Chemistry Reference
In-Depth Information
Appendix B
Local Linearization
Consider a time-invariant nonlinear dynamical system
x
.t
C
1/
D g
.
x
.t//
(B.1)
in discrete time or
x
.t/
D g
.
x
.t//;
(B.2)
n
, with
n
n
. A vector
in continuous time where
g W D 7! R
D R
being a set in
R
x 2 D
is an equilibrium of system (B.1) if and only if
x D g
.
x
/, and it is an equi-
librium of system (B.2) if and only if
g
.
x
/
D
0
.Let
x
2 D
be an arbitrary point. If
x
is interior, then we assume that
g
is differentiable at
x
,andif
x
is on the boundary,
then we assume that
g
can be extended outside
to an open neighborhood of
x
,and
this extension is differentiable at
x
. The Jacobian of
g
at the point
x
is defined as the
matrix
D
0
1
@g
1
@x
1
.
x
/:::
@g
1
@x
n
.
x
/
@
A
:
:
:
J
.
x
/
D
:
@g
n
@x
1
.
x
/:::
@g
n
@x
n
.
x
/
In this Appendix the relation between the local and global asymptotic stability of
the equilibrium and some properties of the Jacobian will be summarized. Some of
the conditions will be based on the computation of matrix norms. Let
n
denote
the set of all n
n real matrices. A real valued function
A
7! k
A
k
, defined for all
A
2 R
n
R
n
n
, is a matrix norm if it satisfies the following conditions:
nn
; and
k
A
kD
0 if and only if
A
is the zero matrix with
all elements being equal to zero
1.
k
A
k
0 for all
A
2 R
nn
2.
k
˛
A
kDj
˛
jk
A
k
for all
A
2 R
and all real numbers ˛
nn
:
3.
k
A
C
B
kk
A
kCk
B
k
for all
A
;
B
2 R
Similarly to the case of vector norms it is easy to prove that matrix norms are also
continuous matrix functions.
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