@article{Krastanov2025, 
author = {Mikhail Ivanov Krastanov and Margarita Nikolaeva Nikolova},
title = {On the "good" Lie brackets related to a polynomial system},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {12},
pages = {29703-29731},
keywords = {small-time local controllability, polynomial control systems},
url = {https://www.sciopen.com/article/10.3934/math.20251306},
doi = {10.3934/math.20251306},
abstract = {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.}
}