Converse Lyapunov Results

  • Iasson KarafyllisEmail author
  • Zhong-Ping Jiang
Part of the Communications and Control Engineering book series (CCE)


Chapter 3 is devoted to answering the following question: do Lyapunov functionals always exist for a robustly globally asymptotically output stable system? The previous Chap. 2 showed that one of the most important ways of proving stability is the derivation of estimates which guarantee appropriate stability properties by means of Lyapunov functionals. The converse Lyapunov results obtained in this chapter show that such Lyapunov functionals always exist.


  1. 1.
    Angeli, D., Sontag, E.D.: Forward completeness, unbounded observability and their Lyapunov characterizations. Systems and Control Letters 38, 209–217 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bacciotti, A., Rosier, L.: Lyapunov stability and Lagrange stability: Inverse theorems for discontinuous systems. Mathematics of Control, Signals and Systems 11, 101–125 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bacciotti, A., Rosier, L.: On the converse of first Lyapunov theorem: The regularity issue. Systems and Control Letters 41, 265–270 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Lecture Notes in Control and Information Sciences, vol. 267. Springer, London (2001) zbMATHGoogle Scholar
  5. 5.
    Bernfeld, S.A., Corduneanu, C., Ignatyev, A.O.: On the stability of invariant sets of functional differential equations. Nonlinear Analysis: Theory Methods and Applications 55, 641–656 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Burton, T.A.: Uniform asymptotic stability in functional differential equations. Proceedings of the American Mathematical Society 68, 195–199 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Grüne, L.: Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Transactions on Automatic Control 47, 1499–1504 (2002) CrossRefGoogle Scholar
  8. 8.
    Hahn, W.: Stability of Motion. Springer, Berlin (1967) zbMATHGoogle Scholar
  9. 9.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) zbMATHGoogle Scholar
  10. 10.
    Ignatyev, A.O.: On the partial equiasymptotic stability in functional differential equations. Journal of Mathematical Analysis and Applications 268, 615–628 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Jiang, Z.P., Wang, Y.: A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Karafyllis, I.: Non-uniform in time robust global asymptotic output stability. Systems and Control Letters 54, 181–193 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Karafyllis, I.: Non-uniform in time robust global asymptotic output stability for discrete-time systems. International Journal of Robust and Nonlinear Control 16, 191–214 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Karafyllis, I.: A system-theoretic framework for a wide class of systems I: Applications to numerical analysis. Journal of Mathematical Analysis and Applications 328, 876–899 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Karafyllis, I., Tsinias, J.: A converse Lyapunov theorem for non-uniform in time global asymptotic stability and its application to feedback stabilization. SIAM Journal Control and Optimization 42, 936–965 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Karafyllis, I., Pepe, P., Jiang, Z.-P.: Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations. European Journal of Control 14, 516–536 (2008) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kellett, C.M., Teel, A.R.: Smooth Lyapunov functions and robustness of stability for difference inclusions. Systems and Control Letters 52, 395–405 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Khalil, H.K.: Nonlinear Systems, 2nd edn. Prentice-Hall, New York (1996) Google Scholar
  19. 19.
    Kharitonov, V.L.: Lyapunov–Krasovskii functionals for scalar time delay equations. Systems and Control Letters 51, 133–149 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kharitonov, V.L., Melchor-Aguilar, D.: On delay dependent stability conditions for time-varying systems. Systems and Control Letters 46, 173–180 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Krasovskii, N.N.: Stability of Motion. Stanford University Press, Stanford (1963) zbMATHGoogle Scholar
  22. 22.
    Kurzweil, J.: On the inversion of Lyapunov’s second theorem on stability of motion. Amer. Math. Soc. Trans. 2(24), 19–77 (1956) Google Scholar
  23. 23.
    Lin, Y., Sontag, E.D., Wang, Y.: A smooth converse Lyapunov theorem for robust stability. SIAM Journal on Control and Optimization 34, 124–160 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Rosier, L.: Smooth Lyapunov functions for discontinuous stable systems. Set-Valued Analysis 7, 375–405 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Sontag, E.D.: Comments on integral variants of ISS. Systems and Control Letters 34, 93–100 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Sontag, E.D., Wang, Y.: Lyapunov characterizations of input-to-output stability. SIAM Journal on Control and Optimization 39, 226–249 (2001) CrossRefGoogle Scholar
  27. 27.
    Teel, A.R., Praly, L.: A Smooth Lyapunov function from a class-KL estimate involving two positive functions. ESAIM Control Optimisation and Calculus of Variations 5, 313–367 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Tsinias, J.: A converse Lyapunov theorem for non-uniform in time global exponential robust stability. Systems and Control Letters 44, 373–384 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Yoshizawa, T.: Asymptotic behavior of solutions in nonautonomous systems. In: Lakshmikantham, V. (ed.) Trends in Theory and Practice of Nonlinear Differential Equations, pp. 553–562. Dekker, New York (1984) Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

There are no affiliations available

Personalised recommendations