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

Approximation of Sweeping Processes and Controllability for a Set Valued Evolution