Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential

TL;DR

This paper develops a distributed optimal control framework for a complex nonlocal phase field model with double obstacle potential, addressing nonlinear couplings and nonsmooth energy terms using approximation techniques.

Contribution

It introduces a novel approach to handle the nonsmooth double obstacle potential in a nonlocal phase field system through deep quench approximation.

Findings

Established existence of solutions for the control problem

Derived first-order necessary optimality conditions

Extended analysis to nonsmooth double obstacle potentials

Abstract

This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors; the nonlocal model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called `deep quench' approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.