Optimal control of elliptic variational inequalities with bounded and unbounded operators
This paper examines optimal control problems governed by elliptic variational inequalities of the second kind with bounded and unbounded operators. To tackle the bounded case, we exploit the dual formulation of the governing variational inequality, which turns out to be an obstacle-type variational inequality featuring a polyhedric structure. Based on the polyhedricity, we are able to prove the directional differentiability of the associated solution operator, which leads to a strong stationary optimality system. These results correspond to the ones obtained recently by De los Reyes and Meyer [JOTA, 2016]. Differently from their work, our results benefit from the L²-boundedness property such that we do not require any additional regularity or structural assumption on the unknown solution and the slack variable. The second part of the paper deals with the unbounded case. Due to the non-smoothness of the variational inequality and the unboundedness of the governing elliptic operator, the directional differentiability of the solution operator becomes highly difficult to tackle. Our strategy is to apply the Yosida approximation to the unbounded operator, while the non-smoothness of the variational inequality is still preserved. Based on the developed strong stationary result for the bounded case, we are able to derive optimality conditions for the unbounded case by passing to the limit in the Yosida approximation. As an application, we apply the developed results to Maxwell-type variational inequalities arising in superconductivity.