Conditions to the density of accessible sets

Given a control system $\dot{p}=X_0(p)+{\sum }_i{u}_i(t)X_i(p)$ on a compact manifold M we study conditions for the foliation defined by the accessible sets be dense in M . To do this we relate the control system to a stochastic differential equation and, using the support theorem, we give a characterization of the density in terms of the infinitesimal generator of the diffusion and its invariant measures. Also we give a different proof of Krener’s theorem.