Optimal Control of Partial Differential Equations

34 CHAPTER 3. CONTROL OF SEMILINEAR ELLIPTIC EQUATIONS
Theorem 3.4.4 Let ū and u be in Uad and let λ > 0. Set wλ = 1 (z(f, ū+λ(u−ū))−z(f, ū)).
λ
Then (wλ)λ tends to w for the weak topology of H1 (Ω), where w is the solution to equation
Aw = 0 in Ω,
∂w
+ ψ
∂nA
′ (z(f, ū))w = u − ū on Γ. (3.4.9)
Proof. Set zλ = z(f, ū + λ(u − ū)) and ¯z = z(f, ū). Writing the equation satisfied by zλ − z,
we can easily prove that zλ tends to ¯z, as λ tends to zero. Now we write the equation satisfied
by wλ = 1
λ (zλ − ¯z):
∂w
Aw = 0 in Ω, + bλw = u − ū on Γ,
∂nA
with bλ = 1
0 ψ′ (¯z + θ(zλ − ¯z))dθ. Since zλ tends to ¯z, bλ tends to ψ ′ (¯z) as λ tends to zero.
Thus we can apply Theorem 3.4.3 to complete the proof.
Theorem 3.4.5 If (¯z, ū) is a solution to (P1) then
(βs|ū| s−2 ū + p)(u − ū) ≥ 0 for all u ∈ Uad,
where p is the solution to equation
Γ
A ∗ p = ¯z − zd in Ω,
∂p
∂nA ∗
+ ψ ′ (¯z)p = 0 on Γ. (3.4.10)
(A ∗ is the formal adjoint of A, that is A ∗ p = −Σ N i,j=1∂j(aij∂ip)+a0p, and ∂p
∂n A ∗ = ΣN i,j=1(aij∂ipnj).)
Proof. With the notation used in the proof of Theorem 3.4.4, for all λ > 0, we have
0 ≤ J1(zλ,
uλ) − J1(¯z, ū) 1
=
λ
2 (zλ + ¯z − 2zd)wλ + β
(|uλ|
λ
s − |u| s ),
Ω
with uλ = ū + λ(u − ū). From the convexity of the mapping u ↦→ |u| s it follows that
(|uλ| s − |u| s
) ≤ s|uλ| s−2 uλ λ(u − ū).
Γ
Thus we have
1
0 ≤
Ω 2 (zλ
+ ¯z − 2zd)wλ + βs|uλ|
Γ
s−2 uλ (u − ū).
Passing to the limit when λ tends to zero, it yields
0 ≤ (¯z − zd)w + βs|ū| s−2 ū(u − ū),
Ω
Γ
where w is the solution to equation (3.4.9). Now using a Green formula between p and w we
obtain
(¯z − zd)w = A
Ω
Ω
∗
∂w ∂p
pw = p −
Γ ∂nA Γ ∂nA∗
w = p(u − ū).
Γ
This completes the proof.
Γ
Γ