First-Order Sweeping Processes and Extended Projected Dynamical Systems: Equivalence, Time-Discretization and Numerical Optimal Control

TL;DR

This paper establishes the equivalence between extended Projected Dynamical Systems and perturbed Moreau sweeping processes, enabling accurate time-discretization and efficient optimal control of constrained dynamical systems.

Contribution

It demonstrates the equivalence between ePDS and sweeping processes, and develops a finite element discretization method for optimal control applications.

Findings

Solutions to ePDS correspond to dynamic complementarity systems.

Perturbed sweeping processes can be reformulated as ePDS.

Finite element discretization enables efficient optimal control.

Abstract

Constrained dynamical systems are systems such that, by some means, the state stays within a given set. Two such systems are the (perturbed) Moreau sweeping process and the recently proposed extended Projected Dynamical System (ePDS). We show that under certain conditions solutions to the ePDS correspond to the solutions of a dynamic complementarity system, similar to the one equivalent to ordinary PDS. We then show that the perturbed sweeping process with time varying set can, under similar conditions, be reformulated as an ePDS. In this paper, we leverage these equivalences to develop an accurate discretization method for perturbed first-order Moreau sweeping processes via the finite elements with switch detection method. This allows the efficient optimal control of systems governed by ePDS and perturbed first-order sweeping processes.