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Representation and Control of Infinite Dimensional Systems (Systems & Control

The quadratic cost optimal control problem for systems described by linear ordinary differential equations occupies a central role in the study of control systems both from the theoretical and design points of view. The study of this problem over an infinite time horizon shows the beautiful interplay between optimality and the qualitative properties of systems such as controllability, observability and stability. This theory is far more difficult for infinite-dimensional systems such as systems with time delay and distributed parameter systems. In the first place, the difficulty stems from the essential unboundedness of the system operator. Secondly, when control and observation are exercised through the boundary of the domain, the operator representing the sensor and actuator are also often unbounded. The present book, in two volumes, is in some sense a self-contained account of this theory of quadratic cost optimal control for a large class of infinite-dimensional systems. Volume I deals with the theory of time evolution of controlled infinite-dimensional systems. It contains a reasonably complete account of the necessary semigroup theory and the theory of delay-differential and partial differential equations. Volume II deals with the optimal control of such systems when performance is measured via a quadratic cost. It covers recent work on the boundary control of hyperbolic systems and exact controllability. Some of the material covered here appears for the first time in book form. The book should be useful for mathematicians and theoretical engineers interested in the field of control.
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Table of Contents

I Qualitative Properties of Linear Dynamical Systems.- 1 Control of Linear Finite Dimensional Differential Systems Revisited.- 1 Introduction.- 2 Controllability and observability of finite dimensional linear systems.- 2.1 Controllability.- 2.2 Observability.- 2.3 Duality.- 2.4 Canonical structure for linear systems.- 2.5 The pole-assigment theorem.- 2.6 Stabilizability and detectability.- 2.7 Applications of controllability and observability.- 3 Optimal control.- 3.1 Optimal control in a finite time interval.- 3.2 Optimal control over an infinite time interval.- 4 A glimpse into H?-theory: state feedback case.- 4.1 Introduction.- 4.2 Main results.- 5 Final remarks.- Notes.- 2 Controllability and Observability for a Class of Infinite Dimensional Systems.- 1 Introduction.- 2 Main definitions.- 2.1 Notation.- 2.2 Definitions.- 3 Criteria for approximate and exact controllability.- 3.1 Criterion for approximate controllability.- 3.2 Criteria for exact controllability and continuous observability.- 3.3 Approximation.- 4 Finite dimensional control space.- 4.1 Finite dimensional case.- 4.2 General state space.- 5 Controllability for the heat equation.- 5.1 Distributed control.- 5.2 Boundary control.- 5.3 Neumann boundary control.- 5.4 Pointwise control.- 6 Controllability for skew-symmetric operators.- 6.1 Notation and general comments.- 6.2 Dynamical system.- 6.3 Approximation.- 6.4 Exact controllability for T arbitrarily small.- 7 General framework: skew-symmetric operators.- 7.1 Operator A.- 7.2 Operator B.- 7.3 Dynamical system.- 7.4 Exact controllability.- 8 Exact controllability of hyperbolic equations.- 8.1 Wave equation with Dirichlet boundary control.- 8.2 Wave equation with Neumann boundary control.- 8.3 Maxwell equations.- 8.4 Plate equation.- References to Part I.- II Quadratic Optimal Control: Finite Time Horizon.- 1 Systems with Bounded Control Operators: Control Inside the Domain.- 1 Introduction and setting of the problem.- 2 Solution of Riccati equation.- 2.1 Notation and preliminaries.- 2.2 Riccati equation.- 2.3 Representation formulas for the solution of the Riccati equation.- 3 Strict and classical solutions of the Riccati equation.- 3.1 The general case.- 3.2 The analytic case.- 3.3 The variational case.- 4 The case of the unbounded observation.- 4.1 The analytic case.- 4.2 The variational case.- 5 The case when A generates a group.- 6 The linear quadratic control problem with finite horizon.- 6.1 The main result.- 6.2 The case of unbounded observation.- 6.3 Regularity properties of the optimal control.- 6.4 Hamiltonian systems.- 7 Some generalizations and complements.- 7.1 Non homogeneous state equation.- 7.2 Time dependent state equation and cost function.- 7.3 Dual Riccati equation.- 8 Examples of controlled systems.- 8.1 Parabolic equations.- 8.2 Wave equation.- 8.3 Delay equations.- 8.4 Evolution equations in noncylindrical domains.- 2 Systems with Unbounded Control Operators: Parabolic Equations with Control on the Boundary.- 1 Introduction.- 2 Riccati equation.- 2.1 Notation.- 2.2 Riccati equation for ? >1/2.- 2.3 Solution of the Riccati equation for ? >1/2.- 3 Dynamic Programming.- 3 Systems with Unbounded Control Operators: Hyperbolic Equations with Control on the Boundary.- 1 Introduction.- 2 Riccati equation.- 3 Dynamic Programming.- 4 Examples of controlled hyperbolic systems.- 5 Some result for general semigroups.- References to Part II.- III Quadratic Optimal Control: Infinite Time Horizon.- 1 Systems with Bounded Control Operators: Control Inside the Domain.- 1 Introduction and setting of the problem.- 2 The algebraic Riccati equation.- 3 Solution of the control problem.- 3.1 Feedback operator and detectability.- 3.2 Stabilizability and stability of the closed loop operator F in the point spectrum case.- 3.3 Stabilizability.- 3.4 Exponential stability of F.- 4 Qualitative properties of the solutions of the Riccati equation.- 4.1 Local stability results.- 4.2 Attractivity properties of a stationary solution.- 4.3 Maximal solutions.- 4.4 Continuous dependence of stationary solutions with respect to the data.- 4.5 Periodic solutions of the Riccati equation.- 5 Some generalizations and complements.- 5.1 Non homogeneous state equation.- 5.2 Time dependent state equation and cost function.- 5.3 Periodic control problems.- 6 Examples of controlled systems.- 6.1 Parabolic equations.- 6.2 Wave equation.- 6.3 Strongly damped wave equation.- 2 Systems with Unbounded Control Operators: Parabolic Equations with Control on the Boundary.- 1 Introduction and setting of the problem.- 2 The algebraic Riccati equation.- 3 Dynamic programming.- 3.1 Existence and uniqueness of the optimal control.- 3.2 Feedback operator and detect ability.- 3.3 Stabilizability and stability of F in the point spectrum case.- 3 Systems with Unbounded Control Operators: Hyperbolic Equations with Control on the Boundary.- 1 Introduction and setting of the problem.- 2 Main results.- 3 Some result for general semigroups.- References to Part III.- Appendix A.- An Isomorphism Result.- Index to Volume II.- Corrections to Volume I.

