Optimal Control of Partial Differential Equations

90 CHAPTER 8. EQUATIONS WITH UNBOUNDED CONTROL OPERATORS
Since G ∈ L(L2 (Γ); D((−A) α )) for all α ∈]0, 1[,
the operator (−A)G can be decomposed
4
in the form (−A)G = (−A) 1−αB1, where B1 = (−A) αG belongs to L(L2 (Γ); L2 (Ω)). We shall
see that this situation (with 0 < α < 1)
is more complicated than the previous one where α
4
was allowed to take values greater than 1
2 .
8.5 The wave equation
We only treat the case of a Dirichlet boundary control. We first define the unbounded operator
Λ in H −1 (Ω) by
D(Λ) = H 1 0(Ω), Λz = ∆z.
We set Z = L 2 (Ω) × H −1 (Ω). We define the unbounded operator A in Z by
D(A) = H 1 0(Ω) × L 2 (Ω), A =
0 I
Λ 0
Using the Dirichlet operator G introduced in section 8.4.2, equation (6.6.15) may be written
in the form
d2z = Λz − ΛGu + f,
dt2 z(0) = z0,
dz
(0) = z1.
dt
Now setting y = (z, dz ), we have
dt
with
Bu =
dy
dt = Ay + Bu + F, y(0) = y0, (8.5.19)
0
−ΛGu
0
, F =
f
.
z0
, and y0 =
z1
The adjoint operator of A for the Z-topology is defined by
D(A ∗ ) = H 1 0(Ω) × L 2 (Ω), A ∗
0
=
−Λ
−I
0
The operator (A, D(A)) is a strongly continuous group of contractions on Z. Set
C(t)z0 = e tA
z0
0
and S(t)z1 = e tA
0
.
Since (e tA )t≥0 is a group, we can verify that C(t) = 1
2 (etA + e −tA ). Using equation (8.5.19), we
can prove that S(t)z = t
0
We can also check that
C(τ)z dτ and
e tA =
e tA∗
= e −tA =
C(t) S(t)
ΛS(t) C(t)
.
C(t) −S(t)
−ΛS(t) C(t)
We denote by B ∗ the adjoint of B, where B is an unbounded operator from L 2 (Γ) into Z.
Thus B ∗ is the adjoint of B with respect to the L 2 (Γ)-topology and the Z-topology.
.
.
z1
.