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On the Geometry of Goursat Structures

Published online by Cambridge University Press:  15 August 2002

William Pasillas-Lépine  and
Witold Respondek
William Pasillas-Lépine
Affiliation:
Laboratoire des Signaux et Systèmes, CNRS – Supélec – Université Paris-Sud XI, Plateau du Moulon, 91192 Gif-sur-Yvette, France; pasillas@lss.supelec.fr.
Witold Respondek
Affiliation:
Laboratoire de Mathématiques de l'INSA, Institut National des Sciences Appliquées de Rouen, Place E. Blondel, 76130 Mont-Saint-Aignan, France; wresp@lmi.insa-rouen.fr.
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Abstract

A Goursat structure on a manifold of dimension n is a rank two distribution Ɗ such that dim Ɗ(i) = i + 2, for 0 ≤ i ≤ n-2, where Ɗ(i) denote the elements of the derived flag of Ɗ, defined by Ɗ(0) = Ɗ and Ɗ(i+1) = Ɗ(i) + [Ɗ(i),Ɗ(i)] . Goursat structures appeared first in the work of von Weber and Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. In the paper, we introduce a new local invariant for Goursat structures, called the singularity type, and prove that the growth vector and the abnormal curves of all elements of the derived flag are determined by this invariant. We provide a detailed analysis of all abnormal and rigid curves of Goursat structures. We show that neither abnormal curves, if n ≥ 6, nor abnormal curves of all elements of the derived flag, if n ≥ 9, determine the local equivalence class of a Goursat structure. The latter observation is deduced from a generalized version of Bäcklund's theorem. We also propose a new proof of a classical theorem of Kumpera and Ruiz. All results are illustrated by the n-trailer system, which, as we show, turns out to be a universal model for all local Goursat structures.

Keywords

Goursat structuresKumpera-Ruiz normal formsabnormal curvesnonholonomic control systemstrailer systems.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 119 - 181
Copyright
© EDP Sciences, SMAI, 2001

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