Chemistry Reference
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therefore
.
x
/
D V
.
x
/:
V
(A.7)
The continuity of the function
g
implies that
x
D
i !1
x
.t
k
i
C
1/
D
i !1
g
.
x
.t
k
i
//
lim
lim
i !1
x
.t
k
i
//
D g
.
x
/;
D g
. lim
which contradicts relation (A.7) and the strict monotonicity of the Lyapunov func-
tion.
Theorem A.3.
Assume that a Lyapunov function is defined on the entire state space
D
. Assume also that
V
.
x
.t//
strictly decreases in t unless
x
.t/
Dx
,furthermore
V
.
x
/
!1
as
kxk!1
.Then
x
is globally asymptotically stable.
Proof.
We show again only the discrete time case. Let
x
.0/
2 D
be arbitrary, then
for all t
0,
.
x
.0//. Therefore the sequence
fx
.t/
g
is bounded, and
the proof can continue along the lines of the proof of the previous theorem.
V
.
x
.t//
V
The particular choice of the Lyapunov function depends on the special properties
of the dynamical system being examined. The most popular choice is
2
;
V
.
x
/
Dkx xk
where the Euclidean norm is selected. This function clearly satisfies properties (a)
and (b) of Definition A.4, so only the monotonicity condition has to be established.
In the discrete case we have to prove that
kx
.t
C
1/
xkkx
.t/
xk
for marginal stability and the corresponding strict inequality for asymptotic stabil-
ity. Notice that the additional condition of Theorem A.3 is also satisfied. In the
continuous case we have to show the monotonicity of the function
.
x
.t//
D
.
x
.t/
x
/
T
.
x
.t/
x
/
V
by showing that its derivative is non-positive or negative. It is easy to see that this
derivative can be expressed as
d
dt
V
.
x
.t//
Dx
.t/
T
.
x
.t/
x
/
C
.
x
.t/
x
/
T
x
.t/
D
2.
x
.t/
x
/
T
g
.
x
.t//:
In proving that this expression is non-positive or negative, the particular form of
the function
g
has to be used. Unfortunately this function is not always monotonic,
and even if it is, then the actual proof is different for different cases.
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