Using infinitesimal symmetries for determining the first Maxwell time of geometric control problem on SH(2)

Abstract

In this work, we utilize infinitesimal symmetries to compute Maxwell points which play a crucial role in studying sub-Riemannian control problems. By examining the infinitesimal symmetries of the geometric control problem on the SH(2) group, particularly through its Lie algebraic structure, we identify invariant quantities and constraints that streamline the Maxwell point computation.

Keywords

Geometric control theory / Lie algebra / Maxwell time / Special hyperbolic group / Sub-Riemannian geometry

Funding

None.

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of this article.

References

[1] Sachkov Y. Introduction to geometric control. In: Springer Optimization and Its Applications. Moscow:URSS; 2019:192.https://doi.org/10.1007/978-3-031-02070-4

[2] Moiseev I, Sachkov YL. Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COV 2010; 16(2): 380-399.https://doi.org/10.1051/cocv/2009030

[3] Sachkov YL. Complete description of the Maxwell strata in the generalized Dido problem. Sb. Math. 2006; 197(6):901-950. https://doi.org/10.1070/SM2006v197n06ABEH003783

[4] Agrachev AA, Barilari D, Boscain U. A Comprehensive Introduction to Sub-Riemannian Geome-try. Cambridge University Press; 2019. https://doi.org/10.1017/9781108677325

[5] Butt YA. Sub-Riemannian Problem on Lie Group of Motions of Pseudo-Euclidean Plane; 2015: 53-91. http://142.54.178.187:9060/xmlui/handle/123456789/1122

[6] Gallier J, Quaintance J. Notes on Differential Geometry and Lie Groups [Lecture Notes]; 2016.

[7] Hrdina J, Zalabová L. Local Geometric Control of a Certain Mechanism with the Growth Vector (4,7). J Dyn Control Syst.; 2020:26;199-216. https://doi.org/10.1007/s10883-019-09460-7

[8] Agrachev AA, Barilari D. Sub-Riemannian structures on 3D Lie groups. J Dyn Control Syst. 2012; 18(1):21-44. https://doi.org/10.1007/s10883-012-9133-8

[9] Jean F. Control of Nonholonomic Systems: From Sub-Riemannian Geometry to Motion Planning. In: Springer Briefs in Mathematics. Springer; 2014.

[10] Hill CD, Nurowski P. A Car as Parabolic Geometry; 2019. arXiv:1908.01169.

[11] Hermans J. A symmetric sphere rolling on a surface. Nonlinearity 1995;8:1-23. https://doi.org/10.1088/0951-7715/8/4/003

[12] Bloch AM. Nonholonomic Mechanics and Control. Springer; 2015. https://doi.org/10.1007/978-1-4939-3017-3

[13] Mao Z, Kobayashi R, Nabae H, Suzumori K. Multimodal strain sensing system for shape recognition of tensegrity structures by combining traditional regression and deep learning approaches. IEEE Robot Automation Lett. 2024; 9(11):1-6. https://doi.org/10.1109/LRA.2024.3469811

[14] Peng Y, Yang X, Li D, et al. Predicting flow status of a flexible rectifier using cognitive computing. Exp Syst Appl. 2025;264: 125878. https://doi.org/10.1016/j.eswa.2024.125878

[15] Jurdjevic V. Optimal Control and Geometry: Integrable Systems, Vol. 154. Cambridge University Press; 2016. https://doi.org/10.1017/CBO9781316286852

[16] Agrachev A, Barilari D. Sub-Riemannian Structures on 3D Lie Groups. SISSA & MIAN; 2011. https://doi.org/10.48550/arXiv.1007.4970

[17] Peter J.Olver, Application of Lie Groups to Differential Equations; 1993. https://doi.org/10.1007/978-1-4612-4350-2

[18] Olver FWJ, Lozier DW, Boisvert RF, Clark CW. (2010). NIST Handbook of Mathematical Functions. National Institute of Standards and Technology, Cambridge University Press; 2010.

[19] Bonnard B, Cots O. Geometric numerical methods and results in the contrast imaging problem in nuclear magnetic resonance. Math Models Methods Appl Sci. 2014; 24(1):187-212. https://doi.org/10.1142/S0218202513500504