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68 Introduction to Control Theory

EXAMPLE 2.7–1: Class K Functions

2
2
The function α(x)=x is a class K function. The function α(x)=x +1 is not a
2
class K function because (1) fails. On the other hand, α(x)=-x is not a class
K function because (2) and (3) fail.

+
DEFINITION 2.7–2 In the following, ℜ =[0,∞).
+
n
1. Locally Positive Definite: A continuous function V:ℜ ×ℜ →R is locally
positive definite (l.p.d) if there exists a class K function a(.) and a
n
neighborhood N of the origin of ℜ such that
V(t, x) α(||x||)
for all t 0, and all x∈N.

2. Positive Definite: The function V is said to be positive definite (p.d) if
n
N=ℜ .
3. Negative and Local Negative Definite: We say that V is (locally) negative
definite (n.d) if -V is (locally) positive definite.

EXAMPLE 2.7–2: Locally Positive Definite Functions
[Vidyasagar 1992] The function is l.p.d but not
p.d, since V(t, x)=0 at x=(0, π/2). On the other hand,
is not even l.p.d because V(t, x)→0 as t→∞ for any x. The function
is p.d.

DEFINITION 2.7–3

In the following,

1. Locally Decrescent: A continuous function is locally
decrescent if There exists a class K function ß(.) and a neighborhood N
of the origin of such that

V(t, x) ß(||x||)

for all t 0 and all x∈Ν.

Copyright © 2004 by Marcel Dekker, Inc.