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1995 | 32 | 1 | 301-317
Tytuł artykułu

Abnormality of trajectory in sub-Riemannian structure

Treść / Zawartość
Warianty tytułu
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.
Słowa kluczowe
Rocznik
Tom
32
Numer
1
Strony
301-317
Opis fizyczny
Daty
wydano
1995
Twórcy
autor
  • LAMA, Université de Savoie, Campus Scientifique, 73376 Le Bourget du Lac Cedex, France
  • 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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv32z1p301bwm
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