Sub-riemannian sphere in Martinet flat case

A. Agrachev; B. Bonnard; M. Chyba; I. Kupka

ESAIM: Control, Optimisation and Calculus of Variations (1997)

  • Volume: 2, page 377-448
  • ISSN: 1292-8119

How to cite


Agrachev, A., et al. "Sub-riemannian sphere in Martinet flat case." ESAIM: Control, Optimisation and Calculus of Variations 2 (1997): 377-448. <>.

author = {Agrachev, A., Bonnard, B., Chyba, M., Kupka, I.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {sub-Riemannian geometry; Martinet distribution; cut locus; conjugate loci},
language = {eng},
pages = {377-448},
publisher = {EDP Sciences},
title = {Sub-riemannian sphere in Martinet flat case},
url = {},
volume = {2},
year = {1997},

AU - Agrachev, A.
AU - Bonnard, B.
AU - Chyba, M.
AU - Kupka, I.
TI - Sub-riemannian sphere in Martinet flat case
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 1997
PB - EDP Sciences
VL - 2
SP - 377
EP - 448
LA - eng
KW - sub-Riemannian geometry; Martinet distribution; cut locus; conjugate loci
UR -
ER -


  1. [1] A. Agrachev, A. V. Sarychev: Strong minimality of abnormal geodesics for 2-distributions, Journal of Dynamical and control Systems, 2, 1995, 139-176. Zbl0951.53029MR1333769
  2. [2] A. Agrachev: Exponential mappings for contact sub-Riemannian structures, Journal of dynamical and Control Systems, 2, 1996, 321-358. Zbl0941.53022MR1403262
  3. [3] A. Agrachev: Any smooth simple H1-local length minimizer in the Carnot-Caratheodory space is a C0-local minimizer, Preprint of Laboratoire de Topologie, Dijon, 1996. 
  4. [4] V. I. Arnold: Méthodes mathématiques pour la mécanique classique, Éditions MIR, Moscou, 1976. Zbl0385.70001MR474391
  5. [5] G.A. Bliss: Lectures on the calculus of variations, The University of Chicago Press, 1946. Zbl0063.00459MR17881
  6. [6] B. Bonnard: Feedback equivalence for nonlinear systems and the time optimal control problem, SIAM J. on Control and Opt., 29, 1991, 1300-1321. Zbl0744.93033MR1132184
  7. [7] B. Bonnard, M. Chyba: Exponential mapping, sphere and waves front in SR-geometry: the generic integrable Martinet case, Preprint of Laboratoire de Topologie, Dijon, 1997. 
  8. [8] B. Bonnard, M. Chyba, H. Heutte: Contrôle optimal géométrique appliqué, Preprint of Laboratoire de Topologie, Dijon, 1995. 
  9. [9] B. Bonnard, M. Chyba, I. Kupka: Non-integrable geodesics in SR Martinet geometry, in Proceedings AMS conference, Boulder, 1997. Zbl0963.53015
  10. [10] B. Bonnard, M. Chyba, E. Trélat: Sub-Riemannian geometry: one parameter deformation of the Martinet flat case, to appear in Journal of Dynamical and Control Systems. Zbl0967.53020MR1605346
  11. [11] R. W. Brockett: Control theory and singular Riemannian geometry, in New directions in applied Math., Springer-Verlag, New-York, 1981. Zbl0483.49035MR661282
  12. [12] E. Cartan: Leçons sur la géométrie des espaces de Riemann, Ed. J. Gabay, Paris, 1988. Zbl0060.38101MR1191392
  13. [13] H. Davis: Introduction to non linear differential and integral equation, Dover, New-York, 1962. Zbl0106.28904
  14. [14] J. Dieudonné: Calcul Infinitésimal, Hermann, Paris, 1980. Zbl0497.26004MR226971
  15. [15] M. Do Carmo: Riemannian geometry, Birkhauser, Boston, 1992. Zbl0752.53001MR1138207
  16. [16] L. V .D. Dries, A. Macintyre, D. Marker: The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics, 140, 1994, 183-205. Zbl0837.12006MR1289495
  17. [17] C. El Alaoui, J. P. Gauthier, I. Kupka: Small sub-Riemannian balls on R3, Journal of dynamical and Control Systems, 2, 1996, 359-421. Zbl0941.53024MR1403263
  18. [18] R. Gérard, H. Tahora: Singular nonlinear PDE, Vieweg-Verlag, Germany, 1996. 
