Stability of nonautonomous differential equations in Hilbert spaces

We introduce a large class of nonautonomous linear differential equations v′=A(t)v in Hilbert spaces, for which the asymptotic stability of the zero solution, with all Lyapunov exponents of the linear equation negative, persists in v′=A(t)v+f(t,v) 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 v′=A(t)v is Lyapunov regular if for every k the limit of Γ(t)1/t as t→∞ exists, where Γ(t) is any k-volume defined by solutions v1(t),…,vk(t). We note that the class of Lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations.