In Carnot groups of step ≤ 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carnot groups.
Mots-clés : subriemannian geometry, geodesics, calculus of variations, Goh condition, generalized Legendre-Jacobi condition
@article{COCV_2013__19_1_274_0,
author = {Tan, Kanghai and Yang, Xiaoping},
title = {Subriemannian geodesics of {Carnot} groups of step 3},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {274--287},
publisher = {EDP-Sciences},
volume = {19},
number = {1},
year = {2013},
doi = {10.1051/cocv/2012006},
mrnumber = {3023070},
zbl = {1276.53041},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2012006/}
}
TY - JOUR AU - Tan, Kanghai AU - Yang, Xiaoping TI - Subriemannian geodesics of Carnot groups of step 3 JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 274 EP - 287 VL - 19 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012006/ DO - 10.1051/cocv/2012006 LA - en ID - COCV_2013__19_1_274_0 ER -
%0 Journal Article %A Tan, Kanghai %A Yang, Xiaoping %T Subriemannian geodesics of Carnot groups of step 3 %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 274-287 %V 19 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012006/ %R 10.1051/cocv/2012006 %G en %F COCV_2013__19_1_274_0
Tan, Kanghai; Yang, Xiaoping. Subriemannian geodesics of Carnot groups of step 3. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 274-287. doi : 10.1051/cocv/2012006. http://www.numdam.org/articles/10.1051/cocv/2012006/
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