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2.7 Lyapunov Stability Theorems 71

EXAMPLE 2.7–5: Stability via Lyapunov Functions

1. Consider the system described by

and choose a Lyapunov function candidate

Then the origin may be shown to be a stable equilibrium point.

2. Consider the Mathieu equation described in Example 2.6.3. Let the
Lyapunov function candidate be given by

The origin is then shown to be a US equilibrium point.

3. The system given in Example 2.6.3–5, has a GES equilibrium point at
the origin. This may be shown by considering a Lyapunov function
candidate

V(x)=x 2

which leads to

Then,

2
0.5x V(x) 2x 2
and

The above inequalities hold for any x∈ℜ .
n

Copyright © 2004 by Marcel Dekker, Inc.