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Optimal control. Theory and applications. (Contrôle optimal. Théorie et applications.) (French) Zbl 1112.49001

Mathématiques Concrètes. Paris: Vuibert (ISBN 2-7117-7175-X/pbk). vi, 246 p. (2005).
This work of Emmanuel Trélat, published in 2005, is an excellent new text book that guides the interested reader into the world of optimal control. To be more precise, it is the optimal control of ordinary differential equations (ODEs) or of systems of ODEs, that this work is concerned with. In literature, there are many possible approaches to optimal control yet: theoretical ones which stay inside of the frames of convex or nonsmooth analysis and optimization, of functional analysis or topology, or numerical ones with a focus on the resolution of the mathematical task, or practical ones which show the problems applied at concrete real-word problems.
The present book is a classical one in so far as it carefully presents and classifies the optimal control problems, presents the analytical problems related and tackles them. Here the underlying perspective is a one from optimization theory and the theory of dynamical systems, including the algorithmical and simulation parts. But the book is also a modern one in so far as it addresses real-world problems, it presents the whole circle from these motivations to the numerical solution. All of this lets the book become a self-contained entity. It is well-structured, equipped with an appropriate design; both writing style and layout are “light” and inviting. Indeed the whole books invites to do exercises, to apply the knowledge obtained to small tasks and problems, and beyond this text book, to become an expert, practitioner and, maybe, researcher in the areas of optimal control. This book could become a stable basis for more advanced studies done in the future. The proofs are given carefully. Many nice photos, illustrations, codes and tables make the reading and studying of this book a pleasure and interesting. The applications range between, e.g., mechanics and aerodynamics to chemical engineering and ecology.
This book begins with notations, opening remarks and an introduction. Then, it contains of three parts. Part 1 is about optimal control of linear systems: controllability, time optimality and linear-quadratic theory. Part 2 is on nonlinear optimal control theory: definitions, preliminaries and problem presentation of optimal control, Pontryagin’s maximum principle, Hamilton-Jacobi theory and numerical methods. Part 3 contains appendices about tools from linear algebra, the theorem of Cauchy-Lipschitz, modeling of a linear control system, stabilization and observability of control systems.
One can truly thank and congratulate the author to this valuable contribution to learning and teaching, and to a support and stimulation for scientific enterprises. Indeed, even those who will be concerned with optimal control of discrete systems, of partial differential equations or stochastic differential equations can benefit from this book, from the foundations given by it and in view of various similarities with those other areas of optimal control.
One final word on the language: Today, most text books are written in English, the present one in French. There reviewer is convinced that there should not remain a big obstacle for those who do not know French. Mathematics and its terminology are so international that the reader will get used to the language after some days, he or she will enjoy reading and gain a lot from this precious work!

MSC:

49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control

Software:

Maple; Matlab
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