Optimal control problems for 1D parabolic state-systems of KWC types with dynamic boundary conditions

TL;DR

This paper studies optimal control problems for 1D parabolic phase-field models of grain boundary motion with dynamic boundary conditions, analyzing solvability, dependence on regularization parameter, and optimality conditions.

Contribution

It establishes solvability, continuous dependence, and optimality conditions for both regularized and limiting cases of the control problems in phase-field models.

Findings

Proved solvability and continuous dependence of the state-systems.

Derived optimality conditions for regularized problems.

Analyzed the limiting behavior as regularization parameter approaches zero.

Abstract

In this paper, we consider a class of optimal control problems governed by 1D parabolic state-systems of KWC types with dynamic boundary conditions. The state-systems are based on a phase-field model of grain boundary motion, proposed in [Kobayashi--Warren--Carter, Physica D, 140, 141--150, 2000], and in the context, the dynamic boundary conditions are supposed to reproduce the transmitted heat exchanges between interior and boundary of a polycrystal body. Our optimal control problems are labeled by using a constant $ \varepsilon \geq 0 $, and roughly summarized, the case when $ \varepsilon = 0 $ and the cases when $ \varepsilon > 0 $ correspond to the physically realistic setting, and its regularized approximating ones, respectively. Under suitable assumptions, the mathematical results concerned with: the solvability and continuous dependence for the state-systems; the solvability and $ \varepsilon $-dependence of optimal control problems; and the first order necessary optimality conditions in the problems when $ \varepsilon > 0 $ and the limiting optimality condition as $ \varepsilon \downarrow 0 $; will be obtained in forms of three Main Theorems of this paper.