A nonlinear inequality and application to global asymptotic stability of perturbed systems

Abstract

The goal of this work is to present a new nonlinear inequality which is used in a study of the Lyapunov uniform stability and uniform asymptotic stability of solutions to time‐varying perturbed differential equations. New sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov‐like functions for nonlinear time‐varying systems is obtained. Our conditions are expressed as relation between the Lyapunov function and the existence of specific function which appear in our analysis through the solution of a scalar differential equation. Moreover, an example in dimensional two is given to illustrate the applicability of the main result. Copyright © 2014 John Wiley & Sons, Ltd.

Input‐to‐state practical stability for nonautonomous nonlinear infinite‐dimensional systems

“…To prove the main results, we need the following nonlinear inequality, which can be found in Ben Makhlouf and Hammami. 26 Proposition 2. Let's consider the following nonlinear inequality:…”

“…The Lie derivative of $$$ W $$$ at $$$ \left(t,x\right)\in {\mathbb{R}}_{+}\times X $$$ corresponding to the input $$$ u $$$ along the corresponding trajectory of $$$ \Sigma, $$$ is defined by $$$$ {\dot{W}}_u\left(t,x\right):= \underset{h\to {0}^{+}}{\lim\ \sup}\frac{1}{h}\left(W\left(t+h,\phi \left(t+h,t,x,u\right)\right)-W\left(t,x\right)\right). $$$$ To prove the main results, we need the following nonlinear inequality, which can be found in Ben Makhlouf and Hammami 26 …”

h-stability and boundedness results for solutions to certain nonlinear perturbed systems

“…which always have negative real parts, hence the system is globally uniformly h-stable. Moreover, 2 , which is non-negative continuous and integrable on R + . Therefore, from Corollary 3.3, one can conclude that the system (3.6) is globally uniformly h-stable.…”

“…This theory has been developed very intensively and several works are published, see [4][5][6]. However, when the origin is not necessarily an equilibrium point, the desired system may be unstable and yet the system may oscillate sufficiently near this state so that its performance is acceptable and still possible to analyze the asymptotic of solutions with respect to a small neighborhood of the origin, which yields to the concept of practical stability, see [1,2,10].…”

A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality

“…The usual property of the solutions that can be deduced for such systems is ultimate boundedness. That means that the solutions remain in some neighborhoods of the origin after a sufficiently large time, (see [3], [10]- [13]). In different cases, the linearized system is independent of the control.…”