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70 Introduction to Control Theory

with an equilibrium point at the origin, i.e. f(t, 0)=0, and let N be a
neighborhood of the origin of size i. e.

Then

1. Stability: The origin is stable in the sense of Lyapunov, if for x∈Ν, there
exists a scalar function V(t, x) with continuous partial derivative such
that

(a) V(t, x) is positive definite
(b) V is negative semi-definite

2. Uniform Stability: The origin is uniformly stable if in addition to (a) and
(b) V(t, x) is decrescent for x∈Ν.
3. Asymptotic Stability: The origin is asymptotically stable if V(t, x) satisfies
(a) and is negative definite for x∈Ν.
4. Global Asymptotic Stability: The origin is globally, asymptotically stable
n
if V(t, x) verifies (a), and V(t, x) is negative definite for all x∈ℜ i.e. if
N=ℜ .
n
5. Uniform Asymptotic Stability: The origin is UAS if V(t, x) satisfies (a),
V(t, x) is decrescent, and V(t,x) is negative definite for x∈Ν.
n
6. Global Uniform Asymptotic Stability: The origin is GUAS if N=ℜ , and
if V(t, x) satisfies (a), V(t,x) is decrescent, V(t,x) is negative definite and
V(t, x) is radially unbounded, i.e. if it goes to infinity uniformly in time
as ||x||→∞.
7. Exponential Stability: The origin is exponentially stable if there exists
positive constants α, ß, γ such that

8. Global Exponential Stability: The origin is globally exponential stable if
n
it is exponentially stable for all x∈ℜ .

The function V(t, x) in the theorem is called a Lyapunov function. Note that
the theorem provides sufficient conditions for the stability of the origin and
that the inability to provide a Lyapunov function candidate has no indication
on the stability of the origin for a particular system.

Copyright © 2004 by Marcel Dekker, Inc.