A condition equivalent to global controllability in systems of vector fields
Let S be a family of \(C^ 1\) vector fields on a finite-dimensional differentiable manifold M of class \(C^ k\) with \(k\geq 2\). Let A(x,S) be the set of all points in M that are reachable from x via a finite number of integral curves of vector fields from S. The family S is said to be globally controllable if \(A(x,S)=M\) for every \(x\in M\). When M is compact, a theorem of \textit{I. Kupka} and \textit{G. Sallet} [ibid. 47, 462- 470 (1983; Zbl 0527.93029)] states that S is globally controllable if and only if S has property (P), i.e. \(x\in int A(x,S)\) for every \(x\in M\). The author shows that this equivalence continues to hold when M is noncompact.
scientific article; zbMATH DE number 3934083