Optimal Control of Partial Differential Equations

44 CHAPTER 4. EVOLUTION EQUATIONS
with ζz = (I − A ∗ 1) −1 yz. We can take
ζ ↦−→ (I − A ∗ 1) −1 ζZ ′
as a norm on D(A ∗ ). For such a choice (I − A ∗ 1) −1 is an isometry from Z ′ to (D(A ∗ )) ′ . Thus
Since
one has
〈z, (I − A
supζ∈D(A∗ )
∗ 1)ζ〉Z,Z ′
ζD(A∗ )
zD((A ∗ 1 )∗ ) = sup ζ∈D(A ∗ )
= sup y∈Z ′
〈z, y〉Z,Z ′
yZ ′
〈z, (I − A∗ 1)ζ〉Z,Z ′
,
ζD(A ∗ )
The equality D((A ∗ 1) ∗ ) = Z follows from (4.3.4) and (4.3.5).
For all z ∈ D(A), and all y ∈ D(A ∗ 1), we have
Thus, (A ∗ 1) ∗ z = Az for all z ∈ D(A).
zD((A ∗ 1 )∗ ) ≤ zZ. (4.3.5)
〈(A ∗ 1) ∗ z, y〉 = 〈z, A ∗ 1y〉 = 〈z, A ∗ y〉 = 〈Az, y〉.
From Theorem 4.2.3 we deduce that ((S ∗ 1) ∗ (t))t≥0 is the semigroup on (D(A ∗ )) ′ generated by
(A ∗ 1) ∗ . To prove that (S ∗ 1) ∗ (t)z = S(t)z for all z ∈ Z and all t ≥ 0, it is sufficient to observe
that
〈(S ∗ 1) ∗ (t)z, y〉 = 〈z, S ∗ 1(t)y〉 = 〈z, S ∗ (t)y〉 = 〈S(t)z, y〉,
for all z ∈ Z, all y ∈ D(A ∗ ), and all t ≥ 0.
Remark. Therefore we can extend the notion of solution for the equation (4.2.2) in the case
where x0 ∈ (D(A ∗ )) ′ and f ∈ L p (0, T ; (D(A ∗ )) ′ ), by considering the equation
z ′ (t) = (A ∗ 1) ∗ z(t) + f(t) dans (0, T ), z(0) = z0. (4.3.6)
It is a usual abuse of notation to replace A ∗ 1 by A ∗ and to write equation (4.3.6) in the form
(cf [2, page 160])
z ′ (t) = (A ∗ ) ∗ z(t) + f(t) dans (0, T ), z(0) = z0. (4.3.7)
Since (A ∗ 1) ∗ is an extension of the operator A, sometimes equations (4.3.6) or (4.3.7) are written
in the form (4.2.2) even if z0 ∈ (D(A ∗ )) ′ and f ∈ L p (0, T ; (D(A ∗ )) ′ ).
Theorem 4.3.2 For every z0 ∈ (D(A∗ )) ′ and every f ∈ Lp (0, T ; (D(A∗ )) ′ ), with 1 ≤ p < ∞,
equation (4.3.6) admits a unique solution z(f, z0) ∈ Lp (0, T ; (D(A∗ )) ′ ), this solution belongs
to C([0, T ]; (D(A∗ )) ′ ) and is defined by
z(t) = e t(A∗ 1 )∗
z0 +
t
0
e (t−s)(A∗ 1 )∗
f(s) ds.
The mapping (f, z0) ↦→ z(f, z0) is linear and continuous from L p (0, T ; (D(A ∗ )) ′ ) × (D(A ∗ )) ′
into C([0, T ]; (D(A ∗ )) ′ ).
.