Optimal Control of Partial Differential Equations

118 CHAPTER 10. ALGORITHMS FOR SOLVING OPTIMAL CONTROL PROBLEMS
Now, by adding the different equalities, we find the adjoint equation by identifying
with
∆t
M
(C ¯z n − y n d , Cw n u)Y + (D¯z M − yT , Dw M u )YT
n=1
∆t
M
n=1
More precisely, if p = (p 0 , . . . , p M ) is defined by
then
(u n , B ∗ p n−1 )U.
p M = D ∗ (D¯z M − yT ),
for n = 1, . . . , M, p n is the solution to
1
∆t (−pn + p n−1 ) = A ∗ p n−1 + C ∗ (C ¯z n − y n d ),
F ′ M(ū)u = ∆t
M
n=1
(u n , B ∗ p n−1 )U + ∆t
M
n=1
(ū n , u n )U.
(10.6.17)
Observe that the above identification is not justified since D ∗ (D¯z M − yT ) does not necessarily
belong to D(A ∗ ). In practice, a ’space-discretization’ is also performed. This means that
equation (10.6.15) is replaced by a system of ordinary differential equations, the operator A
is replaced by an operator belonging to L(R ℓ ), where ℓ is the dimension of the discrete space,
and the above calculations are justified for the corresponding discrete problem.
10.7 Exercises
Exercise 10.7.1
Apply the conjugate gradient method to problem (P4) of chapter 5. In particular identify the
bounded operator Λ from L 2 (Σ) into L 2 (Ω), and its adjoint Λ ∗ , such that
where wu is the solution to equation
Exercise 10.7.2
∂w
∂t
|wu(T )| 2
H −1 (Ω) =
|Λu|
Ω
2
− ∆w = 0 in Q, w = u on Σ, w(0) = 0 in Ω.
Apply the conjugate gradient method to problem (P5) of chapter 6.
Exercise 10.7.3
Apply the SQP method to problem (P1) of chapter 3. In particular, prove that the Linear-
Quadratic problem (QPn+1) of the SQP method is well posed.