Journal of Dynamical and Control Systems

, Volume 1, Issue 2, pp 139–176 | Cite as

Strong minimality of abnormal geodesics for 2-distributions

  • A. A. Agrachev
  • A. V. Sarychev


We investigate the local length minimality (by theW1,1- orH1-topology) of abnormal sub-Riemannian geodesics for rank 2 distributions. In particular, we demonstrate that this kind of local minimality is equivalent to the rigidity for generic abnormal geodesics, and introduce an appropriateJacobi equation in order to computeconjugate points. As a corollary, we obtain a recent result of Sussmann and Liu about the global length minimality of short pieces of the abnormal geodesics.

1991 Mathematics Subject Classification

49K30 53C22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. A. Agrachev, Quadratic mappings in geometric control theory. (Russian)Itogi Nauki i Tekhniki; Problemy Geometrii, VINITI, Akad. Nauk SSSR, Vol. 20Moscow, 1988, 11–205. English translation:J. Sov. Math. 51 (1990), 2667–2734.Google Scholar
  2. 2.
    A. A. Agrachev and R.V. Gamkrelidze, Second-order optimality condition for the time-optimal problem. (Russian)Mat. Sb. 100 (1976), 610–643. English translation:Math. USSR Sb. 29 (1976), 547–576.Google Scholar
  3. 3.
    —, Exponential representation of flows and chronological calculus. (Russian)Mat. Sb. 107 (1978), 467–532. English translation:Math. USSR Sb.,35 (1979), 727–785.Google Scholar
  4. 4.
    A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, Local invariants of smooth control systems.Acta Appl. Math. 14 (1989), 191–237.CrossRefGoogle Scholar
  5. 5.
    A. A. Agrachev and A. V. Sarychev On abnormal extremals for lagrange variational problems.J. Math. Syst., Estimation and Control (to appear).Google Scholar
  6. 6.
    A. A. Agrachev, Abnormal sub-Riemannian geodesics: Morse index and rigidity. Submitted toAnn. Inst. H. Poincaré, Anal. Nonlinéaire'.Google Scholar
  7. 7.
    R. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions.Invent. Math. 114 (1993), 435–461.CrossRefGoogle Scholar
  8. 8.
    V. Gershkovich, Engel structures on four-dimensional manifolds.Univ. Melbourne, Depart. Math., Preprint Series, 1992, No. 10.Google Scholar
  9. 9.
    V. Guillemin and S. Sternberg, Geometric asymptotics.Am. Math. Soc., Providence, Rhode Island, 1977.Google Scholar
  10. 10.
    M. Hestenes, Application of the theory of quadratic forms in Hilbert space to the calculus of variations.Pac. J. Math. 1 (1951), 525–582.Google Scholar
  11. 11.
    H. J. Kelley R. Kopp, and H. G. Moyer, Singular extremals. In G. Leitmann, ed., Topics in Optimization.Academic Press, New York, 1967, 63–101.Google Scholar
  12. 12.
    A. J. Krener, The high-order maximum principle and its applications to singular extremals.SIAM J. Control Optim. 15 (1977), 256–293.CrossRefGoogle Scholar
  13. 13.
    I. Kupka Abnormal extremals.Preprint, 1992.Google Scholar
  14. 14.
    C. Lobry, Dynamical polysystems and control theory. In D. Q. Mayne and R. W. Brockett, eds., Geometric Methods in Systems Theory,Reidel, Dordrecht-Boston, 1973, 1–42.Google Scholar
  15. 15.
    R. Montgomery, Abnormal minimizers,SIAM J. Control 32 (1994), 1605–1620.CrossRefGoogle Scholar
  16. 16.
    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, The mathematical theory of optimal processes.Pergamon Press, Oxford, 1964.Google Scholar
  17. 17.
    A. V. Sarychev, The index of the second variation of a control system. (Russian)Mat. Sb. 113 (1980), 464–486. English translation:Math. USSR Sb. 41 (1982), 383–401.Google Scholar
  18. 18.
    A. V. Sarychev, On Legendre-Jacobi-Morse-type theory of second variation for optimal control problems. Schwerpunktprogramm “Anwendungsbezogene Optimierung und Steuerung” der Deutschen Forschungsgemeinschaft.Würzburg, 1992, Report No. 382.Google Scholar
  19. 19.
    H. J. Sussmann, A cornucopia of abnormal sub-Riemannian minimizers. Part I. The four-dimensional caseIMA Preprint Series, 1992, No. 1073.Google Scholar
  20. 20.
    H. J. Sussmann, Wensheng Liu, Shortest paths for sub-Riemannian metrics on rank 2 distributions.Rutgers Center for System and Control, 1993, Report SYCON-93-08.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • A. A. Agrachev
    • 1
  • A. V. Sarychev
    • 2
  1. 1.Steklov Mathematics InstituteMoscowRussia
  2. 2.Departament of MathematicsUniversity of AveiroAveiroPortugal

Personalised recommendations