Journal of Dynamical and Control Systems

, Volume 21, Issue 2, pp 155–171 | Cite as

Duality of Singular Paths for (2, 3, 5)-Distributions

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Abstract

We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the viewpoint of geometric control theory. In fact, we consider the space of singular (or abnormal) paths on a given five-dimensional space endowed with a Cartan distribution, which form another five-dimensional space with a cone structure. We regard the cone structure as a control system and show that the space of singular paths of the cone structure is naturally identified with the original space. Moreover, we observe an asymmetry on this duality in terms of singular paths.

Keywords

Singular control Cartan prolongation Cone structure 

Mathematics Subject Classifications (2010)

58A30 53A55 53C17 
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Notes

Acknowledgments

This work was supported by KAKENHI no. 22340030 and no. 23654058.

References

  1. 1.
    Agrachev AA. Geometry of optimal control problems and Hamiltonian systems. In: Nonlinear and optimal control theory. Berlin: Springer; 2004. p. 1–59.Google Scholar
  2. 2.
    Agrachev AA, Sarychev AV. Control theory from geometric viewpoint. In: Encyclopaedia of mathematical sciences, vol. 87. Berlin: Springer; 2004.CrossRefGoogle Scholar
  3. 3.
    Agrachev A, Zelenko I. Nurowski’s conformal structures for (2, 5)-distributions via dynamics of abnormal extremals. In: Proceedings of RIMS symposium on developments of cartan geometry and related mathematical problems, RIMS Kokyuroku. 2006. vol. 1502, p. 204–218. arXiv:math.DG/0.605059.
  4. 4.
    Bonnard B, Chyba M. Singular trajectories and the role in control theory. Berlin: Springer; 2003.Google Scholar
  5. 5.
    Bryant RL. Élie Cartan and geometric duality. A lecture given at the Institut d’Élie Cartan on 19 June; 1998.Google Scholar
  6. 6.
    Bryant RL, Hsu L. Rigidity of integral curves of rank 2 distributions. Invent Math. 1993;114:435–61.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cartan E. Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre. Ann Sci Ecole Norm Sup. 1910;27(3):109–92.MATHMathSciNetGoogle Scholar
  8. 8.
    Doubrov B, Zelenko I. On local geometry of nonholonomic rank 2 distributions. J Lond Math Soc. 2009;80(3):545–66.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Doubrov B, Zelenko I. Equivalence of variational problems of higher order. Differ Geom Appl. 2011;29:255–70.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Doubrov B, Zelenko I. Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds. preprint, arXiv:math.DG/1210.7334v2.
  11. 11.
    Hsu L. Calculus of variations via the Griffiths formalism. J Differ Geom. 1992;36:551–89.MATHGoogle Scholar
  12. 12.
    Ishikawa G, Machida Y, Takahashi M. Asymmetry in singularities of tangent surfaces in contact-cone Legendre-null duality. J Singul. 2011;3:126–43.MATHMathSciNetGoogle Scholar
  13. 13.
    Ishikawa G, Machida Y, Takahashi M. Singularities of tangent surfaces in Cartan’s split G 2-geometry. Hokkaido University Preprint Series in Mathematics #1020; 2012.Google Scholar
  14. 14.
    Kitagawa Y. The infinitesimal automorphisms of a homogeneous subriemannian contact manifold. Thesis, Nara Women’s University; 2005.Google Scholar
  15. 15.
    Liu W, Sussman HJ. Shortest paths for sub-Riemannian metrics on rank-two distributions. Memoirs of American mathematical society, vol. 118–564. American Mathematical Society; 1995.Google Scholar
  16. 16.
    Montgomery R. A tour of Subriemannian geometries, their geodesics and applications. Mathematical surveys and monographs, vol. 91. Am Math Soc. 2002.Google Scholar
  17. 17.
    Nurowski P. Differential equations and conformal structures. J Geom Phys. 2005;55(1):19–49.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF. The mathematical theory of optimal processes. New York: Pergamon Press; 1964.MATHGoogle Scholar
  19. 19.
    Tanaka N. On affine symmetric spaces and the automorphism groups of product manifolds. Hokkaido Math J. 1985;14:277–351.CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Yatsui T. On pseudo-product graded Lie algebras. Hokkaido Math J. 1988;17:333–43.CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Zelenko I. Nonregular abnormal extremals of 2-distribution: existence, second variation, and rigidity. J Dyn Control Syst. 1999;5(3):347–383.CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Zelenko I. Fundamental form and the Cartan tensor of (2, 5)-distributions coincide. J Dyn Control Syst. 2006;12(2):247–76.CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Zhitomirskii M. Exact normal form for (2, 5) distributions. RIMS Kokyuroku. 2006;1502:16–28.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan
  2. 2.Oita National College of TechnologyOitaJapan

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