# Hölder equivalence of the value function for control-affine systems

• Volume: 20, Issue: 4, page 1224-1248
• ISSN: 1292-8119

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## Abstract

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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 ujfj(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).

## How to cite

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Prandi, Dario. "Hölder equivalence of the value function for control-affine systems." ESAIM: Control, Optimisation and Calculus of Variations 20.4 (2014): 1224-1248. <http://eudml.org/doc/272783>.

@article{Prandi2014,
abstract = {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 ujfj(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).},
author = {Prandi, Dario},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {control-affine systems; time-dependent systems; sub-riemannian geometry; value function; Ball–Box theorem; nilpotent approximation; sub-Riemannian geometry; ball-box theorem},
language = {eng},
number = {4},
pages = {1224-1248},
publisher = {EDP-Sciences},
title = {Hölder equivalence of the value function for control-affine systems},
url = {http://eudml.org/doc/272783},
volume = {20},
year = {2014},
}

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
PB - EDP-Sciences
VL - 20
IS - 4
SP - 1224
EP - 1248
AB - 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 ujfj(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).
LA - eng
KW - control-affine systems; time-dependent systems; sub-riemannian geometry; value function; Ball–Box theorem; nilpotent approximation; sub-Riemannian geometry; ball-box theorem
UR - http://eudml.org/doc/272783
ER -

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