Temperature Control of PDE Constrained Optimization Problems Governed by Kobayashi--Warren--Carter Type Models of Grain Boundary Motions

TL;DR

This paper investigates temperature-controlled PDE optimal control problems related to grain boundary motion models, establishing solvability, optimality conditions, and analyzing limiting behaviors in up to four dimensions.

Contribution

It introduces new solvability and optimality results for Kobayashi--Warren--Carter type models with temperature control, including limiting systems and well-posedness analysis.

Findings

Existence and parameter dependence of solutions

First order necessary optimality conditions derived

Analysis of limiting systems and their well-posedness

Abstract

In this paper, we consider a class of optimal control problems governed by state-equations of Kobayashi--Warren--Carter type. The control is given by physical temperature. The focus is on problems in dimensions less than equal to 4. The results are divided in four Main Theorems, concerned with: solvability and parameter-dependence of state-equations and optimal control problems; the first order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.