Over some 120 pages, the reader is introduced to control theory. In mathematical terms, it describes an investigation of extremes of real functions and functionals. The book is well arranged and offers a systematic approach to the problem, which is classified in detail (e.g. linear programming, convex minimization and differentiable and non-differentiable optimization). Then the conditions of minimization without constraints are treated. Results on existence and uniqueness and the 1st and 2nd order necessary and sufficient conditions of minimization are given. The same topics on minimization under constraints follow. The author looks for simplicity and applicability instead of dry mathematic rigour and maximal generality. However, the proofs of some theorems are original and the contents of the book is rich and inspiring. Accordingly, there are a lot of interesting and useful examples and applications in the book; it is well written and reads well. It covers the topics for students after the first two years of a university mathematics course and contains extended and revised material of the author’s book L’optimization (1996). As the title shows, it is written in French and is followed by the 2nd volume, which is devoted to extremes of functionals.

Reviewer:

jdr