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control-affine systems

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Hölder equivalence of the value functionfor control-affine systems ∗
Dario Prandi
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 20 / Issue 4 / October 2014

Published online by Cambridge University Press:

08 August 2014, pp. 1224-1248

Print publication:

October 2014
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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).