Abnormality of trajectory in sub-Riemannian structure

• Autorzy: F. Pelletier , L. Valère Bouche
• Języki publikacji: EN

Abstrakty

EN

In the sub-Riemannian framework, we give geometric necessary and sufficient conditions for the existence of abnormal extremals of the Maximum Principle. We give relations between abnormality, $C^1$-rigidity and length minimizing. In particular, in the case of three dimensional manifolds we show that, if there exist abnormal extremals, generically, they are locally length minimizing and in the case of four dimensional manifolds we exhibit abnormal extremals which are not $C^1$-rigid and which can be minimizing or non minimizing according to different metrics.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1995
• Tom: 32
• Numer: 1
• Strony: 301-317

Daty

wydano

1995

Twórcy

autor

F. Pelletier

• LAMA, Université de Savoie, Campus Scientifique, 73376 Le Bourget du Lac Cedex, France

autor

L. Valère Bouche

• LAMA, Université de Savoie, Campus Scientifique, 73376 Le Bourget du Lac Cedex, France

Bibliografia

• [A] A. Agrachev, On abnormal geodesics in sub-Riemannian geometry, preprint.
• [A-S] A. Agrachev and A. Sarychev, Strong minimality of abnormal geodesics for 2-distributions, J. Dynam. Control Systems, to appear.
• [BA] G. Ben Arous, Noyau de la chaleur hypoelliptique et géométrie sous-riemannienne, to appear.
• [Bi] J. M. Bismut, Large Deviations and the Malliavin Calculus, Progress in Math. 45, Birkhäuser, Basel, 1984.
• [B-H] R. L. Bryant and L. Hsu, Rigid trajectory in systems with two linear controls, preprint, 1993.
• [He] R. Hermann, On the accessibility problem of control theory, in: Proc. Sympos. Differential Equations, Colorado Springs, J. La Salle and S. Lefschetz (eds.), Academic Press, 1961.
• [Hs] L. Hsu, Calculus of variations via Griffith's formalism, J. Differential Geom. 36, 1992, 551-585.
• [J-P] B. Jakubczyk and F. Przytycki, Singularities of k-tuples of vector fields, Dissertationes Math. 213 (1984).
• [K1] I. Kupka, Abnormal extremals, preprint, 1992.
• [K2] I. Kupka, Caractérisations d'une courbe anormale, preprint.
• [L] R. Léandre, Développements asymptotiques de la densité de diffusions dégénérées, to appear.
• [L-S] W. Liu and H. J. Sussmann, Abnormal sub-Riemannian minimizers, preprint, 1992.
• [Mar] J. Martinet, Sur les singularités de formes différentielles, Ann. Inst. Fourier (Grenoble) 20 (1) (1970), 95-178.
• [May] A. Mayer, Begründung der Lagrangenschen Multiplicatorenmethode in der Variationsrechnung, Math. Ann. 26 (1886), 74.
• [Mo] R. Montgomery, Geodesics which do not satisfy the geodesic equations, preprint, 1991.
• [P-V 1] F. Pelletier et L. Valère Bouche, Le problème des géodésiques en géométrie sous-riemannienne singulière, C. R. Acad. Sci. Paris Sér. I 317 (1993), 71-76.
• [P-V 2] F. Pelletier et L. Valère Bouche, Le problème des géodésiques, dérivation intrinsèque et utilisation de la théorie du contrôle en géométrie sous-riemannienne singulière, in: Table Ronde en l'Honneur de M. Berger, Luminy, 1992, to appear.
• [St] R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), 221-263, and 30 (2) (1989), 595-596.
• [Su] H. Sussmann, A cornucopia of abnormal sub-Riemannian minimizers, part 1: the four dimensional case, preprint, 1993.
• [V] L. Valère Bouche, The geodesics' problem in sub-Riemannian geometry, about the R. Montgomery's example simplified by I. Kupka, les prépublications du L.A.M.A. (92-06), 1992.