Distributed optimal control of a nonstandard system of phase field equations

TL;DR

This paper studies a distributed optimal control problem for a complex, coupled phase field PDE system modeling atomic lattice segregation, establishing existence of solutions and deriving necessary optimality conditions.

Contribution

It introduces a new control framework for a highly nonlinear phase field model and proves the existence of optimal controls and first-order optimality conditions.

Findings

Existence of optimal control solutions established

First-order necessary conditions derived

Applicable to complex coupled PDE systems

Abstract

We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been recently introduced by the same authors in arXiv:1103.4585v1 [math.AP] and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.