Partial Differential Equations - Modelling and ... - ResearchGate

242 P. Neittaanmäki and D. Tiba
Theorem 2. The gradient of the cost functional (19) with respect to u ∈
L 2 (D) is given by
∇J(u ε )=u ε +(1− H(g))p ε in D, (26)
where p ε ∈ L 2 (D) is the unique solution of the adjoint equation
⎡
⎤
∫
d∑
( )
∂ ∂z
p ε
⎣− a ij + a 0 z⎦ dx = 1 ∫
F (y ε )F (z) dσ
∂x j ∂x i ε
D
i,j=1
[g≡0]
∀z ∈ H 2 (D),
∂z
=0on ∂D,
∂n A
(27)
in the sense of transpositions.
Proof. We discuss first the existence of the unique transposition solution
to (27).
The equation in variations corresponding to (20), (21) is
d∑
( )
∂ ∂z
− a ij + a 0 z =(1− H(g))v
∂x
i,j=1 j ∂x i
in D, (28)
∂z
=0
∂n A
on∂D, (29)
for any v ∈ L 2 (D). By regularity theory for differential equations, the unique
solution of (28), (29) satisfies z ∈ H 2 (D).
We perturb this equation by adding δv, δ > 0, in the right-hand side
and we denote by z δ the corresponding solution, z δ ∈ H 2 (D). The mapping
v → z δ , as constructed above, is an isomorphism T δ : L 2 (D) → W = {z ∈
H 2 ∂z
(D) |
∂n A
=0on∂D}.
We define the linear continuous functional on L 2 (D) by
v −→ 1 ∫
F (y ε )F (T δ v) dσ
ε [g≡0]
∀v ∈ L 2 (D). (30)
The Riesz representation theorem applied to (30) ensures the existence of a
unique ˜p δ ∈ L 2 (D) such that
∫
˜p δ v = 1 ∫
ε
F (y ε )F (T δ v) dσ ∀v ∈ L 2 (D). (31)
D
[g≡0]
Choosing v = T −1
δ
z, z ∈ W arbitrary, the relation (31) gives
⎛
⎞
∫
d∑
( )
˜p δ (1 − H(g)+δ) −1 ∂ ∂z
⎝− a ij + a 0 z⎠ dx
∂x j ∂x i
D
i,j=1
= 1 ε
∫
[g≡0]
F (y ε )F (z) dσ ∀z ∈ W. (32)