On fractional extensions of Barbalat Lemma

•Barbalat-like lemmas for fractional integral are posited.•Qualitative analysis of asymptotic properties for some types of fractional differential equations is shown to indicate applicability of those lemmas.•Error Model of Type I with fractional derivative is comparatively analyzed.

This paper presents Barbalat-like lemmas for fractional order integrals, which can be used to conclude about the convergence of a function to zero, based on some conditions upon its fractional integral. Some examples in the context of asymptotic behaviour of solutions of fractional order differential equations, indicate the potential application of these lemmas in control theory.