Optimal Control of Partial Differential Equations

54 CHAPTER 5. CONTROL OF THE HEAT EQUATION
Since (z0 − wn(0)) ∈ L2 (Ω) and f − ∂wn
∂t − ∆wn belongs to L2 (Q), yn
W (0, T ; H
and zn belong to
1 (Ω), (H1 (Ω)) ′ ). Thus, for every t ∈]0, T ], we have
We first get
Ω
|zn(t)| 2 + 2
t
0
Ω
|∇zn| 2 = 2
t
0
fzn + 2
Ω
t
0
unzn +
Γ
Ω
|z0| 2 .
y 2
C([0,T ];L2 (Ω)) + 2∇y2L
2 (0,T ;L2 (Ω)) ≤ 2fL2 (Q)yL2 (Q) + 2uL2 (Σ)yL2 (Σ) + y0 2
L2 (Ω) .
Thus with Young’s inequality, we obtain
y C([0,T ];L 2 (Ω)) + y L 2 (0,T ;H 1 (Ω)) ≤ C
In the same way, we can prove
fL2 (Q) + uL2 (Σ) + y0L2 (Ω) .
zn − zm C([0,T ];L 2 (Ω)) + zn − zm L 2 (0,T ;H 1 (Ω)) ≤ Cun − um L 2 (Σ).
Hence the sequence (zn)n converges to some z in C([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 1 (Ω)). Due to
Theorem 5.3.3, we can also prove that the sequence (zn)n converges to the solution of equation
(5.3.10) in C([0, T ]; L 2 (Ω)). By using the same arguments as for Theorem 5.2.2, we can next
prove an estimate in W (0, T ; H 1 (Ω), (H 1 (Ω)) ′ ). Therefore we have established the following
theorem.
Theorem 5.3.4 For every f ∈ L 2 (Q), every u ∈ L 2 (Σ), and every z0 ∈ L 2 (Ω), equation
(5.3.10) admits a unique weak solution z(f, u, z0) in L 2 (0, T ; L 2 (Ω)), moreover the operator
(f, u, z0) ↦→ z(f, u, z0)
is linear and continuous from L 2 (Q) × L 2 (Σ) × L 2 (Ω) into W (0, T ; H 1 (Ω), (H 1 (Ω)) ′ ).
We now consider the control problem
(P2) inf{J2(z, u) | (z, u) ∈ C([0, T ]; L 2 (Ω)) × L 2 (0, T ; L 2 (Γ)), (z, u) satisfies (5.3.10)},
where
J2(z, u) = 1
(z − zd)
2 Q
2 + 1
(z(T ) − zd(T ))
2 Ω
2 + β
u
2 Σ
2 .
We assume that f ∈ L 2 (Q), z0 ∈ L 2 (Ω), and zd ∈ C([0, T ]; L 2 (Ω)). Problem (P2) admits a
unique solution (¯z, ū) (see exercise 5.5.2). The adjoint equation for (P2) is of the form
− ∂p
∂t
− ∆p = g in Q,
∂p
∂n = 0 on Σ, p(x, T ) = pT in Ω. (5.3.16)
Theorem 5.3.5 Suppose that u ∈ L2 (Σ), g ∈ L2 (Q), pT ∈ L2 equation
(Ω). Then the solution z of
∂z
− ∆z = 0
∂t
in Q,
∂z
= u
∂n
on Σ, z(0) = 0 in Ω,
and the solution p of (5.3.16) satisfy the following formula
up = z g + z(T )pT . (5.3.17)
Σ
Q
Ω