31
Except for the one-dimensional case, the solution of the corresponding problem no longer
belongs to the space L2(0, T, Hl(n)) so the theory given in section 3.1 does not apply.
However similar results to the previous ones can be obtained.
When discharges take place at points rather than in subsets of non zero measure, it
is reasonable to try to optimize the system with respect to the position of these points
too. This should be the case when a wastewater system is going to be built. In the
next section we consider a simplified version of this problem. Further results concerning
pointwise control problems can be found in Saguez[1974], Lions[1979] and Simon[1982].
3.5 Optimal placement of an outfall
3.5.1 Statement of the problem and theoretical results
Optimal placement of an outfall can be formulated as an optimal control problem where
the control is the position of the outlet, the cost function is the distance to the wastewater
treatment plant and the state equations are those modelling pollutant concentration. For
the sake of simplicity we consider here a model for a passive biological pollutant as, for
instance, phaecal coliphorms. Assuming a first order kinetics for coliphorms decease, we
have the following state equation
8c
at + u.Vc - f3t!.c+ ac
8c
an
c(z, 0) =
m(t)o(z - b) in Q=n x (0, T)
(3.73)
° on r x (O,T)
(3.74)
° in n
(3.75)
We suppose velocity u is given in (L""(Q))N (for example, it could be computed from
the shallow water model) and m is a given function of time representing the flow rate of
phaecal coliphorms in discharge.
For a given discharge point b, this system has unique solution to be denoted by q"
which can be defined by transposition techniques (see Lions and Magenes [1968]). Let us
consider the following problem (P):
To find c E L2(Q) such that
k crpdzdt = loT tfJ(b,t)v(t)dt, \fVJ E L2(Q),
(3.76)
where tfJ is the unique solution of the initial-boundary value problem
8tfJ
- 8t - V.(utfJ) - f3t!.tfJ + atfJ =,p, in Q,
8tfJ
8n - u.ntfJ = 0, on~,
tfJ(z, T) = ° on n.
The existence of solution for this problem is guaranteed by the following
(3.77)
(3.78)
(3.79)
Except for the one-dimensional case, the solution of the corresponding problem no longer
belongs to the space L2(0, T, Hl(n)) so the theory given in section 3.1 does not apply.
However similar results to the previous ones can be obtained.
When discharges take place at points rather than in subsets of non zero measure, it
is reasonable to try to optimize the system with respect to the position of these points
too. This should be the case when a wastewater system is going to be built. In the
next section we consider a simplified version of this problem. Further results concerning
pointwise control problems can be found in Saguez[1974], Lions[1979] and Simon[1982].
3.5 Optimal placement of an outfall
3.5.1 Statement of the problem and theoretical results
Optimal placement of an outfall can be formulated as an optimal control problem where
the control is the position of the outlet, the cost function is the distance to the wastewater
treatment plant and the state equations are those modelling pollutant concentration. For
the sake of simplicity we consider here a model for a passive biological pollutant as, for
instance, phaecal coliphorms. Assuming a first order kinetics for coliphorms decease, we
have the following state equation
8c
at + u.Vc - f3t!.c+ ac
8c
an
c(z, 0) =
m(t)o(z - b) in Q=n x (0, T)
(3.73)
° on r x (O,T)
(3.74)
° in n
(3.75)
We suppose velocity u is given in (L""(Q))N (for example, it could be computed from
the shallow water model) and m is a given function of time representing the flow rate of
phaecal coliphorms in discharge.
For a given discharge point b, this system has unique solution to be denoted by q"
which can be defined by transposition techniques (see Lions and Magenes [1968]). Let us
consider the following problem (P):
To find c E L2(Q) such that
k crpdzdt = loT tfJ(b,t)v(t)dt, \fVJ E L2(Q),
(3.76)
where tfJ is the unique solution of the initial-boundary value problem
8tfJ
- 8t - V.(utfJ) - f3t!.tfJ + atfJ =,p, in Q,
8tfJ
8n - u.ntfJ = 0, on~,
tfJ(z, T) = ° on n.
The existence of solution for this problem is guaranteed by the following
(3.77)
(3.78)
(3.79)