Volume 20, Number 4, October-December 2014
|Page(s)||1224 - 1248|
|Published online||08 August 2014|
Hölder equivalence of the value function for control-affine systems∗
LSIS, Université de Toulon, 83957
La Garde cedex,
2 Centre National de Recherche Scientifique (CNRS), CMAP, École Polytechnique, Route de Saclay, 91128 Palaiseau cedex, France
3 Team GECO, INRIA-Centre de Recherche Saclay, France
Received: 24 April 2013
Revised: 22 November 2013
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).
Mathematics Subject Classification: 53C17 / 53C17
Key words: Control-affine systems / time-dependent systems / sub-Riemannian geometry / value function / Ball–Box theorem / nilpotent approximation
© EDP Sciences, SMAI, 2014
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