# Entropy and complexity of a path in sub-riemannian geometry

ESAIM: Control, Optimisation and Calculus of Variations (2003)

- Volume: 9, page 485-508
- ISSN: 1292-8119

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topJean, Frédéric. "Entropy and complexity of a path in sub-riemannian geometry." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 485-508. <http://eudml.org/doc/245107>.

@article{Jean2003,

abstract = {We characterize the geometry of a path in a sub-riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset $A$ of a metric space is the minimum number of balls of a given radius $\{\varepsilon \}$ needed to cover $A$. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-riemannian manifold as the infimum of the lengths of all trajectories contained in an $\{\varepsilon \}$-neighborhood of the path, having the same extremities as the path. The concept of complexity for paths was first developed to model the algorithmic complexity of the nonholonomic motion planning problem in robotics. In this paper, our aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-riemannian manifold. We construct first a norm $\Vert \cdot \Vert _\{\{\varepsilon \}\}$ on the tangent space that depends on a parameter $\{\varepsilon \}>0$. Our main result states then that the entropy of a path is equivalent to the integral of this $\{\varepsilon \}$-norm along the path. As a corollary we obtain upper and lower bounds for the Hausdorff dimension of a path. Our second main result is that complexity and entropy are equivalent for generic paths. We give also a computable sufficient condition on the path for this equivalence to happen.},

author = {Jean, Frédéric},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {complexity; Hausdorff dimension; metric entropy; non-linear control; nonholonomic systems; sub-riemannian geometry; Complexity; sub-Riemannian geometry},

language = {eng},

pages = {485-508},

publisher = {EDP-Sciences},

title = {Entropy and complexity of a path in sub-riemannian geometry},

url = {http://eudml.org/doc/245107},

volume = {9},

year = {2003},

}

TY - JOUR

AU - Jean, Frédéric

TI - Entropy and complexity of a path in sub-riemannian geometry

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2003

PB - EDP-Sciences

VL - 9

SP - 485

EP - 508

AB - We characterize the geometry of a path in a sub-riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset $A$ of a metric space is the minimum number of balls of a given radius ${\varepsilon }$ needed to cover $A$. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-riemannian manifold as the infimum of the lengths of all trajectories contained in an ${\varepsilon }$-neighborhood of the path, having the same extremities as the path. The concept of complexity for paths was first developed to model the algorithmic complexity of the nonholonomic motion planning problem in robotics. In this paper, our aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-riemannian manifold. We construct first a norm $\Vert \cdot \Vert _{{\varepsilon }}$ on the tangent space that depends on a parameter ${\varepsilon }>0$. Our main result states then that the entropy of a path is equivalent to the integral of this ${\varepsilon }$-norm along the path. As a corollary we obtain upper and lower bounds for the Hausdorff dimension of a path. Our second main result is that complexity and entropy are equivalent for generic paths. We give also a computable sufficient condition on the path for this equivalence to happen.

LA - eng

KW - complexity; Hausdorff dimension; metric entropy; non-linear control; nonholonomic systems; sub-riemannian geometry; Complexity; sub-Riemannian geometry

UR - http://eudml.org/doc/245107

ER -

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## Citations in EuDML Documents

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- Cutberto Romero-Meléndez, Jean Paul Gauthier, Felipe Monroy-Pérez, On complexity and motion planning for co-rank one sub-riemannian metrics
- Dario Prandi, Hölder equivalence of the value function for control-affine systems

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