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Second-order optimality conditions for bang-bang control problems. (English) Zbl 1127.49019

Summary: Second-order necessary and sufficient optimality conditions for bang-bang control problems have been studied by A. A. Milyutin and N. P. Osmolovskij [Calculus of variations and optimal control. Transl. from the original Russian manuscript by Dimitrii Chibisov. Providence, RI: American Mathematical Society (AMS) (1998; Zbl 0911.49001)]. These conditions amount to testing the positive (semi-)definiteness of a quadratic form on a critical cone. The assumptions are appropriate for numerical verification only in some special cases. In this paper, we study various transformations (of the quadratic form and the critical cone which will be tailored to different types of control problems in practice. In particular, by means of a solution to a linear matrix differential equation, the quadratic form can be converted to perfect squares. We demonstrate by three practical examples that the conditions obtained can be verified numerically.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)

Citations:

Zbl 0911.49001
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