Optimal control for doubly nonlinear evolutionary inclusions

We study the optimal control of systems for a class of nonlinear hemivariational inequalities which are in the form of evolutionary inclusions involving Clarke's generalized gradient. The control variables are introduced both in the generalized gradient and in the source terms. We first establish the existence of weak solutions to nonlinear inclusions and prove the upper semicontinuity property of their solution sets. Then, we present the minimization problem and show the existence of optimal admissible state-control pairs. Finally, some examples of our abstract results which appear in applications are discussed.