| Issue |
ESAIM: COCV
Volume 28, 2022
|
|
|---|---|---|
| Article Number | 12 | |
| Number of page(s) | 19 | |
| DOI | https://doi.org/10.1051/cocv/2022006 | |
| Published online | 14 February 2022 | |
Cut time in the sub-Riemannian problem on the Cartan group*
1
Ailamazyan Program Systems Institute, Russian Academy of Sciences,
Pereslavl-Zalessky, Russia.
2
SISSA,
Via Bonomea 265,
34136
Trieste, Italy.
** Corresponding author: aaa@pereslavl.ru
Received:
2
September
2021
Accepted:
24
January
2022
We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the conjectured cut times of Yu. Sachkov for the sub-Riemannian Cartan problem. Along the proof, we obtain a comparison with the known cut times in the sub-Riemannian Engel group, and a sufficient (generic) condition for the uniqueness of the length minimizer between two points. Hence we reduce the optimal synthesis to solving a certain system of equations in elliptic functions.
Mathematics Subject Classification: 22E25 / 49K15 / 53C17
Key words: Cartan group / sub-Riemannian problem / nilpotent approximation / Carnot groups / Euler elastica / optimal synthesis
© The authors. Published by EDP Sciences, SMAI 2022
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