Observation and Control for Operator Semigroups

By: George Weiss, Marius Tucsnak

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Hardcover | 13 March 2009

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The evolution of the state of many systems modeled by linear partial di?erential equations (PDEs) or linear delay di?erential equations can be described by ope- torsemigroups.Thestate ofsucha systemis anelementin anin?nite-dimensional normed space, whence the name "in?nite-dimensional linear system". The study of operator semigroups is a mature area of functional analysis, which is still very active. The study of observation and control operators for such semigroups is relatively more recent. These operators are needed to model the - teraction of a system with the surrounding world via outputs or inputs. The main topicsofinterestaboutobservationandcontroloperatorsareadmissibility,obse- ability, controllability, stabilizability and detectability. Observation and control operators are an essential ingredient of well-posed linear systems (or more gen- ally system nodes). In this book we deal only with admissibility, observability and controllability. We deal only with operator semigroups acting on Hilbert spaces. This book is meant to be an elementary introduction into the topics m- tioned above. By "elementary" we mean that we assume no prior knowledge of ?nite-dimensional control theory, and no prior knowledge of operator semigroups or of unbounded operators. We introduce everything needed from these areas. We do assume that the reader has a basic understanding of bounded operators on Hilbert spaces, di?erential equations, Fourier and Laplace transforms, dist- butions and Sobolev spaces on n-dimensional domains. Much of the background needed in these areas is summarized in the appendices, often with proofs.

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From the reviews:

"This book deals with the observation and control of linear systems described by semigroups of operators in Hilbert spaces. ... In particular, in this book the authors are able to use their ideas to establish the exact observability or exact controllability of various systems. ... The book is written in an appropriate style for graduate students, mathematicians with interests in control theory and, also, researchers in control engineering." (Enrique Fernandez Cara, Mathematical Reviews, Issue 2010 d)

"This book, written at the level of a graduate textbook, is devoted to observation and control operators for operator semigroups. ... provide a self-contained introduction into the topics admissibility, observability and controllability. ... Each chapter concludes with a section containing bibliographic notes and further results. ... In summary, the book is a welcome addition to infinite-dimensional systems theory. The book is suitable for researchers in other areas who wish to enter the field and also for specialists in the field as a general reference." (Birgit Jacob, Zentralblatt MATH, Vol. 1188, 2010)

"This book is an introduction for applied mathematics and control graduate students and researchers interested in systems whose dynamics can be described by an operator semigroup on a Hilbert space. The book aims to unify two traditionally different approaches, one using abstract functional analysis and the other based on partial differential equation techniques such as multipliers, Ingham-Beurling-type inequalities, and Carleman estimates." (IEEE Control Systems Magazine, Vol. 30, June, 2010)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Differential Calculus & Equations Applied Mathematics Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory

