Stability of nonautonomous equations and Lyapunov functions

Abstract

We consider nonautonomous linear equations $x'=A(t)x$ in a Banach space, and we give a complete characterization of those admitting nonuniform exponential contractions in terms of strict Lyapunov functions. The uniform contractions are a very particular case of nonuniform exponential contractions. In addition, we establish ``inverse theorems'' that give explicitly a strict Lyapunov function for each nonuniform contraction. These functions are constructed in terms of Lyapunov norms, which transform the nonuniform behavior of the contraction into a uniform exponential behavior. Moreover, we use the characterization of nonuniformexponential contractions in terms of strict Lyapunov functions to establish in a very simple manner, in comparison with former works,the persistence of the asymptotic stability under sufficiently smalllinear and nonlinear perturbations.