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

Mikhail Ivanov Krastanov; Margarita Nikolaeva Nikolova; Mikhail Ivanov Krastanov; Margarita Nikolaeva Nikolova

doi:10.3934/math.20251306

AIMS Mathematics

2025, Volume 10, Issue 12: 29703-29731. doi: 10.3934/math.20251306

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

Mikhail Ivanov Krastanov ^{1,2
,
,},
Margarita Nikolaeva Nikolova ²

1.
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., block 8, 1113 Sofia, Bulgaria
2.
Faculty of Mathematics and Informatics, University of Sofia "St. Kliment Ohridski" 5 James Bourchier blvd., 1126 Sofia, Bulgaria

Received: 07 July 2025 Revised: 24 November 2025 Accepted: 01 December 2025 Published: 17 December 2025
MSC : 93B05, 93C10, 93C15

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.
- small-time local controllability,
- polynomial control systems
Citation: Mikhail Ivanov Krastanov, Margarita Nikolaeva Nikolova. On the 'good' Lie brackets related to a polynomial system[J]. AIMS Mathematics, 2025, 10(12): 29703-29731. doi: 10.3934/math.20251306

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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.

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