Optimal Control of Partial Differential Equations

14 CHAPTER 2. CONTROL OF ELLIPTIC EQUATIONS
is linear and continuous from L 2 (Ω) × L 2 (Γ) into H 3/2 (Ω).
We want to write optimality conditions for the control problem
(P1) inf{J1(z, u) | (z, u) ∈ H 1 (Ω) × L 2 (Γ), (z, u) satisfies (2.2.1)}.
We suppose that (P1) admits a unique solution (¯z, ū) (this result is established in section 2.6).
We set F1(u) = J1(z(f, u), u), where z(f, u) is the solution to equation (2.2.1). From the
optimality of (¯z, ū), it follows that
1
F1(ū + λu) − F1(ū) ≥ 0
λ
for all λ > 0 and all u ∈ L2 (Γ). By an easy calculation we obtain
F1(ū + λu) − F1(ū) = 1
2
(zλ − ¯z)(zλ + ¯z − 2zd) +
Ω
β
2
(2λuū + λ
Γ
2 u 2 ),
where zλ = z(f, ū + λu). The function wλ = zλ − ¯z is the solution to equation
Due to Theorem 2.2.1 we have
−∆w + w = 0 in Ω,
∂w
∂n
wλ H 1 (Ω) ≤ C|λ|u L 2 (Γ).
= λu on Γ.
Thus the sequence (zλ)λ converges to ¯z in H 1 (Ω) when λ tends to zero. Set 1
λ wλ = wu, the
function wu is the solution to equation
−∆w + w = 0 in Ω,
∂w
= u
∂n
on Γ. (2.2.2)
By passing to the limit when λ tends to zero, we finally obtain:
1
0 ≤ limλ→0 F1(ū + λu) − F1(ū) = F
λ
′
1(ū)u = (¯z − zd)wu + βuū.
Here F ′ 1(ū)u denotes the derivative of F1 at ū in the direction u. It can be easily checked that
F1 is differentiable in L 2 (Γ). Since F ′ 1(ū)u ≥ 0 for every u ∈ L 2 (Γ), we deduce
Ω
F ′ 1(ū)u = 0 for all u ∈ L 2 (Γ). (2.2.3)
In this form the optimality condition (2.2.3) is not usable. For the computation of optimal
controls, we need the expression of F ′ 1(ū). Since F1 is a differentiable mapping from L2 (Γ) into
R, F ′ 1(ū) may be identified with a function of L2 (Γ). Hence, we look for a function π ∈ L2 (Γ)
such that
(¯z − zd)wu = πu for all u ∈ L
Ω
Γ
2 (Γ).
This identity is clearly related to a Green formula. We observe that if p ∈ H1 solution to the equation
(Ω) is the
−∆p + p = ¯z − zd in Ω,
∂p
= 0
∂n
on Γ, (2.2.4)
then we have
(¯z − zd)wu = (−∆p + p)wu = p
Ω
Ω
Γ
∂wu
∂n =
pu.
Γ
This means that F ′ 1(ū) = p|Γ + βū, where p is the solution to equation (2.2.4).
Γ