Global minima for optimal control of the obstacle problem

TL;DR

This paper investigates the global minima in optimal control problems constrained by obstacle problems, proposing a discretisation approach and conditions to verify global optimality, supported by numerical examples.

Contribution

It introduces a novel condition to determine global optimality in discretised obstacle control problems and extends this to the limit problem.

Findings

A condition for global optimality in discrete control problems.

The condition can be transferred to the continuous limit under uniform convergence.

Numerical examples demonstrate the effectiveness of the proposed approach.

Abstract

An optimal control problem subject to an elliptic obstacle problem is studied. We obtain a numerical approximation of this problem by discretising the PDE obtained via a Moreau--Yosida type penalisation. For the resulting discrete control problem we provide a condition that allows to decide whether a solution of the necessary first order conditions is a global minimum. In addition we show that the corresponding result can be transferred to the limit problem provided that the above condition holds uniformly in the penalisation and discretisation parameters. Numerical examples with unique global solutions are presented.