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Mat. Sb., 2018, Volume 209, Number 11, Pages 3–31 (Mi msb9017)  

Extremal trajectories in the sub-Lorentzian problem on the Engel group

A. A. Ardentova, Yu. L. Sachkova, T. Huangb, X. Yangc

a Ailamazyan Program Systems Institute of Russian Academy of Sciences
b Zhejiang Sci-Tech University, Hangzhou, The People's Republic of China
c Nanjing University of Science and Technology, The People's Republic of China

Abstract: Let $\mathbb{E}$ be the Engel group and let $D$ be a rank-two left-invariant distribution with Lorentzian metric on $\mathbb{E}$. The sub-Lorentzian problem is stated as the problem of maximizing the sub-Lorentzian distance. A parametrization of timelike and spacelike normal extremal trajectories is obtained in terms of Jacobi elliptic functions. Discrete symmetry groups are described in the cases of timelike and spacelike trajectories; in both cases the fixed points and the corresponding Maxwell points are calculated for each symmetry. These calculations underlie estimates for the cut time (when the trajectory ceases to be globally optimal).
Bibliography: 17 titles.

Keywords: Engel group, extremal trajectories, sub-Lorentzian metric, Jacobi functions.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation АААА-А17-117040610374-8
Russian Science Foundation 17-11-01387
The research by A. A. Ardentov and Yu. L. Sachkov in § 4 was carried out as part of the implementation of a state assignment of the Ministry of Education and Science of the Russian Federation (project no. AAAA-A17-117040610374-8) and in §§ 5 and 6 it was carried out with the support of the Russian Science Foundation under grant no. 17-11-01387. Sections 2 and 3 are due to T. Huang and X. Yang, and §§ 4, 5 and 6 are due to A. A. Ardentov and Yu. L. Sachkov.


DOI: https://doi.org/10.4213/sm9017

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English version:
Sbornik: Mathematics, 2018, 209:11, 1547–1574

Bibliographic databases:

Document Type: Article
UDC: 517.977
MSC: Primary 53C17, 53C50; Secondary 22E25
Received: 16.10.2017

Citation: A. A. Ardentov, Yu. L. Sachkov, T. Huang, X. Yang, “Extremal trajectories in the sub-Lorentzian problem on the Engel group”, Mat. Sb., 209:11 (2018), 3–31; Sb. Math., 209:11 (2018), 1547–1574

Citation in format AMSBIB
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