Abstract
In this paper we present a converse Lyapunov theorem for global practical uniform exponential stability of nonlinear time-varying systems. The main result shows that the system is practically globally uniformly exponentially stable if and only if it admits a Lyapunov function which satisfies some conditions. An example is also discussed to illustrate the advantage of the proposed result.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Xingu Song, Senlin Li, An Li, “Practical Stability of nonlinear differential equation with initial time difference”, AppliedMathematics and Compulation, 203 (1), 157–162, 2008.
Stephen R. Bernfeld, V. LakshmiKantham, “Practical Stability and Lyapunov functions”, TohokuMath.J., 32 (4), 607–613, 1980.
Coşkun Yakar, Michael D. Shaw, “Practical stability in terms of two measures with initial time difference”, Nonlinear Analysis: Theory,Methods and Applications, 71 (12), 781–785, 2009.
A. Benabdallah, I. Ellouze, M. A. Hammami, “Practical Exponential Stability of perturbed Triangular systems and a separation principle”, Asian Journal of Control, 13 (3), 445–448, 2011.
M. Dlala, M. A. Hammami, “Uniform Exponential Practical Stability of Implusive Perturbed Systems”, Journal of Dynamical and Control Systems, 13 (3), 373–386, 2007.
A. Benabdallah, I. Ellouze, M. A. Hammami, “Practical stability of nonlinear time-varying cascade systems”, Journal of Dynamical and Control Systems, 15 (1), 45–62, 2009.
M. Gil, “Norm estimates for fundamental solutions of neutral type functional differential equations”, Int. J. of Dynamical Systems and Differential Equations, 4 (3), 255–273, 2012.
M. A. Hammami, “On the stability of nonlinear control systems with uncertainty”, Journal of Dynamical and Control Systems, 7 (2), 171–179, 2001.
J. Hoffacker, B. Jackson, “Stability results for higher dimensional equations on time scales”, Int. J. of Dynamical Systems and Differential Equations, 3 (1), 48–58, 2011.
H. K. Khalil, Nonlinear systems, (Prentice-Hall, New York, 2002).
M. Corless, L. Glielmo, “New converse Lyapunov theorems and related results on exponential stability”, Mathematics of Control, Signals, and Systems, 11, 79–100, 1998.
J. L. Mancilla-Aguilar, R. A. Garcia, “On converse Lyapunov theorems for ISS and iISS switched nonlinear systems”, Systems and Control Letters, 42, 47–53, 2001.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © M. Errebii, I. Ellouze, M. A. Hammami, 2015, published in Izvestiya NAN Armenii. Matematika, 2015, No. 4, pp. 23-35.
About this article
Cite this article
Errebii, M., Ellouze, I. & Hammami, M.A. Exponential convergence of nonlinear time-varying differential equations. J. Contemp. Mathemat. Anal. 50, 167–175 (2015). https://doi.org/10.3103/S1068362315040020
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1068362315040020