Distributed optimal control of a nonstandard nonlocal phase field system

TL;DR

This paper studies a complex nonlocal phase field model involving nonlinear parabolic and differential equations, establishing the existence of optimal controls and deriving necessary optimality conditions.

Contribution

It introduces a novel optimal control framework for a highly nonlinear nonlocal phase field system with singular terms, extending existing models.

Findings

Existence of an optimal control solution.

Derivation of first-order necessary optimality conditions.

Handling of nonlocal and singular terms in the analysis.

Abstract

We investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. 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.