We describe precisely, under generic conditions, the contact of the accessibility set at time with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-riemannian system of rank 2. As a consequence we obtain in sub-riemannian geometry a new splitting-up of the sphere near an abnormal minimizer into two sectors, bordered by the first Pontryagin’s cone along , called the -sector and the -sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.
Keywords: accessibility set, abnormal trajectory, end-point mapping, single-input affine control system, sub-riemannian geometry
@article{COCV_2001__6__387_0, author = {Tr\'elat, Emmanuel}, title = {Asymptotics of accessibility sets along an abnormal trajectory}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {387--414}, publisher = {EDP-Sciences}, volume = {6}, year = {2001}, mrnumber = {1836049}, zbl = {0996.93009}, language = {en}, url = {http://www.numdam.org/item/COCV_2001__6__387_0/} }
TY - JOUR AU - Trélat, Emmanuel TI - Asymptotics of accessibility sets along an abnormal trajectory JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2001 SP - 387 EP - 414 VL - 6 PB - EDP-Sciences UR - http://www.numdam.org/item/COCV_2001__6__387_0/ LA - en ID - COCV_2001__6__387_0 ER -
Trélat, Emmanuel. Asymptotics of accessibility sets along an abnormal trajectory. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 387-414. http://www.numdam.org/item/COCV_2001__6__387_0/
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