On the geometry of the reachability set of vector fields

We study the geometry of the reachability set of a family of vector fields on a C ∞ manifold. We show that, for each real number T, the T-reachability set is a smooth submanifold of an orbit of codimension zero or one and that, on an arbitrary connected C ∞ manifold of dimension greater than one, there exists a system of three vector fields such that each 0-reachability set coincides with the manifold itself.

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