Observation and Control for Operator Semigroups: (Birkhauser Advanced Texts / Basler Lehrbucher)

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.

Table of Contents:
Observability and Controllability for Finite-dimensional Systems.- Operator Semigroups.- Semigroups of Contractions.- Control and Observation Operators.- Testing Admissibility.- Observability.- Observation for the Wave Equation.- Non-harmonic Fourier Series and Exact Observability.- Observability for Parabolic Equations.- Boundary Control Systems.- Controllability.- Appendix I: Some Background on Functional Analysis.- Appendix II: Some Background on Sobolev Spaces.- Appendix III: Some Background on Differential Calculus.- Appendix IV: Unique Continuation for Elliptic Operators.

Review :
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 Fernández 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)