Limitations and applications in a fractional Barbalat's Lemma

Barbalat's Lemma is a mathematical result that can lead to the solution of many asymptotic stability problems. On the other hand, Fractional Calculus has been widely used in mathematical modeling, mainly due to its potential to make explicit the dependence of previous stages through nonlocal operators. In this work, we present a fractional Barbalat's Lemma and its proof, as proposed in [31]. The proof is analyzed in order to show an imprecision. In fact, for orders $$ , we are not able to get the supreme limit of the integrand. Then, a counterexample and a corrected version of the lemma are presented, according to [9]. The objective of this work is to draw attention to the potential and limitations of a fractional Barbalat's Lemma, given its wide use in recent articles. In a fractional SIR model, we exhibit the constraint of the result by introducing a non-periodic relapse. So, the supreme limit could not be verified. Also in this context, we provide a general discussion of the classical Calculus' properties that are not inherited if we change the integer orders to fractional ones.