Chemistry Reference
In-Depth Information
Let
A
bearealn
n matrix and an eigenvalue of
A
.If
kk
is a matrix norm which
is compatible with a vector norm, then
j
jk
A
k
:
This relation is a simple consequence of the eigenvalue equation of matrix
A
,
Av
D
v
;
where
v
ยค
0
is an associated eigenvector to : Then
k
Av
kk
A
kk
v
k
and
k
v
kDj
jk
v
k
;
so
j
jk
v
kk
A
kk
v
k
:
The assertion is obtained by dividing both sides by
k
v
k
>0:
Consider first the discrete time system (B.1).
Theorem B.1.
L
et
x 2 D
be an equilibrium and assume that
J
.
x
/
exists in an op
en
neigh
bo
rhood of
x
,
kJ
.
x
/
k
<1
with some matrix norm and
J
is continuous at
x
.
Then
x
is locally asymptotically stable.
Proof.
Since
J
.
x
/ is continuous at
x
,thereisan">0such that
J
.
x
/ exists for all
x 2 U Dfx jkx xk
<"
g
and
kJ
.
x
/
k
q with some 0
q<1. Then with any
x 2 U
,
Z
1
0
J
.
x C
t.
x x
//.
x x
/dt:
g
.
x
/
x D g
.
x
/
g
.
x
/
D
Therefore
Z
1
kg
.
x
/
xk
kJ
.
x C
t.
x x
//
kkx xk
dt
q
kx xk
:
(B.3)
0
Starting with arbitrary initial state
x
.0/
2 U
, the entire state sequence generated by
(B.1) remains in
U
, furthermore for all t
0,
kx
.t
C
1/
xkDkg
.
x
.t//
xk
q
kx
.t/
xk
:
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