, heat equations, etc.) will reveal many new properties and characteristics, which is also quite meaningful. 10. To extend the concept of synchronization to the case of components with different time delays will be more challenging and may expose quite different features. 11. The study above focuses on the exact or approximate synchronization for a coupled system of wave equations in finite time through boundary controls. It is also worthwhile to study the asymptotic synchronization in the linear and nonlinear situations without any boundary control when t → +∞ (cf. [56]). This should be a meaningful extension of the research on the asymptotic stability of the solution to the coupled system of wave equations. 12. To dig into more practical applications of related results will further enrich the new field of study and lead to greater promotion and influence of the subject.

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Index

A Adjoint problem, 10, 12, 35, 84, 87, 89, 94, 103, 108, 112, 113, 123, 130, 140, 156, 158, 167, 174, 202, 204, 208, 211, 213, 217, 225, 234, 236 Adjoint system, 4, 35, 87, 88, 90, 96, 151, 209, 219, 234, 281, 282, 302, 303, 310, 318 Adjoint variables, 156 Admissible set of all boundary controls, 38 Approximate boundary controllability of nodal profile, 322 Approximate boundary null controllability, 9–12, 16, 81, 86, 87, 97, 114, 124, 126, 130, 140, 148, 201, 202, 207, 209, 321 Approximate boundary synchronization, 9, 12, 106, 112, 217, 218, 297, 305, 313, 322 Approximate boundary synchronization by groups, 150 Approximate boundary synchronization by 2-groups, 147 Approximate boundary synchronization by p-groups, 13, 116, 119, 125, 133, 135, 138, 151, 223, 226, 305, 313 Approximately null controllable, 10, 11, 81, 83, 84, 86, 103, 106, 114, 130, 132, 204, 282, 301, 308, 310, 318 Approximately synchronizable, 12, 13, 106, 108, 110, 126, 215, 218, 219 Approximately synchronizable by p groups, 15, 117, 119, 130, 134, 226, 305, 315, 319

Approximately synchronizable state, 13, 218, 219, 302, 323 Approximately synchronizable state by p groups, 122, 124, 185, 315 Asymptotically synchronizable state, 3 Asymptotic stability, 323 Asymptotic synchronization, 2, 323 Asymptotic synchronization in the consensus sense, 3 Asymptotic synchronization in the pinning sense, 3 Attainable set, 7, 49, 69, 177, 184 Attainable set of exactly synchronizable states, 49, 177 Attainable set of exactly synchronizable states by p groups, 183 Average controllability, 322 B Backward problem, 39, 40, 47, 49, 82, 168, 177, 204 Banach’s theorem of closed graph, 40 Beam equations, 323 Bi-orthonormal, 20, 29, 67, 75, 112, 122, 140, 187, 192, 196, 197, 227, 267, 272, 309, 313 Bi-orthonormal basis, 54 Bi-orthonormality, 19, 123, 187 Boundary control, 3, 5, 6, 9–11, 13, 15, 34, 40, 44, 45, 47, 52, 57, 60, 63, 66, 70, 71, 75, 77, 83, 84, 87, 106, 123–125, 133, 138, 157, 170, 174, 175, 179, 181, 186–188, 197, 202, 204, 215, 219, 223, 243, 245, 247, 255, 258, 267, 283, 297, 305, 309, 319, 321

© Springer Nature Switzerland AG 2019 T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications 94, https://doi.org/10.1007/978-3-030-32849-8

329

330 Boundary control matrix, 2, 6, 9, 13, 14, 47, 53, 56, 63, 66, 67, 70, 77, 86, 91, 108, 123, 138, 141, 145, 171, 187, 196, 197, 219, 220, 226, 247, 257, 267, 271, 309, 317 Boundary coupling matrix, 17, 261, 286, 322

