A class of parabolic systems associated with optimal controls of grain boundary motions
Abstract
We propose a semi-discrete numerical scheme and establish well-posedness of a class of parabolic systems. Such systems naturally arise while studying the optimal control of grain boundary motions. The latter is typically described using a set of parabolic variational inequalities. We use a regularization approach to deal with the variational inequality. The resulting optimization problem is a nonsmooth, nonconvex, and nonlinear programming problem. This is a long term project where in the current work we are first analyzing systems of PDEs associated with the regularized optimal control problem. Such a system is a set of highly coupled parabolic equations, and proposes significant analytical and numerical challenges. We establish well-posedness of this system. In addition, we design a provably convergent semi-discrete (time discrete spatially continuous) numerical scheme to solve the system. We have developed several new tools during the course of this paper that can be applied to a wider class of coupled systems.
Keywords
Cite
@article{arxiv.1809.06419,
title = {A class of parabolic systems associated with optimal controls of grain boundary motions},
author = {Harbir Antil and Ken Shirakawa and Noriaki Yamazaki},
journal= {arXiv preprint arXiv:1809.06419},
year = {2018}
}