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

TL;DR

This paper investigates optimal control problems for conserved phase-field systems, addressing both viscous Cahn-Hilliard dynamics with singular potentials and standard Cahn-Hilliard equations with regular potentials, deriving optimality conditions.

Contribution

It provides the first-order optimality conditions for controlling phase-field systems with singular and regular potentials, expanding the theoretical understanding of such control problems.

Findings

Derived necessary optimality conditions for singular potentials.

Established control framework for regular potentials.

Analyzed two distinct phase-field models under control.

Abstract

In this paper we study a distributed control problem for a phase-field system of conserved type with a possibly singular potential. We mainly handle two cases: the case of a viscous Cahn-Hilliard type dynamics for the phase variable in case of a logarithmic-type potential with bounded domain and the case of a standard Cahn-Hilliard equation in case of a regular potential with unbounded domain, like the classical double-well potential, for example. Necessary first order conditions of optimality are derived under natural assumptions on the data.