Control in the coefficients of an elliptic differential operator: topological derivatives and Pontryagin maximum principle

TL;DR

This paper develops a new approach to optimal control problems involving elliptic operators by deriving the Pontryagin maximum principle using topological derivatives, without requiring continuity assumptions on coefficients.

Contribution

It introduces a novel method combining topological derivatives with Pontryagin's principle for elliptic PDE control problems, relaxing previous regularity constraints.

Findings

Derived Pontryagin maximum principle for elliptic control problems

Established stronger optimality conditions via shape optimization in 2D

Removed need for continuity assumptions on coefficients or solution gradients

Abstract

We consider optimal control problems, where the control appears in the main part of the operator. We derive the Pontryagin maximum principle as a necessary optimality condition. The proof uses the concept of topological derivatives. In contrast to earlier works, we do not need continuity assumptions for the coefficient or gradients of solutions of partial differential equations. Following classical proofs, we consider perturbations of optimal controls by multiples of characteristic functions of sets, whose scaling factor is send to zero. For $2d$ problems, we can perform an optimization over the elliptic shapes of such sets leading to stronger optimality conditions involving a variational inequality of a new type.