Geodesic
Asymptotics of accessibility sets along an abnormal trajectory
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 387-414

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We describe precisely, under generic conditions, the contact of the accessibility set at time T 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 γ into two sectors, bordered by the first Pontryagin’s cone along γ, called the L ∞ -sector and the L 2 -sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.

Classification : 93B03, 49K15
Keywords: accessibility set, abnormal trajectory, end-point mapping, single-input affine control system, sub-riemannian geometry 
Trélat, Emmanuel. Asymptotics of accessibility sets along an abnormal trajectory. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 387-414. http://geodesic.mathdoc.fr/item/COCV_2001__6__387_0/
@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},
     year = {2001},
     publisher = {EDP-Sciences},
     volume = {6},
     mrnumber = {1836049},
     zbl = {0996.93009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COCV_2001__6__387_0/}
}
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