Skip to main content
SpringerLink
Log in

Stratification du secteur anormal dans la sphère de Martinet de petit rayon

  • Conference paper
  • First Online:
Book cover Nonlinear control in the Year 2000

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 258))

Abstract

L’objectif de cet article est de fournir le cadre géométrique pour faire une analyse de la singularité de l’application exponentielle le long d’une direction anormale en géométrie sous-Riemannienne. Il utilise les calculs de [9], [12], et conduit dans le cas Martinet à une stratification de la singularité en secteurs Lagrangiens.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agrachev A. (1999) Compactness for sub-Riemannian length minimizers and subanalyticity, Report SISSA, Trieste.

    Google Scholar 

  2. Agrachev A., Sarychev A. V. (1995) Strong minimality of abnormal geodesics for 2-distributions, Journal of Dynamical and Control Systems, Vol. 1, No. 2, 139–176.

    Article  MATH  MathSciNet  Google Scholar 

  3. Agrachev A., Gamkrelidze R. V. (1997) Feedback invariant control theory and differential geometry I, Regular extremals, Journal of Dynamical and Control Systems, Vol. 3, No. 3, 343–390.

    Article  MATH  MathSciNet  Google Scholar 

  4. Agrachev A. and al. (1997) Sub-Riemannian spheres in the Martinet flat case, ESAIM/COCV, Vol. 2, 377–448.

    MATH  MathSciNet  Google Scholar 

  5. Arnold V. (1976) Méthodes mathématiques de la mécanique classique, Eds Mir, Moscou.

    MATH  Google Scholar 

  6. Arnold V. and al., Singularities of differentiable mappings, Eds Mir, Moscou.

    Google Scholar 

  7. Bellaïche A. (1996) Tangent space in sub-Riemannian geometry, Sub-Riemannian Geometry, Birkhäuser.

    Google Scholar 

  8. Bliss G. A. (1946) Lectures on the calculus of variations, U. of Chicago Press, Chicago.

    MATH  Google Scholar 

  9. Bonnard B., Chyba M. (1999) Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-Riemannienne de Martinet, ESAIM/COCV, Vol. 4, 245–334.

    MATH  MathSciNet  Google Scholar 

  10. Bonnard B., Kupka I. (1993) Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, Forum Math. 5, 111–159.

    Article  MATH  MathSciNet  Google Scholar 

  11. Bonnard B., de Morant J. (1995) Towards a geometric theory in the time-minimal control of chemical batch reactors, SIAM Journal on Control and Optimization, Vol. 33, No. 5, 1279–1311.

    Article  MATH  MathSciNet  Google Scholar 

  12. Bonnard B., Trélat E. (1999) Role of abnormal minimizers in sub-Riemannian geometry, PrePrint Dijon.

    Google Scholar 

  13. Gromov M. (1996) Carnot-Carathéodory spaces seen from within, Sub-Riemannian Geometry, Birkhäuser.

    Google Scholar 

  14. Guillemin V., Sternberg S. (1984) Symplectic techniques in physics, Cambridge University Press.

    Google Scholar 

  15. Hrmander L. (1983) The analysis of linear partial differential operators, Springer-Verlag, New-York.

    Google Scholar 

  16. Jean F. (1999) Entropy and complexity of a path in sub-Riemannian geometry, rapport ENSTA.

    Google Scholar 

  17. Kerkovian J., Cole J. D. (1981) Perturbation methods in applied mathematics, Springer-Verlag.

    Google Scholar 

  18. Kupka I. (1992) Abnormal extremals, Preprint.

    Google Scholar 

  19. Kupka I. (1996) Géométrie sous-Riemannienne, Séminaire Bourbaki, Paris.

    Google Scholar 

  20. Lawden D. F. (1989) Elliptic functions and applications, Springer-Verlag, New-York.

    MATH  Google Scholar 

  21. Liu W. S., Sussmann H. J. (1995) Shortest paths for sub-Riemannian metrics of rank two distributions, Memoirs AMS, N564, Vol. 118.

    Google Scholar 

  22. Mischenko A. S. and al. (1990) Lagrangian manifolds and the Maslov operator, Springer-Verlag.

    Google Scholar 

  23. Moyer H. G. (1973) Sufficient conditions for a strong minimum in singular problems, SIAM Journal on Control and Optimization, 11, 620–636.

    Article  MATH  MathSciNet  Google Scholar 

  24. Naimark M. A. (1967) Linear differential operators, Frederick U. Pub. Co.

    Google Scholar 

  25. Nikiforov A., Ouranov V. (1982) Fontions spéciales de la physique mathématique, Eds Mir.

    Google Scholar 

  26. Ramis J. P., Séries divergentes et théorie asymptotique, Mémoires de la SMF.

    Google Scholar 

  27. Roussarie R. (1968) Bifurcations of planar vector fields and Hilbert's 16th problem, Birkhäuser, Berlin.

    Google Scholar 

  28. Trélat E. (2000) Some properties of the value function and its level sets for affine control systems with quadratic cost, to appear in Journal of Dynamical and Control Systems.

    Google Scholar 

  29. Treves F., Symplectic geometry and analytic hypo-ellipticity, Preprint.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dpt de Mathématiques, UFR Sciences et Techniques, Université de Bourgogne, BP 47870, 21078, Dijon Cedex, France

    Bernard Bonnard & Emmanuel Trélat

Authors
  1. Bernard Bonnard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emmanuel Trélat
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Alberto Isidori (Professor)Françoise Lamnabhi-Lagarrigue (Docteur D’état)Witold Respondek (Professor)

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag London Limited

About this paper

Cite this paper

Bonnard, B., Trélat, E. (2001). Stratification du secteur anormal dans la sphère de Martinet de petit rayon. In: Isidori, A., Lamnabhi-Lagarrigue, F., Respondek, W. (eds) Nonlinear control in the Year 2000. Lecture Notes in Control and Information Sciences, vol 258. Springer, London. https://doi.org/10.1007/BFb0110218

Download citation

  • DOI: https://doi.org/10.1007/BFb0110218

  • Published:

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-363-8

  • Online ISBN: 978-1-84628-568-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation