*Accepted Paper*

**Inserted:** 1 dec 2021

**Last Updated:** 7 may 2022

**Journal:** Springer Proceedings dedicated to DCO 2021

**Year:** 2021

**Abstract:**

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.

**Keywords:**
global controllability, Poisson stable points, Poincar\'{e} recurrentce theorem

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