On the "good" Lie brackets related to a polynomial system

Small-time local controllability (STLC) at a point $x_0$ is a fundamental property of control systems, and is intimately connected to the local structure of their reachable sets. This study built upon the notion of a tangent vector field to the reachable set of a control system, a concept introduced by Hermes in ^[7], based on an idea of Krener (cf. ^[18]). The importance of this concept stemed from the fact that the set $E^{+}(x_0)$, consisting of all tangent vector fields to the reachable set at $x_0$, formed a convex cone. If the zero vector lay in the interior of this cone, the system is STLC at $x_0$. A long-standing open question concerns the precise characterization of the set $E^{+}(x_0)$. In this paper, we studied the Lie algebra generated by the drift term—a vector field homogeneous of degree two—and the constant vector fields of a polynomial control system. By applying the classical Campbell–Baker–Hausdorff formula from Lie group theory, along with symmetries inherent to the control system, we derived new elements of the set $E^{+}(x_0)$. Our results showed that certain "bad" Lie brackets (in the sense of Sussmann) do not obstruct the STLC property. As a corollary, we provided a sufficient condition for STLC.