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 v 1 ( t ) , … , v k ( t ) . We note that the class of Lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations.