Variational analysis of optimal control problems for history-dependent elliptic inclusions

In this paper we examine a Lagrange optimal control problem for a new class of elliptic time-dependent operator inclusions involving history-dependent operators. First, results on existence and variational sensitivity of optimal control problem are proved. Next, the lower and upper Kuratowski semicontinuity of the admissible sets, sets of optimal solutions, and the continuity of the value function are shown with respect to a parameter which appears in the inclusion and the integrand of the cost functional. Finally, we illustrate the applicability of the results to a quasi-static viscoelastic frictional contact problem in mechanics.