The left-invariant sub-riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.
Mots-clés : optimal control, sub-riemannian geometry, differential-geometric methods, left-invariant problem, group of motions of a plane, rototranslations, conjugate time, cut time
@article{COCV_2010__16_4_1018_0,
author = {Sachkov, Yuri L.},
title = {Conjugate and cut time in the sub-riemannian problem on the group of motions of a plane},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1018--1039},
publisher = {EDP-Sciences},
volume = {16},
number = {4},
year = {2010},
doi = {10.1051/cocv/2009031},
mrnumber = {2744160},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2009031/}
}
TY - JOUR AU - Sachkov, Yuri L. TI - Conjugate and cut time in the sub-riemannian problem on the group of motions of a plane JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 1018 EP - 1039 VL - 16 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009031/ DO - 10.1051/cocv/2009031 LA - en ID - COCV_2010__16_4_1018_0 ER -
%0 Journal Article %A Sachkov, Yuri L. %T Conjugate and cut time in the sub-riemannian problem on the group of motions of a plane %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 1018-1039 %V 16 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2009031/ %R 10.1051/cocv/2009031 %G en %F COCV_2010__16_4_1018_0
Sachkov, Yuri L. Conjugate and cut time in the sub-riemannian problem on the group of motions of a plane. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 1018-1039. doi : 10.1051/cocv/2009031. http://www.numdam.org/articles/10.1051/cocv/2009031/
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