Partial Differential Equations - Modelling and ... - ResearchGate

274 Y. Achdou
∫
B (α)
X
v(x) =− k ∗ (z)log(|z|)
R
∫
B (ψ,κ)
X
v(x) =
R
(
x(e z − 1) ∂v
∂x (x)
+ e z (1 {z>− log( X
x )}v(xe−z ) − v(x))
(
κ(z)
x(e z − 1) ∂v
|z| 1+2α∗ ∂x (x)
+ e z (1 {z>− log( X
x )}v(xe−z ) − v(x))
)
,
)
dz.
One can check that G (σ) (u ∗ ,p), G (α) (u ∗ ,p)and 〈 G (ψ) (u ∗ ,p),κ 〉 are well defined
and do not depend of the particular choice of φ.
We are now ready to state some necessary optimality for the least square
problem (42):
Theorem 2. Let a subsequence (σε ∗ n
,αε ∗ n
,ψε ∗ n
,u ∗ ε n
) of solutions of (42) converge
to (σ ∗ ,α ∗ ,ψ ∗ ,u ∗ ) as in Proposition 8 (we know that (σ ∗ ,α ∗ ,ψ ∗ ,u ∗ ) is
a solution of (41)). We assume that u ∗ (T i ,x i ) >u ◦ (x i ), for all i ∈ I.
There exists a function p ∗ ∈ L 2 ((0,T) × (0,X)) and a Radon measure ξ ∗
such that for all v ∈ Z (Z is defined by (43))
∫ T ∫ X
( )
∂v
0 0 ∂t + A Xv p ∗ + 〈ξ ∗ ,v〉 =2 ∑ ω i (u ∗ (T i ,x i ) − ū i )v((T i ,x i )), (47)
i∈I
and
µ ∗ |p ∗ | =0, (48)
|u ∗ |ξ ∗ =0. (49)
Furthermore, with φ defined above, φp ∗ ∈ L 2 (0,T,V X ), and for all
(σ, α, ψ) ∈H,
(
)
(σ − σ ∗ ) 2(σ ∗ − σ ◦ )+σ ∗ G (σ) (u ∗ ,p ∗ ) ≥ 0, (50)
(
)
(α − α ∗ ) α ∗ − α ◦ + G (α) (u ∗ ,p ∗ ) ≥ 0, (51)
〈
〈DJ ψ (ψ ∗ ),ψ− ψ ∗ 〉 + G (ψ) (u ∗ ,p ∗ ),ψ− ψ ∗〉 ≥ 0. (52)
with G (σ) , G (α) and G (ψ) defined respectively by (44), (45) and (46).
Proof. The proof consists of first finding the optimality conditions for (42),
then passing to the limit as the penalty parameter tends to zero. It is written
in [Ach06]. Optimality conditions for (42) can be obtained in a now classical
way (see, e.g., the pioneering book of O. Pironneau [Pir84], he was among the
first to understand the potentiality of optimal control techniques in relation
with partial differential equations and optimum design).