  It is not hard to prove that a uniformly continuous real function, whose
integral up to infinity exists, vanishes at infinity, and it is probably little
known that this statement runs under the name "Barbalat's Lemma." In fact, the
latter name is frequently used in control theory, where the lemma is used to
obtain Lyapunov-like stability theorems for non-linear and non-autonomous
systems. Barbalat's Lemma is qualitative in the sense that it asserts that a
function has certain properties, here convergence to zero. Such qualitative
statements can typically be proved by "soft analysis", such as indirect proofs.
Indeed, in the original 1959 paper by Barbalat, the lemma was proved by
contradiction and this proof prevails in the control theory textbooks. In this
short note we first give a direct, "hard analyis" proof of the lemma, yielding
quantitative results, i.e. rates of convergence to zero. This proof allows also
for immediate generalizations. Finally, we unify three different versions which
recently appeared and discuss their relation to the original lemma.
