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
Access Full Article
topHow to cite
topAgrachev, A., et al. "Sub-riemannian sphere in Martinet flat case." ESAIM: Control, Optimisation and Calculus of Variations 2 (1997): 377-448. <http://eudml.org/doc/90514>.
@article{Agrachev1997,
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 = {http://eudml.org/doc/90514},
volume = {2},
year = {1997},
}
TY - JOUR
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 - http://eudml.org/doc/90514
ER -
References
top- [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] A. Agrachev: Exponential mappings for contact sub-Riemannian structures, Journal of dynamical and Control Systems, 2, 1996, 321-358. Zbl0941.53022MR1403262
- [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. Zbl0385.70001MR474391
- [5] G.A. Bliss: Lectures on the calculus of variations, The University of Chicago Press, 1946. Zbl0063.00459MR17881
- [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] 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. Zbl0963.53015
- [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] R. W. Brockett: Control theory and singular Riemannian geometry, in New directions in applied Math., Springer-Verlag, New-York, 1981. Zbl0483.49035MR661282
- [12] E. Cartan: Leçons sur la géométrie des espaces de Riemann, Ed. J. Gabay, Paris, 1988. Zbl0060.38101MR1191392
- [13] H. Davis: Introduction to non linear differential and integral equation, Dover, New-York, 1962. Zbl0106.28904
- [14] J. Dieudonné: Calcul Infinitésimal, Hermann, Paris, 1980. Zbl0497.26004MR226971
- [15] M. Do Carmo: Riemannian geometry, Birkhauser, Boston, 1992. Zbl0752.53001MR1138207
- [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] 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] 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. Zbl0468.15015MR599362
- [20] U. Hamenstadt: Some regularity theorem for Carnot-Caratheodory metries, J. Differential geometry, 32, 1991, 819-850. Zbl0687.53041MR1078163
- [21] F. John: Partial differential equations, Springer-Verlag, New-York, 1971. Zbl0209.40001
- [22] A. G. Khovanskii: Fewnomials, Trans. AMS, 88, 1991. Zbl0728.12002MR1108621
- [23] I. Kupka: Abnormal extremals, Preprint, 1992.
- [24] I. Kupka: Géométrie sous-Riemannienne, in Séminaire Bourbaki, 1996. MR1472545
- [25] D.F. Lawden: Elliptic functions and applications, Springer-Verlag, New-York, 1989. Zbl0689.33001MR1007595
- [26] E. B. Lee, L. Markus: Foundations of optimal control theory, John Wiley and Sons, New-York, 1967. Zbl0159.13201MR220537
- [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] 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] 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] A. E. H. Love: A treatise of the mathematical theory of elasticity, Dover, 1944. Zbl0063.03651MR10851JFM47.0750.09
- [31] S. B. Myers: Connections between differential geometry and topology, Duke Math. J., 1, 1935, 376-391. MR1545884JFM61.0787.02
- [32] L. Pontriaguineet al.: Théorie mathématique des processus optimaux, Éditions MIR, Moscou, 1974. Zbl0289.49002MR358482
- [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] M. Spivak: Differential geometry, Publish on Perish, Inc., Berkeley, 1979.
- [35] R. Strichartz: Sub-Riemannian geometry, J. Differential geometry, 24, 1986, 221-263. Zbl0609.53021MR862049
- [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] A. Weinstein: The cut-locus and conjugate-locus of a Riemannian manifold, Annals of Maths, 87, 1968, 29-41. Zbl0159.23902MR221434
- [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
top- Monique Chyba, Le front d'onde en géométrie sous-riemannienne : le cas Martinet
- 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
- Andrei A. Grachev, Andrei V. Sarychev, Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity
- Kanghai Tan, Xiaoping Yang, Subriemannian geodesics of Carnot groups of step 3
- Andrei Agrachev, Jean-Paul Gauthier, On the subanalyticity of Carnot–Caratheodory distances
- 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
- Andrei A. Agrachev, Andrei V. Sarychev, Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity
- Emmanuel Trélat, Global subanalytic solutions of Hamilton–Jacobi type equations
- Roberta Ghezzi, On almost-Riemannian surfaces
- B. Bonnard, E. Trélat, On the role of abnormal minimizers in sub-riemannian geometry
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.