We prove the continuity and the Hölder equivalence w.r.t. an Euclidean distance of the value function associated with the L1 cost of the control-affine system q̇ = f0(q) + ∑j=1m uj fj(q), satisfying the strong Hörmander condition. This is done by proving a result in the same spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the L1 cost of time-dependent systems of the form q̇ = ∑j=1m uj fjt(q).
Mots clés : control-affine systems, time-dependent systems, sub-riemannian geometry, value function, Ball-Box theorem, nilpotent approximation
@article{COCV_2014__20_4_1224_0,
author = {Prandi, Dario},
title = {H\"older equivalence of the value function for control-affine systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1224--1248},
publisher = {EDP-Sciences},
volume = {20},
number = {4},
year = {2014},
doi = {10.1051/cocv/2014014},
mrnumber = {3264241},
zbl = {1301.53029},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2014014/}
}
TY - JOUR AU - Prandi, Dario TI - Hölder equivalence of the value function for control-affine systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 1224 EP - 1248 VL - 20 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014014/ DO - 10.1051/cocv/2014014 LA - en ID - COCV_2014__20_4_1224_0 ER -
%0 Journal Article %A Prandi, Dario %T Hölder equivalence of the value function for control-affine systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 1224-1248 %V 20 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014014/ %R 10.1051/cocv/2014014 %G en %F COCV_2014__20_4_1224_0
Prandi, Dario. Hölder equivalence of the value function for control-affine systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1224-1248. doi : 10.1051/cocv/2014014. http://www.numdam.org/articles/10.1051/cocv/2014014/
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