Boundary Synchronization for Hyperbolic Systems (Progress in Nonlinear Differential Equations and Their Applications)

Description

Within this carefully presented monograph, the authors extend the universal phenomenon of synchronization from finite-dimensional dynamical systems of ordinary differential equations (ODEs) to infinite-dimensional dynamical systems of partial differential equations (PDEs). By combining synchronization with controllability, they introduce the study of synchronization to the field of control and add new perspectives to the investigation of synchronization for systems of PDEs. With a focus on synchronization for a coupled system of wave equations, the text is divided into three parts corresponding to Dirichlet, Neumann, and coupled Robin boundary controls. Each part is then subdivided into chapters detailing exact boundary synchronization and approximate boundary synchronization, respectively. The core intention is to give artificial intervention to the evolution of state variables through appropriate boundary controls for realizing the synchronization in a finite time, creating a novel viewpoint into the investigation of synchronization for systems of partial differential equations, and revealing some essentially dissimilar characteristics from systems of ordinary differential equations.  Primarily aimed at researchers and graduate students of applied mathematics and applied sciences, this text will particularly appeal to those interested in applied PDEs and control theory for distributed parameter systems.  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