Approximation of Sweeping Processes and Controllability for a Set Valued Evolution

TL;DR

This paper studies controllability of set-valued evolutions inspired by a flocking model and demonstrates how to approximate sweeping processes through controlled set evolutions.

Contribution

It introduces conditions for complete controllability of set evolutions and provides a method to approximate sweeping processes via controlled evolutions.

Findings

Established necessary and sufficient controllability conditions.

Proved the existence of controls approximating sweeping processes.

Applied techniques to set-valued evolution models.

Abstract

We consider a controlled evolution problem for a set $\Omega(t)\in\mathbb{R}^d$, originally motivated by a model where a dog controls a flock of sheep. Necessary conditions and sufficient conditions are given, in order that the evolution be completely controllable. Similar techniques are then applied to the approximation of a sweeping process. Under suitable assumptions, we prove that there exists a control function such that the corresponding evolution of the set $\Omega(t)$ is arbitrarily close to the one determined by the sweeping process.