Differentiability and Control of a Model for Granular Material Accumulation

TL;DR

This paper investigates the differentiability and control of a model for granular material accumulation, introducing a regularization approach and establishing Newton differentiability of the control-to-state map for optimization.

Contribution

It presents a novel regularization method for the problem and proves Newton differentiability of the control-to-state map, enabling advanced solution algorithms.

Findings

Established Newton differentiability of the control-to-state map.

Developed solution algorithms for the state equation and optimization.

Proposed a regularization approach via nonlinear PDEs.

Abstract

We consider differentiability issues associated to the problem of minimizing the accumulation of a granular cohesionless material on a certain surface. The design variable or control is determined by source locations and intensity thereof. The control problem is described by an optimization problem in function space and 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.