Sub-riemannian sphere in Martinet flat case
ESAIM: Control, Optimisation and Calculus of Variations, Tome 2 (1997), pp. 377-448 
@article{COCV_1997__2__377_0,
     author = {Agrachev, A. and Bonnard, B. and Chyba, M. and Kupka, I.},
     title = {Sub-riemannian sphere in {Martinet} flat case},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {377--448},
     year = {1997},
     publisher = {EDP Sciences},
     volume = {2},
     mrnumber = {1483765},
     zbl = {0902.53033},
     language = {en},
     url = {https://www.numdam.org/item/COCV_1997__2__377_0/}
}
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Agrachev, A.; Bonnard, B.; Chyba, M.; Kupka, I. Sub-riemannian sphere in Martinet flat case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 2 (1997), pp. 377-448. https://www.numdam.org/item/COCV_1997__2__377_0/

[1] A. Agrachev, A. V. Sarychev: Strong minimality of abnormal geodesics for 2-distributions, Journal of Dynamical and control Systems, 2, 1995, 139-176. | Zbl | MR

[2] A. Agrachev: Exponential mappings for contact sub-Riemannian structures, Journal of dynamical and Control Systems, 2, 1996, 321-358. | Zbl | MR

[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] V. I. Arnold: Méthodes mathématiques pour la mécanique classique, Éditions MIR, Moscou, 1976. | Zbl | MR

[5] G.A. Bliss: Lectures on the calculus of variations, The University of Chicago Press, 1946. | Zbl | MR

[6] B. Bonnard: Feedback equivalence for nonlinear systems and the time optimal control problem, SIAM J. on Control and Opt., 29, 1991, 1300-1321. | Zbl | MR

[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] B. Bonnard, M. Chyba, H. Heutte: Contrôle optimal géométrique appliqué, Preprint of Laboratoire de Topologie, Dijon, 1995.

[9] B. Bonnard, M. Chyba, I. Kupka: Non-integrable geodesics in SR Martinet geometry, in Proceedings AMS conference, Boulder, 1997. | Zbl

[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. | Zbl | MR

[11] R. W. Brockett: Control theory and singular Riemannian geometry, in New directions in applied Math., Springer-Verlag, New-York, 1981. | Zbl | MR

[12] E. Cartan: Leçons sur la géométrie des espaces de Riemann, Ed. J. Gabay, Paris, 1988. | Zbl | MR

[13] H. Davis: Introduction to non linear differential and integral equation, Dover, New-York, 1962. | Zbl

[14] J. Dieudonné: Calcul Infinitésimal, Hermann, Paris, 1980. | Zbl | MR

[15] M. Do Carmo: Riemannian geometry, Birkhauser, Boston, 1992. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[18] R. Gérard, H. Tahora: Singular nonlinear PDE, Vieweg-Verlag, Germany, 1996.

[19] J. Gregory: Quadratic form theory and differential equation, Academic Press, New-York, 1980. | Zbl | MR

[20] U. Hamenstadt: Some regularity theorem for Carnot-Caratheodory metries, J. Differential geometry, 32, 1991, 819-850. | Zbl | MR

[21] F. John: Partial differential equations, Springer-Verlag, New-York, 1971. | Zbl

[22] A. G. Khovanskii: Fewnomials, Trans. AMS, 88, 1991. | Zbl | MR

[23] I. Kupka: Abnormal extremals, Preprint, 1992.

[24] I. Kupka: Géométrie sous-Riemannienne, in Séminaire Bourbaki, 1996. | MR | Numdam

[25] D.F. Lawden: Elliptic functions and applications, Springer-Verlag, New-York, 1989. | Zbl | MR

[26] E. B. Lee, L. Markus: Foundations of optimal control theory, John Wiley and Sons, New-York, 1967. | Zbl | MR

[27] J. M. Lion, J. P. Rolin: Théorèmes de préparation pour les fonctions logarithmo-exponentielles, Annales de l'Institut Fourier, 47, 1997, 859-884. | Zbl | MR | Numdam

[28] W. S. Liu and H. J. Susmann: Shortest paths for sub-Riemannian metries of rank two distributions, to appear in Trans. AMS. | Zbl

[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. | Zbl | MR

[30] A. E. H. Love: A treatise of the mathematical theory of elasticity, Dover, 1944. | Zbl | MR | JFM

[31] S. B. Myers: Connections between differential geometry and topology, Duke Math. J., 1, 1935, 376-391. | MR | JFM

[32] L. Pontriaguineet al.: Théorie mathématique des processus optimaux, Éditions MIR, Moscou, 1974. | Zbl | MR

[33] W. Respondek, M. Zhitomirskii: Feedback classification of nonlinear control Systems on 3-manifolds, to appear in Math. Control Systems and Signals. | Zbl | MR

[34] M. Spivak: Differential geometry, Publish on Perish, Inc., Berkeley, 1979.

[35] R. Strichartz: Sub-Riemannian geometry, J. Differential geometry, 24, 1986, 221-263. | Zbl | MR

[36] J. Tannery, J. Molk: Eléments de la théorie des fonctions elliptiques, Tomes I à IV, Gauthier-Villars, Paris, 1896. | JFM

[37] A. Weinstein: The cut-locus and conjugate-locus of a Riemannian manifold, Annals of Maths, 87, 1968, 29-41. | Zbl | MR

[38] E. T. Whittaker, G. N. Watson: A course of modern analysis, Cambridge U. Press, New York, 1927. | MR | JFM