Optimal Control of Partial Differential Equations

2.8. EXERCISES 25
2 - Let (un)n be a sequence in L 2 (Γ1) converging to û for the weak topology of L 2 (Γ1). Let
z(un) be the solution to equation (2.8.16) corresponding to un. What can we say about the
sequence (z(un))n ? Prove that problem (P6) admits a unique weak solution.
Characterize the optimal control by writing first order optimality conditions.
Exercise 2.8.2
Let Ω be a bounded domain in R N , with a Lipschitz boundary Γ. Consider the elliptic equation
−∆z + z = f + χωu in Ω,
∂z
∂n
= 0 on Γ. (2.8.17)
The function f belongs to L 2 (Ω), the control u ∈ L 2 (ω), and ω is an open subset in Ω. We
want to study the control problem
(P7) inf{J7(z, u) | (z, u) ∈ H 1 (Ω) × L 2 (ω), (z, u) satisfies (2.8.17)}.
with
the function zd belongs to H 1 (Ω).
J7(z, u) = 1
(z − zd)
2 Γ
2 + β
u
2 ω
2 ,
1 - Prove that problem (P7) admits a unique weak solution.
2 - Characterize the optimal control by writing first order optimality conditions.