Optimal control for a phase field system with a possibly singular potential

TL;DR

This paper addresses an optimal control problem for a phase field system with a possibly singular potential, focusing on approximating a discontinuous cost functional and deriving necessary optimality conditions.

Contribution

It introduces a method to handle singular potentials in phase field systems and derives first order optimality conditions for the control problem.

Findings

Successfully approximates the discontinuous cost functional

Derives first order necessary optimality conditions

Provides a framework for controlling phase interfaces

Abstract

In this paper we study a distributed control problem for a phase field system of Caginalp type with logarithmic potential. The main aim of this work would be to force the location of the diffuse interface to be as close as possible to a prescribed set. However, due to the discontinuous character of the cost functional, we have to approximate it by a regular one and, in this case, we solve the associated control problem and derive the related first order necessary optimality conditions.