SUB-LORETZIAN EXTREMALS DEFINED BY AN ANTINORM
- Authors: Podobryaev A.V1
-
Affiliations:
- A.K. Ailamazyan Program Systems Institute of RAS
- Issue: Vol 60, No 3 (2024)
- Pages: 386-398
- Section: Articles
- URL: https://journals.rcsi.science/0374-0641/article/view/257617
- DOI: https://doi.org/10.31857/S0374064124030089
- EDN: https://elibrary.ru/PLBQDM
- ID: 257617
Cite item
Abstract
We consider a left-invariant sub-Lorentzian structure on a Lie group. We assume that this structure is defined by a closed convex salient cone in the corresponding Lie algebra and a continuous antinorm associated with this cone. We derive the Hamiltonian system for sub-Lorentzian extremals and give conditions under that normal extremal trajectories keep their causal type. Tangent vectors of abnormal extremal trajectories are either light-like or tangent vectors of sub-Riemannian extremal trajectories for the sub-Riemannian distribution spanned by the cone.
About the authors
A. V Podobryaev
A.K. Ailamazyan Program Systems Institute of RAS
Email: alex@alex.botik.ru
Pereslavl-Zalesskiy, Russia
References
- Grochowski, M. On the Heisenberg sub-Lorentzian metric on R / M. Grochowski // Geometric Singularity Theory. Banach Center Publications. — Warszawa : Institute of Mathematics. Polish Academy of Sciences, 2004. — V. 65. — P. 57–65.
- Grochowski, M. Reachable sets for the Heisenberg sub-Lorentzian structure on R. An estimate for the distance function / M. Grochowski // J. Dyn. Control Syst. — 2006. — V. 12, № 2. — P. 145–160.
- Сачков, Ю.Л. Сублоренцева задача на группе Гейзенберга / Ю.Л. Сачков, Е.Ф. Сачкова // Мат. заметки. — 2023. — Т. 113, № 1. — С. 154–157.
- Sachkov, Yu.L. Sub-Lorentzian distance and spheres on the Heisenberg group / Yu.L. Sachkov, E.F. Sachkova // J. Dyn. Control Syst. — 2023. — V. 29. — P. 1129–1159.
- Grong, E. Sub-Riemannian and sub-Lorentzian geometry on ????????(1, 1) and on its universal cover / E. Grong, A. Vasil’ev // J. Geom. Mech. — 2011. — V. 3, № 2. — P. 225–260.
- Сачков, Ю.Л. Лоренцева геометрия на плоскости Лобачевского / Ю.Л. Сачков // Мат. заметки. — 2023. — T. 114, № 1. — С. 154–157.
- Sachkov, Yu.L. Lorentzian distance on the Lobachevsky plane / Yu.L. Sachkov // arXiv:2307.07706. — 2023.
- Agrachev, A. A Comprehensive Introduction to Sub-Riemannian Geometry / A. Agrachev, D. Barilari, U. Boscain. — Cambridge; New York : Cambridge University Press, 2019. — 745 p.
- Сачков, Ю.Л. Введение в геометрическую теорию управления / Ю.Л. Сачков. — М. : Ленанд, 2021. — 160 с.
- Локуциевский, Л.В. Выпуклая тригонометрия с приложениями к субфинслеровой геометрии / Л.В. Локуциевский // Мат. сборник. — 2019. – Т. 210, № 8. — С. 120–148.
- Ардентов, А.А. Решение серии задач оптимального управления с 2-мерным управлением на основе выпуклой тригонометрии / А.А. Ардентов, Л.В. Локуциевский, Ю.Л. Сачков // Докл. РАН. Математика, информатика, процессы управления. — 2020. — Т. 494. — С. 86–92.
- Ardentov, A.A. Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry / A.A. Ardentov, L.V. Lokutsievskiy, Yu.L. Sachkov // ESAIM: Control, Optimization and Calculus of Variations. — 2021. — V. 27, № 32. — P. 32–52.
- Lokutsievskiy, L.V. Explicit formulae for geodesics in left-invariant sub-Finsler problems on Heisenberg groups via convex trigonometry / L.V. Lokutsievskiy // J. Dyn. Control Syst. — 2021. — V. 27. — P. 661–681.
