Journal of Mathematical Sciences

, Volume 144, Issue 1, pp 3841–3847 | Cite as

On curvature and feedback classification of two-dimensional optimal control systems

  • Ulysse Serres


The goal of this paper is to extend the classical notion of Gaussian curvature of a two-dimensional Riemannian surface to two-dimensional optimal control systems with scalar input using Cartan’s moving frame method. This notion was already introduced by A. A. Agrachev and R. V. Gamkrelidze for more general control systems using a purely variational approach. Further, we will see that the “control” analogue of Gaussian curvature reflects similar intrinsic properties of the extremal flow. In particular, if the curvature is negative, arbitrarily long segments of extremals are locally optimal. Finally, we will define and characterize flat control systems.


Optimal Control Problem Gaussian Curvature Riemannian Problem Jacobi Equation Cotangent Bundle 
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    A. A. Agrachev, Yu. L. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, 2004.Google Scholar
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    U. Serres, “On the curvature of two-dimensional optimal control systems and Zermelo’s navigation problem,” J. Math. Sci. (in press).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Ulysse Serres
    • 1
    • 2
  1. 1.UFR des Sciences et des TechniquesUniversité de BourgogneDijon cedexFrance
  2. 2.SISSA/ISASTriesteItaly

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