Differentiability and Control of a Model for Granular Material Accumulation

We consider differentiability properties associated to the problem of minimizing the accumulation of a granular cohesionless material on a given surface. The design variable or control is determined by source locations and intensity thereof. The control problem is described by an optimization one in function space constrained by a variational inequality or a non-smooth equation. We address a regularization approach via a family of nonlinear partial differential equations, and provide a novel result of Newton differentiability of the control-to-state map. Further, we discuss solution algorithms for the state equation as well as for the optimization problem.