Optimal Control of Partial Differential Equations

84 CHAPTER 8. EQUATIONS WITH UNBOUNDED CONTROL OPERATORS
Using estimates on analytic semigroups we obtain
|(−A ∗ ) 1−α p(t)| ≤
C
(T − t) 1−α−β |(−A∗ ) β pT |Z +
T
t
C
|g(s)|Zds.
(s − t) 1−α
The mapping t ↦→ C
t1−α−β belongs to L2 (0, T ). Moreover the mapping t ↦→ |g(t)| belongs to
L∞ (0, T ) and the mapping t ↦→ C
t1−α belongs to Ls (0, T ) for all s < 1 . Thus the mapping
1−α
t ↦→ T C
t (s−t) 1−α |g(s)|Zds belongs to L∞ (0, T ).
Theorem 8.2.9 Assume that (HD) and (HP) are satisfied with α ≤ 1
2 . Problem ( P ) admits
a unique solution (z, u).
To prove this theorem we need the following lemma.
Lemma 8.2.2 Let (un)n be a sequence in L 2 (0, T ; U) converging to u for the weak topology of
L 2 (0, T ; U). Then (z(f, un, z0))n (the sequence of solutions to equation (8.1.1) corresponding to
(f, un, z0)) converges to z(f, u, z0) for the weak topology of L 2 (0, T ; Z), and (z(f, un, z0)(T ))n
converges to z(f, u, z0)(T ) for the weak topology of D((−A) −β ).
Proof. The lemma is a direct consequence of Theorems 8.2.7 and 2.6.2.
Proof of Theorem 8.2.9. We leave the reader adapt the proof of Theorem 8.2.4.
Theorem 8.2.10 If (¯z, ū) is the solution to ( P ) then ū = −B ∗ 1(−A ∗ ) 1−α p, where p is the
solution to equation
−p ′ = A ∗ p + C ∗ (Cz − yd), p(T ) = D ∗ D(−A) β ((−A) −β z(T ) − zT ). (8.2.9)
Conversely, if a pair (˜z, ˜p) ∈ C([0, T ]; Z) × C([0, T ]; Z) ∩ L 2 (0, T ; D((−A ∗ ) 1−α )) obeys the
system
˜z ′ = A˜z − (−A) 1−α B1B ∗ 1(−A ∗ ) 1−α ˜p + f, ˜z(0) = z0,
−˜p ′ = A ∗ ˜p + C ∗ (C ˜z − yd), ˜p(T ) = D ∗ D(−A) β ((−A) −β z(T ) − zT ),
then the pair (˜z, −B ∗ 1(−A ∗ ) 1−α ˜p) is the optimal solution to problem ( P ).
(8.2.10)
Proof. Let (¯z, ū) be the optimal solution to problem (P ). Set F (u) = J(z(f, u), u). For every
u ∈ L2 (0, T ; U), we have
F ′ (ū)u =
T
0
(C ∗ (Cz(t) − yd), w(t))Z
+((−A ∗ ) β D ∗ D(−A) β ((−A) −β z(T ) − zT ), (−A) −β w(T ))Z +
where w is the solution to
w ′ = Aw + Bu, w(0) = 0.
Applying formula (8.2.8) to p and w, we obtain
F ′ (ū)u =
T
((−A
0
∗ ) 1−α T
p(t), B1u(t))Z +
0
(ū(t), u(t))U =
T
0
(ū(t), u(t))U,
T
(B
0
∗ 1(−A ∗ ) 1−α p(t)+ū(t), u(t))U,
for all u ∈ L 2 (0, T ; U). The first part of the Theorem is established. The second part follows
from Theorem 2.2.3.