Optimal Control of Partial Differential Equations

116 CHAPTER 10. ALGORITHMS FOR SOLVING OPTIMAL CONTROL PROBLEMS
Uad is a closed convex subset of L 2 (0, T ; U), and the state equation is of the form
z ′ = Az + φ(z) + Bu, z(0) = z0. (10.5.11)
Assumptions on C, D, Y , YT are the ones of chapter 7. We suupse that (A, D(A)) is the
generator of a strongly continuous semigroup on the Hilbert space Y , and (for simplicity) that
φ is Lipschitz on Y .
The optimality system for (P6) satisfied by a solution (¯z, ū) consists of the equations
−¯p ′ = A ∗ ¯p + φ ′ (¯z) ∗ ¯p + C ∗ (C ¯z − yd), ¯p(T ) = D ∗ (D¯z(T ) − yT ),
¯z ′ = A¯z + φ(¯z) + Bū, ¯z(0) = z0,
T
0 (B∗ ¯p + ū)(u − ū) ≥ 0 for all u ∈ Uad.
(10.5.12)
The Newton method applied to the system (10.5.12) corresponds to the following algorithm:
Algorithm 8.
Initialization. Set n = 0. Choose u0 in Uad, compute ˆz0 the solution to the state equation for
u = u0, and ˆp0 the solution to the adjoint equation
−p ′ = A ∗ p + φ ′ (ˆz0) ∗ p + C ∗ (C ˆz0 − yd), p(T ) = D ∗ (Dˆz0(T ) − yT ).
Step 1. Compute (un+1, ˆzn+1, ˆpn+1) ∈ Uad × C([0, T ]; Z) × C([0, T ]; Z) the solution to the
system
−p ′ = A ∗ p + φ ′ (ˆzn) ∗ p + (φ ′′ (ˆzn)(z − ˆzn)) ∗ ˆpn + C ∗ (Cz − yd),
p(T ) = D∗ (Dz(T ) − yT ),
z ′ = Az + φ(ˆzn) + φ ′ (ˆzn)(z − ˆzn) + Bu, z(0) = z0,
T
0 (B∗ (10.5.13)
p + u, v − u) U ≥ 0 for all v ∈ Uad.
Step 2. If |un+1 − un|U ≤ ε, stop the algorithm, else replace n by n + 1 and go to step 1.
Observe that the mapping φ must necessarily be of class C 2 . The convergence of the Newton
method is studied in [33]. Roughly speaking, if φ ′ satisfy some Lipschitz property, and if the
optimality system (10.5.12) is strongly regular in the sense of Robinson (see [33]), then there
exists a neighborhood ¯ V of (¯z, ū, ¯p) such that for any starting point in ¯ V the Newton algorithm
is quadratically convergent.
The SQP method corresponds to the previous algorithm in which (un+1, ˆzn+1) is computed by
solving the ’Linear-Quadratic’ problem
(QP n+1)
Minimize J ′ (ˆzn, un)(z − ˆzn, u − un) + 1
2 〈ˆpn, φ ′′ (ˆzn)(z − ˆzn) 2 〉,
subject to z ′ = Az + φ(ˆzn) + φ ′ (ˆzn)(z − ˆzn) + Bu, z(0) = z0,
u ∈ Uad,
and ˆpn+1 is the solution to the adjoint equation for (QP n+1) associated with (un+1, ˆzn+1). For
problems with bound constraints this ’Linear-Quadratic’ problem may be solved by Algorithm
6.