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Book review

Author(s) Bensoussan, Alain
Prato, Giuseppe Da
Delfour, Michel C.
Mitter, Sanjoy K.
Title Representation and Control of Infinite Dimensional Systems, Vol. 2
Publisher Birkhäuser
Year of publication 1993
Reviewed by Vasile Postolica

This volume is concerned with the optimal control of certain classes of infinite-dimensional systems with a quadratic cost criterion both over a finite and an infinite-time horizon, being organized as follows.

Part I, entitled "Qualitative Properties of Linear Dynamical Systems", contains in Chapter 1 the control of linear-finite dimensional differential systems revizited. Chapter 2 of this part is devoted to the controllability and observability for an infinite-dimensional abstract linear dynamical system which can be specialized to obtain the controllability of parabolic and hyperbolic partial differential equations, both when control is exercised in the interior of the domain and also when the control is exercised through the boundary. Here it is also discussed in detail the problem of exact controllability for hyperbolic equations in appropriate spaces.

Part II "Quadratic Optimal Control: Finite Time Horizon" contains: systems with bounded control operators (control inside the domain) (Chapter 1); systems with unbounded control operators (parabolic equations with control on the boundary) (Chapter 2); and systems with unbounded control operators for hyperbolic equations with control on the boundary (Chapter 3).

Part II "Quadratic Optimal Control: Infinite Time Horizon" deals with the concepts of stabilizability and detectability for the parabolic and hyperbolic cases. Thus Chapter 1 of this part is devoted to the control inside the domain for systems with bounded control operators. Chapter 2 deals with systems having unbounded control operators, especially for parabolic equations with control on the boundary. Finally, one discusses the dynamic programming. The systems with unbounded control operators (hyperbolic equations with control on the boundary) are treated in Chapter 3. An isomorphism result is given in the Appendix A. Each part of the book is completed by important comments and/or references.

This book in two volumes has been written by experts in the field of Control of Infinite Dimensional Systems. It represents at the same time a remarkable contribution to the development of this scientific field very useful for mathematicians, theoretical engineers, and, in general, for all the scientists interested in control of infinite dimensional systems.