Coefficient Control of Variational Inequalities

TL;DR

This paper investigates coefficient control in obstacle problems, establishing existence, optimality conditions, and convergence results, with numerical examples demonstrating the effectiveness and potential extensions to phase-field fracture models.

Contribution

It introduces a novel approach to coefficient control in obstacle problems using H-convergence, deriving optimality conditions and demonstrating convergence through regularization and numerical experiments.

Findings

Existence of optimal solutions established.

First order necessary optimality conditions derived.

Numerical examples confirm convergence and potential for extension.

Abstract

Within this chapter, we discuss control in the coefficients of an obstacle problem. Utilizing tools from H-convergence, we show existence of optimal solutions. First order necessary optimality conditions are obtained after deriving directional differentiability of the coefficient to solution mapping for the obstacle problem. Further, considering a regularized obstacle problem as a constraint yields a limiting optimality system after proving, strong, convergence of the regularized control and state variables. Numerical examples underline convergence with respect to the regularization. Finally, some numerical experiments highlight the possible extension of the results to coefficient control in phase-field fracture.