Optimal Control of Partial Differential Equations

Chapter 10
Algorithms for solving optimal control
problems
10.1 Introduction
In section 10.2.1, we first recall the Conjugate Gradient Method (CGM in brief) for quadratic
functionals. We next explain how this algorithm can be used for control problems studied in
chapter 7. For functionals which are not necessarily quadratic we introduce the Polak-Ribiere
algorithm, the Fletcher-Reeves algorithm, and Quasi-Newton methods. These algorithms can
be used for control problems governed by semilinear equations such as the ones studied in
chapter 3. For linear-quadratic problems with bound constraints on the control variable we
introduce in section 10.4 a projection method due to Bertsekas. For other problems with
control constraints we describe the Gradient Method with projection in section 10.5.1. We
end this chapter with the Sequential Quadratic Programming Method (SQP method), which
is a particular implementation of the Newton method applied to the optimality system of
control problems.
10.2 Linear-quadratic problems without constraints
10.2.1 The conjugate gradient method for quadratic functionals
In chapter 2 we have applied the Conjugate Gradient Method to control problems governed
by elliptic equations. In this section, we want to apply the CGM to control problems governed
by evolution equations. Let us recall the algorithm for quadratic functionals. Consider the
optimization problem
(P1) inf{F (u) | u ∈ U},
where U is a Hilbert space and F is a quadratic functional
F (u) = 1
2 (u, Qu)U − (b, u)U.
In this setting Q ∈ L(U), Q = Q ∗ > 0, b ∈ U, and (·, ·)U denotes the scalar product in U. For
simplicity we write (·, ·) in place of (·, ·)U.
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