Asymptotic behavior of a class of perturbed differential equations

  • A. Dorgham Univ. Sfax, Tunisia
  • M. Hammi Univ. Sfax, Tunisia
  • M. A. Hammami Univ. Sfax, Tunisia
Keywords: Differential equations, perturbations, Lyapunov function, stability


UDC 517.9

This paper deals with the problem of stability of nonlinear differential equations with perturbations.
Sufficient conditions for global uniform asymptotic stability in terms of Lyapunov-like functions and integral inequality are obtained.
The asymptotic behavior is studied in the sense that the trajectories converge to a small ball centered at the origin.
Furthermore, an illustrative example in the plane is given to verify the effectiveness of the theoretical results.



Author Biography

M. A. Hammami , Univ. Sfax, Tunisia




D. Aeyels, J. Peuteman, A new asymptotic stability criterion for nonlinear time-variant differential equations, IEEE Trans. Automat. Control, 43, № 7, 968 – 971 (1998), DOI:

N. S. Bay, V. N. Phat, Stability of nonlinear difference time-varying systems with delays, Vietnam J. Math., 4, 129 – 136 (1999).

A. Ben Abdallah, I. Ellouze, M. A. Hammami, Practical stability of nonlinear time-varying cascade systems, J. Dyn. and Control Syst., 15, № 1, 45 – 62 (2009), DOI:

A. Ben Abdallah, M. Dlala, M. A. Hammami, A new Lyapunov function for stability of time-varying nonlinear perturbed systems, Systems Control Lett., 56, № 3, 179 – 187 (2007), DOI:

A. Ben Makhlouf, Stability with respect to part of the variables of nonlinear Caputo fractional differential equations, Math. Commun., 23, № 1, 119 – 126 (2018).

A. Ben Makhlouf, M. A. Hammami, A nonlinear inequality and application to global asymptotic stability of perturbed systems, Math. Methods Appl. Sci., 38, № 12, 2496 – 2505 (2015), DOI:

M. Corless, G. Leitmann, Controller design for uncertain systems via Lyapunov functions, Proc. 1988 Amer. Control Conf., Atlanta, Georgia (1988), DOI:

M. Corless, Guaranteed rates of exponential convergence for uncertain systems, J. Optim. Theory and Appl., 64, № 3, 481 – 494 (1990), DOI:

F. Garofalo, G. Leitmann, Guaranteeing ultimate boundedness and exponential rate of convergence for a class of nominally linear uncertain systems, J. Dyn. Syst., Measurement and Control, 111, 584 – 588 (1989). DOI:

B. Ghanmi, N. Hadj Taieb, M. A. Hammami, Growth conditions for exponential stability of time-varying perturbed systems, Internat. J. Control, 86, № 6, 1086 – 1097 (2013), DOI:

W. Hahn, Stability of motion, Springer, New York (1967). DOI:

Z. HajSalem, M. A. Hammami, M. Mabrouk, On the global uniform asymptotic stability of time-varying dynamical systems, Stud. Univ. Babe¸s-Bolyai Math., 59, № 1, 57 – 67 (2014).

M. A. Hammami, On the stability of nonlinear control systems with uncertainty, J. Dyn. and Control Syst., 7, № 2, 171 – 179 (2001), DOI:

M. Hammi, M. A. Hammami, Nonlinear integral inequalities and applications to asymptotic stability, IMA J. Math. Control and Inform., 32, № 4, 717 – 735 (2015), DOI:

W. G. Kelley, A. C. Peterson, The theory of differential equations. Classical and Qualitative, Springer (2010), DOI:

H. Khalil, Nonlinear systems, Prentice Hall (2002).

A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55, № 3, 521 – 790 (1992), DOI:

Xiaoxin Liao, Liqiu Wang, Pei Yu, Stability of dynamical systems, Monogr. Ser. Nonlinear Sci., Elsevier, Amsterdam, Netherlands (2007). DOI:

T. Yoshizawa, Stability theory by Lyapunov’s second method, Math. Soc. Jap. (1996).

How to Cite
DorghamA., HammiM., and Hammami M. A. “Asymptotic Behavior of a Class of Perturbed Differential Equations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 5, May 2021, pp. 627 -39, doi:10.37863/umzh.v73i5.232.
Research articles