Partial Differential Equations - Modelling and ... - ResearchGate

238 P. Neittaanmäki and D. Tiba
Proof. The existence and the uniqueness of the optimal pair [u ε ,y ε ] ∈
L 2 (Ω 0 ) × H 1 (Ω 0 ) of the control problem (8), (9) is obvious. The pair [0,0]
is clearly admissible and, for any ε>0, we obtain
1
2 |y ε − w| 2 H 1/2 (Γ ) + ε 2 |u ε| 2 L 2 (Ω 0) ≤ 1 2 |w|2 H 1/2 (Γ ) .
Therefore, {y ε } and {ε 1/2 u ε } are bounded respectively in H 1/2 (Γ ), L 2 (Ω 0 ).
We denote by l ∈ H 1/2 (Γ ) the weak limit (on a subsequence) of {y ε − w}.
Let us define the adjoint system by:
⎡
⎤
∫ d∑
∫
⎣
∂z ∂p ε
a ij + a 0 zp ε
⎦ dx = (y ε − w)zdσ ∀z ∈ H 1 (Ω 0 ), (11)
Ω 0
∂x i ∂x j Γ
i,j=1
which is a non-homogeneous Neumann problem and p ε ∈ H 1 (Ω 0 ). We also
introduce the equation in variations
⎡
⎤
∫ d∑
∫
⎣
∂µ ∂z
a ij + a 0 µz⎦ dx = νz dx ∀z ∈ H 1 (Ω 0 ), (12)
Ω 0
∂x i ∂x j Ω 0
i,j=1
which defines the variations y ε + λµ, u ε + λν for any ν ∈ L 2 (Ω 0 )andλ ∈ R.
A standard computation using (11), (12) and the optimality of [u ε ,y ε ]
gives
0=ε(u ε ,ν) L 2 (Ω 0) +(y ε − w, µ) H 1/2 (Γ )
⎡
⎤
∫ d∑
= ε(u ε ,ν) L2 (Ω 0) + ⎣
∂µ ∂p ε
a ij + a 0 µp ε
⎦ dx
Ω 0
∂x
i,j=1 i ∂x j
= ε(u ε ,ν) L 2 (Ω 0) +(p ε ,ν) L 2 (Ω 0). (13)
Due to the convergence properties of the right-hand side in (11), {p ε } is
bounded in H 1 (Ω 0 ) and we can pass to the limit (on a subsequence) p ε → p
weakly in H 1 (Ω 0 ), to obtain
⎤
∫ d∑
∫
⎣
∂z ∂p
a ij + a 0 zp⎦ dx = lz dσ ∀z ∈ H 1 (Ω 0 ). (14)
∂x i ∂x j
Ω 0
⎡
i,j=1
The passage to the limit in (13), as {ε 1/2 u ε } is bounded, gives that p ≡ 0in
Ω 0 and (14) shows that l =0inΓ .
We have proved (10) in the weak topology of H 1/2 (Γ ). The strong convergence
is a consequence of the Mazur theorem [Yos80] and of a variational
argument.
Γ