Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9501573 | Journal of Differential Equations | 2005 | 45 Pages |
Abstract
We introduce a large class of nonautonomous linear differential equations vâ²=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 vâ²=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 vâ²=A(t)v is Lyapunov regular if for every k the limit of Î(t)1/t as tââ exists, where Î(t) is any k-volume defined by solutions v1(t),â¦,vk(t). We note that the class of Lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Luis Barreira, Claudia Valls,