Stability of nonlinear time-varying perturbed differential equations
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O R I G I N A L PA P E R
Stability of nonlinear time-varying perturbed differential equations Bassem Ben Hamed · Zaineb Haj Salem · Mohamed Ali Hammami
Received: 22 May 2012 / Accepted: 13 March 2013 © Springer Science+Business Media Dordrecht 2013
Abstract The goal of this paper is twofold. The first part presents a converse Lyapunov theorem for the notion of uniform practical exponential stability of nonlinear differential equations in presence of small perturbation. This class of nonlinear differential equations can be viewed as parametric differential equations. The second part provides the classical perturbation method of seeking an approximate solution as a finite Taylor expansion of the exact solution. The practical asymptotic validity on the approximate is established on infinite-time interval. Finally, we give a numerical example to prove the validity of our methods. Keywords Lyapunov stability · Practical exponential stability · Perturbed differential equations
B. Ben Hamed () Department of Mathematics, Higher Institute of Electronics and Communication of Sfax, Street Menzel Chaker km 0.5, B.P. 261, 3038 Sfax, Tunisia e-mail: [email protected] Z. Haj Salem Department of Mathematics, Higher Institute of Applied Science and Technology of Gabès, Street Amor Ben El Khatab, 6029 Gabès, Tunisia M.A. Hammami Department of Mathematics, Faculty of Science of Sfax, BP 1171, Street Soukra, 3000 Sfax, Tunisia
1 Introduction The question addressed in this paper is related to the study of the preservation of uniform asymptotic stability of time-varying differential equations, particularly when considering a new system with a perturbation, see [1, 11, 12, 24, 25]. In [15], a converse theorem for uniform asymptotic stability is established, and confirms that if the origin is uniformly asymptotically stable, then there is a Lyapunov function that meets some conditions. Motivated by robust control analysis and design, the authors in [17] establish a smooth converse Lyapunov theorem for uniform global asymptotic robust stability. The author in [23], give a Lyapunov characterization of a concept of, non-uniform in time, global exponential robust stability of the origin. In [14], a converse Lyapunov theorem of a concept of global asymptotic robust non-uniform stability of the origin is shown. In [5], the author studied the concept of robust global practical stability (RpGES) for nonlinear time varying systems. The authors in [2] studied this problem when the origin is not necessary an equilibrium point. When the perturbed term is small, then the trajectory will be ultimately bound and tends to the origin when the ultimate bound approaches to zero. Motivated by problems of nonlinear stabilization of uncertain systems [9, 10, 16, 18–20] (see also the references therein), the notion of practical stability was introduced. The uncertainties were represented by an additive term on the right-hand side of the state equa-
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tion and the origin was not supposed to be an equilibrium point of the system
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