Abnormality of trajectory in sub-Riemannian structure

F. Pelletier; L. Bouche

Banach Center Publications (1995)

  • Volume: 32, Issue: 1, page 301-317
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Pelletier, F., and Bouche, L.. "Abnormality of trajectory in sub-Riemannian structure." Banach Center Publications 32.1 (1995): 301-317. <http://eudml.org/doc/262701>.

@article{Pelletier1995,
abstract = {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.},
author = {Pelletier, F., Bouche, L.},
journal = {Banach Center Publications},
keywords = {rigidity; sub-Riemannian manifold; arc minimizing curves; abnormal extremal; Pontryagin maximum principle},
language = {eng},
number = {1},
pages = {301-317},
title = {Abnormality of trajectory in sub-Riemannian structure},
url = {http://eudml.org/doc/262701},
volume = {32},
year = {1995},
}

TY - JOUR
AU - Pelletier, F.
AU - Bouche, L.
TI - Abnormality of trajectory in sub-Riemannian structure
JO - Banach Center Publications
PY - 1995
VL - 32
IS - 1
SP - 301
EP - 317
AB - 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.
LA - eng
KW - rigidity; sub-Riemannian manifold; arc minimizing curves; abnormal extremal; Pontryagin maximum principle
UR - http://eudml.org/doc/262701
ER -

References

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  9. [K1] I. Kupka, Abnormal extremals, preprint, 1992. 
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  11. [L] R. Léandre, Développements asymptotiques de la densité de diffusions dégénérées, to appear. 
  12. [L-S] W. Liu and H. J. Sussmann, Abnormal sub-Riemannian minimizers, preprint, 1992. Zbl0823.49026
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  14. [May] A. Mayer, Begründung der Lagrangenschen Multiplicatorenmethode in der Variationsrechnung, Math. Ann. 26 (1886), 74. 
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  16. [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. 
  17. [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. 
  18. [St] R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), 221-263, and 30 (2) (1989), 595-596. Zbl0609.53021
  19. [Su] H. Sussmann, A cornucopia of abnormal sub-Riemannian minimizers, part 1: the four dimensional case, preprint, 1993. 
  20. [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. 

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