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2.6 Stability Theory 59

will be equivalent to the stability concepts defined next. On the other hand,
and even though the system (2.6.1) is time-dependent, we would like to have
its stability properties not depend on t 0 since that would later provide us
with a desired degree of robustness. This leads us to define the uniform
stability concepts [Khalil 2001].

DEFINITION 2.6–2 In all parts of this definition, x e is an equilibrium point
at time t 0 .

1. Uniform Stability: x e is uniformly stable (US) over [t 0 , ∞) if δ( , t 0 )in
definition 2.6.1 is independent of t 0 .
2. Uniform Convergence: x e is uniformly convergent (UC) over [t 0 , ∞) if
δ 1 (t 0 ) and T( 1 ,x 0 , t 0 ) of definition 2.6.1 can be chosen independent of
t 0 .
3. Uniform Asymptotic Stability: x e is uniformly, asymptotically stable (UAS)
over [t 0 , ∞), if it is both US and UC.
4. Global Uniform Asymptotic Stability: x e is globally, uniformly,
asymptotically stable (GUAS) if it is US, and UC.
5. Global Exponential Stability: x e is globally exponentially stable (GES) if
there exists α>0, and ß 0 such that for all x 0 ∈ℜ ,
n

Note that GES implies GUAS, and see Figure 2.6.9 for an illustration of
uniform stability concepts.

EXAMPLE 2.6–3: Uniform stability
1. Consider the damped Mathieu equation,

The origin is a US equilibrium point as shown in Figure 2.6.10
2. The scalar system

has an equilibrium point at the origin which is UC.

Copyright © 2004 by Marcel Dekker, Inc.