Lyapunov conditions for uniform asymptotic output stability and a relaxation of Barb?lat’s lemma

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