The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic rank two sub-Riemannian problems on four-dimensional manifolds. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping.
DOI: 10.1051/cocv/2015027
Keywords: Sub-Riemannian geometry, optimal control, Engel group, Lie algebra, Maxwell time, cut time, exponential mapping, Euler’s elastica
@article{COCV_2015__21_4_958_0, author = {Ardentov, A.A. and Sachkov, Yu.L.}, title = {Cut time in sub-riemannian problem on engel group}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {958--988}, publisher = {EDP-Sciences}, volume = {21}, number = {4}, year = {2015}, doi = {10.1051/cocv/2015027}, mrnumber = {3395751}, zbl = {1330.53044}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2015027/} }
TY - JOUR AU - Ardentov, A.A. AU - Sachkov, Yu.L. TI - Cut time in sub-riemannian problem on engel group JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 958 EP - 988 VL - 21 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2015027/ DO - 10.1051/cocv/2015027 LA - en ID - COCV_2015__21_4_958_0 ER -
%0 Journal Article %A Ardentov, A.A. %A Sachkov, Yu.L. %T Cut time in sub-riemannian problem on engel group %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 958-988 %V 21 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2015027/ %R 10.1051/cocv/2015027 %G en %F COCV_2015__21_4_958_0
Ardentov, A.A.; Sachkov, Yu.L. Cut time in sub-riemannian problem on engel group. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 958-988. doi : 10.1051/cocv/2015027. http://www.numdam.org/articles/10.1051/cocv/2015027/
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