On stability of totally controlled systems (Q1876431)

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 \(\eta\). The author obtains sufficient conditions under which a system totally controlled on a fixed leaf is totally controlled on the leaves close to this leaf. He studies also a control system with a parameter totally controlled for some value of the parameter and obtains sufficient conditions under which the system is totally controlled for the values of the parameter close to a given value. This is an english translation of the author's article [Mat. Tr. 4, No. 1, 94-110 (2001; Zbl 0977.37048)].