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arXiv · 1809.06419

A class of parabolic systems associated with optimal controls of grain boundary motions

Harbir Antil·Ken Shirakawa·Noriaki Yamazaki
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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.

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Harbir Antil, Ken Shirakawa, Noriaki Yamazaki. 2018-09-17. A class of parabolic systems associated with optimal controls of grain boundary motions. https://arxiv.org/abs/1809.06419

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