Optimal Control of Partial Differential Equations

104 CHAPTER 9. CONTROL OF A SEMILINEAR PARABOLIC EQUATION
Theorem 9.5.2 Let z(u) be the solution to equation (9.2.1). The mapping
u ↦−→ z(u),
is of class C1 from L2 (0, T ; L2 (ω)) into W (0, T ; H1 0(Ω), H−1 (Ω)), and for all ū and u in
L2 (0, T ; L2 (ω)), the function w = dz (ū)u is the solution to equation
du
∂w
∂t − ∆w + 2∂x1(z(ū)w) = χωu in Q, w = 0 on Σ, w(0) = 0 in Ω. (9.5.9)
Proof. Let ū ∈ L 2 (0, T ; L 2 (ω)). We have G(z(ū), ū) = 0. From Theorem 9.5.1 and
from the implicit function theorem, it follows that there exists a neighborhood V (ū) of ū
in L 2 (0, T ; L 2 (ω)), such that the mapping u ↦→ z(u) is of class C 1 from V (ū) to Z, and
G ′ z(z(ū), ū) ◦ dz
du (ū) u + G′ u(z(ū), ū) u = 0,
for all u ∈ L2 (0, T ; L2 (ω)). If we set w = dz (ū)u, we have
du
G ′ z(z(ū), ū)w = ( ∂w
∂t − ∆w + 2∂x1(z(ū)w), w(0))
and G ′ u(z(ū), ū) u = −χωu. The proof is complete.
Theorem 9.5.3 If (¯z, ū) is a solution to (P1) then
χω(βū + p)(u − ū) ≥ 0 for all u ∈ Uad,
where p is the solution to equation
Q
− ∂p
∂t − ∆p − 2¯z∂xi p = ¯z − zd in Q, p = 0 on Σ, p(x, T ) = ¯z(T ) − zT in Ω. (9.5.10)
Proof. Set F1(u) = J1(z(u), u), where z(u) is the solution to equation (9.5.9). From Theorem
9.5.2 it follows that F1 is of class C 1 on L 2 (0, T ; L 2 (ω)), and that
F ′
1(ū)u =
Q
(¯z − zd)w +
Ω
(¯z(T ) − zd(T ))w(T ) +
Ω
χωβūu,
where w is the solution to equation (9.5.9). Since (¯z, ū) is a solution to (P1), F ′ 1(ū)(u − ū) ≥ 0
for all u ∈ Uad. Using a Green formula between p and w we obtain
(¯z(T ) − zd(T ))w(T ) + (¯z − zd)w = χωpu,
and
This completes the proof.
Ω
F ′
1(ū)u =
Q
Q
χωpu +
Ω
χωβūu.
Q