Stability of nonautonomous differential equations in Hilbert spaces

We introduce a large class of nonautonomous linear differential equations upsilon'=A(t)upsilon in Hilbert spaces, for which the asymptotic stability of the zero solution, with all Lyapunov exponents of the linear equation negative, persists in upsilon'=A(t)upsilon + f(t, upsilon) under sufficiently small perturbations f This class of equations, which we call Lyapunov regular, is introduced here inspired in the classical regularity theory of Lyapunov developed for finite-dimensional spaces, that is nowadays apparently overlooked in the theory of differential equations. Our study is based on a detailed analysis of the Lyapunov exponents. Essentially, the equation upsilon' = A(t)upsilon is Lyapunov regular if for every k the limit of Gamma(t)(1/t) as t -> infinity exists, where Gamma(t) is any k-volume defined by solutions upsilon(1)(t), ... , upsilon(k)(t). We note that the class of Lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations. (c) 2005 Elsevier Inc. All rights reserved.