A new edition in a single volume Over the past decade, more and more sophisticaced mathematical tools and approaches have been incorporated in the ?eld of Control of in?nite dim- sional systems. This was motivated by a whole range of challenging appli- tions arising from new phenomenologicalstudies, technologicaldevelopments, and more stringent design requirements. At the same time, researchers and advanced engineers have been steadily using an impressive amount of very sophisticated mathematics in their analysis, synthesis, and design of systems. Whatwasregardedastooabstract,specialized,ortheoreticalin1990hasnow become a standard part of the toolkit. The decision to produce a second edition of the original 1992-1993 t- volume edition is further motivated by several other factors. Over the years the book has been recognizedas a key referencein the ?eld, and a revised and corrected edition was desirable. Even if some good books on the control of in?nite dimensional linear systems have appeared since then, we felt that the original material has not aged too much and that the breadth of its presen- tion is still attractiveand very competitive. The result is a completely revised and corrected second edition in a single convenient volume with integrated bibliography and index.

"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."

"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 (reviews of Volumes I and II of the First Edition)

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 a theoretical and design point 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, stabilizability, and detectability. This theory is far more difficult for infinite dimensional systems such as those with time delays and distributed parameter systems.

This reorganized, revised, and expanded edition of a two-volume set is a self-contained account of quadratic cost optimal control for a large class of infinite dimensional systems. The book is structured into five parts. Part I reviews basic optimal control and game theory of finite dimensional systems, which serves as an introduction to the book. Part II deals with time evolution of some generic controlled infinite dimensional systems and contains a fairly complete account of semigroup theory. It incorporates interpolation theory and exhibits the role of semigroup theory in delay differential and partial differential equations. Part III studies the generic qualitative properties of controlled systems. Parts IV and V examine theoptimal control of systems when performance is measured via a quadratic cost. Boundary control of parabolic and hyperbolic systems and exact controllability are also covered.

New material and original features of the Second Edition:

* Part I on finite dimensional controlled dynamical systems contains new material: an expanded chapter on the control of linear systems including a glimpse into H^-infinity theory and dissipative systems, and a new chapter on linear quadratic two-person zero-sum differential games.

* A unique chapter on semigroup theory and interpolation of linear operators brings together advanced concepts and techniques that are usually treated independently.

* The material on delay systems and structural operators is not available elsewhere in book form.

Control of infinite dimensional systems has a wide range and growing number of challenging applications. This book is a key reference for anyone working on these applications, which arise from new phenomenological studies, new technological developments, and more stringent design requirements. It will be useful for mathematicians, graduate students, and engineers interested in the field and in the underlying conceptual ideas of systems and control.

"I found myself browsing almost every time I looked in the book to work on the review, distracted by some exposition of one item or another of fascinating material...What is presented [in the work] is presented well, and I will continue to find the book valuable as a reference. Such a good treatment of this material is extremely welcome. That was indeed the case for the well-received original two-volume version of 1992-1993, and is now again the case for this second edition."   -Thomas I. Seidman, IEEE Control Systems Magazine (Review of the Second Edition)

"The monograph presents a broad review of the existing results in control theory of infinite-dimensional dynamical systems. Theoretical results are illustrated by many examples and additional comments. The monograph contains an extensive list of recent references concerning the theory of abstract linear systems. Finally, it should be pointed out that the monograph is the second edition of a two-volume monograph published by the same authors in 1993. Many results in the one-volume second edition are completely revised and corrected. Moreover, the second edition contains many new research results."   -Zentralblatt MATH (Review of the Second 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 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)

"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)