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