  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.
