29
By assuming that none of subsets Pj, j = 1, ... , M intersects the area to be protected
A, if u E (L""( Q))N and Pia E 0(0), (i = 1,2) we can prove the continuity of the state of
the system (PI, P2) in the subset A x [0, T] by using the results in Casas[1994].
Then we can take the space W = (C(A X [0,T]))2 to be the Banach space for state
constraints and its convex subset
as the set for admissible states.
Thus the proposition 3 allows us to prove the following result:
Proposition 4 Let u E Uad be an optimal control. Then there ezist A ~ 0, two measures
~i E O(A X [O,T])' (i = 1,2) and two functions Pi E L'(O,T, wI''l(n» (i = 1,2),
V8,q E [1,2) with 2/8 + N/q > N + 1 S'Uch that
,X+ \I ~l \I + II ~211> 0,
~
1M
1
7ft + u.Y' PI - f3tllpI + klPl = h.t; m;(t) , Pj ,XPi in Q,
OPI = ° ~
On
on~,
PI(X,O) = PIO on n,
OP2
1 .
7ft + u.Y' P2 - fM),·P2 + kl P2 = hk2(d. - P2) ln Q
OP2 = ° ~
On
on~,
P2(X,0)=Pao onn
OPI
-7ft - Y'.(UPl) - (3lllpI + kl(PI + P2) = ~IIQ in Q,
OPl
On + u.npI = ~IIE on~,
Pl(x, T) = ~llnxT on n,
OP2
1 .
-7ft - Y'.(Up2) - (32llp2 + hk2P) = ~21Q ln Q
OP2
On + u.nP2 = ~21E on~,
Pa(X,T)=~2InxT onn
(PI, Pa) E C,
(/LI,W - PI) ~ 0, Vw E O(A x [0, T)), w ~ U,
(~2'W - P2) ~ 0, Vw E C(A x [O,T]), w ~ 5,
A loT fj(mj(t»(nj(t) - mj(t»dt + loT (I ;; , k i ~Pldx)(nj(t) - mj(t»dt ~ 0,
Vnj E L2(0, T), ° ~ nj(t) ~ nj, Vj = 1, ... , M.
(3.50)
(3.51)
(3.52)
(3.53)
(3.54)
(3.55)
(3.56)
(3.57)
(3.58)
(3.59)
(3.60)
(3.61)
(3.62)
(3.63)
(3.64)
(3.65)
(3.66)