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We propose a semi-discrete numerical scheme and establish well-posedness of aclass of parabolic systems. Such systems naturally arise while studying theoptimal control of grain boundary motions. The latter is typically describedusing a set of parabolic variational inequalities. We use a regularizationapproach to deal with the variational inequality. The resulting optimizationproblem is a nonsmooth, nonconvex, and nonlinear programming problem. This is along term project where in the current work we are first analyzing systems ofPDEs associated with the regularized optimal control problem. Such a system isa set of highly coupled parabolic equations, and proposes significantanalytical and numerical challenges. We establish well-posedness of thissystem. In addition, we design a provably convergent semi-discrete (timediscrete spatially continuous) numerical scheme to solve the system. We havedeveloped several new tools during the course of this paper that can be appliedto a wider class of coupled systems.

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