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).

Introduction and Overview.- Preliminaries.- Part 1: Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls: Exact Boundary Synchronization.- Exact boundary controllability and non-exact boundary controllability.- Exact boundary synchronization and non-exact boundary synchronization.- Exactly synchronizable states.- Exact boundary synchronization by groups.- Exactly synchronizable states by p-groups.- Part 2: Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls: Approximate Boundary Synchronization.- Approximate boundary synchronization.- Approximate boundary synchronization by p-groups.- Induced approximate boundary synchronization.- Part 3: Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls: Exact Boundary Synchronization.- Exact boundary controllability and non-exact boundary controllability.- Exact boundary synchronization and non-exactly boundary synchronization.- Exact boundary synchronization by p-groups.- Determination of exactly synchronizable states by p-groups.- Part 4: Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls: Approximate Boundary Synchronization.- Approximate boundary null controllability.- Approximate boundary synchronization.- Approximate Boundary Synchronization by p-groups.- Part 5: Synchronization for a Coupled System of Wave Equations with Coupled Robin Boundary Controls: Exact Boundary Synchronization.- Preliminaries on problem (III) and (III0).- Exact boundary controllability and non-exact boundary controllability.- Exact boundary synchronization.- Determination of exactly synchronizable states.- Exact boundary synchronization by p-groups.- Necessity of the conditions of Cp-compatibility.- Determination of exactly synchronizable states by p-groups.- Part 6. Synchronization for a Coupled System of Wave Equations with Coupled Boundary Controls: Approximate Boundary Synchronization.- Some algebraic lemmas.- Approximate boundary null controllability.- Unique continuation for Robin problem.- Approximate boundary synchronization.- Approximate boundary synchronization by p-groups.- Approximately synchronizable states by p-groups.- Closing remarks.

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.

“The book is well organized and presents the most important notions of boundary synchronization for hyperbolic systems. The book is suitable for senior undergraduate and graduate students as well as practical engineers, scientist and researchers interested in boundary synchronization for hyperbolic systems.” (Seenith Sivasundaram, zbMATH 1443.93002, 2020)

Opens a wide research subject by incorporating the study of synchronization into the field of control Introduces a complete theory for treating the exact and approximate boundary synchronizations for a coupled system of wave equations Outlines the solutions not only for exact boundary synchronization, but also for situations where there is further lack of boundary controls for approximate boundary synchronization