Advertisement

General Classes of Control-Lyapunov Functions

  • Eduardo D. Sontag
  • Héctor J. Sussmann
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 121)

Abstract

The main result of this paper establishes the equivalence between null asymptotic controllability of nonlinear finite-dimensional control systems and the existence of continuous control-Lyapunov functions (CLF’s) defined by means of generalized derivatives. In this manner, one obtains a complete characterization of asymptotic controllability, applying in principle to a far wider class of systems than Artstein’s Theorem (which relates closed-loop feedback stabilization to the existence of smooth CLF’s). The proof relies on viability theory and optimal control techniques.

Keywords

Directional Derivative Differential Inclusion Nonlinear Control System Closed Convex Hull Viability Theory 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Artstein, Z., “Stabilization with relaxed controls,” Nonlinear Analysis, Theory, Methods & Applications 7 (1983): 1163–1173.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Aubin, J.-P., and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984.zbMATHGoogle Scholar
  3. [3]
    Aubin, J.-P., Viability Theory, Birkhäuser, Boston, 1991.zbMATHGoogle Scholar
  4. [4]
    Bacciotti, A., Local Stabilizability of Nonlinear Control Systems, World Scientific, London, 1991.Google Scholar
  5. [5]
    Coron, J.M., L. Praly, and A. Teel, “Feedback stabilization of nonlinear systems: sufficient conditions and Lyapunov and input-output techniques,” in Trends in Control: A European Perspective ( A. Isidori, Ed.), Springer, London, 1995 (pp. 293–348 ).Google Scholar
  6. [6]
    Deimling, K., Multivalued Differential Equations, de Gruyter, Berlin, 1992.Google Scholar
  7. [7]
    Krstic, M., I. Kanellakopoulos, and P. Kokotovic, Nonlinear and adaptive control design, John Wiley & Sons, New York, 1995.Google Scholar
  8. [8]
    Long, Y., and M. M. Bayoumi, “Feedback stabilization: Control Lyapunov functions modeled by neural networks,” in Proc. IEEE Conf. Decision and Control, San Antonio, Dec. 1993, IEEE Publications, 1993, pp. 2812–2814.Google Scholar
  9. [9]
    Isidori, A., Nonlinear Control Systems: An Introduction, Springer-Verlag, Berlin, third ed., 1995.zbMATHGoogle Scholar
  10. [10]
    Lafferriere, G. A., “Discontinuous stabilizing feedback using partially defined Lyapunov functions,” in Proc. IEEE Conf. Decision and Control, Lake Buena Vista, Dec. 1994, IEEE Publications, 1994, pp. 3487–3491.Google Scholar
  11. [11]
    Sontag, E.D., Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer, New York, 1990.zbMATHGoogle Scholar
  12. [12]
    Sontag, E.D., “A Lyapunov-like characterization of asymptotic controllability,” SIAM J. Control & Opt. 21(1983): 462–471. (See also “A characterization of asymptotic controllability,” in Dynamical Systems II (A. Bednarek and L. Cesari, eds.), Academic Press, NY, 1982, pp. 645–648.)Google Scholar
  13. [13]
    Sontag, E.D., “A ‘universal’ construction of Artstein’s theorem on nonlinear stabilization,” Systems and Control Letters, 13 (1989): 117–123.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Sontag, E.D., and H.J. Sussmann, “Non-smooth control-Lyapunov functions,” Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1996

Authors and Affiliations

  • Eduardo D. Sontag
    • 1
  • Héctor J. Sussmann
    • 2
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Dept. of MathematicsRutgers UniversityNew BrunswickUSA

Personalised recommendations