27
which contradicts (3.10). Then A cannot be 0 which finishes the proof.
In what follows we apply the previous results to obtain a system of first order optimality conditions for the optimal control problem. Suppose that for each JL E W' there
exists a solution p, in some Banach space Z and in a sense to be precised, for the following
backward in time evolution equation:
Then we have the following
dp A"
--+ p
dt
p(T)
JL+AW'(y)
O.
(3.26)
(3.27)
Proposition 3 Let ' 1. £ E Uad be an optimal control and y the corresponding state. Then
there ezist a nonnegative real number A, p. E W' and a function p E Z such that the
following equations hold:
A+ \I p.11> 0,
dy
dt + Ay = f + Bu
y(O) = Yo.
- : +A"P=P.+AW'(y)
peT) = o.
y E C, (I',W - y) $ 0, Vw E C,
(A~'( ' 1.£) + B"p, v - ' 1. £) ~ 0, "Iv E U"d'
(3.28)
(3.29)
(3.30)
(3.31 )
(3.32)
(3.33)
(3.34)
Sketch of the proof.- We apply the proposition 2 for the choice G(v) = y(v). Then we
have
A[(~'(U),V - ' 1. £) + (w'(y(u)),DG(u)(v - '1.£))]+ (I', DG(u)(v-u)) ~ 0, "Iv E U"d' (3.35)
Let p be defined by (3.31). Then
(AW'(y(U)) + 1', DG(u)(v - ' 1.£)) = (-: + A'p, y(v) - y(u)) = (p, B(v - ' 1.£)) =
(B'p, v - ' 1. £) (3.36)
which imply (3.34).
3.4 Optimal management of a wastewater treatment system
3.4.1 Statement of the problem and theoretical results
Suppose we have a wastewater treatment system including M plants toghether with their
respective outfalls discharging in some areas Pi, j = 1 ... M, included in il, the region