Optimal Control of Partial Differential Equations

64 CHAPTER 6. CONTROL OF THE WAVE EQUATION
Theorem 6.2.1 The operator (A, D(A)) is the infinitesimal generator of a strongly continuous
semigroup of contractions on Y .
Proof. The theorem relies on the Hille-Yosida theorem.
(i) The domain D(A) is dense in Y . Prove that A is a closed operator. Let (yn)n be a sequence
converging to y = (y1, y2) in H 1 0(Ω) × L 2 (Ω), and such that (Ayn)n = (y2,n, ∆y1,n)n converges
to (f, g) in H 1 0(Ω)×L 2 (Ω). We have y2 = f, and ∆y1 = g because (∆y1,n)n converges to ∆y1 in
the sense of distributions in Ω. Due to Theorem 3.2.1, we have y1,n −y1,m H 2 (Ω) ≤ C∆y1,n −
∆y1,m L 2 (Ω) . Thus (y1,n)n is a Cauchy sequence in H 2 (Ω)∩H 1 0(Ω). Hence y1 ∈ H 2 (Ω)∩H 1 0(Ω).
The first condition of Theorem 4.1.1 is satisfied.
(ii) For λ > 0, f ∈ H1 0(Ω), g ∈ L2 (Ω), consider the equation
y1 y1 f
λ − A = ,
g
that is
We have
y2
y2
λy1 − y2 = f in Ω,
λy2 − ∆y1 = g in Ω.
λ 2 y1 − ∆y1 = λf + g in Ω.
(6.2.2)
This equation admits a unique solution in H2 (Ω) ∩ H1 0(Ω). Thus the system (6.2.2) admits a
unique solution y ∈ D(A). From the equation λy2 − ∆y1 = g , we deduce
λ y
Ω
2
2 + ∇y1∇y2 =
Ω
gy2.
Ω
Replacing y2 by λy1 − f in the second term, we obtain
λ y
Ω
2
2 + λ |∇y1|
Ω
2
= gy2 +
Ω
≤
|∇y1| 2
1/2
and
We can choose y ↦→
y
Ω
2 2 +
Ω
g
Ω
2 +
Ω
Ω
∇y1∇f
|∇f| 2
1/2 ,
λ y
Ω
2
2 + |∇y1|
Ω
2
1/2
≤ g
Ω
2
+ |∇f|
Ω
2
1/2
Ω
.
y2 2 +
1/2 2 |∇y1| as a norm on Y and the proof is complete.
Ω
Theorem 6.2.2 For every f ∈ L 2 (Q), every z0 ∈ H 1 0(Ω), every z1 ∈ L 2 (Ω), equation (6.2.1)
admits a unique weak solution z(f, z0, z1), moreover the operator
(f, z0, z1) ↦→ z(f, z0, z1)
is linear and continuous from L 2 (Q)×H 1 0(Ω)×L 2 (Ω) into C([0, T ]; H 1 0(Ω))∩C 1 ([0, T ]; L 2 (Ω)).
Proof. The theorem is a direct consequence of Theorem 4.2.1 and Theorem 6.2.1.