Optimal Control of Partial Differential Equations

2.2. NEUMANN BOUNDARY CONTROL 15
Theorem 2.2.2 If (¯z, ū) is the solution to (P1) then ū = − 1
β p|Γ, where p is the solution to
equation (2.2.4).
Conversely, if a pair (˜z, ˜p) ∈ H 1 (Ω) × H 1 (Ω) obeys the system
−∆˜z + ˜z = f in Ω,
−∆˜p + ˜p = ˜z − zd in Ω,
∂˜z
∂n
∂ ˜p
∂n
then the pair (˜z, − 1 ˜p) is the optimal solution to problem (P1).
β
1 = − ˜p on Γ,
β
= 0 on Γ,
(2.2.5)
Proof. The first part of the theorem is already proved. Suppose that (˜z, ˜p) ∈ H 1 (Ω) × H 1 (Ω)
obeys the system (2.2.5). Set ũ = − 1
β ˜p. For every u ∈ L2 (Γ), we have
F1(ũ + u) − F1(ũ) = 1
2
= 1
(zu − ˜z)
2 Ω
2 + β
2
Γ
(zu − ˜z)(zu + ˜z − 2zd) +
Ω
β
2
Γ
u 2
+ (zu − ˜z)(˜z − zd) + β
Ω
Γ
(2uũ + u 2 )
with zu = z(f, ũ + u). From the equation satisfied by ˜p and a Green formula it follows that
(zu − ˜z)(˜z − zd) = (zu − ˜z)(−∆˜p + ˜p)
Ω
Ω
=
∂zu ∂˜z
− ˜p = −β
∂n ∂n
uũ.
We finally obtain
F1(ũ + u) − F1(ũ) = 1
(zu − ˜z)
2 Ω
2 + β
u
2 Γ
2 ≥ 0.
Thus (˜z, − 1 ˜p) is the optimal solution to problem (P1).
β
We give another proof of the second part of Theorem 2.2.2 by using a general result stated
below.
Theorem 2.2.3 Let F be a differentiable mapping from a Banach space U into R. Suppose
that F is convex.
(i) If ū ∈ U and F ′ (ū) = 0, then F (ū) ≤ F (u) for all u ∈ U.
(ii) If Uad is a closed convex subset in U, F is convex, ū ∈ Uad and if F ′ (ū)(u − ū) ≥ 0 for all
u ∈ Uad, then F (ū) ≤ F (u) for all u ∈ Uad.
Proof. The theorem follows from the convexity inequality
Γ
Γ
uũ,
F (u) − F (v) ≥ F ′ (v)(u − v) for all u ∈ U, and all v ∈ U.
The second statement of the theorem will be used for problems with control constraints.