First Order Limiting Optimality Conditions in the Coefficient Control of an Obstacle Problem

Abstract

We consider an optimal control problem governed by an obstacle problem. The novelty and focus of this work will be the introduction of a control variable into the coefficients of this optimal control problem. In particular, we discuss the effects of the multiplicative coupling between control and, derivatives of, the state. This prevents the use of standard weak convergence arguments in limit arguments. As it is known, H‐convergence needs to be utilized, e.g., to prove existence of a solution to the problem. To obtain optimality conditions, we present a regularization approach to circumvent the non‐differentiability of the solution operator of the obstacle problems. We then discuss the limiting optimality conditions obtained by the proposed regularization.