Some new results on the global uniform asymptotic stability of time-varying dynamical systems

Abstract

The objective of this work is twofold. In the first part, we present sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov-like functions for nonlinear time varying systems. Furthermore, an illustrative numerical example is presented.

Partial stability criteria for time-varying nonlinear systems

“…But this technique is very restrictive when the term of perturbation is more complexity. Hence some researchers attempt to use the Lyapunov function associated to the nominal system to construct a new Lyapunov function that guarantees the stability of the considered system (4.1), see ( [1], [4], [5], [6], [7], [8]). Thus, we continue in this direction and we answer to the question mentioned in the Section 2. by building a Lyapunov function for a more general class of perturbed systems.…”

“…For standard state-space systems, [12], presented a systematic study of the theory of practical stability and collected most valuable results. The question addressed in this paper is related to the study of the preservation of stability when considering a new system with a perturbation term ( [1], [2], [3], [4], [6]). Recently, for the time-varying perturbed systems ẋ = f (t, x) + g(t, x) (1.1) where f, g : R + × R n → R n are piecewise continuous in t, locally Lipschitz in x, the authors in ( [4], [7], [8]) studied the asymptotic and exponential stability of a class of system (1.1) with respect to a neighborhood of the origin approximated by a small ball of radius r > 0 centered at the origin in the sense that the trajectories approach a small compact set containing the origin based on the stability of the nominal system…”

Practical Input-to-State Stability of Nonlinear Systems with Time Delay

“…Te characterizations of ISpS have been developed for a class of ordinary diferential equations [17]. Nonlinear systems are usually stabilized using LFs [14,18,19]. In addition, they provide tools for designing a more robust controller and/or verifying its robustness.…”

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

“…The stability of nonautonomous perturbed dynamical systems has attracted the attention of many researchers and has produced several important results. [1][2][3][4][5] It turns out that time-varying differential equations appear as a natural description of observed evolution phenomena of various real-world problems where the study of asymptotic stability is more interesting than stability. Indeed, the study of asymptotic stability of dynamical systems is one of the most important research area in system design.…”