Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential

TL;DR

This paper develops an optimal control framework for a nonconserved phase field model with thermal memory and a non-smooth double obstacle potential, extending previous differentiable potential results.

Contribution

It introduces a novel approach using deep quench approximation to handle nonsmooth potentials in optimal control of phase field models with thermal memory.

Findings

Established existence of optimal controls for the nonsmooth potential case.

Derived first-order necessary optimality conditions using the deep quench approach.

Extended previous differentiable potential results to the double obstacle potential case.

Abstract

In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. It consists of two nonlinearly coupled partial differential equations that govern the phase dynamics and the universal balance law for internal energy, written in terms of the phase variable and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. We extend recent results obtained for optimal control problems in which the free energy governing the phase transition was differentiable (i.e., of regular or logarithmic type) to the nonsmooth case of a double obstacle potential. As is well known, in this nondifferentiable case standard methods to establish the existence of appropriate Lagrange multipliers fail. This difficulty is overcome utilizing of the so-called deep quench approach. Namely,the double obstacle potential is approximated by a family of (differentiable) logarithmic ones for which the existence of optimal controls and first-order necessary conditions of optimality in terms of the adjoint state variables and a variational inequality are known. By proving appropriate bounds for the adjoint states of the approximating systems, we can pass to the limit in the corresponding first-order necessary conditions, thereby establishing meaningful first-order necessary optimality conditions also for the case of the double obstacle potential.