| Issue |
ESAIM: COCV
Volume 21, Number 4, October-December 2015
|
|
|---|---|---|
| Page(s) | 958 - 988 | |
| DOI | https://doi.org/10.1051/cocv/2015027 | |
| Published online | 30 June 2015 | |
Cut time in sub-riemannian problem on engel group∗
Program Systems Institute of RAS, Pereslavl-Zalessky
152020,
Russia
aaa@pereslavl.ru; yusachkov@gmail.com
Received:
28
August
2014
Revised:
23
March
2015
The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic rank two sub-Riemannian problems on four-dimensional manifolds. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping.
Mathematics Subject Classification: 22E25 / 58E25
Key words: Sub-Riemannian geometry / optimal control / Engel group / Lie algebra / Maxwell time / cut time / exponential mapping / Euler’s elastica
© EDP Sciences, SMAI, 2015
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