Environmental Engineering Reference
In-Depth Information
2.6.1.1 Existence and Uniqueness
In this section the existence and uniqueness of solution to variational problem (
2.50
)-
(
2.52
) is proved. To this end, we remind some properties of the Hilbert space
H
=
L
2
(
0
,
T
)
together with a strong result of approximation theory (a minimum distance
theorem).
Theorem 2.1
([
5
])
A non-empty closed convex set in a uniformly convex Banach
space possesses a unique point closest to a given point.
Lemma 2.1
([
5
])
The space H
=
L
2
(
0
,
T
)
is a uniformly convex Banach space.
We point out that the meaning of condition (
2.50
) is the minimization of the norm
(distance) in the space
H
. It is for this reason Theorem
2.1
is useful in proving the
existence and uniqueness. We now consider the specific set and point in space
H
for
which Theorem
2.1
is applied.
Definition 2.1
The feasible space
F
for variation problem (
2.50
)-(
2.53
) is given as
follows
Q
c
1
T
F
=
∈
H
;
Q
(
t
)
≥
0
,
0
≤
t
≤
T
,
and
Q
(
t
)
g
1
(
r
1
,
t
)
dt
=
(2.54)
0
Lemma 2.2
The feasible space F is a non-empty set in space H .
Proof
Because the adjoint solution
g
1
(
r
1
,
t
)
is a non-negative square-integrable func-
tion, we have that
c
1
g
1
(
r
1
,
t
)
Q
∗
(
t
)
=
(2.55)
T
0
g
1
(
r
1
,
t
)
dt
is a function in
H
that fulfils constraints (
2.51
) and (
2.52
). Therefore,
Q
∗
(
t
)
belongs
to the feasible space
F
. The lemma is proved.
The meaning and usefulness of function
Q
∗
(
t
)
defined by (
2.55
) is established in
the next section.
Lemma 2.3
The feasible space F is a convex set in H .
Proof
In fact, let
Q
1
,
Q
2
∈
F
and
ʻ
∈
(
0
,
1
)
. Then, evidently,
ʻ
Q
1
+
(
1
−
ʻ)
Q
2
≥
0
,
0
≤
t
≤
T
. Besides,
T
0
(ʻ
Q
1
+
(
1
−
ʻ)
Q
2
)
g
1
(
r
1
,
t
)
dt
=
ʻ
c
1
+
(
1
−
ʻ)
c
1
=
c
1
and hence,
F
is a convex set in
H
. The lemma is proved.
Lemma 2.4
The feasible space F is a closed set in H .
Search WWH ::
Custom Search