Optimal Control of Partial Differential Equations

60 CHAPTER 5. CONTROL OF THE HEAT EQUATION
The function f belongs to L 2 (Q), χω is the characteristic function of ω, ω is an open subset
of Ω, and the function u is a control variable. We suppose that V ∈ (L ∞ (Q)) N . We want to
study the control problem
(P5) inf{J5(z, u) | (z, u) ∈ C([0, T ]; L 2 (Ω)) × L 2 (0, T ; L 2 (ω)), (z, u) satisfies (5.5.28)},
where
J5(z, u) = 1
(z − zd)
2 Q
2 + 1
(z(T ) − zd(T ))
2 Ω
2 + β
χωu
2 Q
2 ,
and β > 0. We assume that zd ∈ C([0, T ]; L 2 (Ω)).
We first study equation (5.5.28) by a fixed point method. For that we need a regularity for
the heat equation that we state below.
Regularity result. For any 1 < q < ∞, there exists a constant C(q) such that the solution
z to the heat equation
satisfies
∂z
∂t
− ∆z = f in Q, z = 0 on Σ, z(x, 0) = 0 in Ω,
z C([0,T ];L 2 (Ω)) + z L 2 (0,T ;H 1 0 (Ω)) ≤ C(q)f L q (0,T ;L 2 (Ω))
1 - Now we choose 1 < q < 2. Let r be defined by 1
2
+ 1
r
for all f ∈ L q (0, T ; L 2 (Ω)).
= 1
q , and ¯t ∈]0, T ] such that
C(q)¯t 1/r V (L ∞ (Q)) N ≤ 1
2 . Let φ ∈ C([0, ¯t]; L 2 (Ω)) ∩ L 2 (0, ¯t; H 1 0(Ω)), and denote by zφ the
solution to equation
∂z
∂t − ∆z = f + χωu − V · ∇φ in Q, z = 0 on Σ, z(x, 0) = z0 in Ω. (5.5.29)
Prove that the mapping
φ ↦−→ zφ
is a contraction in C([0, ¯t]; L 2 (Ω)) ∩ L 2 (0, ¯t; H 1 0(Ω)).
2 - Let ˆz be the solution in C([0, ¯t]; L 2 (Ω)) ∩ L 2 (0, ¯t; H 1 0(Ω)) to equation
∂z
∂t − ∆z + V · ∇z = f + χωu in Ω × (0, ¯t), z = 0 on Γ × (0, ¯t), z(x, 0) = z0 in Ω.
The existence of ˆz follows from the previous question. Let φ ∈ C([0, 2¯t]; L 2 (Ω))∩L 2 (0, 2¯t; H 1 0(Ω))
such that φ = ˆz on [0, ¯t], and denote by zφ the solution to equation
∂z
∂t − ∆z = f + χωu − V · ∇φ in Q, z = 0 on Σ, z(x, 0) = z0 in Ω. (5.5.30)
Prove that the mapping
is a contraction in the metric space
φ ↦−→ zφ
{φ ∈ C([0, 2¯t]; L 2 (Ω)) ∩ L 2 (0, 2¯t; H 1 0(Ω)) | φ = ˆz on [0, ¯t]},