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

Mohamad Cheaito; Piotr Mormul

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 4, page 137-158
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Cheaito, Mohamad, and Mormul, Piotr. "Rank-2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 137-158. <http://eudml.org/doc/197324>.

@article{Cheaito2010,
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. },
author = {Cheaito, Mohamad, Mormul, Piotr},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Distribution; Goursat condition; flag; derived system; small growth vector.; distribution; small growth vector},
language = {eng},
month = {3},
pages = {137-158},
publisher = {EDP Sciences},
title = {Rank-2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8},
url = {http://eudml.org/doc/197324},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Cheaito, Mohamad
AU - Mormul, Piotr
TI - Rank-2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 137
EP - 158
AB - 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.
LA - eng
KW - Distribution; Goursat condition; flag; derived system; small growth vector.; distribution; small growth vector
UR - http://eudml.org/doc/197324
ER -

References

top
  1. R. Bryant, S. Chern, R. Gardner, H. Goldschmidt and P. Griffiths, Exterior Differential Systems, MSRI Publications 18, Springer-Verlag, New York (1991).  
  2. 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.  Zbl41.0417.01
  3. 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).  
  4. M. Gaspar, A. Kumpera and C. Ruiz, Sur les systèmes de Pfaff en drapeau. An. Acad. Brasil. Cienc.55 (1983) 225-229.  
  5. A. Giaro, A. Kumpera and C. Ruiz, Sur la lecture correcte d'un résultat d'Elie Cartan. C. R. Acad. Sci. Paris287 (1978) 241-244.  Zbl0398.58003
  6. F. Jean, 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.  Zbl0874.93033
  7. 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.  Zbl0516.58004
  8. J.- P. Laumond, Controllability of a multibody mobile robot. in: Proc. of the International Conference on Advanced Robotics and Automation, Pisa (1991) 1033-1038.  
  9. 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).  
  10. P. Mormul, Local models of 2-distributions in 5 dimensions everywhere fulfilling the Goursat condition (preprint Rouen, 1994).  
  11. 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).  
  12. R. Murray, Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Math. Control Signals Systems7 (1994) 58-75.  Zbl0825.93319
  13. M. Zhitomirskii, Normal forms of germs of distributions with a fixed segment of growth vector (English translation). Leningrad Math. J.2 (1991) 1043-1065.  
  14. 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.  
  15. M. Zhitomirskii, Rigid and abnormal line subdistributions of 2-distributions. J. Dyn. Control Systems1 (1995) 253-294.  Zbl0970.53020

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.