  The paper addresses the study of a class of evolutionary quasi-variational
inequalities of the parabolic type arising in the formation and growth models
of granular and cohensionless materials. Such models and their mathematical
descriptions are highly challenging and require powerful tools of their
analysis and implementation. We formulate a space-time continuous optimal
control problem for a basic model of this type, develop several regularization
and approximation procedures, and establish the existence of optimal solutions
for the time-continuous and space-discrete problem. Viewing a version of this
problem as a controlled quasi-variational sweeping process leads us to deriving
necessary optimality conditions for the fully discrete problem by using the
advanced machinery of variational analysis and generalized differentiation.