- Protasov, V.Yu. Antinorms on cones: duality and applications / V.Yu. Protasov // Linear and Multilinear Algebra. — 2021. — V. 70, № 22. — P. 7387–7413.
- Математическая теория оптимальных процессов / Л.С. Понтрягин, В.Г. Болтянский, Р.В. Гамкрелидзе, Е.Ф. Мищенко. — 4-е изд., стереотип. — М. : Наука, 1983. — 393 с.
- Аграчев, А.А. Геометрическая теория управления / А.А. Аграчев, Ю.Л. Сачков. — М. : Физматлит, 2004. — 392 с.
- Рокафеллар, Р. Выпуклый анализ / Р. Рокафеллар ; пер. с англ. А.Д. Иоффе и В.М. Тихомирова. — М. : Мир, 1973. — 469 с.
- Сачков, Ю.Л. Структура анормальных экстремалей в субримановой задаче с вектором роста (2, 3, 5, 8) / Ю.Л. Сачков, Е.Ф. Сачкова // Мат. сб. — 2020. — Т. 211, № 10. — С. 112–138.
- Grochowski, M., On the Heisenberg sub-Lorentzian metric on R, in Geometric singularity theory. Banach Center publications, Warszawa: Institute of Mathematics. Polish Academy of Sciences, 2004, vol. 65, pp. 57–65.
- Grochowski, M., Reachable sets for the Heisenberg sub-Lorentzian structure on R. An estimate for the distance function, J. Dyn. Control Syst., 2006, vol. 12, no. 2, pp. 145–160.
- Sachkov, Yu.L. and Sachkova, E.F., Sub-Lorentzian problem on the Heisenberg group, Math. Notes, 2023, vol. 113, pp. 159–162.
- Sachkov, Yu.L. and Sachkova, E.F., Sub-Lorentzian distance and spheres on the Heisenberg group, J. Dyn. Control Syst., 2023, vol. 29, pp. 1129–1159.
- Grong, E. and Vasil’ev, A., Sub-Riemannian and sub-Lorentzian geometry on ???????? (1, 1) and on its universal cover, J. Geom. Mech., 2011, vol. 3, no. 2, pp. 225–260.
- Sachkov, Yu.L., Lorentzian geometry on the Lobachevsky plane, Math. Notes, 2023, vol. 114, pp. 127–130.
- Sachkov, Yu.L., Lorentzian distance on the Lobachevsky plane, 2023, arXiv:2307.07706.
- Agrachev, A., Barilari, D., and Boscain, U., A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge–New York: Cambridge University Press, 2019.
- Sachkov, Yu.L., Introduction to Geometric Control, Cham: Springer, 2021.
- Lokutsievskiy, L.V., Convex trigonometry with applications to sub-Finsler geometry, Sb. Math., 2019, vol. 210, no. 8, pp. 1179–1205.
- Ardentov, A.A., Lokutsievskiy, L.V., and Sachkov, Y.L., Explicit solutions for a series of optimization problems with 2-dimensional control via convex trigonometry, Dokl. Math., 2020, vol. 210, pp. 427–432.
- Ardentov, A.A., Lokutsievskiy, L.V., and Sachkov, Yu.L., Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry, ESAIM: Control, Optimization and Calculus of Variations, 2021, vol. 27, no. 32, pp. 32–52.
- Lokutsievskiy, L.V., Explicit formulae for geodesics in left-invariant sub-Finsler problems on Heisenberg groups via convex trigonometry, J. Dyn. Control Syst., 2021, vol. 27, pp. 661–681.
- Protasov, V.Yu., Antinorms on cones: duality and applications, Linear and Multilinear Algebra, 2021, vol. 70, no. 22, pp. 7387–7413.
- Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko E.F., The Mathematical Theory of Optimal Processes, Oxford: Pergamon Press, 1964.
- Agrachev, A.A. and Sachkov, Yu.L., Control Theory from the Geometric Viewpoint, Berlin–Heidelberg–New York: Springer-Verlag, 2004.
- Rockafellar, R., Convex Analysis, Princeton: Princeton University Press, 1970.
- Sachkov, Yu.L. and Sachkova, E.F., The structure of abnormal extremals in a sub-Riemannian problem with growth vector (2, 3, 5, 8), Sb. Math., 2020, vol. 211, no. 10, pp. 1460–1485.