  In this paper we deal with infinite-dimensional nonlinear forward complete
dynamical systems which are subject to external disturbances. We first extend
the well-known Datko lemma to the framework of the considered class of systems.
Thanks to this generalization, we provide characterizations of the uniform
(with respect to disturbances) local, semi-global, and global exponential
stability, through the existence of coercive and non-coercive Lyapunov
functionals. The importance of the obtained results is underlined through some
applications concerning 1) exponential stability of nonlinear retarded systems
with piecewise constant delays, 2) exponential stability preservation under
sampling for semilinear control switching systems, and 3) the link between
input-to-state stability and exponential stability of semilinear switching
systems.