Preface	p. ix
Observability and Controllability for Finite-dimensional Systems
Norms and inner products	p. 1
Operators on finite-dimensional spaces	p. 5
Matrix exponentials	p. 7
Observability and controllability for finite-dimensional linear systems	p. 11
The Hautus test and Gramians	p. 15
Operator Semigroups
Strongly continuous semigroups and their generators	p. 20
The spectrum and the resolvents of an operator	p. 24
The resolvents of a semigroup generator and the space	p. 28
Invariant subspaces for semigroups	p. 33
Riesz bases	p. 36
Diagonalizable operators and semigroups	p. 40
Strongly continuous groups	p. 47
The adjoint semigroup	p. 53
The embeddings	p. 56
The spaces and	p. 59
Bounded perturbations of a generator	p. 65
Semigroups of Contractions
Dissipative and m-dissipative operators	p. 69
Self-adjoint operators	p. 73
Positive operators	p. 78
The spaces and	p. 81
Sturm-Liouville operators	p. 88
The Dirichlet Laplacian	p. 92
Skew-adjoint operators	p. 98
The theorems of Lumer-Phillips and Stone	p. 102
The wave equation with boundary damping	p. 106
Control and Observation Operators
Solutions of non-homogeneous equations	p. 112
Admissible control operators	p. 116
Admissible observation operators	p. 121
The duality between the admissibility concepts	p. 126
Two representation theorems	p. 128
Infinite-time admissibility	p. 134
Remarks and bibliographical notes on Chapter 4	p. 136
Testing Admissibility
Gramians and Lyapunov inequalities	p. 139
Admissible control operators for left-invertible semigroups	p. 144
Admissibility for diagonal semigroups	p. 147
Some unbounded perturbations of generators	p. 157
Admissible control operators for perturbed semigroups	p. 164
Remarks and bibliographical notes on Chapter 5	p. 168
Observability
Some observability concepts	p. 173
Some examples based on the string equation	p. 179
Robustness of exact observability with respect to admissible perturbations of the generator	p. 184
Simultaneous exact observability	p. 190
A Hautus-type necessary condition for exact observability	p. 194
Hautus-type tests for exact observability with a skew-adjoint generator	p. 197
From to	p. 200
From first- to second-order equations	p. 205
Spectral conditions for exact observability with a skew-adjoint generator	p. 211
The clamped Euler-Bernoulli beam with torque observation at an endpoint	p. 217
Remarks and bibliographical notes on Chapter 6	p. 220
Observation for the Wave Equation
An admissibility result for boundary observation	p. 226
Boundary exact observability	p. 231
A perturbed wave equation	p. 234
The wave equation with distributed observation	p. 240
Some consequences for the Schrödinger and plate equations	p. 247
The wave equation with boundary damping and boundary velocity observation	p. 251
Remarks and bibliographical notes on Chapter 7	p. 257
Non-harmonic Fourier Series and Exact Observability
A theorem of Ingham	p. 261
Variable coefficients PDEs in one space dimension with boundary observation	p. 266
Domains associated with a sequence	p. 270
The results of Kahane and Beurling	p. 276
The Schrödinger and plate equations in a rectangular domain with distributed observation	p. 280
Remarks and bibliographical notes on Chapter 8	p. 284
Observability for Parabolic Equations
Preliminary results	p. 287
From to	p. 289
Final state observability with geometric conditions	p. 295
A global Carleman estimate for the heat operator	p. 298
Final state observability without geometric conditions	p. 312
Remarks and bibliographical notes on Chapter 9	p. 314
Boundary Control Systems
What is a boundary control system?	p. 317
Two simple examples in one space dimension	p. 322
A one-dimensional heat equation with Neumann boundary control	p. 323
A string equation with Neumann boundary control	p. 324
A string equation with variable coefficients	p. 326
An Euler-Bernoulli beam with torque control	p. 330
An Euler-Bernoulli beam with angular velocity control	p. 334
The Dirichlet map on an n-dimensional domain	p. 337
Heat and Schrödinger equations with boundary control	p. 341
The convection-diffusion equation with boundary control	p. 344
The wave equation with Dirichlet boundary control	p. 347
Remarks and bibliographical notes on Chapter 10	p. 352
Controllability
Some controllability concepts	p. 355
The duality between controllability and observability	p. 357
Simultaneous controllability and the reachable space with inputs	p. 364
An example of a coupled system	p. 371
Null-controllability for heat and convection-diffusion equations	p. 375
Boundary controllability for Schrödinger and wave equations	p. 378
Boundary controllability for the Schrödinger equation	p. 379
Boundary controllability for the wave equation	p. 380
Remarks and bibliographical notes on Chapter 11	p. 381
Appendix I
The closed-graph theorem and some consequences	p. 385
Compact operators	p. 387
The square root of a positive operator	p. 391
The Fourier and Laplace transformations	p. 394
Banach space-valued functions	p. 399
Appendix II
Test functions	p. 404
Distributions on a domain	p. 408
The operators div, grad, rot and ¿	p. 412
Definition and first properties of Sobolev spaces	p. 415
Regularity of the boundary and Sobolev spaces on manifolds	p. 420
Trace operators and the space (¿)	p. 424
The Green formulas and extensions of trace operators	p. 430
Appendix III
Critical points and Sard's theorem	p. 435
Existence of Morse functions on ¿	p. 437
Proof of Theorem 9.4.3	p. 441
Appendix IV
A Carleman estimate for elliptic operators	p. 445
The unique continuation results	p. 454
Bibliography	p. 459
List of Notation	p. 475
Index	p. 477
Table of Contents provided by Ingram. All Rights Reserved.

Observation and Control for Operator Semigroups

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Hardcover

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