Optimal Control of Partial Differential Equations

38 CHAPTER 3. CONTROL OF SEMILINEAR ELLIPTIC EQUATIONS
where fi,ε = (¯z(xi)−zd(xi))
|B(xi,ε)|
χB(xi,ε). We can easily verify that w and pε satisfy the Green formula
Σ k i=1
fi,εwu =
Ω
The sequence (pε)ε is bounded in W 1,τ (Ω) for all τ < N/(N − 1), and when ε tends to zero,
(pε)ε tends to p for the weak topology of W 1,τ (Ω) for all 1 < τ < N/(N − 1). By passing to
the limit when ε tends to zero in the previous formula, we obtain
Σ k
i=1(¯z(xi) − zd(xi))wu(xi) = pu.
Γ
pεu.
On the other hand, setting J2(z(f, u), u) = F2(u), we have
F ′ 2(ū)u = Σ k
i=1(¯z(xi) − zd(xi))wu(xi) + β
Using the previous Green formula, we deduce
F ′
2(ū)u = pu + β
This completes the proof.
3.6 Exercises
Exercise 3.6.1
Γ
Γ
Γ
|ū| s−2 ūu.
Γ
s|ū| s−2 ūu.
We study the optimal control problem of section 1.2.1. We suppose that the electrical potential
φ in a bounded domain Ω satisfies the elliptic equation
−div(σ∇φ) = 0 in Ω,
−σ ∂φ
∂n = i on Γa, −σ ∂φ
∂n = 0 on Γi, −σ ∂φ
∂n
= f(φ) on Γc,
(3.6.18)
where Γa is a part of the boundary of Ω occupied by the anode, Γc is a part of the boundary
of Ω occupied by the cathode, Γi is the rest of the boundary Γ, Γi = Γ \ (Γa ∪ Γc). The control
function is the current density i, the constant σ is the conductivity of the electrolyte, the
function f is supposed to be of class C 1 , globally Lipschitz in R, and such that f(r) ≥ c1 > 0
for all r ∈ R. We study the control problem
(P3) inf{J3(φ, i) | (φ, i) ∈ H 1 (Ω) × L 2 (Γa), (φ, i) satisfies (3.6.18), a ≤ i ≤ b},
where
J3(φ, i) =
Γc
(φ − ¯ φ) 2
+ β
a ∈ L 2 (Γa) and b ∈ L 2 (Γa) are some bounds on the current i, and β is a positive constant.
Prove that (P3) has at least one solution. Write the first order optimality condition for the
solutions to (P3).
Γa
i 2 ,