On Stability of Totally Controlled Systems

It is known that a smooth control system whose phase space is a smooth manifold $M$ generates some smooth foliation with singularities. This foliation has the property that, for every point $\eta\in M$, the controllability set with goal point $\eta$ is a subset of the leaf passing through $\eta$. In the first part of the article, we obtain sufficient conditions under which a system totally controlled on a fixed leaf is totally controlled on the leaves close to this leaf. In the second part, we consider a control system with a parameter totally controlled for some value of the parameter. We obtain sufficient conditions under which the system is totally controlled for the values of the parameter close to a given value.