Cut time in the sub-Riemannian problem on the Cartan group
Abstract
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 Y. 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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2021
- DOI:
- 10.48550/arXiv.2107.06730
- arXiv:
- arXiv:2107.06730
- Bibcode:
- 2021arXiv210706730A
- Keywords:
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- Mathematics - Optimization and Control;
- Mathematics - Differential Geometry;
- 22E25;
- 49K15;
- 53C17
- E-Print:
- 23 pages, 3 figures