Geoscience Reference
In-Depth Information
norm in a Banach space. For PDEs,
A
is a differential operator with respect to
x
,
for instance
A.
u
/.t/
D
U
x
.t;x/
. Under a Lipschitz condition for
f.
u
/
and some
assumptions about
A
and its spectrum, there exists a finite dimensional subspace
e
tA
. The subspace is generated by a set of
V
E
that is invariant under
eigenvectors of
A
. Over this subspace is an invariant manifold of (
1.31
), denoted
by
M
, which can be defined as a graph
M
Df
.p;p
C
ˆ.p//
j
p
2
V
g
where
ˆ
is a mapping from
V
to its orthogonal complement in
E
. It can be proved
that (
1.31
) induces an ODE in
V
,
dp
dt
f .p
C
ˆ.p//; p.0/
D
p
0
2
V:
(1.32)
C
Ap
D
f
for some function
derived from the original PDE. The most important property of
M
is that it is
exponentially attractive
, that is, for any solution
u
.t/
of (
1.31
), there
exists an induced trajectory
u
.t/
D
p.t/
C
ˆ.p.t//
2
M
such that
u
exponentially. Thus, the finite dimensional dynam-
ics (
1.32
) determines the long-term behavior of the original system (
1.31
). An
observer designed for (
1.32
) can be used to estimate
u
.t/
approaches
u
.t/
.
For example, let us consider the system of reaction-diffusion equations
.t/
ˇ
ˇ
ˇ
@
D
0;
@
u
@t
@
u
@n
D
u
C
f.
u
;
r
u
/;
(1.33)
R
d
.Here
u
in a bounded domain
D
.
u
1
;:::;
u
m
/
. The function
f.
u
;
w
/
satisfies the global Lipschitz condition:
w
2
/
j
L
p
j
u
v
j
2
Cj
w
1
w
2
j
2
;
j
f.
u
;
w
1
/
f.
v
;
(1.34)
R
m
,
w
1
;
R
md
,and
where
u
. It can be verified that the
system satisfies all the conditions to guarantee the existence of an inertial manifold
(
Chueshov 2002
). In fact, in this example we have
;
v
2
w
2
2
L>0
ˆ.p/
D
0
. The manifold is the
same as the invariant space
V
. The induced ODE in
V
has the following form.
d
u
dt
D
f.
u
;0/;
u
.t/
2
V:
(1.35)
Thus for any PDE solution
u
to (
1.33
), there exists an ODE solution
u
to (
1.35
)such
that
.t/
k
1
Ce
t
;
.t/
u
t
0;
k
u
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