Optimal Control, Geometry, and Mechanics

  • V. Jurdjevic


This essay highlights the contributions of geometric control theory to the calculus of variations. The Maximum Principle is recognized as a covariant necessary condition of optimality valid for variational problems defined on subsets C of the tangent bundle of the ambient manifold, rather than the entire tangent bundle, as was commonly assumed by the classical theory. Control-theoretic interpretations provide a uniform framework for variational problems with non-holonomic constraints and reveal the significance of controllability theory for such problems. The merits of this theory are then illustrated through solutions of variational problems on Lie groups having either left or right invariance. By focusing on problems fromm geometry and mechanics, the essay visibly demonstrates the innovative ideas that geometric control brings to the calculus of variations.


Vector Field Symmetric Space Heisenberg Group Elastic Problem Hamiltonian Vector Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Agrachev, Exponential mappings for contact sub-Riemannian Structures, Journal Dyn. and Control Systems 2(3), (1996), pages 321–358.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    A. Agrachev, C. El Alaoui, and J.P. Gauthier, Sub-Riemannian Metrics on ℝ3, to appear in Geometric Control and Nonholonomic Problems in Mechanics (V. Jurdjevic and R. Sharpe, eds.), Conference Proceedings Series, Canad. Math. Soc.Google Scholar
  3. [3]
    V.I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York, 1978.zbMATHGoogle Scholar
  4. [4]
    B. Bonnard and M. Chyba, Sub-Riemannian geometry: the Martinet case, to appear in Geometric Control and Nonholonomic Problems in Mechanics (V. Jurdjevic and R. Sharpe, eds), Conference Proceedings Series, Canad. Math. Soc.Google Scholar
  5. [5]
    R.W. Brockett, System theory on group manifolds and coset spaces, SIAM J. Control Optim., 10(2), 1972, pages 265–284.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    R.W. Brockett, Lie Theory and Control Systems defined on spheres, SIAM J. Appl. Math., 25, 1973, pages 213–225.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    R.W. Brockett, Lie algebras and Lie groups in control theory, in Geometric Methods in System Theory (D.Q. Mayne and R.W. Brockett, eds.), Reidel, Dordrecht, 1973, pages 43–82.CrossRefGoogle Scholar
  8. [8]
    R.W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Applied Mathematics (P. Hilton and G. Young, eds.), Springer-Verlag, New York, 1981, pages 11–27.Google Scholar
  9. [9]
    R.W. Brockett, Nonlinear control theory and differential geometry, Proceedings of the International Congress of Mathematicians, August 16–24, Warszawa, 1983, pages 1357–1368.Google Scholar
  10. [10]
    El-Alaoui C, J.P. Gauthier, and I. Kupka, Small sub-Riemannian balls on ℝ3, J. Dynamics Control Systems, 2(30), 1996, pages 359–421.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    V.Y. Gershkovich and A.M. Vershik, Nonholonomic dynamical systems, geometry of distributions and variational problems, in Dynamical Systems VII, Vol. 16 (V.I. Arnold and S.P. Novikov, eds.), Encyclopedia of Mathematical Sciences, Springer-Verlag, New York, 1991.Google Scholar
  12. [12]
    P. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, Boston, 1983.zbMATHGoogle Scholar
  13. [13]
    S. Helgason, Differential Geometry and Symmetric Spaces Academic Press, New York, 1962.zbMATHGoogle Scholar
  14. [14]
    V. Jurdjevic, The geometry of the plate-ball problem, Arch. Rational Mech. Anal., 124, 1993, pages 305–328.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    V. Jurdjevic, Non-Euclidean elastica, Amer. J. Math., 117, 1995, pages 93–125.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    V. Jurdjevic, Geometric Control Theory, Cambridge Studies in Advanced Mathematics, Vol. 52, Cambridge University Press, 1997.Google Scholar
  17. [17]
    V. Jurdjevic, Integrable Hamiltonian systems on Lie groups: Kowalevska type, 1997, preprint.Google Scholar
  18. [18]
    A.A. Kirillov, Elements of the Theory of Representations, Grundlehren der Mathematischen Wissenshaften, Vol. 220, Springer-Verlag, New York, 1976.Google Scholar
  19. [19]
    W. Liu and H.J. Sussmann, Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions, Memoirs AMS, Vol. 118(564), 1995.Google Scholar
  20. [20]
    A.E. Love, A Treatise on the Mathematical Theory of Elasticity, 4th Edition, Dover, New York, 1927.zbMATHGoogle Scholar
  21. [21]
    R. Montgomery, Abnormal minimizers, SIAM J. Control Optim., 32(6), 1994, pages 605–620.CrossRefGoogle Scholar
  22. [22]
    H. Poincaré, Sur une form nouvelle des equations de la mechanique, Comptes Rendus des Sciences, 132, 1901, pages 369–371.zbMATHGoogle Scholar
  23. [23]
    R. Sharpe, Differential Geometry, Graduate Texts in Mathematics, Vol. 166, Springer-Verlag, New York, 1997.Google Scholar
  24. [24]
    A. Weil, Euler and Jacobians of elliptic curves, in Arithmetic and Geometry, papers dedicated to Shafarevich, Prog. Math., Birkhauser, Boston, 1983, pages 353–359.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • V. Jurdjevic

There are no affiliations available

Personalised recommendations