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E-raamat: Optimal Boundary Control and Boundary Stabilization of Hyperbolic Systems

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Optimal Boundary Control and Boundary Stabilization of Hyperbolic SystemsSuurem pilt
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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.

Arvustused

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 Aniculesei, zbMATH 1328.49001, 2016)

1 Introduction
1(2)
2 Systems governed by the wave equation
3(26)
2.1 Dirichlet boundary control
4(12)
2.2 Neumann boundary control
16(6)
2.3 Robin boundary control
22(7)
3 Exact Controllability
29(18)
3.1 Dirichlet boundary control
30(9)
3.2 Neumann boundary control
39(6)
3.2.1 A traveling waves solution
39(5)
3.2.2 Exact Controllability with Neumann boundary control
44(1)
3.3 Robin boundary control
45(2)
4 Optimal Exact Control
47(22)
4.1 Optimal Dirichlet control
47(7)
4.2 Optimal Neumann control
54(9)
4.3 An example for the destabilizing effect of delay
63(6)
5 Boundary Stabilization
69(20)
5.1 Telegraph equation
69(4)
5.2 Neumann control
73(6)
5.3 Time Delay
79(10)
5.3.1 Definition of System S
79(1)
5.3.2 Well-posedness of System S
79(3)
5.3.3 Exponential Stability of System S
82(3)
5.3.4 Destabilization of System S by small delays
85(4)
6 Nonlinear Systems
89(38)
6.1 The Korteweg-de Vries Equation (KdV)
89(16)
6.1.1 Well-posedness of the linearized system
90(4)
6.1.2 A traveling waves solution for the linearized system
94(1)
6.1.3 Well-posedness for the nonlinear system
95(2)
6.1.4 A traveling wave solution for the nonlinear system
97(2)
6.1.5 The linearized system with critical length: An example for a system that is not exactly controllable
99(6)
6.2 The isothermal Euler equations
105(3)
6.3 An initial boundary value problem for the Burgers equation
108(4)
6.4 The Burgers equation with source term
112(15)
7 Distributions
127(8)
7.1 Distributional derivatives
131(4)
Bibliography 135(4)
Index 139
Martin Gugat is Professor in the Department of Mathematics at Friedrich-Alexander-University, Erlangen-Nürnberg, Germany.
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