Lyapunov functions for trichotomies with growth rates

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:248

Article

Barreira, Luis¹  Valls, Claudia¹ 

[1] Univ Tecn Lisboa, Dept Matemat, Inst Super Tecn, P-1049001 Lisbon, Portugal

关键词: Lyapunov functions;    Nonuniform exponential trichotomies;   

DOI  :  10.1016/j.jde.2009.07.001

来源: Elsevier

【 摘 要 】

We consider linear equations x' = A(t) x that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates e(c rho(t)) determined by a function rho(t). For example, the usual exponential behavior with rho(t) = t is included as a very special case, and when rho(t) = log t we obtain a polynomial behavior. We emphasize that we also consider the general case of nonuniform exponential behavior, which corresponds to the existence of what we call rho-nonuniform exponential trichotomy.This is known to occur in a large class of nonautonomous linear equations. Our main objective is to give a complete characterization in terms of strict Lyapunov functions of the linear equations admitting rho-nonuniform exponential trichotomy. This includes criteria for the existence of rho-nonuniform exponential trichotomy, as well as inverse theorems providing explicit strict Lyapunov functions for each given exponential trichotomy. In the particular case of quadratic Lyapunov functions we show that the existence of strict Lyapunov sequences can be deduced from more algebraic relations between the quadratic forms de. ning the Lyapunov functions. As an application of the characterization of nonuniform exponential trichotomies in terms of strict Lyapunov functions, we establish the robustness of rho-nonuniform exponential trichotomies under sufficiently small linear perturbations. We emphasize that in comparison with former works, our proof of the robustness is much simpler even when rho(t) = t. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free