C C 0 groups, 37 C1 D-observability, 112 C1 D-observable, 12, 108, 217 C1 D-observation, 113, 221, 302 C p D-observability, 123 C p D-observable, 117, 131, 225, 310, 319 C p D-observation, 123, 307 Cq∗ D-observability, 141 Cq∗ D-observable, 141 Cq∗ D-observation, 140 Carleman’s unique continuation, 36 Cascade matrix, 89, 146 Cascade system, 11, 171, 212, 287, 321 Cauchy–Schwarz inequality, 289 Cayley–Hamilton’s theorem, 24, 26, 118, 277 Classical solutions, 4, 38, 322 Compactness-uniqueness argument, 156, 166 Compact operator, 37 Compact perturbation, 4, 166, 170, 242 Compact support, 3, 8, 10, 12, 13, 34, 43, 59, 66, 81, 83, 105, 115, 125, 127, 134, 174, 179, 201, 203, 240, 243, 282, 297 Condition of C1 -compatibility, 6, 12, 107, 126, 176, 177, 192, 216, 218, 219, 226, 307 Condition of C2 -compatibility, 146, 263 Condition of C3 -compatibility, 193 Condition of C p -compatibility, 8, 14, 25, 63, 68–71, 75, 117, 119, 127, 131, 133, 135, 138, 182, 187, 188, 225, 226, 259, 306 Condition of compatibility, 5, 45, 48, 49, 53, 61, 63, 107, 175, 182 Consensus sense, 2, 13, 14, 16, 112, 122, 147, 218, 227, 313 Conservation of energy, 288 Controllability, 5, 9, 81, 321 Controllable part, 51, 74, 190, 191 Cosine operator, 232 Coupled Robin boundary conditions, 2, 17, 231, 234, 261

Index Coupled Robin boundary controls, 17, 171, 240, 243, 244, 322 Coupled system of 1-D wave equations, 38 Coupled system of first-order linear or quasilinear hyperbolic systems, 323 Coupled system of quasilinear wave equations, 322 Coupled system of wave equations, 2, 4, 16, 33, 42, 69, 174, 183, 207, 234, 321, 322 Coupled system of wave equations with different wave speeds, 323 Coupling matrix, 2, 4, 5, 8, 11, 12, 17, 34, 44, 49, 61, 66, 73, 86, 90, 100, 108, 119, 135, 144, 145, 156, 165, 171, 180, 182, 186, 207, 216, 218, 220, 224, 259, 263, 287, 322 Criterion of Kalman’s type, 217, 226

D D0 -observable, 91, 95 D-observability, 10, 12, 82, 89, 97, 202, 208, 281, 285 D-observable, 10, 84, 85, 87, 91, 96, 101, 103, 128, 202, 205, 212, 281, 319 D-observation, 85, 92, 131, 211, 281, 282, 318 Diagonalizable by blocks, 22, 54, 112, 123 Diagonalizable systems, 97 Dimension equality, 19 Direct boundary controls, 113, 124 Direct inequality, 170 Dirichlet boundary conditions, 2, 17, 231 Dirichlet boundary controls, 16, 59, 157, 171, 173, 185, 189, 207, 209 Dirichlet observation, 212

E Energy, 160, 171, 288 Energy estimates, 160 Energy space, 37, 157, 166, 171 Exact boundary controllability, 34, 128, 157, 166, 231, 239, 243, 249, 269, 321, 322 Exact boundary controllability of nodal profile, 322 Exact boundary null controllability, 4, 6, 10, 34, 40, 44, 48, 52, 57, 64, 66, 72, 86, 113, 166, 176, 201, 246, 258 Exact boundary synchronization, 3, 4, 7, 12, 14, 44, 47, 48, 52, 59, 75, 107, 174,

