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Optimal control: nonlocal conditions, computational methods, and the variational principle of maximum. (English. Russian original) Zbl 1183.49003

Russ. Math. 53, No. 1, 1-35 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 1, 3-43 (2009).
Summary: This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions to the Hamilton-Jacobi inequalities). We pay special attention to the conversion of the maximum principle to a sufficient condition for the global and strong minimum without assumptions of the linear convexity, normality, or controllability. We give a survey of computational methods for solving classical optimal control problems and describe nonstandard procedures of nonlocal improvement of admissible processes in linear and quadratic problems. Furthermore, we cite some recent results on the variational principle of maximum in hyperbolic control systems. This principle is the strongest first order necessary optimality condition; it implies the classical maximum principle as a consequence.

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49K15 Optimality conditions for problems involving ordinary differential equations
49K20 Optimality conditions for problems involving partial differential equations
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