Approximation of Sweeping Processes and Controllability for a Set Valued Evolution

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.

28 pages, 3 figures