  Asymptotic output stability (AOS) is an interesting property when addressing
control applications in which not all state variables are requested to converge
to the origin. AOS is often established by invoking classical tools such as
Barbashin-Krasovskii-LaSalle's invariance principle or Barbalat's lemma.
Nevertheless, none of these tools allow to predict whether the output
convergence is uniform on bounded sets of initial conditions, which may lead to
practical issues related to convergence speed and robustness. The contribution
of this paper is twofold. First, we provide a testable sufficient condition
under which this uniform convergence holds. Second, we provide an extension of
Barbalat's lemma, which relaxes the uniform continuity requirement. Both these
results are first stated in a finite-dimensional context and then extended to
infinite-dimensional systems. We provide academic examples to illustrate the
usefulness of these results and show that they can be invoked to establish
uniform AOS for systems under adaptive control.
