Control in the Coefficients of an Obstacle Problem

TL;DR

This paper develops a regularization-based method to derive optimality conditions for a control problem involving obstacle problems with matrix-valued coefficients, overcoming non-differentiability issues using $H$-convergence techniques.

Contribution

It introduces a novel regularization approach and $H$-convergence analysis to establish optimality conditions for obstacle problems with matrix controls, a previously challenging problem.

Findings

Established a regularization framework for obstacle control problems.

Proved strong $L^p$-convergence of controls via $H$-convergence.

Derived optimality conditions through limit analysis of regularized problems.

Abstract

In this work, we consider optimality conditions of an optimal control problem governed by an obstacle problem. Here, we focus on introducing a, matrix valued, control variable as the coefficients of the obstacle problem. As it is well known, obstacle problems can be formulated as a complementarity system and consequently the associated solution operator is not Gateaux differentiable. As a consequence, we utilize a regularization approach to obtain optimality conditions as the limit of optimality conditions of a family of regularized problems. Due to the coupling of the controlled coefficient with the gradients of the solution to the obstacle problem, weak convergence arguments can not be applied and we need to argue by $H$-convergence. We show, that, based on initial $H$-convergence, a bootstrapping argument can be utilized to prove strong $L^p$-convergence of the control and thus enable the passage to the limit in the optimality conditions.