33
Proposition 7 Assume there is at least a point b in Uad such that the corresponding state
q, satisfy the constraint (9.89). Then there ezists solution for the optimal control problem
(9.84)·
Proo/.- Let {b,.} a minimizing sequence, i. e. such that
lim J(b,,) = inf{J(b) : q, S u, in CA) x [0, Tn
"->00
(3.85)
with Cf.,. S u in CA) x [0, T]. Then {b,,} is bounded so there exists a limit point b and a
subsequence, still denoted by {b,,} such that
(3.86)
In order to check that b is an admisible control, i.e. that crb) S u in A x [0, T] it is enough
to take into account the continuity of the mapping
bE U --+ q, E C(A x [O,T])
(3.87)
3.5.3 Optimality conditions
Since this problem has pointwise state constraints, in order to get optimality conditions
we need to replace the state space L2(Q) by another one in which the set of admissible
states has a nonvoid interior. This is why a regularity result for the solution of the state
equation is required at least in the set A x [0, T].
The next proposition furnishes an optimality system to be satisfied by an optimal
control
Proposition 8 Let b an optimal control. Then there ezist a nonnegative real number >.,
c E L2(Q), P E L'([O,T], wl,q(n)), Vs,q E [1,2) with 2/s + N/q > N + 1 and a measure
I/- E C(A x [0, T])' with >.+ II 1/-11> 0 such that
&
-
at + u.Vc - ,8ll.c + ac = m(t)c5(z - b)
ac = 0
an
c(z,O) = eo(z)
ap
- at - V.(up) - ,8ll.p + ap = I/-IQ
op
an + u.n = I/-Il:n(Ax[o,T])
p(:z:, T) = I/-nxT
in Q,
on E,
onn,
inQ,
on E
onn
(3.88)
(3.89)
(3.90)
(3.91)
(3.92)
(3.93)
c S u, (I/-, w - c) SO, Vw E C(A x [0, T]) with w S u
(3.94)
>.(b - a, b - b) + loT V",p(b, t).(b - b)dt ~ 0, Vb E Uad.
(3.95)