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2.7 Lyapunov Stability Theorems 73

With the above simplification, the Lyapunov results hold except that no
distinction is made between uniform and regular stability results.

EXAMPLE 2.7–6: Uniform Stability via Lyapunov Functions
Consider again the damped pendulum described by

and obtain a state-space by choosing x 1=θ and then a Lyapunov
function candidate is

so that and the origin is SL and actually USL.

EXAMPLE 2.7–7: Uniform Stability via Lyapunov Functions

This example illustrates the local asymptotic, uniform stability of the origin
for the system

(1)

Choose then

which is strictly less than zero for all

Sometimes, and although V(x) is only non-positive, LaSalle’s theorem
[LaSale and Lefschetz 1961], [Khalil 2001] may be used to guarantee the
global asymptotic stability of the equilibrium point as described in the
next theorem.

Copyright © 2004 by Marcel Dekker, Inc.