  19. [19] J. Gregory: Quadratic form theory and differential equation, Academic Press, New-York, 1980. Zbl0468.15015MR599362
  20. [20] U. Hamenstadt: Some regularity theorem for Carnot-Caratheodory metries, J. Differential geometry, 32, 1991, 819-850. Zbl0687.53041MR1078163
  21. [21] F. John: Partial differential equations, Springer-Verlag, New-York, 1971. Zbl0209.40001
  22. [22] A. G. Khovanskii: Fewnomials, Trans. AMS, 88, 1991. Zbl0728.12002MR1108621
  23. [23] I. Kupka: Abnormal extremals, Preprint, 1992. 
  24. [24] I. Kupka: Géométrie sous-Riemannienne, in Séminaire Bourbaki, 1996. MR1472545
  25. [25] D.F. Lawden: Elliptic functions and applications, Springer-Verlag, New-York, 1989. Zbl0689.33001MR1007595
  26. [26] E. B. Lee, L. Markus: Foundations of optimal control theory, John Wiley and Sons, New-York, 1967. Zbl0159.13201MR220537
  27. [27] J. M. Lion, J. P. Rolin: Théorèmes de préparation pour les fonctions logarithmo-exponentielles, Annales de l'Institut Fourier, 47, 1997, 859-884. Zbl0873.32004MR1465789
  28. [28] W. S. Liu and H. J. Susmann: Shortest paths for sub-Riemannian metries of rank two distributions, to appear in Trans. AMS. Zbl0843.53038
  29. [29] S. Lojasiewicz, H. J. Sussmann: Some examples of reachable sets and optimal cost functions that fail to be subanalytic, SIAM J. Control and Optimization, 23, 1985, 584-598. Zbl0569.49029MR791889
  30. [30] A. E. H. Love: A treatise of the mathematical theory of elasticity, Dover, 1944. Zbl0063.03651MR10851JFM47.0750.09
  31. [31] S. B. Myers: Connections between differential geometry and topology, Duke Math. J., 1, 1935, 376-391. MR1545884JFM61.0787.02
  32. [32] L. Pontriaguineet al.: Théorie mathématique des processus optimaux, Éditions MIR, Moscou, 1974. Zbl0289.49002MR358482
  33. [33] W. Respondek, M. Zhitomirskii: Feedback classification of nonlinear control Systems on 3-manifolds, to appear in Math. Control Systems and Signals. Zbl0925.93367MR1403291
  34. [34] M. Spivak: Differential geometry, Publish on Perish, Inc., Berkeley, 1979. 
  35. [35] R. Strichartz: Sub-Riemannian geometry, J. Differential geometry, 24, 1986, 221-263. Zbl0609.53021MR862049
  36. [36] J. Tannery, J. Molk: Eléments de la théorie des fonctions elliptiques, Tomes I à IV, Gauthier-Villars, Paris, 1896. Zbl25.0758.01JFM27.0335.01
  37. [37] A. Weinstein: The cut-locus and conjugate-locus of a Riemannian manifold, Annals of Maths, 87, 1968, 29-41. Zbl0159.23902MR221434
  38. [38] E. T. Whittaker, G. N. Watson: A course of modern analysis, Cambridge U. Press, New York, 1927. MR1424469JFM53.0180.04

Citations in EuDML Documents

  1. Monique Chyba, Le front d'onde en géométrie sous-riemannienne : le cas Martinet
  2. Bernard Bonnard, Monique Chyba, Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet
  3. Andrei A. Grachev, Andrei V. Sarychev, Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity
  4. Kanghai Tan, Xiaoping Yang, Subriemannian geodesics of Carnot groups of step 3
  5. Bernard Bonnard, Monique Chyba, Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet
  6. Andrei Agrachev, Jean-Paul Gauthier, On the subanalyticity of Carnot–Caratheodory distances
  7. Andrei A. Agrachev, Andrei V. Sarychev, Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity
  8. Roberta Ghezzi, On almost-Riemannian surfaces
  9. B. Bonnard, E. Trélat, On the role of abnormal minimizers in sub-riemannian geometry
  10. Emmanuel Trélat, Global subanalytic solutions of Hamilton–Jacobi type equations

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