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2.7 Lyapunov Stability Theorems 69

n
2. Decrescent: We say that V is decrescent if N=ℜ .

EXAMPLE 2.7–3: Decrescent Functions
[Vidyasagar 1992] The function is locally but
not globally decrescent. On the other hand, is globally
decrescent.

DEFINITION 2.7–4 Given a continuously differentiate function V: ℜ ×ℜ →
+
n
R together with a system of differential equations (2.7.1), the derivative of V
+
n
along (2.7.1) is defined as a function V: ℜ ×ℜ →R given by

EXAMPLE 2.7–4: Lyapunov Functions
Consider the function of Example 2.7.3 and assume
given a system

Then, the derivative of V(t, x) along this system is

Lyapunov Theorems

We are now ready to state Lyapunov Theorems, which we group in Theorem
2.7.1. For the proof, see [Khalil 2001], [Vidyasagar 1992].

THEOREM 2.7–1: Lyapunov

Given the nonlinear system

Copyright © 2004 by Marcel Dekker, Inc.