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We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned singularities are the abnormal extremals associated to the dynamics of the control system and the Wazewski equation is an o.d.e. on the control space called the Path Lifting Equation (PLE). We first show elementary facts relative to the maximal solution of the PLE such as local existence and uniqueness. Then we prove two general results, a finite-dimensional reduction for the PLE on compact time intervals and a regularity preserving theorem. In a second part, if the Strong Bracket Generating Condition holds, we show, for several control spaces, the global existence of the solution of the PLE, extending a previous result of H.J. Sussmann.
@article{COCV_2006__12_1_139_0,
author = {Chitour, Yacine},
title = {A continuation method for motion-planning problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {139--168},
publisher = {EDP-Sciences},
volume = {12},
number = {1},
year = {2006},
doi = {10.1051/cocv:2005035},
mrnumber = {2192072},
zbl = {1105.93030},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2005035/}
}
TY - JOUR AU - Chitour, Yacine TI - A continuation method for motion-planning problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 139 EP - 168 VL - 12 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2005035/ DO - 10.1051/cocv:2005035 LA - en ID - COCV_2006__12_1_139_0 ER -
%0 Journal Article %A Chitour, Yacine %T A continuation method for motion-planning problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 139-168 %V 12 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2005035/ %R 10.1051/cocv:2005035 %G en %F COCV_2006__12_1_139_0
Chitour, Yacine. A continuation method for motion-planning problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 1, pp. 139-168. doi: 10.1051/cocv:2005035
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