Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-09T09:11:07.756Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A. Bellaïche, The tangent space in sub-Riemannian geometry, edited by A. Bellaïche and J.-J. Risler, Sub-Riemannian Geometry. Birkhäuser, Progr. Math. (1996).
A. Bellaïche, F. Jean and J.-J. Risler, Geometry of nonholonomic systems, edited by J.-P. Laumond, Robot Motion Planning and Control. Springer, Lecture Notes Inform. Control Sci. 229 (1998).
A. Bellaïche, J.-P. Laumond and J. Jacobs, Controllability of car-like robots and complexity of the motion planning problem, in International Symposium on Intelligent Robotics. Bangalore, India (1991) 322-337.
J.F. Canny, The Complexity of Robot Motion Planning. MIT Press (1988).
Chow, W.L., Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117 (1940) 98-115. CrossRef
G. Comte and Y. Yomdin, Tame geometry with applications in smooth analysis. Preprint of the IHP-RAAG Network (2002).
M. Gromov, Carnot-Carathéodory spaces seen from within, edited by A. Bellaïche and J.-J. Risler, Sub-Riemannian Geometry. Birkhäuser, Progr. Math. (1996).
W. Hurewicz and H. Wallman, Dimension Theory. Princeton University Press, Princeton (1948).
F. Jean, Paths in sub-Riemannian geometry, edited by A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek, Nonlinear Control in the Year 2000. Springer-Verlag (2000).
Jean, F., Complexity of nonholonomic motion planning. Int. J. Control 74 (2001) 776-782. CrossRef
Jean, F., Uniform estimation of sub-Riemannian balls. J. Dynam. Control Systems 7 (2001) 473-500. CrossRef
Kolmogorov, A.N., On certain asymptotics characteristics of some completely bounded metric spaces. Soviet Math. Dokl. 108 (1956) 385-388.
I. Kupka, Géométrie sous-riemannienne, in Séminaire N. Bourbaki, Vol. 817 (1996).
Laumond, J.-P., Controllability of a multibody mobile robot. IEEE Trans. Robotics Automation 9 (1993) 755-763. CrossRef
J.-P. Laumond, S. Sekhavat and F. Lamiraux, Guidelines in nonholonomic motion planning for mobile robots, edited by J.-P. Laumond, Robot Motion Planning and Control. Springer, Lecture Notes Inform. Control Sci. 229 (1998).
Mitchell, J., Carnot-Carathéodory, On metrics. J. Differential Geom. 21 (1985) 35-45. CrossRef
Nagano, T., Linear differential systems with singularities and an application to transitive Lie algebras. J. Math. Soc. Japan 18 (1966) 398-404. CrossRef
Schwartz, J.T. and Sharir, M., On the ``piano movers" problem II: General techniques for computing topological properties of real algebraic manifolds. Adv. Appl. Math. 4 (1983) 298-351. CrossRef
Sussmann, H.J., An extension of theorem of Nagano on transitive Lie algebras. Proc. Amer. Math. Soc. 45 (1974) 349-356. CrossRef
A.M. Vershik and V.Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems, edited by V.I. Arnold and S.P. Novikov, Dynamical Systems VII. Springer, Encyclopaedia Math. Sci. 16 (1994).