Optimal Control of Partial Differential Equations
Optimal Control of Partial Differential Equations Optimal Control of Partial Differential Equations
108 CHAPTER 9. CONTROL OF A SEMILINEAR PARABOLIC EQUATION with β > 0 and zd ∈ C([0, T ]; L 2 (Ω)). Prove the existence of a solution to problem (P2). Write optimality conditions. 2 - Consider the following variant of problem (P2) (P3) inf{J3(z, u) | (z, u) ∈ W (0, T ; H 1 (Ω), (H 1 (Ω)) ′ ) × Uad, (z, u) satisfies (9.7.15)}, where Uad is a closed convex subset of H 1 1 , 2 4 (Σ), J3(z, u) = 1 |∇z − ∇zd| 2 Q 2 + 1 (z(T ) − zd(T )) 2 Ω 2 + β u 2 Σ 2 , with β > 0 and zd ∈ C([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 1 (Ω)). Prove the existence of a solution to problem (P3). Write optimality conditions.
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. 109
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108 CHAPTER 9. CONTROL OF A SEMILINEAR PARABOLIC EQUATION<br />
with β > 0 and zd ∈ C([0, T ]; L 2 (Ω)). Prove the existence <strong>of</strong> a solution to problem (P2). Write<br />
optimality conditions.<br />
2 - Consider the following variant <strong>of</strong> problem (P2)<br />
(P3) inf{J3(z, u) | (z, u) ∈ W (0, T ; H 1 (Ω), (H 1 (Ω)) ′ ) × Uad, (z, u) satisfies (9.7.15)},<br />
where Uad is a closed convex subset <strong>of</strong> H 1 1<br />
, 2 4 (Σ),<br />
J3(z, u) = 1<br />
<br />
|∇z − ∇zd|<br />
2 Q<br />
2 + 1<br />
<br />
(z(T ) − zd(T ))<br />
2 Ω<br />
2 + β<br />
<br />
u<br />
2 Σ<br />
2 ,<br />
with β > 0 and zd ∈ C([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 1 (Ω)). Prove the existence <strong>of</strong> a solution to<br />
problem (P3). Write optimality conditions.
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