Hostname: page-component-7bb8b95d7b-wpx69 Total loading time: 0 Render date: 2024-09-20T17:04:17.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Méthodes géométriques et analytiques pour étudierl'application exponentielle, la sphère et le front d'onde en géométriesous-riemannienne dans le cas Martinet

Published online by Cambridge University Press:  15 August 2002

Bernard Bonnard  and
Monique Chyba
Bernard Bonnard
Affiliation:
Université de Bourgogne, Laboratoire de Topologie, UMR 5584 du CNRS, BP. 400, 21004 Dijon Cedex France; bbonnard@u-bourgogne.fr.
Monique Chyba
Affiliation:
Université de Paris VI, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France; chyba@sunny.unige.ch.
Get access

Abstract

Consider a sub-riemannian geometry(U,D,g) where U is a neighborhood of 0 in R3, D is a Martinet type distribution identified to ker ω, ω being the 1-form: $\omega=dz-\frac{y^2}{2}dx$, q=(x,y,z) and g is a metric on D which can be taken in the normal form: $g=a(q)dx^2+c(q)dy^2$, a=1+yF(q), c=1+G(q), $G_{|_{x=y=0}}=0$. In a previous article we analyze the flat case: a=c=1; we describe the conjugate and cut loci, the sphere and the wave front. The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three one parameter deformations of the flat case which clarify the role of the three parameters $\alpha,\beta,\gamma$ in the gradated normal form of order 0 where: $a=(1+\alpha y)^2$, $c=(1+\beta x+\gamma y)^2$. More generally this analysis provides an explanation of the role of abnormal minimizers in SR-geometry.

Keywords

Sub-riemannian geometryMartinet caseabnormal geodesicssphere and wave front of small radius. Mots clés : Géométrie sous-riemanniennele cas Martinetgéodésiques anormalessphère et front d'onde de petit rayon.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 245 - 334
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agrachev, A., Bonnard, B., Chyba, M. and Kupka Sub-Riemannian, I. sphere in Martinet flat case. ESAIM:COCV 2 (1997) 377-448. CrossRef
A. Agrachev, C. El Alaoui and J.P. Gauthier, Sub-Riemannian metrics on R 3. Geometric Control and Non-holonomic Problems in Mechanics, Conference Proceedings Series, Canad. Math. Soc. (to appear).
Agrachev, A.A. and Gamkrelidze, R.V., Exponentional representations of flows and chronological calculus. Math. USSR Sb. 35 (1979) 727-785. CrossRef
Agrachev, A.A. and Sarychev, A.V., Strong minimality of abnormal geodesics for 2- distributions. J. Dynamical and Control Systems 1 (1995) 139-176. CrossRef
A.A. Agrachev and A.V. Sarychev, Abnormal geodesics in SR-geometry subanalycity. Preprint (1997).
A.A. Andronov, A.A. de Vitt and S.E. Khaikin, Theory of oscillations, Dover , New-York (1966).
B. Bonnard, M. Chyba and I. Kupka, Non-integrable geodesics in SR-Martinet geometry, Proceedings AMS conference, Boulder (1997).
Bonnard, B., Chyba, M. and Trélat, E., Sub-Riemannian geometry: one parameter deformation of the Martinet flat case. J. Dynamical and Control Systems 4 (1998) 59-76. CrossRef
B. Bonnard, G. Launey and E. Trélat, The transcendence we need to compute the Sphere and the Wave Front in Martinet SR-Geometry. to appear in Proc. of Steklov Institute.
B. Bonnard and E. Trélat, The role of abnormal minimizers in SR-geometry. Preprint (1999).
M. Chyba, Le cas Martinet en géométrie sous-Riemannienne, Thèse de l'Université de Bourgogne (1997).
H. Davis, Introduction to non linear differential and integral equations, Dover, New-York, (1962).
J. Dieudonné, Calcul infinitésimal, Hermann, Paris (1980).
Dries, L.V.D., Macintyre, A. and Marker, D., The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics 140 (1994) 183-205. CrossRef
J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Hermann, Paris (1992).
G.H. Halphen, Traité des fonctions elliptiques, Gauthier-Villars, Tomes I à IV, Paris (1886).
S. Jacquet, Distance sous-riemannienne et sous analycité. Preprint (1997).
A.G. Khovanskii, Fewnomials, Trans. Math. Monographs 88, (1991) AMS.
I. Kupka, Abnormal extremals. Preprint (1992).
I. Kupka, Géométrie sous-Riemannienne, Séminaire Bourbaki (1996).
D.F. Lawden, Elliptic functions and applications, Springer-Verlag, New-York (1989).
E.B. Lee and L. Markus, Foundations of optimal control theory, John Wiley and Sons, New-York (1967).
S. Lefschetz, Differential equations: geometry theory, Dover, New-York (1977).
M.A. Liapounoff, Problème général de la stabilité du mouvement. Annals of Maths. Studies, Princeton University Press (1947).
Lion, J.M. and Rolin, J.P., Théorèmes de préparation pour les fonctions logarithmo-exponentielles. Annales de l'Institut Fourier 47 (1997) 859- 884. CrossRef
W.S. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics of rank-2 distributions. Memoirs of the Americain Math. Society 118, (1995).
Lojasiewicz, S. and Sussmann, H.J., Some examples of reachable sets and optimal cost functions that fail to be subanalytic. SIAM J. Control Optim. 23 (1985) 584-598. CrossRef
A.E.H. Love, A treatise of the mathematical theory of elasticity, Dover (1944).
Montgomery, R., Abnormal minimizers, SIAM J. Control Optim. 32 (1994) 1605-1620. CrossRef
Mourtada, A. and Moussu, R., Applications de Dulac et applications pfaffiennes. Bulletin SMF 125 (1997) 1-13.
Moussu, R. and Roche, A., Théorie de Khovanski et problème de Dulac. Inv. Math. 105 (1991) 431-441. CrossRef
R. Roussarie, Bifurcations of planar vector fields and Hilbert's 16th problem, Birkhauser, Berlin (1998).
J.J Stoker, Nonlinear elasticity, Gordon and Breach, London (1968).
R.A. Struble, Nonlinear differential equations, Mac Graw Hill (1962).
J. Tannery and J. Molk, Éléments de la théorie des fonctions elliptiques, Gauthier-Villars, Tomes I à IV, Paris (1896).
E.T. Whittaker and G.N. Watson, A course of modern analysis, Cambridge U. Press, New York (1927).