Entropy and complexity of a path in sub-riemannian geometry
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 485-508.

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 · ε 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.

DOI : 10.1051/cocv:2003024
Classification : 53C17
Mots clés : complexity, Hausdorff dimension, metric entropy, non-linear control, nonholonomic systems, sub-riemannian geometry
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     title = {Entropy and complexity of a path in sub-riemannian geometry},
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Jean, Frédéric. Entropy and complexity of a path in sub-riemannian geometry. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 485-508. doi : 10.1051/cocv:2003024. http://www.numdam.org/articles/10.1051/cocv:2003024/

[1] A. Bellaïche, The tangent space in sub-Riemannian geometry, edited by A. Bellaïche and J.-J. Risler, Sub-Riemannian Geometry. Birkhäuser, Progr. Math. (1996). | MR | Zbl

[2] 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). | MR

[3] 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.

[4] J.F. Canny, The Complexity of Robot Motion Planning. MIT Press (1988). | MR

[5] W.L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117 (1940) 98-115. | EuDML

[6] G. Comte and Y. Yomdin, Tame geometry with applications in smooth analysis. Preprint of the IHP-RAAG Network (2002). | MR

[7] M. Gromov, Carnot-Carathéodory spaces seen from within, edited by A. Bellaïche and J.-J. Risler, Sub-Riemannian Geometry. Birkhäuser, Progr. Math. (1996). | Zbl

[8] W. Hurewicz and H. Wallman, Dimension Theory. Princeton University Press, Princeton (1948). | JFM | MR | Zbl

[9] 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).

[10] F. Jean, Complexity of nonholonomic motion planning. Int. J. Control 74 (2001) 776-782. | MR | Zbl

[11] F. Jean, Uniform estimation of sub-Riemannian balls. J. Dynam. Control Systems 7 (2001) 473-500. | MR | Zbl

[12] A.N. Kolmogorov, On certain asymptotics characteristics of some completely bounded metric spaces. Soviet Math. Dokl. 108 (1956) 385-388. | MR | Zbl

[13] I. Kupka, Géométrie sous-riemannienne, in Séminaire N. Bourbaki, Vol. 817 (1996). | Numdam | MR

[14] J.-P. Laumond, Controllability of a multibody mobile robot. IEEE Trans. Robotics Automation 9 (1993) 755-763.

[15] 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). | MR

[16] J. Mitchell, On Carnot-Carathéodory metrics. J. Differential Geom. 21 (1985) 35-45. | Zbl

[17] T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras. J. Math. Soc. Japan 18 (1966) 398-404. | MR | Zbl

[18] J.T. Schwartz and M. Sharir, On the “piano movers” problem II: General techniques for computing topological properties of real algebraic manifolds. Adv. Appl. Math. 4 (1983) 298-351. | Zbl

[19] H.J. Sussmann, An extension of theorem of Nagano on transitive Lie algebras. Proc. Amer. Math. Soc. 45 (1974) 349-356. | MR | Zbl

[20] 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). | MR | Zbl

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