We prove that in a class of non-equiregular sub-Riemannian manifolds corners are not length minimizing. This extends the results of [G.P. Leonardi and R. Monti, Geom. Funct. Anal. 18 (2008) 552–582]. As an application of our main result we complete and simplify the analysis in [R. Monti, Ann. Mat. Pura Appl. (2013)], showing that in a -dimensional sub-Riemannian structure suggested by Agrachev and Gauthier all length-minimizing curves are smooth.
DOI : 10.1051/cocv/2014041
Mots-clés : Sub-Riemannian geometry, regularity of geodesics, corners
@article{COCV_2015__21_3_625_0,
author = {Le Donne, Enrico and Leonardi, Gian Paolo and Monti, Roberto and Vittone, Davide},
title = {Corners in non-equiregular {sub-Riemannian} manifolds},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {625--634},
publisher = {EDP-Sciences},
volume = {21},
number = {3},
year = {2015},
doi = {10.1051/cocv/2014041},
mrnumber = {3358624},
zbl = {1333.53045},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2014041/}
}
TY - JOUR AU - Le Donne, Enrico AU - Leonardi, Gian Paolo AU - Monti, Roberto AU - Vittone, Davide TI - Corners in non-equiregular sub-Riemannian manifolds JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 625 EP - 634 VL - 21 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014041/ DO - 10.1051/cocv/2014041 LA - en ID - COCV_2015__21_3_625_0 ER -
%0 Journal Article %A Le Donne, Enrico %A Leonardi, Gian Paolo %A Monti, Roberto %A Vittone, Davide %T Corners in non-equiregular sub-Riemannian manifolds %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 625-634 %V 21 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014041/ %R 10.1051/cocv/2014041 %G en %F COCV_2015__21_3_625_0
Le Donne, Enrico; Leonardi, Gian Paolo; Monti, Roberto; Vittone, Davide. Corners in non-equiregular sub-Riemannian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 625-634. doi : 10.1051/cocv/2014041. http://archive.numdam.org/articles/10.1051/cocv/2014041/
A. Agrachev, Some open problems, Geometric Control Theory and sub-Riemannian Geometry. Edited by G. Stefani, U. Boscain, J.-P. Gauthier, A. Sarychev, M. Sigalotti. Vol. 5 of Springer INdAM Series (2014) 1–14. | MR | Zbl
, , and , Extremal Curves in Nilpotent Lie Groups. Geom. Funct. Anal. 23 (2013) 1371–1401. | DOI | MR | Zbl
E. Le Donne, G.P. Leonardi, R. Monti and D. Vittone, Extremal polynomials in stratified groups. Preprint ArXiv:1307.5235 (2013). | MR
and , End-point equations and regularity of sub-Riemannian geodesics. Geom. Funct. Anal. 18 (2008) 552–582. | DOI | MR | Zbl
and , Some remarks on the definition of tangent cones in a Carnot-Carathéodory space. J. Anal. Math. 80 (2000) 299–317. | DOI | MR | Zbl
R. Monti, A family of nonminimizing abnormal curves. Ann. Mat. Pura Appl. (2013). | MR | Zbl
R. Monti, The regularity problem for sub-Riemannian geodesics, Geometric Control Theory and sub-Riemannian Geometry. Edited by G. Stefani, U. Boscain, J.-P. Gauthier, A. Sarychev, M. Sigalotti. Vol. 5 of Springer INdAM Series (2014) 313–332. | MR | Zbl
, Regularity results for sub-Riemannian geodesics. Calc. Var. Partial Differ. Eqs. 49 (2014) 549–582. | DOI | MR | Zbl
, and , Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155 (1985) 103–147. | DOI | MR | Zbl
D. Vittone, The regularity problem for sub-Riemannian geodesics. Preprint (2013). Available at http://cvgmt.sns.it/. | MR
Cité par Sources :
