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Book review

Author(s) Zabczyk, Jerzy
Title Mathematical Control Theory: An Introduction
Publisher Birkhäuser
Year of publication 1992
Reviewed by Vasile Postolica

A wonderful presentation of basic concepts and results in the field of mathematical control theory and its immediate applications is given in this book which contains four parts (of of them being divided in chapters) and an Appendix preceded by a very persuasive Introduction in which we find basic concepts, questions of the theory and interesting examples.

Part I deals with structural properties of linear systems (controllability, observability, stability and stabilizability). The last chapter of this part is devoted to linear systems with bounded sets of control parameters and the positive systems. Structural properties for nonlinear systems are studies in Part II.

The problem of find optimal controls is tackled in Part III in which one discusses Bellman's opimality principle and its typical applications to the linear regulator problem and to impulse control, ove gives a smart proof of Pontriagin's maximum principle for classical problems with fixed control intervals and for time-optimal and impulse control problems. The final chapters of this part are devoted to the existence problems.

Concerning the infinite dimensional systems, in Part IV we find linear systems without control (a presentation in this way of the theory of semigroups of linear operators, controllability, stability and stabilizability) and the linear regulator problem in Hilbert spaces.

Some advanced mathematical concepts are presented in the Appendix. The book includes also a substantial bibliography from the point of view of contents and an Index.

It is a great merit of the author to present for the first time in a synthesized book form recently published results in: impulsive control (positive systems), stabilization of nonlinear systems using topological methods, realization of nonlinear systems, control of rigid bodies, stabilization of infinite dimensional systems and the minimum energy problem.

Thus, this volume offers to graduate students, researchers and engineers the basic methods in mathematical theory and applications, explains engineering concepts through the agency of mathematical notions and covers in this way all deterministic theory, being very concise.