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

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.