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Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. (English) Zbl 0844.49017

Summary: This paper deals with the Bolza problem (P) for differential inclusions subject to general endpoint constraints. We pursue a twofold goal. First, we develop a finite difference method for studying (P) and construct a discrete approximation to (P) that ensures a strong convergence of optimal solutions. Second, we use this direct method to obtain necessary optimality conditions in a refined Euler-Lagrange form without standard convexity assumptions. In general, we prove necessary conditions for the so-called intermediate local minimum that takes an intermediate place between the classical concepts of strong and weak minima. In the case of a Mayer cost functional or boundary solutions to differential inclusions, this Euler-Lagrange form holds without any relaxation. The results obtained are expressed in terms of nonconvex-valued generalized differentiation constructions for nonsmooth mappings and sets.

MSC:

49K24 Optimal control problems with differential inclusions (nec./ suff.) (MSC2000)
49J52 Nonsmooth analysis
49M25 Discrete approximations in optimal control
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