A Generalized Barbalat Lemma Based on a Persistently Exciting Condition

This paper investigates attractivity based on a new persistently exciting (PE) condition. Under an integral condition, the proposed PE condition is shown to be a sufficient condition to guarantee attractivity. Then, it is applied to derive a generalized Barbalat lemma. Based on this result, several generalizations of Barbalat lemma are then proposed. In particular, the standard assumption that requires uniform continuity on the positive real axis can be relaxed by admitting countable discontinuous points. Moreover, the integrand of the assumed integral condition can be a composition of a targeted function and a time-varying function. An interesting example is provided to illustrate the usefulness of the proposed results.