Optimal Control of Partial Differential Equations

10.4. LINEAR-QUADRATIC PROBLEMS WITH CONTROL CONSTRAINTS 113
and
Replace n by n + 1 and go to step 1.
Hn+1 = Hn + γnγ∗ ∗
n Hnsn(Hnsn)
−
〈sn, γn〉 〈sn, Hnsn〉 .
Comments. The terminology Quasi-Newton method comes from that the update rule of
Hn in step 3 is based on a secant approximation of the Hessian operator F ′′ (un) . In the
initialization procedure H0 = I can be replaced by H0 = F ′′ (x0) if the computation of the
Hessian operator is not too expensive or too complicated. A direct update of the matrix H−1 n
can be performed. It corresponds to a variant of the above method, where in step 3 the update
of Hn is replaced by
H −1
n+1 = H −1
n + (sn − H−1 n γn)s∗ n + sn(sn − H−1 n γn) ∗
〈sn, γn〉
For more details on quasi-Newton methods we refer to [35].
− 〈sn − H−1 n γn, γn〉
〈sn, γn〉 2 sns ∗ n.
These algorithms can be applied to control problems governed by semilinear evolution equations
of the form
z ′ = Az + φ(z) + Bu, z(0) = z0, (10.3.7)
or by semilinear elliptic equations of the form
Az = f,
∂z
+ φ(z) = u, (10.3.8)
∂nA
where A is a uniformly elliptic operator. (Control of semilinear elliptic equations has been
studied in chapter 3.) Let us explain how algorithms 3-5 can be applied to the control problem
(P3) inf{J(z, u) | (z, u) ∈ C([0, T ]; Z) × L 2 (0, T ; U), (z, u) satisfies (10.3.7)}.
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.
Assumptions on C, D, Y , YT are the ones of chapter 7. The nonlinear function φ is for example
the one of chapter 3. For algorithms 3-5, we have to compute the gradient of F (u) = J(zu, u),
where zu is the solution to equation (10.3.7).
For a given u ∈ U, F ′ (u) is computed as follows. We first solve equation (10.3.7). Next we
solve the adjoint equation
We have
−p ′ = A ∗ p + φ ′ (zu) ∗ p + C ∗ (Czu − yd), p(T ) = D ∗ (Dzu(T ) − yT ).
F ′ (u) = B ∗ p + u.
10.4 Linear-quadratic problems with control constraints
We consider the problem
(P4) inf{F (u) | u ∈ Uad},