24
where f is given in 12(0, T; V') and Yo E H.
Let J be the following cost function
J(v) = if>(v) + loT llI(y(v))dt
where if> : U -+ IR and III : H -+ IR are given functions.
(3.4)
Finally, let W be a separable Banach space and C a closed convex subset of W.
Then the optimal control problem consists on finding u E Uad minimizing J(v) under the
constraints
• v E Uad
• y(v)EC.
where Uad is a closed convex subset of U.
3.2 Existence of solution
In this paragrph we prove the following existence result
Proposition 1 Under the assumptions
1. {v EUad, y(v)EC}~0
2. if> and III are lower semicontinuous proper convex functions
3. Either if> is coercive or Uad is bounded
4. C is a closed convex subset of W
5. if {vn } -+ v weakly in U then {y(vn )} -+ y(v) weakly in W
then the optimal control problem has solution.
Proof.- Let {vn } be a minimizing sequence, i.e.
lim J(vn ) = inf{J(v): v E Ur.d, y(v) E C}
n-+oo
(3.5)
Then from assumption 3 {vn } is bounded and hence there exists a subsequence, still
denoted by {vn }, and u E Uad such that
{vn } -+ u weakly in U
and therefore, by using assumption 5
{y(Vn )} -+ y(u) weakly in 12(0, T; V) and in W
As Uad and C are weakly closed (since they are strongly closed and convex) we have
u E Uad and y(u) E C. On the other hand, from assumption 2, J is weakly lower
semi continuous and then
J(u) ~ lim J(vn )
n-+oo
(3.6)
which finishes the proof.