proof of Barbalat’s lemma

We suppose that y(t)↛0 as t. There exists a sequence (tn) in + such that tn as n and |y(tn)|ε for all n. By the uniform continuity of y, there exists a δ>0 such that, for all n and all t,


So, for all t[tn,tn+δ], and for all n we have

|y(t)| = |y(tn)-(y(tn)-y(t))||y(tn)|-|y(tn)-y(t)|



for each n. By the hypothesisMathworldPlanetmathPlanetmath, the improprer Riemann integral 0y(t)𝑑t exists, and thus the left hand side of the inequalityMathworldPlanetmath convergesPlanetmathPlanetmath to 0 as n, yielding a contradictionMathworldPlanetmathPlanetmath.

Title proof of Barbalat’s lemma
Canonical name ProofOfBarbalatsLemma
Date of creation 2013-03-22 15:10:45
Last modified on 2013-03-22 15:10:45
Owner ncrom (8997)
Last modified by ncrom (8997)
Numerical id 7
Author ncrom (8997)
Entry type Proof
Classification msc 26A06