Optimization Methods and Control Theory

Research Article

Lorentzian problem on the group SE(2)

Yuriy Leonidovich Sachkov¹, Ivan Andreevich Galyaev²

1	Ailamazyan Program Systems Institute of RAS, Ves'kovo, Russia
2	V. A. Trapeznikov Institute of Control Sciences of RAS, Moscow, Russia
1	yusachkov@gmail.com

Abstract. From the perspective of global differential geometry, general relativity is described by Lorentzian geometry. This paper explores Lorentzian geometry on the group $SE(2)$ : it solves the problem of finding Lorentzian longest paths that maximize the length functional along admissible curves. For the Lorentzian problem on the group $SE(2)$ , the absence of globally optimal trajectories is proven. Complete controllability of the system under consideration is demonstrated. Pontryagin's maximum principle is applied. Liouville integrability of the Hamiltonian system of Pontryagin's maximum principle is proven. Abnormal and normal extremals are parameterized by Jacobi elliptic functions. (In Russian).

Keywords: Lorentzian length maximizers, attainable set, extremals, sub-Lorentzian geometry, Lie groups

MSC-2020 93B27; 93C20, 49K20

Acknowledgments: This research was carried out under Research Project No. FL-9524115148 of the Ministry of Higher Education, Science and Innovation of the Republic of Uzbekistan

For citation: Yuriy L. Sachkov, Ivan A. Galyaev. Lorentzian problem on the group SE(2). Program Systems: Theory and Applications, 2026, 17:2, pp. 327–341. (In Russ.). https://psta.psiras.ru/2026/2_327-341.

Full text of article (PDF): https://psta.psiras.ru/read/psta2026_2_327-341.pdf.

The article was submitted 26.03.2026; approved after reviewing 11.05.2026; accepted for publication 22.05.2026; published online 27.06.2026.