Optimal Control of Partial Differential Equations

80 CHAPTER 8. EQUATIONS WITH UNBOUNDED CONTROL OPERATORS
exists B1 ∈ L(U; Z) and 0 < α < 1 such that
B = (−A) 1−α B1.
(HH) (The hyperbolic case) The operator B∗etA∗ admits a continuous extension from Z into
L2 (0, T ; U), that is there exists a constant C(T ) such that
T
0
B ∗ e tA∗
ζ 2 U ≤ C(T )ζ 2 Z
for every ζ ∈ D(A ∗ ). In the sequel we denote by [B ∗ e tA∗
]e the extension of B ∗ e tA∗
8.2 The case of analytic semigroups
to Z.
(8.1.3)
We suppose that (HP) is satisfied. We have to distinguish the cases α > 1
1 and α ≤ . We are
2 2
going to see that if α ≤ 1 an additional assumption on D is needed in order that the problem
2
(P ) be well posed.
8.2.1 The case α > 1
2
Theorem 8.2.1 In this section we suppose that (HP) is satisfied with α > 1.
For every
2
z0 ∈ Z, every f ∈ L2 (0, T ; Z) and every u ∈ L2 (0, T ; U), equation (8.1.1) admits a unique
weak solution z(z0, u, f) in L2 (0, T ; Z), this solution belongs to C([0, T ]; Z) and the mapping
(z0, u, f) ↦−→ z(z0, u, f)
is continuous from Z × L 2 (0, T ; U) × L 2 (0, T ; Z) into C([0, T ]; Z).
Proof. Due to Theorem 4.3.1, we have
Thus z(t) satisfies the estimate
z(t) = e tA z0 +
= e tA z0 +
|z(t)|Z ≤ Cz0Z +
t
e
0
(t−τ)A (−A) 1−α B1u(τ)dτ
t
(−A)
0
1−α e (t−τ)A B1u(τ)dτ.
t
0
|t − τ| α−1 |u(τ)|Udτ.
Since the mapping t ↦→ t α−1 belongs to L 2 (0, T ) and the mapping t ↦→ |u(t)| belongs to
L 2 (0, T ), from the above estimate it follows that t ↦→ |z(t)| belongs to L ∞ (0, T ).
The adjoint equation for (P ) is of the form
−p ′ = A ∗ p + g, p(T ) = pT . (8.2.4)
Theorem 8.2.2 For every (g, pT ) ∈ L 2 (0, T ; Z)×Z, the solution p to equation (8.2.4) belongs
to L 2 (0, T ; D((−A ∗ ) 1−α )).