Optimal control of elliptic variational inequalities with bounded and unbounded operators

Abstract

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 employ the polyhedricity of the test set appearing in the dual formulation of the governing variational inequality. Based thereon, we are able to prove the directional differentiability of the associated solution operator, which leads to a strong stationary optimality system. 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 difficult to handle. 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. Finally, we apply the developed results to Maxwell-type variational inequalities arising in superconductivity. 2020 Mathematics Subject Classification. 35J86.

Strong Stationarity Conditions for Optimal Control Problems Governed by a Rate-Independent Evolution Variational Inequality

“…Because of these advantageous properties, strong stationarity conditions have come to play a distinct role in the field of optimal control of nonsmooth systems and have received considerable attention in the recent past. See, e.g., [4,11,13,18,25,26,48,51] for contributions on strong stationarity conditions for optimal control problems governed by various elliptic variational inequalities of the first and the second kind, [3,14,16,37] for extensions to optimal control problems governed by nonsmooth semi-and quasilinear PDEs, and [15] for a generalization to the multiobjective setting. Note that all of these works on the concept of strong stationarity have in common that they are only concerned with elliptic variational inequalities or PDEs involving nonsmooth terms.…”