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Fig. 2.6
Global stability and global asymptotic stability
2.3.2.2
Lyapunov Stability Approach
The Lyapunov method analyzes the stability of a dynamical system without the need
to compute explicitly the trajectories of the state vector x D Œx
1
;x
2
; ;x
n
T
.
Theorem.
The system described by the relation
x D f.x/
is asymptotically stable
in the vicinity of the equilibrium
x
0
D 0
if there is a function
V.x/
such that
(i)
V.x/
to be continuous and to have a continuous first order derivative at
x
0
(ii)
V.x/>0
if
x¤0
and
V.0/D 0
(iii)
V.x/<0
,
8x¤0
.
The Lyapunov function is usually chosen to be a quadratic (and thus positive)
energy function of the system; however, there in no systematic method to define it.
Assume now, that x D f.x/and x
0
D 0 is the equilibrium. Then the system is
globally asymptotically stable if for every >0, 9ı./ > 0, such that if jjx.0/jj <ı
then jjx.t/jj <, 8t0.
This means that if the state vector of the system starts in a disc of radius ı then
as time advances it will remain in the same disc, as shown in Fig.
2.6
. Moreover, if
lim
t!1
jjx.t/jj D x
0
D 0, then the system is globally asymptotically stable.
Example 1.
Consider the system
x
1
D x
2
x
2
Dx
1
x
3
(2.41)
The following Lyapunov function is defined
V.x/D x
1
C x
2
(2.42)
