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Differential inclusions and optimal control. (Russian) Zbl 0595.49026

This is a survey of some results in convex analysis, multifunctions, differential inclusions and optimal control theory in the finite- dimensional case. In the first part, 45 theorems are presented about different classes of multifunctions, selections and integration of set- valued mappings. The second part is devoted to differential inclusions. Here 43 theorems on existence and different properties of solutions of such inclusions (such as, for example, dependence on initial data and right-hand sides, stability, monotonicity, compactness, convexity of the set of all solutions, the existence of bounded and periodic solutions). In the last part, some optimal control problems for differential inclusions are considered. Necessary optimality conditions in form of Pontryagin’s maximum principle are given for a time-optimal problem and for a problem with integral cost functional. Some sufficient conditions of optimality are also presented.
All theorems are formulated without proofs, some of them are illustrated by examples.
The bibliography consists of 119 positions and is not complete.
Reviewer: Z.Wyderka

MSC:

93B05 Controllability
34A60 Ordinary differential inclusions
49K15 Optimality conditions for problems involving ordinary differential equations
26E25 Set-valued functions
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
49J45 Methods involving semicontinuity and convergence; relaxation
54C60 Set-valued maps in general topology
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
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