Optimal Control of Partial Differential Equations

6.2. EXISTENCE AND REGULARITY RESULTS 65
6.2.2 The wave equation in L 2 (Ω) × H −1 (Ω)
We study equation (6.2.1) when (z0, z1) ∈ L 2 (Ω) × H −1 (Ω) and f ∈ L 2 (0, T ; H −1 (Ω)). In that
case we set Y = L 2 (Ω) × H −1 (Ω), D( A) = H 1 0(Ω) × L 2 (Ω) and
where ( Ãy1, ζ) H −1 (Ω) = −
Ay =
y1
A
y2
We have the same kind of result as above.
=
y2
Ãy1
,
Ω ∇y1 · ∇(−∆) −1 ζ and (−∆) −1 ζ is the solution w of the equation
w ∈ H 1 0(Ω), −∆w = ζ in Ω.
Theorem 6.2.3 The operator ( A, D( A)) is the infinitesimal generator of a semigroup of contractions
on Y .
Proof. The theorem still relies on the Hille-Yosida theorem.
(i) The domain D( A) is dense in Y . As for the proof of Theorem 6.2.1, we prove that ( A, D( A))
is a closed operator. The first condition of Theorem 4.1.1 is satisfied.
(ii) For λ > 0, f ∈ L 2 (Ω), g ∈ H −1 (Ω), consider the system
The equation
λy1 − y2 = f in Ω,
λy2 − ∆y1 = g in Ω.
λ 2 y1 − ∆y1 = λf + g in Ω,
(6.2.3)
admits a unique solution in H 1 0(Ω). Thus the system (6.2.3) admits a unique solution y ∈
D(A). The obtention of the estimate is more delicate than previously. We compose the two
members of the first equation by (−∆) −1 , the inverse of the Laplace operator with homogeneous
boundary conditions. We have λ(−∆) −1 y1 − (−∆) −1 y2 = (−∆) −1 f , and we choose
(−∆) −1 y2 as a test function for the second equation:
Recall that the mapping
λ〈y2, (−∆) −1 y2〉 H −1 (Ω),H 1 0 (Ω) + λ〈−∆y1, (−∆) −1 y1〉 H −1 (Ω),H 1 0 (Ω)
= 〈g, (−∆) −1 y2〉 H −1 (Ω),H 1 0 (Ω) + 〈−∆y1, (−∆) −1 f〉 H −1 (Ω),H 1 0 (Ω).
f ↦−→ |f| H −1 (Ω) = 〈f, (−∆) −1 f〉 1/2
H −1 (Ω)×H 1 0 (Ω)
is a norm in H −1 (Ω) equivalent to the usual norm (Theorem 5.4.4). This norm is associated
with the scalar product
(f, g) ↦−→ 〈f, (−∆) −1 g〉 1/2
H −1 (Ω)×H 1 0 (Ω).
Thus we have
〈g, (−∆) −1 y2〉 H −1 (Ω),H 1 0 (Ω) ≤ |g| H −1 (Ω)|y2| H −1 (Ω).