Environmental Engineering Reference
In-Depth Information
zones and the water outflow from region
D
, the concentration of nutrient decreases
in region
D
towards its natural value. Therefore, the following conditions for the
mean concentration of nutrient must be fulfilled since the moment
t
0
>
T
:
1
|
ʩ
i
|
ʩ
i
ˆ(
r
,
t
)
dr
<
c
i
+
ʲ
i
i
=
1
,...,
Nt
≥
t
0
(2.30)
The moment
t
0
can be determined through monitoring the mean concentration of
nutrient in region
D
or by using the solution
ˆ
forecasted by the model (
2.4
)-(
2.11
)
with the forcing
Q
j
(
N
. Once conditions
(
2.30
) are fulfilled, the initial concentration for modelling the next application of
nutrient is chosen as
t
)
equal to zero for
t
>
T
and
j
=
1
,...,
0
˕
(
)
=
ˆ(
,
t
0
)
r
r
(2.31)
and the next time interval for such modelling is
[
t
0
,
t
0
+
T
]
. Due to the conditions
0
(
2.30
), the contribution of the new initial condition
˕
(
r
)
to the mean concentrations
of nutrient during time interval
[
t
0
+
T
−
˄,
t
0
+
T
]
is less than the upper bounds
c
i
+
ʲ
i
in
. Note that without such conditions the feasibility space
for problem (
2.1
)-(
2.3
) is empty and there is no solution to the control problem.
Thus, if the conditions (
2.30
) are fulfilled then one can take
t
0
=
ʩ
i
,
(
i
=
1
,...,
N
)
0 and consider
the problem (
2.27
)-(
2.29
) again for modelling the second discharge of nutrient with
the following positive parameters:
c
i
0
=
c
i
−
g
i
(
r
,
0
)˕
(
r
)
dr
,
i
=
1
,...,
N
.
(2.32)
D
Note that the adjoint functions in (
2.32
) must be calculated in time interval
[
t
0
,
t
0
+
T
. Also we assume, without loss of generality, that negative values, if they appear
on the left side of the constraints (
2.28
), are replaced by zero. With these remarks, the
variational problem (
2.27
)-(
2.29
) represents a general remediation strategy which
can be applied repeatedly.
It is important to note that all the adjoint solutions
g
i
(
]
r
j
,
t
)
which figure in con-
straints (
2.28
) are independent of the discharge rates
Q
j
(
. This non-negative solu-
tions are determined by the dynamical processes in region
D
and serve in constraints
(
2.28
) as the weight functions characterizing the effect of the discharge of nutrient at
a point
r
j
on the mean concentration of nutrient in a zone
t
)
ʩ
i
(see Figs.
2.3
,
2.4
and
2.5
). In other words, the adjoint solutions are the influence functions (or information
functions) in the control theory. That is why the adjoint problem solutions are widely
used in the sensitivity study of various models, and in particular, in the atmosphere
and ocean model, weather forecast and climate theory [
21
,
23
], data assimilation
problems [
24
], problems of identification of unknown pollution sources, like nuclear
accidents [
28
,
34
,
39
,
50
], simulation of oil pollution [
9
,
42
] and optimal control in
pollution problems [
1
,
15
,
16
,
20
,
22
,
30
,
33
,
49
].
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