Optimal control problems governed by 1-D Kobayashi–Warren–Carter type systems

Abstract

This paper is devoted to the study of a class of optimal control problems governed by 1–D Kobayashi–Warren–Carter type systems, which are based on a phase-field model of grain boundary motion, proposed by [Kobayashi et al, Physica D, 140, 141–150, 2000]. The class consists of an optimal control problem for a physically realistic state-system of Kobayashi–Warren–Carter type, and its regularized approximating problems. The results of this paper are stated in three Main Theorems 1–3. The first Main Theorem 1 is concerned with the solvability and continuous dependence for the state-systems. Meanwhile, the second Main Theorem 2 is concerned with the solvability of optimal control problems, and some semi-continuous association in the class of our optimal control problems. Finally, in the third Main Theorem 3, we derive the first order necessary optimality conditions for optimal controls of the regularized approximating problems. By taking the approximating limit, we also derive the optimality conditions for the optimal controls for the physically realistic problem.