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One goal of this paper is to study robustness of stability of nonautonomouslinear ordinary differential equations under integrally small perturbations inan infinite dimensional Banach space. Some applications are obtained to thecase of rapid oscillatory perturbations, with arbitrary small periods, showingthat even in this case the stability is robust. These results extend toinfinite dimensions some results given in Coppel (1970). Based in Rodrigues(1970) and in Kloeden and Rodrigues (2011) we introduce a class of functionsthat we call Generalized Almost Periodic Functions that extend the usual almostperiodic functions and are suitable to deal with oscillatory perturbations. Wealso present an infinite dimensional example of the previous results. We showin another example that it is possible to stabilize an unstable system using aperturbation with large period and small mean value. Finally, we give anexample where we stabilize an unstable linear ODE with small perturbation ininfinite dimensions using some ideas developed in Rodrigues andSol\`a-Morales(2019)} and in an example of Kakutani, see Rickard (1960).

Stability problems in non autonomous linear differential equations in infinite dimensions