Hostname: page-component-7bb8b95d7b-wpx69 Total loading time: 0 Render date: 2024-09-20T13:30:41.035Z Has data issue: false hasContentIssue false
  • English
  • Français

Rank-2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8

Published online by Cambridge University Press:  15 August 2002

Mohamad Cheaito  and
Piotr Mormul
Mohamad Cheaito
Affiliation:
Laboratoire E. Picard, U.M.R. C.N.R.S. 5580, Département de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France; [email protected].
Piotr Mormul
Affiliation:
Instytut Matematyki, Uniwersytet Warszawski, Banacha 2, 02-097 Warszawa, Poland; [email protected].
Get access

Abstract

We study the rank–2 distributions satisfying so-called Goursat condition (GC); that is to say, codimension–2 differential systems forming with their derived systems a flag. Firstly, we restate in a clear way the main result of[7] giving preliminary local forms of such systems. Secondly – and this is the main part of the paper – in dimension 7 and 8 we explain which constants in those local forms can be made 0, normalizing the remaining ones to 1. All constructed equivalences are explicit. The complete list of local models in dimension 7 contains 13 items, and not 14, as written in[7], while the list in dimension 8 consists of 34 models (and not 41, as could be concluded from some statements in[7]). In these dimensions (and in lower dimensions, too) the models are eventually discerned just by their small growth vector at the origin.

Keywords

DistributionGoursat conditionflagderived systemsmall growth vector.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 137 - 158
Copyright
© EDP Sciences, SMAI, 1999

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

R. Bryant, S. Chern, R. Gardner, H. Goldschmidt and P. Griffiths, Exterior Differential Systems, MSRI Publications 18, Springer-Verlag, New York (1991).
E. Cartan, Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre. Ann. Ec. Norm., XXVII. 3 (1910) 109-192.
M. Gaspar, Sobre la clasificacion de sistemas de Pfaff en bandera, in: Proceedings of 10th Spanish-Portuguese Conference on Math., Univ. of Murcia (1985) 67-74 (in Spanish).
Gaspar, M., Kumpera, A. and Ruiz, C., Sur les systèmes de Pfaff en drapeau. An. Acad. Brasil. Cienc. 55 (1983) 225-229.
Giaro, A., Kumpera, A. and Ruiz, C., Sur la lecture correcte d'un résultat d'Elie Cartan. C. R. Acad. Sci. Paris 287 (1978) 241-244.
Jean, F., The car with N trailers: characterisation of the singular configurations. ESAIM: Contr. Optim. Cal. Var. (URL: http://www.emath.fr/cocv/) 1 (1996) 241-266. CrossRef
A. Kumpera and C. Ruiz, Sur l'équivalence locale des systèmes de Pfaff en drapeau, in: Monge -Ampère Equations and Related Topics, Inst. Alta Math., Rome (1982) 201-248.
J.- P. Laumond, Controllability of a multibody mobile robot. in: Proc. of the International Conference on Advanced Robotics and Automation, Pisa (1991) 1033-1038.
J.- P. Laumond and T. Simeon, Motion planning for a two degrees of freedom mobile robot with towing, LAAS/CNRS Report 89 148, Toulouse (1989).
P. Mormul, Local models of 2-distributions in 5 dimensions everywhere fulfilling the Goursat condition (preprint Rouen, 1994).
P. Mormul, Local classification of rank -2 distributions satisfying the Goursat condition in dimension 9, preprint 582, Inst. of Math., Polish Acad. Sci., Warsaw, January (1998).
Murray, R., Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Math. Control Signals Systems 7 (1994) 58-75. CrossRef
Zhitomirskii, M., Normal forms of germs of distributions with a fixed segment of growth vector (English translation). Leningrad Math. J. 2 (1991) 1043-1065.
M. Zhitomirskii, Singularities and normal forms of smooth distributions, in: Geometry in Nonlinear Control and Differential Inclusions, Banach Center Publications, Vol. 32, Warsaw (1995) 395-409.
Zhitomirskii, M., Rigid and abnormal line subdistributions of 2-distributions. J. Dyn. Control Systems 1 (1995) 253-294. CrossRef