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.
Keywords: Cartan group, sub-Riemannian problem, nilpotent approximation, Carnot groups, Euler elastica, optimal synthesis
@article{COCV_2022__28_1_A12_0,
author = {Ardentov, Andrei and Hakavuori, Eero},
title = {Cut time in the {sub-Riemannian} problem on the {Cartan} group},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022006},
mrnumber = {4379606},
zbl = {1507.53028},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022006/}
}
TY - JOUR AU - Ardentov, Andrei AU - Hakavuori, Eero TI - Cut time in the sub-Riemannian problem on the Cartan group JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022006/ DO - 10.1051/cocv/2022006 LA - en ID - COCV_2022__28_1_A12_0 ER -
%0 Journal Article %A Ardentov, Andrei %A Hakavuori, Eero %T Cut time in the sub-Riemannian problem on the Cartan group %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022006/ %R 10.1051/cocv/2022006 %G en %F COCV_2022__28_1_A12_0
Ardentov, Andrei; Hakavuori, Eero. Cut time in the sub-Riemannian problem on the Cartan group. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 12. doi: 10.1051/cocv/2022006
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E.H. was supported by the SISSA project DIP_ECC_MATE_CoordAreaMate_0459 – Dipartimenti di Eccellenza 2018 – 2022 (CUP: G91—18000050006).
