Journal of Mathematical Sciences

, Volume 126, Issue 6, pp 1614–1629 | Cite as

Kumpera-Ruiz algebras in Goursat flags are optimal in small lengths

  • P. Mormul


Kumpera-Ruiz algebras of germs of Goursat distributions are nilpotent algebras with explicitly computable orders of nilpotence. At some points, called tangential, these orders coincide with the nonholonomy degrees (calculated earlier by Jean) of the ambient G. germs. At nontangential points (that mostly occur and are stratified into geometric classes of Jean, Montgomery, and Zhitomirskii), nonholonomy degrees are lesser, sometimes even much lesser. It is a well-known open question in the theory of G. distributions: whether Kumpera-Ruiz algebras realize the minimal possible nilpotence orders or not. In the present paper, in small lengths of the induced G. flags (up to 5 inclusively) and also for 8 nontangential classes in length 6 (among 18 nontangential existing in this length 6), we show that the answer to this question is affirmative: the nilpotence orders of Kumpera-Ruiz algebras cannot be lowered. This makes more probable that a general answer to the question is affirmative in the general case.


Small Length General Answer Computable Order Nilpotent Algebra Geometric Classis 
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  1. 1.
    A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, “Local invariants of smooth control systems,” Acta Appl. Math., 14, 191–237 (1989).CrossRefGoogle Scholar
  2. 2.
    A. A. Agrachev and J.-P. Gauthier, On subanalyticity of Carnot-Carathéodory distances, Ann. Inst. H. Poincaré, 18, 359–382 (2001).Google Scholar
  3. 3.
    A. Bellïche, “The tangent space in sub-Riemannian geometry,” in: Sub-Riemannian Geometry (A. Bellïche and J.-J. Risler, Eds.), Birkhäuser, Basel-Berlin (1996), pp.1–78.Google Scholar
  4. 4.
    R. M. Bianchini and G. Stefani, “Graded approximations and controllability along a trajectory,” SIAM J. Control Optimiz., 28, 903–924 (1990).CrossRefGoogle Scholar
  5. 5.
    A. Bressan, “Nilpotent approximations and optimal trajectories,” in: Analysis of Controlled Dynamical Systems, Progr. Systems Control Theory, 8 Birkhäuser, Basel-Berlin (1991), pp.103–117.Google Scholar
  6. 6.
    R. Bryant and L. Hsu, “Rigidity of integral curves of rank 2 distributions,” Invent. Math., 114, 435–461 (1993).CrossRefGoogle Scholar
  7. 7.
    M. Cheaito and P. Mormul, “Rank-2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8,” ESAIM: Control, Optimization and Calculus of Variations, 4, 137–158 (1999); http://www. Scholar
  8. 8.
    H. Hermes, “Nilpotent approximations of control systems and distributions,” SIAM J. Control Optimiz., 24, 731–736 (1986).CrossRefGoogle Scholar
  9. 9.
    H. Hermes, “Nilpotent and high-order approximations of vector field systems,” SIAM Review, 33, 238–264 (1991).CrossRefGoogle Scholar
  10. 10.
    H. Hermes, A. Lundell, and D. Sullivan, “Nilpotent bases for distributions and control systems,” J. Differ. Equations, 55, 385–400 (1984).CrossRefGoogle Scholar
  11. 11.
    F. Jean, “The car with N trailers: characterisation of the singular configurations,” ESAIM: Control, Optimization and Calculus of Variations, 1, 241–266 (1996); Scholar
  12. 12.
    A. Kumpera and J. L. Rubin, “Multi-flag systems and ordinary differential equations,” Nagoya Math.J., 166, 1–27 (2002).Google Scholar
  13. 13.
    A. Kumpera and C. Ruiz, “Sur l’équivalence locale des systèmes de Pfaff en drapeau,”in: Monge-Ampère Equations and Related Topics (F. Gherardelli, Ed.), Inst. Alta Math. F. Severi, Rome (1982), pp. 201–248.Google Scholar
  14. 14.
    G. Lafferriere and H. J. Sussmann, “A differential geometric approach to motion planning,” in: Nonholonomic Motion Planning (F. J. Canny and Z. Li, Eds.), Kluwer, Dordrecht-London (1993), pp. 235–270.Google Scholar
  15. 15.
    Mathematical Encyclopaedia, Vol.3 [in Russian], I. M. Vinogradov (Main Editor), Soviet Encyclopaedia, Moscow (1982).Google Scholar
  16. 16.
    R. Montgomery and M. Zhitomirskii, “Geometric approach to Goursat flags,” Ann. Inst. H. Poincaré, 18, 459–493 (2001).CrossRefGoogle Scholar
  17. 17.
    P. Mormul, “Local classification of rank-2 distributions satisfying the Goursat condition in dimension 9,” in: Singularités et Géométrie sous-riemannienne (P. Orro et F. Pelletier, Eds.), Collection Travaux en cours, 62, Hermann, Paris (2000), pp.89–119.Google Scholar
  18. 18.
    P. Mormul, Simple codimension-two singularities of Goursat flags, I: One flag’s member in singular position, Preprint 01-01(39), Institute of Mathematics, Warsaw University (2001); Scholar
  19. 19.
    P. Mormul, “Discrete models of codimension-two singularities of Goursat flags of arbitrary length with one flag’s member in singular position,” Proc. Steklov Inst. Math., 236, 478–489 (2002).Google Scholar
  20. 20.
    P. Mormul, “Goursat distributions not strongly nilpotent in dimensions not exceeding seven,” in: Nonlinear and Adaptive Control NCN4 2001 (A. Zinober and D. Owens, Eds.), Lect. Notes Control Information Sci., 281 Springer-Verlag, Berlin-New York (2003), pp. 249–261.Google Scholar
  21. 21.
    P. Mormul, Multi-dimensional Cartan prolongation and special k-flags, Preprint 02-06(58), Institute of Mathematics, Warsaw University (2002); Scholar
  22. 22.
    P. Mormul and F. Pelletier, “Contrôlabilité complète par courbes anormales par morceaux d’une distribution de rang 3 générique sur des variétés connexes de dimension 5 et 6,” Bull. Polish Acad. Sci., Ser. Math., 45, 399–418 (1997).Google Scholar
  23. 23.
    R. M. Murray, “Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems,” Math. Control Signals Syst., 7, 58–75 (1994).CrossRefGoogle Scholar
  24. 24.
    W. Pasillas-Lépine and W. Respondek, “Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus,” Int. J. Control, 74, 628–637 (2001).CrossRefGoogle Scholar
  25. 25.
    W. Pasillas-Lépine and W. Respondek, “Contact systems and corank one involutive subdistributions,” Acta Appl. Math., 69, 105–128 (2001).CrossRefGoogle Scholar
  26. 26.
    C. Rockland, “Intrinsic nilpotent approximations,” Acta Appl. Math., 8, 213–270 (1987).CrossRefGoogle Scholar
  27. 27.
    L. P. Rothschild and E. M. Stein, “Hypoelliptic differential operators and nilpotent groups,” Acta Math., 137, 247–320 (1976).Google Scholar
  28. 28.
    Yu. L. Sachkov, “On nilpotent sub-Riemannian (2, 3, 5) problem,” in: Workshop Math. Control Theory and Robotics, Trieste (2000).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • P. Mormul
    • 1
  1. 1.Institute of MathematicsWarsaw UniversityWarsawPoland

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