  We give a constructive proof of a global controllability result for an
autonomous system of ODEs guided by bounded locally Lipschitz and divergence
free (i.e.\ incompressible) vector field, when the phase space is the whole
Euclidean space and the vector field satisfies so-called vanishing mean drift
condition. For the case when the ODE is defined over some smooth compact
connected Riemannian manifold, we significantly strengthen the assertion of the
known controllability theorem in absence of nonholonomic constraints by proving
that one can find a control steering the state vector from one given point to
another by using the observations of only the state vector, i.e., in other
words, by changing slightly the vector field, and such a change can be made
small not only in uniform, but also in Lipschitz (i.e. $C^1$) topology.

Dynamical Systems