Optimal Control of Partial Differential Equations

10.6. ALGORITHMS FOR DISCRETE PROBLEMS 117
If the optimal solution (¯z, ū, ¯p) satisfies a sufficient second order optimality condition, and
if the optimality system (10.5.12) is strongly regular in the sense of Robinson, then the SQP
method and the Newton method are equivalent ([33]).
10.6 Algorithms for discrete problems
For numerical computations, we have to write discrete approximations to control problems.
Suppose that equation
z ′ = Az + Bu + f, z(0) = z0, (10.6.14)
is approximated by an implicit Euler scheme
where f n = 1
tn
∆t
the functional
we set
z 0 = z0,
for n = 1, . . . , M, z n is the solution to
1
∆t (zn − z n−1 ) = Az n + Bu n + f n ,
tn−1 f(t) dt, un = 1
∆t
J(z, u) = 1
2
T
0
JM(z, u) = 1
2 ∆t
tn
(10.6.15)
tn−1 u(t) dt, tn = n∆t, and T = M∆t. To approximate
|Cz(t) − yd(t)| 2 Y + 1
2 |Dz(T ) − yT | 2 YT
M
n=1
with z = (z 0 , . . . , z M ), u = (u 1 , . . . , u M ), y n d
control problem associated with (P2) as follows:
|Cz n − y n d | 2 Y + 1
2 |DzM − yT | 2 YT
= 1
∆t
tn
T
1
+ |u(t)|
2 0
2 U,
+ 1
2 ∆t
M
|u n | 2 U,
n=1
tn−1 yd(t) dt. We can define a discrete
(PM) inf{JM(z, u) | (z, u) ∈ Z M+1 × U M , (z, u) satisfies (10.6.15)}.
To apply the CGM to problem (PM), we have to compute the gradient of the mapping u ↦→
JM(zu, u), where zu is the solution to (10.6.15) corresponding to u. Set FM(u) = JM(zu, u).
We have
F ′ M
M(ū)u = ∆t (C ¯z n − y n d , Cw n u)Y + (D¯z M − yT , Dw M M
u )YT + ∆t
n=1
where ¯z = zū and w = (w 0 , . . . , w M ) ∈ Z M+1 is defined by
w 0 = 0,
for n = 1, . . . , M, w n is the solution to
1
∆t (wn − w n−1 ) = Aw n + Bu n .
n=1
(ū n , u n )U,
(10.6.16)
To find the expression of F ′ M (ū), we have to introduce an adjoint equation. Let p = (p0 , . . . , p M )
be in Z M+1 , or in D(A ∗ ) M+1 if we want to justify the calculations. Taking a weak formulation
of the different equations in (10.6.16), we can write
1
∆t ((wn − w n−1 ), p n−1 )Z − (w n , A ∗ p n−1 )Z = (Bu n , p n−1 )Z = (u n , B ∗ p n−1 )U.