Chemistry Reference
In-Depth Information
In order to make the notation as simple as possible, let
X
N
1
N
C
1
D
c
k
:
k
c
l
lD1
The Jacobian of the discrete time system (5.91) has the special structure
I C J
where
0
a
1
1
B
1
"
1
A
a
1
B
2
"
2
a
1
B
N
"
2
N
1/B
C
:::
@
A
.N
C
a
2
:::
a
2
B
1
"
1
B
2
"
2
a
2
B
N
"
2
N
A
1/B
C
.N
C
J D
;
:
:
:
:
:
:
:
:
:
:
:
:
:::a
N
a
N
B
1
"
1
a
N
B
2
"
2
B
N
"
2
N
A
1/B
C
.N
C
(5.95)
T
so it can be written as
D C ab
with
D D
diag
1
;
a D
.a
1
B;:::;a
N
B/
T
;
a
1
A
.N
C
1/B
;:::;1
a
N
A
.N
C
1/B
and
1
"
1
:
;:::;
N
"
N
T
b
D
Therefore the characteristic equation of the Jacobian can be rewritten as
T
I
/
D
det
.
D
I
/
det
.
I C
.
D
I
/
1
T
/
det
.
D C ab
ab
.N
C
1/B
2
3
1
a
k
B
k
"
k
Y
N
X
N
a
k
A
4
1
C
5
D
0;
D
(5.96)
a
k
A
.N C1/B
1
kD1
kD1
where we have used the results of Appendix E. Assume that
X
N
1
N
C
1
k
D
c
l
c
k
0
(5.97)
lD1
for all k, which is satisfied if the marginal costs c
l
are close to each other. By repeat-
ing the proof of Theorem 2.1 and noticing that at the steady state "
k
D
B for all k,
we have the following result.
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