Uniform asymptotic stability of perturbed systems of neutral differential equations

Abstract

IN THIS work we discuss the effect of a certain type of perturbations on systems for which the zero solution is uniformly asymptotic stable, using the second method of Lyapunov. With this objective, we give a new alternative for the converse theorems for uniform asymptotic stability of Hale and Cruz [l]. We define a Lyapunov’s functional which together with its ‘derivative’ satisfies some inequalities involving the solution of the differential equation of neutral type and not only the uniformly stable operator D(t, .), defined in the sense of Section 1. On the other hand, our Lemma 1, of very simple demonstration, shows that the hypothesis that Ix,(c, 9) x,(5 491 < eL(+14 YI, t 2 0, where L is a constant and xr(g,& is a solution of a neutral functional differential equation, is unnecessary in Theorem 1 of Iti and dos Reis [2] ; and we can obtain the same and new results. We think that these results are important for studying the stability of perturbed systems of neutral differential equations and applications.