  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.