Index 176, 179, 194, 216, 231, 243, 247, 322 Exact boundary synchronization by groups, 59 Exact boundary synchronization by 2groups, 76, 263 Exact boundary synchronization by 3groups, 194 Exact boundary synchronization by pgroups, 7, 61, 63, 66, 69, 75, 179, 185, 186, 258, 267 Exact controllability of systems of ODEs, 87 Exactly controllable, 157, 239, 242, 269 Exactly null controllable, 34, 41, 55, 59, 66, 168, 175, 183, 185, 192, 242, 244, 258 Exactly synchronizable, 5, 44, 47, 59, 174, 176, 177, 191, 193, 243, 246, 247, 252 Exactly synchronizable by 2-groups, 12, 53, 263 Exactly synchronizable by p-groups, 8, 64, 66, 74, 180, 183, 186, 191, 255, 256, 258, 262, 267, 271 Exactly synchronizable state, 3, 7, 51, 57, 174, 175, 177, 192, 244, 247 Exactly synchronizable state by 2-groups, 77 Exactly synchronizable state by 3-groups, 196 Exactly synchronizable state by p-groups, 8, 69, 179, 183, 186, 187, 193, 255, 267, 271 Existence of the exactly synchronizable state, 323 F Fewer boundary controls, 5, 10, 59, 157, 169, 241, 247, 321 Finite-dimensional dynamical systems of ordinary differential equations, 2 Finite speed of wave propagation, 87 Fractional power, 232 Framework of classical solutions, 4, 38, 322 Fréchet derivative, 249, 315 Fredholm’s alternative, 164 G Gap condition, 114, 124 Generalized approximate boundary synchronization, 323 Generalized exact boundary synchronization, 323

331 Generalized Ingham’s inequality (Inghamtype theorem), 97 Geometrical multiplicity, 143, 144

H Hautus test, 24 Heat equations, 323 Hidden regularity, 128, 231 Higher dimensional case, 101, 287, 294, 323 Hilbert basis, 101, 294 Hilbert Uniqueness Method (HUM), 4, 34, 155, 166 Holmgren’s uniqueness theorem, 10, 85, 89, 113, 123, 131, 159, 206, 217, 245, 262, 282, 286, 301, 310, 318

I Induced approximate boundary synchronization, 15, 133, 138, 148 Induced approximately synchronizable, 16, 133, 137, 148 Induced extension matrix, 15, 27, 133 Infinite-dimensional dynamical systems of partial differential equations (PDEs), 2 Ingham’s theorem, 98 Internal coupling matrix, 259, 261 Interpolation inequality, 235 Invariant subspace, 6, 22–24, 38, 46, 53, 70, 73, 76, 86, 102, 107, 110, 118, 135, 139, 143, 164, 187, 197, 212, 224, 227 Inverse inequality, 170

J Jordan block, 89, 222 Jordan chain, 27, 50, 76, 134, 143, 192, 194 Jordan’s theorem, 28 Jordan subspaces, 28

K Kalman’s criterion, 11, 14, 23, 86, 89, 96, 97, 102, 114, 117, 124, 138, 145, 171, 207, 211, 213, 277, 322 Kalman’s rank condition, 24

L Lack of boundary controls, 5, 43, 103, 173, 242, 255

332 Largest geometrical multiplicity of the eigenvalues, 143 Lax–Milgram’s Lemma, 169 Linear parabolic system, 322 Lions’ compact embedding theorem, 242 M Marked subspace, 26, 74, 77, 194 Matrix of synchronization by 2-groups, 76 Matrix of synchronization by p-groups, 8, 60, 116, 119, 180, 224, 256, 257, 306 Maximal dissipative, 232 Minimal number of total controls, 15, 108, 109, 119, 135, 139, 142, 147, 218 Minimal rank condition, 13–15, 65, 111, 112, 119, 133, 139, 222, 226, 302 Mixed initial-boundary value problem, 60, 174 Moore–Penrose (generalized) inverse, 25, 46 Multiplier approach, 287 Multiplier geometrical condition, 4, 9, 11, 33, 42, 48, 68, 85, 91, 95, 141, 158, 206, 221, 289 N Networks, 322 Neumann, 2, 17, 316, 321 Neumann boundary condition, 174, 189, 231, 237, 287 Neumann boundary controls, 17, 78, 157, 170, 181, 189, 201, 207, 240, 255 Nilpotent, 89 Nilpotent matrix, 89, 141 Nilpotent system, 11, 89, 113 Non-approximate boundary null controllability, 107 Non-exact boundary controllability, 33, 39, 157, 169, 170, 240, 243, 245 Non-exact boundary null controllability, 33, 45 Non-exact boundary synchronization, 43, 47, 173, 176 Non-exact boundary synchronization by pgroups, 182 Nonharmonic series, 97 Number of direct boundary controls, 11, 87, 207 Number of indirect controls, 87 Number of the total (direct and indirect) controls, 11, 118, 133, 138, 142, 151, 207, 218, 223, 226

