Growth conditions for exponential stability of time-varying perturbed systems

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

“…The problem of stability analysis of nonlinear time‐varying systems has attracted the attention of several researchers and has produced a vast body of important results (see , and the references therein). The authors in studied recent results in the stability theory of nonautonomous differential equations under sufficiently small perturbations where they consider a nonuniform exponential behavior of the linear variational equations, given either by the existence of a nonuniform exponential contraction or a nonuniform exponential dichotomy.…”

“…As an important basic tool, inequality technique such as the famous Gronwall inequality is extensively applied in diversity areas including global existence, uniqueness and stability. In the past decades, various inequalities and their generalized forms have been established , . The aim of this paper is to provide new sufficient conditions that ensure the global uniform stability of perturbed system using the solution of a scalar differential equation through a nonlinear inequality.…”

On the practical ψ^γ-exponential asymptotic stability of nonlinear systems of differential equations

“…The condition (2.2) means that the origin x = 0 is not required to be an equilibrium point for the system under consideration. Indeed, this fails in many cases when studying the practical stability (see [8]). Throughout this paper, a solution of system (2.1) through a point (t 0 , x 0 ) ∈ R + × R n will be denoted by such a form as…”

“…Moreover, for R ≥ 0, ψ = I (identity matrix) and γ = 1, the practical ψ γ −stability coincides with a known practical type of stability: a): If R = 0, then the system (2.1) is globally exponentially asymptotically stable (see [10]). b): If R > 0, then the system (2.1) is globally practically exponentially asymptotically stable (see [8]).…”

On Stability and Stabilization of Perturbed Time Scale Systems with Gronwall Inequalities

“…Concerning the continuous type, i.e., T = R, several results related to asymptotic stability using Lyapunov techniques were obtained (see [5,14,15]). At first, one of the methods of the perturbation theory was referred to integral inequalities to quest some type of stability.…”