  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.
