In this paper we prove the existence of strong solutions for evolution inclusions of the form −
(t) ∈ ∂ϕ(x(t))+F(t,x)) defined in a separable Hilbert space, where ∂ϕ(·) denotes the subdifferential of a proper, closed, convex function ϕ(·) andF(t,x) is a multivalued nonconvex, nonmonotone perturbation satisfying a general growth condition.
In this paper we examine optimal control problems governed by maximal monotone integrodifferential inclusions inRN. First we establish the existence of an optimal control. Then we show that the value of the problem depends continuously on a parameter appearing in all the data. Then we introduce the relaxed system, we show that under very general hypotheses it has a solution and that its value equals that of the original problem. Subsequently we show that relaxability and performance stability are equivalent concepts. Finally we specialize our results to the class of controlled differential variational inequalities.