On the Global Uniform Asymptotic Stability of Time-Varying 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.…”

Trajectory convergence from coordinate-wise decrease of general energy functions

“…It can be completely arbitrary provided it satisfies the constraints (1). By contrast, a large proportion of the convergence results based on decrease of energy functions rely on variations or extensions of Lyapunov-Kraskowski-LaSalle Theorems [9], and typically assume that trajectories follow some ordinary differential equation such as ẏ(t) = f (y(t), t) or ẏ(t) = f (y(t)) for an f satisfying some (uniform) continuity conditions [1,3]. For example, LaSalle theorem guarantees (under some conditions) the convergence of ẏ = f (y) to an invariant set, but not to a single point, provided f (x) T ∇V (x) ≤ 0 everywhere [10].…”

converse theorem for practical $h$-stability of time-varying nonlinear systems

“…However, in practice we may only need to stabilize a system into the region of a phase space where the system may oscillate near the state in which the implementation is still acceptable. This concept is called practical stability (see [1,4,7,12]) which is very useful for studying the asymptotic behavior of the system in which the origin is not necessarily an equilibrium point. This work introduces a new notion of practical stability called practical h-stability (see [11]).…”