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This brief considers recent results on optimal control and stabilization of systems governed by hyperbolic partial differential equations, specifically those in which the control action takes place at the boundary. The wave equation is used as a typical example of a linear system, through which the author explores initial boundary value problems, concepts of exact controllability, optimal exact control, and boundary stabilization. Nonlinear systems are also covered, with the Korteweg-de Vries and Burgers Equations serving as standard examples. To keep the presentation as accessible as possible, the author uses the case of a system with a state that is defined on a finite space interval, so that there are only two boundary points where the system can be controlled. Graduate and post-graduate students as well as researchers in the field will find this to be an accessible introduction to problems of optimal control and stabilization.
List of contents
Introduction.- Systems that are Governed by the Wave Equation.- Exact Controllability.- Optimal Exact Control.- Boundary Stabilization.- Nonlinear Systems.- Distributions.- Index
About the author
Martin Gugat is Professor in the Department of Mathematics at Friedrich-Alexander-University, Erlangen-Nürnberg, Germany.
Summary
This brief considers recent results on optimal control and stabilization of systems governed by hyperbolic partial differential equations, specifically those in which the control action takes place at the boundary. The wave equation is used as a typical example of a linear system, through which the author explores initial boundary value problems, concepts of exact controllability, optimal exact control, and boundary stabilization. Nonlinear systems are also covered, with the Korteweg-de Vries and Burgers Equations serving as standard examples. To keep the presentation as accessible as possible, the author uses the case of a system with a state that is defined on a finite space interval, so that there are only two boundary points where the system can be controlled. Graduate and post-graduate students as well as researchers in the field will find this to be an accessible introduction to problems of optimal control and stabilization.
Additional text
“The book presents the subject of controlling and stabilizing PDE systems in a didactic manner, detailing the computations. The book is very well organized … . This textbook will be useful for graduate and Ph.D. students in mathematics and engineering, interested in the subject … . In addition, the Bibliography contains some of the classical references in the literature regarding control and stabilization.” (Valéria N. Domingos Cavalcanti, Mathematical Reviews, August, 2016)
“The book under review treats optimal boundary control problems and stabilizability, where the system dynamics are governed by hyperbolic partial differential equations. … The book is written in an understandable style. The contents of the book, along with several exercises and references, make it an interesting and useful text for a wide group of mathematicians andengineers.” (Gheorghe Aniculăesei, zbMATH 1328.49001, 2016)
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"The book presents the subject of controlling and stabilizing PDE systems in a didactic manner, detailing the computations. The book is very well organized ... . This textbook will be useful for graduate and Ph.D. students in mathematics and engineering, interested in the subject ... . In addition, the Bibliography contains some of the classical references in the literature regarding control and stabilization." (Valéria N. Domingos Cavalcanti, Mathematical Reviews, August, 2016)
"The book under review treats optimal boundary control problems and stabilizability, where the system dynamics are governed by hyperbolic partial differential equations. ... The book is written in an understandable style. The contents of the book, along with several exercises and references, make it an interesting and useful text for a wide group of mathematicians andengineers." (Gheorghe Aniculaesei, zbMATH 1328.49001, 2016)
