25
3.3 Optimality conditions
In this section we obtain necessary conditions for u to be an optimal control. The main
functional tool is the following existence result of Lagrange multipliers which has been
proved in Bonnans and Casas[1988]. We include the proof for the sake of completeness.
Proposition 2 Let U and W be two Banach spaces, Uad C U and C C W two convex
subsets, C having a nonvoid interior. Let u be a solution of the optimization problem
min
J(v)
{VEU.d: G(u)EC}
(3.7)
where J : U -+ lR and G : U -+ Ware Gateaux differentiable in u. Then there exists a
nonnegative real number A and p, E W' such that the following equations hold:
A+ I I p,1I> 0,
(p"w - G(u)) ~ 0, Vw E C,
(U'(u) + [DG(u)]*p"v - u) ~ 0, Vv E Uad .
Proof.- Let us consider the following two sets:
(3.8)
(3.9)
(3.10)
Au = {(w, 'Y) E W x lR : :3v E Uad / w = G(u) + DG(u)(v - u), ' Y = i(u)(v - u)} (3.11)
and
B = int(C) x (-00,0).
(3.12)
which is nonempty due to the assumption on the interior of C.
By using the linearity of i (u) and DG( u) it is easy to check that Au and B are convex
sets. Furthermore Au n B = 0. Indeed, otherwise there might exist Vo E Uad such that
Wo = G(u) + DG(u)(vo - u) = G(u) + lim -hI [G(u + h(vo - u)) - G(u)] E int(C) (3.13)
h--+O
and
' Yo = i(u)(vo - u) = lim -hI [J(u + h(vo - u)) - J(u)] < O.
(3.14)
h--+O
Therefore, it is possible to take ho E (0,1) such that
Wh = G(u)+~[G(u+h(vo)-u))-G(u)lEint(C), VhE(O,ho), (3.15)
1
h[J(u+h(vo-u))-J(u)] <0, VhE(O,ho),
(3.16)
from which it follows that
G(u + h(vo - u)) = hWh + (1 - h)G(u) E C, and J(u + h(vo - u)) < J(u) (3.17)
for all h E (0, ho), which contradicts the fact that u is a solution of (3.7).