Environmental Engineering Reference
In-Depth Information
Lagrange multipliers [
44
]. Let
T
c
1
T
1
2
Q
2
L
(
Q
)
=
(
t
)
dt
−
ʻ
g
1
(
r
1
,
t
)
Q
(
t
)
dt
−
(2.56)
0
0
be the Lagrange functional corresponding to problem (
2.50
)-(
2.52
), where
is the
respective Lagrange multiplier. The first variation of
L
in the sense of Gateaux [
44
]
is calculated as
ʻ
T
Q
)
ʴ
∂
∂ʵ
ʴ
L
(
Q
;
ʴ
Q
)
=
L
(
Q
+
ʵʴ
Q
)
ʵ
=
0
=
(
t
)
−
ʻ
g
1
(
r
1
,
t
Qdt
(2.57)
0
Q
is the variation of
Q
. A necessary condition for
Q
∗
where
ʴ
to be a minimum is
Q
∗
;
ʴ
ʴ
L
(
Q
)
=
0, for any
ʴ
Q
[
44
]. Therefore, from Eq. (
2.57
) we get
Q
∗
(
t
)
=
ʻ
g
1
(
r
1
,
t
),
(2.58)
where the Lagrange multiplier
ʻ
is determined by means of the constraint (
2.51
)in
the way
c
1
ʻ
=
dt
.
(2.59)
T
0
g
1
(
r
1
,
t
)
The final result is obtained by substituting Eq. (
2.59
)in(
2.58
).
Note that, due to Schwarz inequality [
17
],
T
T
2
T
0
dt
1
2
g
1
(
0
<
g
1
(
r
1
,
t
)
dt
≤
r
1
,
t
)
0
and therefore
T
0
g
1
(
0, that is function
Q
∗
r
1
,
t
)
dt
>
is well-defined by the
0, we conclude that
Q
∗
(
Eqs. (
2.58
) and (
2.59
). Besides, since
g
1
(
r
1
,
t
)
≥
t
)
≥
0,
0
≤
T
.
We now show that
Q
∗
,
t
≤
defined by (
2.58
) and (
2.59
), also satisfies the sufficient
condition to be a minimum. Indeed, let
Q
0
=
Q
∗
+
ʴ
Q
be a feasible discharge rate.
From constraint (
2.51
)wehave
T
g
1
(
r
1
,
t
)ʴ
Qdt
=
0
,
(2.60)
0
0 is an arbitrary variation of
Q
∗
. Then,
where
ʴ
Q
=
T
T
1
2
Q
∗
)
=
Q
∗
(
2
Qdt
m
(
Q
0
)
−
m
(
t
)ʴ
Qdt
+
ʴ
.
(2.61)
0
0
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