Document Type

Technical Report

Abstract

We study discrete approximations of nonconvex differential inclusions in Hilbert spaces and dynamic optimization/optimal control problems involving such differential inclusions and their discrete approximations. The underlying feature of the problems under consideration is a modi- fied one-sided Lipschitz condition imposed on the right-hand side (i.e., on the velocity sets) of the differential inclusion, which is a significant improvement of the conventional Lipschitz continuity. Our main attention is paid to establishing efficient conditions that ensure the strong approximation (in the W^1,p-norm as p greater than or equal to 1) of feasible trajectories for the one-sided Lipschitzian differential inclusions under. consideration by those for their discrete approximations and also the strong con- vergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations. To proceed with the latter issue, we derive a new extension of the Bogolyubov-type relaxation/density theorem to the case of differential inclusions satisfying the modified one-sided Lipschitzian condition. All the results obtained are new not only in the infinite-dimensional Hilbert space framework but also in finite-dimensional spaces.

Number in Series

2007.01

Disciplines

Applied Mathematics | Mathematics

AMS Subject Classification

49J24, 49M25, 90C99

Comments

Dedicated to Arrigo Cellina in honor of his 65th birthday. This research was partially supported by the USA National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP.0451168.

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