Optimal Control of Pseudo-Parabolic KWC Systems for Grain Boundary Motion

TL;DR

This paper develops an optimal control framework for a pseudo-parabolic KWC system modeling grain boundary motion, ensuring well-posedness and deriving necessary optimality conditions.

Contribution

It introduces a pseudo-parabolic structure to the KWC system, enabling the analysis of optimal control with proven existence, stability, and first-order optimality conditions.

Findings

Existence of optimal controls established.

Semi-continuous dependence on data proven.

First-order necessary conditions derived.

Abstract

The KWC system is a well-known generic framework for phase-field models of grain boundary motion, whose original formulation is given as a parabolic gradient flow of a free energy. In the original KWC system, the results of uniqueness have been relatively scarce compared to other issues, such as existence, qualitative behavior, and numerics. This lack of progress has posed a significant challenge for more advanced topics, including optimal control. To overcome this, the authors have recently introduced the pseudo-parabolic structure to simultaneously preserve the gradient flow nature of free-energy, and to ensure the well-posedness including the uniqueness. The goal of this paper is to study an optimization problem constrained by pseudo-parabolic KWC system. The theory will be developed through a series of Main Theorems concerning the existence and semi-continuous dependence of optimal controls, and first-order necessary conditions for optimality.