Optimal Control of Two-Phase Membrane Problem

TL;DR

This paper studies an optimal control problem involving a two-phase membrane free boundary problem, addressing the challenges of non-differentiability through regularization and establishing fundamental properties of optimal solutions.

Contribution

It introduces a regularization approach for the non-differentiable control-to-state map and proves existence, uniqueness, and characterization of optimal pairs.

Findings

Existence of optimal controls established

Uniqueness of solutions demonstrated

Characterization of optimal pairs provided

Abstract

We consider an optimal control problem where the state is governed by a free boundary problem called the two-phase membrane problem and the control appears in the coefficients of the characteristic function of the positivity and negativity parts of the solution. Our investigation focuses on various properties associated with the control-to-state map. Due to the non-differentiability of this map, we regularize the state equation. The existence, uniqueness, and characterization of the optimal pairs are established.