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
Applied Mathematics | Mathematics
AMS Subject Classification
49J24, 49M25, 90C99
Donchev, Tzanko; Farkhi, Elza; and Mordukhovich, Boris S., "Discrete Approximations, Relaxation, and Optimization of One-Sided Lipschitzian Differential Inclusions in Hilbert Spaces" (2007). Mathematics Research Reports. 46.