Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Angeli, D., Sontag, E.D.: Forward completeness, unbounded observability and their Lyapunov characterizations. Systems and Control Letters 38, 209–217 (1999)
Bacciotti, A., Rosier, L.: Lyapunov stability and Lagrange stability: Inverse theorems for discontinuous systems. Mathematics of Control, Signals and Systems 11, 101–125 (1998)
Bacciotti, A., Rosier, L.: On the converse of first Lyapunov theorem: The regularity issue. Systems and Control Letters 41, 265–270 (2000)
Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Lecture Notes in Control and Information Sciences, vol. 267. Springer, London (2001)
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)
Burton, T.A.: Uniform asymptotic stability in functional differential equations. Proceedings of the American Mathematical Society 68, 195–199 (1978)
Grüne, L.: Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Transactions on Automatic Control 47, 1499–1504 (2002)
Hahn, W.: Stability of Motion. Springer, Berlin (1967)
Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)
Ignatyev, A.O.: On the partial equiasymptotic stability in functional differential equations. Journal of Mathematical Analysis and Applications 268, 615–628 (2002)
Jiang, Z.P., Wang, Y.: A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58 (2002)
Karafyllis, I.: Non-uniform in time robust global asymptotic output stability. Systems and Control Letters 54, 181–193 (2005)
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)
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)
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)
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)
Kellett, C.M., Teel, A.R.: Smooth Lyapunov functions and robustness of stability for difference inclusions. Systems and Control Letters 52, 395–405 (2004)
Khalil, H.K.: Nonlinear Systems, 2nd edn. Prentice-Hall, New York (1996)
Kharitonov, V.L.: Lyapunov–Krasovskii functionals for scalar time delay equations. Systems and Control Letters 51, 133–149 (2004)
Kharitonov, V.L., Melchor-Aguilar, D.: On delay dependent stability conditions for time-varying systems. Systems and Control Letters 46, 173–180 (2002)
Krasovskii, N.N.: Stability of Motion. Stanford University Press, Stanford (1963)
Kurzweil, J.: On the inversion of Lyapunov’s second theorem on stability of motion. Amer. Math. Soc. Trans. 2(24), 19–77 (1956)
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)
Rosier, L.: Smooth Lyapunov functions for discontinuous stable systems. Set-Valued Analysis 7, 375–405 (1999)
Sontag, E.D.: Comments on integral variants of ISS. Systems and Control Letters 34, 93–100 (1998)
Sontag, E.D., Wang, Y.: Lyapunov characterizations of input-to-output stability. SIAM Journal on Control and Optimization 39, 226–249 (2001)
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)
Tsinias, J.: A converse Lyapunov theorem for non-uniform in time global exponential robust stability. Systems and Control Letters 44, 373–384 (2001)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag London Limited
About this chapter
Cite this chapter
Karafyllis, I., Jiang, ZP. (2011). Converse Lyapunov Results. In: Stability and Stabilization of Nonlinear Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-513-2_3
Download citation
DOI: https://doi.org/10.1007/978-0-85729-513-2_3
Publisher Name: Springer, London
Print ISBN: 978-0-85729-512-5
Online ISBN: 978-0-85729-513-2
eBook Packages: EngineeringEngineering (R0)