Optimal Control of Partial Differential Equations

110 CHAPTER 10. ALGORITHMS FOR SOLVING OPTIMAL CONTROL PROBLEMS
Let us recall the GC algorithm:
Algorithm 1.
Initialization. Choose u0 in U. Compute g0 = Qu0 − b. Set d0 = −g0 and n = 0.
Step 1. Compute
and
Determine
ρn = (gn, gn)/(dn, Qdn),
un+1 = un + ρndn.
gn+1 = Qun+1 − b = gn + ρnQdn.
Step 2. If gn+1U/g0U ≤ ε, stop the algorithm and take u = un+1, else compute
and
Replace n by n + 1 and go to step 1.
βn = (gn+1, gn+1)/(gn, gn),
dn+1 = −gn+1 + βndn.
10.2.2 The conjugate gradient method for control problems
We want to apply the CGM to problems studied in chapter 7. The state equation is of the
form
z ′ = Az + Bu + f, z(0) = z0, (10.2.1)
and the control problem is defined by
(P2) inf{J(z, u) | (z, u) ∈ C([0, T ]; Z) × L 2 (0, T ; U), (z, u) satisfies (10.2.1)}.
with
J(z, u) = 1
T
|Cz(t) − yd(t)|
2 0
2 Y + 1
2 |Dz(T ) − yT | 2 YT
T
1
+ |u(t)|
2 0
2 U. (10.2.2)
Assumptions are the ones of chapter 7. We have to identify problem (P2) with a problem of the
form (P1). Let zu be the solution to equation (10.2.1), and set F (u) = J(zu, u). Observe that
(zu, zu(T )) = (Λ1u, Λ2u) + ζ(f, z0), where Λ1 is a bounded linear operator from L 2 (0, T ; U) to
L 2 (0, T ; Z), and Λ2 is a bounded linear operator from L 2 (0, T ; U) to Z. We must determine
the quadratic form Q such that
1
2
T
0
|Czu(t) − yd(t)| 2 Y + 1
2 |Dzu(T ) − yT | 2 YT
Since (zu, zu(T )) = (Λ1u, Λ2u) + ζ(f, z0), we have
Q = Λ ∗ 1 C ∗ CΛ1 + Λ ∗ 2D ∗ DΛ2 + I,
T
1
+ |u(t)|
2 0
2 U = 1
2 (u, Qu)U − (b, u)U + c.
where C ∈ L(L 2 (0, T ; Z); L 2 (0, T ; Y )) is defined by ( Cz)(t) = Cz(t) for all z ∈ L 2 (0, T ; Z),
and C ∗ ∈ L(L 2 (0, T ; Y ); L 2 (0, T ; Z)) is the adjoint of C. In the CGM we have to compute