Index O Observability, 37, 156, 166 Observability inequality, 4, 158, 160 Observability of compactly perturbed systems, 34 Observation, 85, 87, 89, 95, 101, 103, 113, 123, 128, 140, 206, 209, 213, 217, 225, 281, 307, 310 Observation of finite horizon, 292 Observation on the infinite horizon/ time interval, 208, 265, 288 ω-linearly independent, 97, 98, 295 One-space-dimensional case, 4, 97, 100, 209, 292, 322 One-space-dimensional systems, 11, 113, 124 Orthogonal complement, 21, 166, 204

P Parallelepiped, 232, 239–241, 244, 256, 260, 316, 322 Phenomenon of synchronization, 1 Pinning sense, 2, 13, 14, 16, 112, 122, 123, 138, 218, 302, 313 Plate equations, 323

R Rank-nullity theorem, 23, 278 Real symmetric matrix, 29, 234, 236, 239, 242, 244, 251, 263, 267, 286, 302, 307 Reduced adjoint problem, 12, 108, 123, 131, 140, 217, 225 Reduced matrix, 6, 8, 14, 141, 144, 146, 246, 302, 306 Reduced matrix of A by C1 , 6, 46, 107, 175, 298 Reduced matrix of A by C p , 25, 65, 116, 225, 306 Reduced system, 6, 12, 16, 48, 57, 64, 66, 72, 107, 113, 117, 123, 132, 140, 147, 176, 183, 185, 216, 221, 246, 249, 269, 302, 307, 308, 310, 316 Rellich’s identity, 288 Riesz basis, 37, 101, 166, 210 Riesz–Fréchet’s representation theorem, 167, 236 Riesz sequence, 98 Robin boundary conditions, 2, 17, 231, 234, 261, 287

Index Row-sum condition, 5, 45, 65, 106, 174, 176, 182, 216, 298 Row-sum condition by blocks, 8, 44, 65, 182, 261

S Schur’s theorem, 159 Semi-simple, 26 Skew-adjoint operator with compact, 37 Stability of the exactly synchronizable state, 323 State variables, 2–4, 7, 11, 15, 86, 87, 157, 171, 239, 321 Stronger version of the approximate boundary synchronization by p-groups, 125 Strongly marked subspace, 26 Supplement, 14, 15, 19, 73, 112, 123, 133, 137, 139, 145, 227 Synchronizable part, 52–54, 74, 190–193 Synchronizable state, 45, 122 Synchronization, 1, 2, 4–7, 44, 45, 321, 322 Synchronization for systems governed by ODEs, 7 Synchronization matrix, 6, 46, 105, 109, 215, 218, 297 Synchronization on a tree-like network, 322 System of root vectors, 29, 37, 38, 141, 147, 149

333 Systems described by ordinary differential equations (ODEs), 1 Systems of parabolic equations, 322

T 2 × 2 systems, 11, 95, 113, 124 Total (direct and indirect) controls, 13, 14, 86, 87, 108, 109, 118, 119, 133, 138, 141, 147, 207, 218, 226

U Unique continuation, 35, 85, 97, 212, 265, 285–287 Upper density of a sequence, 98, 102

V Variational form, 235 Variational problem, 235

W Weak exact boundary controllability, 321

Z Zero-sum condition by blocks, 261