Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory

TL;DR

This paper extends the Caginalp phase field model to include thermal memory effects, providing existence results and developing an optimal control framework with necessary conditions for controlling heat sources and initial temperature.

Contribution

It introduces a thermodynamically consistent nonlinear phase field model with thermal memory and derives optimal control conditions for the system.

Findings

Existence and continuous dependence of solutions are established.

Fréchet differentiability of the control-to-state operator is proved.

First-order optimality conditions for control are derived.

Abstract

A nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. Two equations, resulting from phase dynamics and the universal balance law for internal energy, are written in terms of the phase variable (representing a non-conserved order parameter) and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. Existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem. Then, an optimal control problem is investigated for a suitable cost functional, in which two data act as controls, namely, the distributed heat source and the initial temperature. Fr\'echet differentiability between suitable Banach spaces is shown for the control-to-state operator, and meaningful first-order necessary optimality conditions are derived in terms of variational inequalities involving the adjoint variables. Eventually, characterizations of the optimal controls are given.