Stability problems in nonautonomous linear differential equations in infinite dimensions

In this paper we study the robustness of the stability in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel [3]. Based in Rodrigues [11] and in Kloeden & Rodrigues [10] we introduce a class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.As counterparts, e show first in another example that it is possible to stabilize an unstable system by using a perturbation with a large period and a small mean value, and finally we give an example where we stabilize an unstable linear ODE with a small perturbation in infinite dimensions by using some ideas developed in Rodrigues & Solà-Morales [21] after an example due to Kakutani (see [13]).