Families of vector fields which generate the group of diffeomorphisms (Q630207)

In the paper under review, the author gives a simple sufficient condition for a family of flows on a smooth compact manifold \(M\) to generate the group \(Diff_0(M)\) of all diffeomorphisms of \(M\) that are isotopic to the identity. The problem of realizing a diffeomorphism as a composition of exponentials of smooth vector fields arises in the framework of control theory. In the interesting cases, the system cannot evolve along all possible directions but only along a prescribed vector distribution. Given a family of vector fields \(\mathcal{F}\) such that the group generated by \(\mathcal{F}\) acts transitively on \(M\), the author proves that the group \(Diff_0(M)\) is generated by the exponentials of vector fields in \(\mathcal{F}\) rescaled by smooth functions. More precisely, the author proves the following main theorem. Theorem 1. Let \(\mathcal{F} \subset Vec~M\) be a family of smooth vector fields and let \(Gr \mathcal{F} = \{e^{t_1f_1} \circ \cdots \circ e^{t_kf_k}: t_i \in \mathbb{R}, f_i \in \mathcal{F}, k \in \mathbb{N}\}\). If \(Gr \mathcal{F}\) acts transitively on \(M\), then there exists a neighbourhood \(\mathcal{O}\) of the identity in \(Diff_0(M)\) and a positive integer \(m\) such that every \(P \in \mathcal{O}\) can be presented in the form \(P= e^{t_1f_1} \circ \cdots \circ e^{t_mf_m}\) for some \(f_1,\dots, f_m \in \mathcal{F} \) and \(a_1,\dots, a_m \in C^{\infty}(M)\).