A class of parabolic systems associated with optimal controls of grain boundary motions
A class of parabolic systems associated with optimal controls of grain boundary motions
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发表时间：
2018-09
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通讯作者：
Harbir Antil;K. Shirakawa;N. Yamazaki
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作者：
Harbir Antil;K. Shirakawa;N. Yamazaki
我们提出了一个半离散的数值格式，并建立了一类抛物型方程组的适定性。这样的系统自然出现，而研究晶界运动的最佳控制。后者通常使用一组抛物型变分不等式来描述。我们使用正则化方法来处理变分不等式。由此产生的优化问题是一个非光滑，非凸，非线性规划问题。这是一个长期的项目，在目前的工作中，我们首先分析与正则化最优控制问题相关的偏微分方程系统。这样的系统是一组高度耦合的抛物方程，并提出了重大的分析和数值挑战。我们建立了这个系统的适定性。此外，我们设计了一个可证明收敛的半离散（时间离散空间连续）数值格式来求解该系统。在本文的过程中，我们已经开发了几个新的工具，可以应用到更广泛的一类耦合系统。
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.