Optimal Control of Partial Differential Equations

8.2. THE CASE OF ANALYTIC SEMIGROUPS 81
Proof. We have
(−A ∗ ) 1−α p(t) = (−A ∗ ) 1−α (T −t)A∗
e pT +
Using estimates on analytic semigroups (Theorem 4.4.1), we obtain
|(−A ∗ ) 1−α p(t)|Z ≤
C
(T − t) 1−α |pT |Z +
T
t
T
t
(−A ∗ ) 1−α e (s−t)A∗
g(s)ds.
C
|g(s)|Zds.
(s − t) 1−α
Since the mapping t ↦→ C
t1−α belongs to L2 (0, T ) (because α > 1),
and the mapping t ↦→ |g(t)|
2
belongs to L2 (0, T ), the mapping t ↦→ T C
t (s−t) 1−α |g(s)|Zds belongs to L∞ (0, T ). Moreover the
mapping t ↦→ C
(T −t) 1−α |pT |Z belongs to L2 (0, T ).
Theorem 8.2.3 For every u ∈ L 2 (0, T ; U), and every (g, pT ) ∈ L 2 (0, T ; Z) × Z, the solution
z to equation
z ′ = Az + Bu, z(0) = 0,
and the solution p to equation (8.2.4) satisfy the following formula
T
(B1u(t), (−A
0
∗ ) 1−α T
p(t))Z dt =
0
(z(t), g(t))Z dt + (z(T ), pT )Z. (8.2.5)
Proof. This formula can be proved if p and z are regular enough to justify integration by
parts and tranposition of the operator A (for example if p and z belong to C 1 ([0, T ]); Z) ∩
C([0, T ]; D(A))). Due to Theorem 8.2.2, formula (8.2.5) can be next obtained by a passage to
the limit.
Theorem 8.2.4 Assume that (HP) is satisfied with α > 1.
Problem (P ) admits a unique
2
solution (z, u).
To prove this theorem we need the following lemma.
Lemma 8.2.1 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 Z.
Proof. The lemma is a direct consequence of Theorems 8.2.1 and 2.6.2.
Proof of Theorem 8.2.4. The proof is completely analogous to that of Theorem 7.3.1.
Theorem 8.2.5 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 ∗ (Dz(T ) − yT ). (8.2.6)
Conversely, if a pair (˜z, ˜p) ∈ C([0, T ]; Z) × C([0, T ]; Z) 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˜z(T ) − yT ),
then the pair (˜z, −B ∗ 1(−A ∗ ) 1−α ˜p) is the optimal solution to problem (P ).
(8.2.7)