Journal of Mathematical Sciences

, Volume 135, Issue 4, pp 3224–3243 | Cite as

On the curvature of two-dimensional optimal control systems and Zermelo’s navigation problem

  • U. Serres


The goal of this paper is to extend the notion of Gaussian curvature of two-dimensional surfaces to nonlinear time-optimal control systems in the plane by applying the moving frame method. This notion of curvature was introduced earlier by A. A. Agrachev and R. V. Gamkrelidze by means of Jacobi curves. Here we give a self-contained presentation of its two-dimensional version and apply the results to the well-known Zermelo navigation problem.


Control System Gaussian Curvature Navigation Problem Optimal Control System Frame Method 
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.


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  1. 1.
    A. A. Agrachev, “On the curvature of control systems,” available on < http//>.Google Scholar
  2. 2.
    A. A. Agrachev and R. V. Gamkrelidze, “The exponential representation of flows and the chronological calculus,” Math. USSR. Sb., 35, 727–785 (1979).Google Scholar
  3. 3.
    A. A. Agrachev and R. V. Gamkrelidze, “Feedback-invariant optimal control theory and differential geometry-I. Regular extremals,” J. Dyn. Contr. Syst., 3, 343–389 (1997).MathSciNetGoogle Scholar
  4. 4.
    A. A. Agrachev and Yu. L. Sachkov, Control Theory from the Geometric Viewpoint, Preprint SISSA 77/2002/M, November 2002, SISSA, Trieste, Italy, to appear in Springer.Google Scholar
  5. 5.
    V. I. Arnold, Catastrophe Theory, Springer-Verlag (1992).Google Scholar
  6. 6.
    V. I. Arnold, V. S. Afrajmovich, Yu. S. Il’yashenko, and L. P. Shil’nikov, Bifurcation Theory and Catastrophe Theory, Springer-Verlag (1999).Google Scholar
  7. 7.
    V. I. Arnold, H. Brézis, P. Cartier, J. Coates, Y. Colin de Verdière, H. Helson, J.-P. Kahane, P.-L. Lions, B. Malgrange, Y. Meyer, F. Pham, and D. Zagier, Leçons de Mathématiques d’Aujourd’hui, Cassini, Paris, (2000), Leçon 11, F. Pham, Caustiques: Aspects Géométriques et Ondulatoires.Google Scholar
  8. 8.
    R. Berndt, An Introduction to Symplectic Geometry, American Mathematical Society, 26 (2000).Google Scholar
  9. 9.
    C. Carathéodory, Calculus of Variations, Chelsea Publishing Company, New York (1989).Google Scholar
  10. 10.
    H. Cartan, Cours de Calcul Différentiel, Hermann, Paris (1967).Google Scholar
  11. 11.
    V. Jurdjevic, Geometric Control Theory, Cambridge University Press, New York (1997).Google Scholar

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© Springer Science+Business Media, Inc. 2006

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  • U. Serres

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