32
Proposition 5 Ifu. E (Loo(Q))N, m E L 2 (0,T) andeo E L2(O) then there erists au.niqu.e
solution c E L 2 (Q) for problem (P).
Proof.- Let F be the mapping from L2(Q) into III defined by
F(t/J) = loT tf>(b,t)v(t)dt,
(3.80)
tf> being the solution of (3.77)-(3.79). Then F is well defined because tf> E L2(0, Tj H2(O))
and, for N ::; 3, H2(O) C C(O).
Moreover, from the Sobolev imbedding theorem we have
(3.81)
and then F is a bounded linear operator.
Now the result follows from the Riesz theorem: there must exist a unique c E L 2 (Q)
such that F(t/J) = (C,t/J)L'(q), which completes the proof.
Next we give a regularity result for the solution of problem (P) which will be essential
to get first order optimality conditions for the optimal control problem.
Proposition 6 AS8'Ume b ¢ A and eo E C(A). Then c E C(A x [0, Tn
The optimum design problem is to choose the discharge point P in such a way that
the length of the outfall be minimized, while respecting limitations for phaecal coliphorms
concentration in some areas. More precisely, suppose a is the point where the wastewater
treatment plant has been already placed, then the cost function is taken to be
(3.82)
while the state constraints are
Cf.(:r:,t)::; (J' in A x [O,T]
(3.83)
where A C 0 represents the region to be protected and (J' is the maximum level of phaecal
coliphorms concentration permitted in this region.
Finally, let UDAl C 0 \ II be a compact convex set representing all admisible points to
place the outlet of the outfall. Then the problem is to determine b E U"rl solution of the
following state constrained optimal control problem
min
J(b).
{LEUn • c.~.,}
(3.84)
3.5.2 Existence of solution
The following existence result can be proved