# New results on the uniform exponential stability of non-autonomous perturbed dynamical systems

## TL;DR

This paper establishes conditions under which nonlinear non-autonomous perturbed systems maintain uniform exponential stability, using Gronwall-Bellman inequalities instead of Lyapunov functions, with practical numerical illustrations.

## Contribution

It provides new stability criteria for perturbed systems that do not rely on Lyapunov functions, broadening applicability in complex scenarios.

## Key findings

- Equilibrium remains stable under certain perturbation estimates.
- Stability criteria are derived using Gronwall-Bellman inequalities.
- Numerical examples demonstrate practical applicability.

## Abstract

In this paper, we investigate the asymptotic behaviors of the solutions of nonlinear dynamic systems nearby an equilibrium point, when the nominal parts are subject to non necessarily small perturbations. We show that, under some estimates on the perturbation terms, the equilibrium point remains (globally) uniformly exponentially stable. The results we obtained can easily be applied in practice since they are based on the Gronwall-Bellman inequalities rather than the classical Lyapunov methods that require the knowledge of a Lyapunov function. Several numerical examples are presented in order to illustrate the validity of our study, especially when the standard Lyapunov approaches are useless.