Approximation of sweeping processes and controllability for a set-valued evolution

We consider a controlled evolution problem for a set \Omega (t) \subset \BbbR ^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.