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

ESAIM: Control, Optimisation and Calculus of Variations (2014)

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

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topPrandi, 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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