Optimal Control of Partial Differential Equations

114 CHAPTER 10. ALGORITHMS FOR SOLVING OPTIMAL CONTROL PROBLEMS
where Uad is a closed convex subset of U and F is a quadratic functional on U. For computational
considerations, we have to approximate the control set Uad by a finite dimensional set.
Let U m ad be such a finite dimensional approximation of Uad, and suppose that U m ad is a closed
convex subset in Rm . Let us denote by (P m ) the corresponding finite dimensional optimization
problem
(P m ) inf{F (u) | u ∈ U m ad}.
We only treat the case where U m ad
is defined by bound constraints, that is
U m ad = {v ∈ R m | u j a ≤ v j ≤ u j
b
for all j = 1, . . . , m}.
A projection algorithm due to Bertsekas [32] is an efficient method for solving problem with
bound constraints. The algorithm is the following.
Algorithm 6. Choose two fixed positive numbers ε and σ. We denote by un = (u 1 n, · · · , u m n ) T
the vector representing the current iterate, and let I = {1, · · · , m} be the index set associated
with un.
1 - Choose u0 = (u 1 0, · · · , u m 0 ) T , and set n = 0.
2 - Compute F ′ (un) = (∂1F (un), . . . , ∂nF (un)).
3 - Define the sets of strongly active inequalities
I σ a = {j ∈ I | u j n = u j a and ∂jF (un) > σ},
I σ b = {i ∈ I | u j n = u j
b and ∂jF (un) < −σ}.
4 - Set û j n = u j n for all j ∈ I σ a ∪ I σ b .
5 - Solve the unconstrained problem
(Paux) inf{F (u) | u ∈ R m and u j = û j for all j ∈ I σ a ∪ I σ b }.
Denote by vn the vector solution to (Paux).
6 - Set un+1 = P[ua,ub]vn, where P[ua,ub] denotes the projection onto [u 1 a, u 1 b ] × . . . [um a , u m b ].
7 - If un+1 − un ≥ ε, then replace n by n + 1 and go to 2. Otherwise stop the iteration.
The auxiliary problem (Paux) may be solved by the CGM.
10.5 General problems with control constraints
10.5.1 Gradient method with projection
We consider the problem
(P5) inf{F (u) | u ∈ Uad},