This work is a revised and enlarged edition of a book with the same title published in Romanian by the Publishing House of the Romanian Academy in 1989. It grew out of lecture notes for a graduate course given by the author at the University if Ia~i and was initially intended for students and readers primarily interested in applications of optimal control of ordinary differential equations. In this vision the book had to contain an elementary description of the Pontryagin maximum principle and a large number of examples and applications from various fields of science. The evolution of control science in the last decades has shown that its meth� ods and tools are drawn from a large spectrum of mathematical results which go beyond the classical theory of ordinary differential equations and real analy� ses. Mathematical areas such as functional analysis, topology, partial differential equations and infinite dimensional dynamical systems, geometry, played and will continue to play an increasing role in the development of the control sciences. On the other hand, control problems is a rich source of deep mathematical problems. Any presentation of control theory which for the sake of accessibility ignores these facts is incomplete and unable to attain its goals. This is the reason we considered necessary to widen the initial perspective of the book and to include a rigorous mathematical treatment of optimal control theory of processes governed by ordi� nary differential equations and some typical problems from theory of distributed parameter systems.

Synopsis:

This volume is concerned with optimal control problems governed by ordinary differential systems and partial differential equations. The emphasis is on first-order necessary conditions of optimality and the construction of optimal controllers in feedback forms. These subjects are treated using some new concepts and techniques in modern optimization theory, such as Clarke's generalized gradient, Ekeland's variational principle, viscosity solution to the Hamilton-Jacobi equation, and smoothing processes for optimal control problems governed by variational inequalities. A substantial part of this book is devoted to applications and examples. A background in advanced calculus should enable readers to understand most of this book, including the statement of the Pontriagin maximum principle and many of the applications. This work should be of interest to graduate students in mathematics and engineering, and researchers in applied mathematics, control theory and systems theory.

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