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"This book will undoubtedly prove [to be] a very valuable text to researchers familiar with finite-dimensional control theory and methods of functional analysis/semigroup theory who are interested in learning more about PDE systems and their control. This task is greatly facilitated by exploiting analogies with finite-dimensional theory and relying for the most part on operator/semigroup methods, thus reducing to a minimum the necessity of PDE background. The book presents, or refers to, the most recent and updated results in the field. For this reason, it should serve as an excellent asset to anyone pursuing a research career in the field." -Mathematical Reviews (Review of Volume II of the First Edition) "We state at the outset that this book is a most welcome addition to the literature of this field, where it serves the need for a modern treatment on topics that only very recently have found a satisfactory solution. ...Many readers will appreciate the concise exposition.... The book makes a worthwhile effort to be accessible and relatively self-contained. [It] should prove to be a valuable source for mathematicians who want to learn more about aspects of deterministic control theory as well as theoretical engineers willing to learn the mathematical tools necessary to give precise formulations and solutions to problems arising from applications." -Mathematical Reviews (Review of Volume I of the First Edition) "This is a book which people in the field have been waiting for since the late seventies.... The difference [in this book] lies in the scope of the classes of systems which are covered, which is much wider than that covered in earlier texts.... This book is a welcome addition to the literature. It presents a unified, up-to-date treatment of the main approaches to the representation of partial and differential delay systems.... The book is recommended both as an advanced graduate text for mathematicians and as a valuable reference guide to the literature." -Journal of Mathematical Systems, Estimation, and Control (Review of Volume I of the First Edition)

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