Environmental Engineering Reference
In-Depth Information
=
Proof
To show this we must prove that
F
F
[
10
]. Let
Q
0
be an element of
F
.
{
Q
k
}
k
=
1
in
F
such that
Then there is a sequence
Q
k
−
Q
0
ₒ
0as
k
ₒ∞
Assume that
Q
0
(
t
)<
0 in some interval
I
ↂ
(
0
,
T
)
of positive measure
|
I
|
>
0.
Then
T
2
2
2
Q
0
dt
Q
k
−
Q
0
=
(
Q
k
−
Q
0
)
dt
≥
I
(
Q
k
−
Q
0
)
dt
≥
=
l
>
0
0
I
Q
k
}
k
=
1
in
H
, and hence,
The last inequality contradicts the convergence of sequence
{
(
,
)
Q
0
is a non-negative function in
.
On the other hand, applying the Schwarz inequality we get
0
T
dt
=
dt
T
T
c
1
−
Q
0
g
1
(
r
1
,
t
)
(
Q
k
−
Q
0
)
g
1
(
r
1
,
t
)
0
0
≤
Q
k
−
Q
0
g
1
(
r
1
,
t
)
ₒ
0as
k
ₒ∞
and therefore
0
Q
0
g
1
(
r
1
,
t
)
dt
=
c
1
, that is
Q
0
∈
F
. The lemma is proved.
does not belong to the feasible
set
F
. Indeed, the constraint (
2.51
) is not satisfied for such function because
c
1
>
Note that the zero function
q
(
t
)
≡
0
,
0
≤
t
≤
T
,
0.
This remark allows us to establish the most important result of this section.
Theorem 2.2
The variational problem
(
2.50
)
-
(
2.52
)
has non-trivial unique solution
in the space H .
Proof
By Lemma
2.1
, the space
H
is a uniformly convex Banach space. Besides,
by Lemmas
2.1
,
2.2
and
2.3
, the feasibility space
F
is a non-empty closed convex
set in
H
. Therefore, due to Theorem
2.1
, there is a unique function
Q
∗
∈
F
that
minimizes the distance between the set
F
and the point
q
≡
0. That is according to
(
2.50
), function
Q
∗
minimizes the objective functional
m
(
Q
)
. Finally, we observe
that
Q
∗
=
F
, and hence, the unique solution of problem (
2.50
)-(
2.52
)
is non-trivial. The theorem is proved.
0 because
q
∈
It is shown in the next section that function
Q
∗
, mentioned in Theorem
2.2
,is
precisely the function (
2.55
).
2.6.1.2 Optimal Discharge Parameters and the Adjoint Functions
The analytical expression for the optimal discharge rate
Q
∗
, namely, the solution
of variational problem (
2.50
)-(
2.52
), can be obtained by means of the method of
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