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92 Introduction to Control Theory

or finally

The next Lemma may be used in the case of non-autonomous systems and
leads to results similar to LaSalle’s theorem. For a proof see for example
[Khalil 2001]

LEMMA 2.10–2: Barbalat Let f(t) be a differentiate function of t.

• First Version: If is uniformly continuous and
, then
• Second Version: If f(t) 0, and bounded, then

EXAMPLE 2.10–4: Barbalat Lemma Example
-t
1. Consider f(t)=e +1, then dot f(t)=-e which is uniformly continuous. On
-t
-t
the other hand lim t→∞(e +1)=1 therefore .
2. As a second example, consider f(t)=1/(1+t), with t>0. Using the second
version of Barbalat’s lemma, we can show that .

LEMMA 2.10–3: Consider the quadratic equation

where a, b, and c are positive constants. Then P(x)<0 if

for all x 1<x<x 2 where

Copyright © 2004 by Marcel Dekker, Inc.