Proving the Chow-Rashevskii Theorem \`a la Rashevskii

Abstract

We give a new independent proof of a generalised version of the theorem by Rashevskii, which appeared in [Uch. Zapiski Ped. Inst. K. 2 (1938), 83 -- 94] and from which the classical Chow-Rashevskii Theorem follows as a corollary. The proof is structured to allow generalisations to the case of orbits of compositions of flows in absence of group structures, thus appropriate for applications in Control Theory. In fact, the same structure of the proof has been successfully exploited in [C. Giannotti, A. Spiro and M. Zoppello, arXiv 2401.07555 \& 2401.07560 (2024)] to determine new controllability criteria for real analytic non-linear control systems. It also yields a corollary, which can be used to derive results under lower regularity assumptions, as it is illustrated by a simple explicit example.