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2.10 Useful Theorems and Lemmas 91

LEMMA 2.10–1: Bellman-Gronwall Let x(.), a(.), b(.): [0, 8) → [0, 8), and
T 0. Suppose that for all t [0,T], the following inequality holds

Then for all t [0,T]

If b(t) is further a constant, the following holds

EXAMPLE 2.10–3: Bellman-Gronwall Lemma Example
Consider a LTI system given by

and assume that A is a stable matrix. Then the solution of the state equation
is given by

or

which, when taking the norm of each side becomes

We can then use the Bellman-Gronwall lemma by letting
to obtain

Copyright © 2004 by Marcel Dekker, Inc.