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Optimal control. Transl. from the Russian by V. M. Volosov. (English) Zbl 0689.49001

Contemporary Soviet Mathematics. New York etc.: Consultants Bureau. xiii, 309 p. $ 75.00 (1987).
This book, written more than ten years ago, is centered around necessary and sufficient optimality conditions. They are developed in chapter 3 for the mathematical programming problem in a Banach space with an infinite dimensional equality and a finite dimensional inequality constraint for both the convex and the differential case. In chapter 4, the authors apply them to problems of the calculus of variations of one variable and to the optimal control problem for ordinary differential equations. In both cases, they include boundary conditions and isoperimetric inequality constraints. Since the authors want the book “to serve as a text in various courses in optimization that are offered in universities and colleges” and also want to address “engineers, economists, and mathematicians involved in the solution of extremal problems”, they separately (in chapter 1) present the cases for which the proofs are much easier, namely the Kuhn Tucker conditions in \(R^ n\), the simplest problem of the calculus of variations, and the optimal control problem with free end point. For the same reason, the prerequisites from functional analysis, ODE theory and convex analysis are treated in chapter 2. Everything is done in a detailed, self-contained and mathematically rigorous manner. At the end of the book, 100 problems from optimization, optimal control and the calculus of variations (with a strong emphasis on the latter) are formulated and the solutions are stated (sometimes with explanations).
This book is definitely neither a research monograph nor a reference text. It is a textbook on the undergraduate/graduate level. Compared to its contents as described above, its title is somewhat misleading. There is some discussion of the Hamilton Jacobi equation (smooth case) in the context of integral invariants, but all other aspects of optimal control theory (existence theory, numerical methods, singular and switching surfaces, controllability, LQ problems, additional pointwise constraints) receive no or almost no treatment. It is particularly disappointing that the only optimal control problems which are solved in the text are Newton’s aerodynamic problem and the problem of time optimal control to the zero state in the equation \(d^ 2x/dt^ 2=u\). As applications are offered the fundamental theorem of algebra, Sylvester’s theorem, the theorem of Hestenes on the index of quadratic forms in Hilbert space, and at the very end two short paragraphs on Noether’s theorem on the existence of first integrals and its applications to mechanics.
Reviewer: M.Brokate

MSC:

49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
49K15 Optimality conditions for problems involving ordinary differential equations
49K27 Optimality conditions for problems in abstract spaces
49K40 Sensitivity, stability, well-posedness
34H05 Control problems involving ordinary differential equations
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