Sensitivity analysis of optimal control problems for differential hemivariational inequalities in steady thermistor problem

The paper is concerned with a new class of differential hemivariational inequalities which appears as the weak formulation of steady thermistor problems with mixed boundary conditions. First, we show the existence of solution to this kind of inequality problems combining the theory of pseudomonotone operators and a fixed point argument. Then, an optimal control problem is considered where the control is represented by the heat source. We introduce parameter perturbations of the electric conductivity and the boundary temperature in the system to examine their impact on the sensitivity properties of the optimal control problem. We prove that the optimal state-control set is nonempty and the value function of the optimal control problem is continuous. Finally, the multivalued map induced by optimal state-control set is established to be weakly upper semicontinuous in the weak topology.