Control in the Coefficients of an Obstacle Problem
Nicolai Simon, Winnifried Wollner
TL;DR
The paper addresses optimal control of an obstacle problem with coefficient control in the main PDE, where the solution operator is not Gateaux differentiable due to the variational inequality. It adopts a regularization strategy together with $H$-convergence to handle the coupling between the control and the state, establishing existence for the regularized problems and deriving Clarke-type limiting first-order optimality conditions. Through a bootstrapping argument, it proves strong convergence of the control and state and passes to the limit to obtain a limiting obstacle MPCC system that characterizes optimality for the original problem, including a projection formula for the control. This provides a rigorous framework for coefficient identification in variational inequality settings and offers a pathway to inverse problems where the matrix of coefficients is the control.
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.