We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems, that play the role of the constant curvature spaces. As an application, we prove a version of sub-Riemannian Bonnet−Myers theorem and we obtain some new results on conjugate points for three dimensional left-invariant sub-Riemannian structures.
DOI : 10.1051/cocv/2015013
Mots clés : Sub-Riemannian geometry, curvature, comparison theorems, conjugate points
@article{COCV_2016__22_2_439_0, author = {Barilari, D. and Rizzi, L.}, title = {Comparison theorems for conjugate points in {sub-Riemannian} geometry}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {439--472}, publisher = {EDP-Sciences}, volume = {22}, number = {2}, year = {2016}, doi = {10.1051/cocv/2015013}, mrnumber = {3491778}, zbl = {1344.53023}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2015013/} }
TY - JOUR AU - Barilari, D. AU - Rizzi, L. TI - Comparison theorems for conjugate points in sub-Riemannian geometry JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 439 EP - 472 VL - 22 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2015013/ DO - 10.1051/cocv/2015013 LA - en ID - COCV_2016__22_2_439_0 ER -
%0 Journal Article %A Barilari, D. %A Rizzi, L. %T Comparison theorems for conjugate points in sub-Riemannian geometry %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 439-472 %V 22 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2015013/ %R 10.1051/cocv/2015013 %G en %F COCV_2016__22_2_439_0
Barilari, D.; Rizzi, L. Comparison theorems for conjugate points in sub-Riemannian geometry. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 439-472. doi : 10.1051/cocv/2015013. http://www.numdam.org/articles/10.1051/cocv/2015013/
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