  It is first shown that a smooth controllable system on a compact manifold is
finite time controllable. The technique of proof is close to the one of
Sussmann's orbit theorem, and no rank condition is required. This technique is
also used to give a new and elementary proof of the equivalence between
controllability for essentially bounded inputs and for piecewise constant ones.
Two sufficient conditions for controllability at exact time on a compact
manifold are then stated. Some applications, in particular to linear systems on
Lie groups, are provided.

Optimization and Control