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Entropy and complexity of a path in sub-Riemannian geometry

Published online by Cambridge University Press:  15 September 2003

Frédéric Jean
Frédéric Jean*
Affiliation:
ENSTA, Unité de Mathématiques Appliquées, 32 boulevard Victor, 75739 Paris, France; fjean@ensta.fr.
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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 ε 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 ε-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 $\| \cdot \|_{\varepsilon}$ on the tangent space that depends on a parameter ε > 0. Our main result states then that the entropy of a path is equivalent to the integral of this ε-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.

Keywords

ComplexityHausdorff dimensionmetric entropynon-linear controlnonholonomic systemssub-Riemannian geometry.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 485 - 508
Copyright
© EDP Sciences, SMAI, 2003

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