In this paper, we give a characterization of the set of curves that may be approached by trajectories of a smooth control-ane nonlinear system, in the topology of uniform convergence. This characterization is in terms of the drift vector field and the distribution spanned by the Lie algebra generated by the control vector fields. The characterization is valid on open sets where this distribution has constant rank. These results also characterize the support of di↵usion processes with smooth coecients. ⇤