Optimal control of a class of boundary hemivariational inequalities of hyperbolic type

In this paper we examine a class of nonlinear hyperbolic initial boundary value problems with nonmonotone multivalued boundary conditions characterized by the Clarke subdifferential. First we prove an existence result for multidimensional hemivariational inequalities with relation between reaction and velocity. The existence of weak solutions is established by using a surjectivity result for pseudomonotone operators and a priori estimates. Next, we deal with the Bolza optimal control problem for systems monitored by the boundary hemivariational inequalities. We deliver a result on the dependence of the solution set on the control parameter and finally we provide conditions that guarantee the existence of optimal solutions.