Observation and Control for Operator Semigroups

This book studies observation and control operators for linear systems where the free evolution of the state can be described by an operator semigroup on a Hilbert space. It includes a large number of examples coming mostly from partial differential equations.

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.

Book is to large extent self-contained Attempts to unify two approaches traditionally different, one using essentially abstract functional analysis and the other based on techniques coming from partial differential equations Main exact observability results are obtained by combining functional analysis with various methods such as multipliers, Ingham-Beurling-type inequalities and Carleman estimates Elliptic and the parabolic Carleman-type estimates used in this book are worked out in detail, with an elementary construction of the necessary weight function Includes supplementary material: sn.pub/extras

Contenu
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.