Optimal Control of Partial Differential Equations

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4 CONTENTS 4.3 Weak solutions in L p (0, T ; (D(A ∗ )) ′ ) . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Analytic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.4.1 Fractional powers of infinitesimal generators . . . . . . . . . . . . . . . 45 4.4.2 Analytic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Control of the heat equation 47 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Distributed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Neumann boundary control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4 Dirichlet boundary control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.4.1 Observation in L 2 (Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.4.2 Observation in C([0, T ]; H −1 (Ω)) . . . . . . . . . . . . . . . . . . . . . 57 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6 Control of the wave equation 63 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 Existence and regularity results . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2.1 The wave equation in H 1 0(Ω) × L 2 (Ω) . . . . . . . . . . . . . . . . . . . 63 6.2.2 The wave equation in L 2 (Ω) × H −1 (Ω) . . . . . . . . . . . . . . . . . . 65 6.3 Distributed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 Neumann boundary control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.5 Trace regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.6 Dirichlet boundary control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7 Equations with bounded control operators 75 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2 Adjoint equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.3 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8 Equations with unbounded control operators 79 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.2 The case of analytic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.2.1 The case α > 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.2.2 The case α ≤ 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.3 The hyperbolic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.4 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.4.1 Neumann boundary control . . . . . . . . . . . . . . . . . . . . . . . . 88 8.4.2 Dirichlet boundary control . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.5 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.6 A first order hyperbolic system . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.6.1 State equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.6.2 Control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
CONTENTS 5 9 Control of a semilinear parabolic equation 99 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.2 Distributed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.3 Existence of solutions for L 2p -initial data, p > 2 . . . . . . . . . . . . . . . . . 100 9.3.1 Existence of a local solution . . . . . . . . . . . . . . . . . . . . . . . . 100 9.3.2 Initial data in D((−A) α ) . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.3.3 Existence of a global solution . . . . . . . . . . . . . . . . . . . . . . . 101 9.4 Existence of a global weak solution for L 2 -initilal data . . . . . . . . . . . . . . 101 9.5 Optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 10 Algorithms for solving optimal control problems 109 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.2 Linear-quadratic problems without constraints . . . . . . . . . . . . . . . . . . 109 10.2.1 The conjugate gradient method for quadratic functionals . . . . . . . . 109 10.2.2 The conjugate gradient method for control problems . . . . . . . . . . 110 10.3 Control problems governed by semilinear equations . . . . . . . . . . . . . . . 112 10.4 Linear-quadratic problems with control constraints . . . . . . . . . . . . . . . 113 10.5 General problems with control constraints . . . . . . . . . . . . . . . . . . . . 114 10.5.1 Gradient method with projection . . . . . . . . . . . . . . . . . . . . . 114 10.5.2 The sequential quadratic programming method . . . . . . . . . . . . . 115 10.6 Algorithms for discrete problems . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Page 1: Université Paul Sabatier Optimal C
Page 6 and 7: 6 CONTENTS
Page 8 and 9: 8 CHAPTER 1. EXAMPLES OF CONTROL PR
Page 10 and 11: 10 CHAPTER 1. EXAMPLES OF CONTROL P
Page 12 and 13: 12 CHAPTER 1. EXAMPLES OF CONTROL P
Page 14 and 15: 14 CHAPTER 2. CONTROL OF ELLIPTIC E
Page 28 and 29: 28 CHAPTER 3. CONTROL OF SEMILINEAR
Page 42 and 43: 42 CHAPTER 4. EVOLUTION EQUATIONS 4
Page 44 and 45: 44 CHAPTER 4. EVOLUTION EQUATIONS w
Page 46 and 47: 46 CHAPTER 4. EVOLUTION EQUATIONS D
Page 48 and 49: 48 CHAPTER 5. CONTROL OF THE HEAT E
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54 CHAPTER 5. CONTROL OF THE HEAT E
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64 CHAPTER 6. CONTROL OF THE WAVE E
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76 CHAPTER 7. EQUATIONS WITH BOUNDE
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78 CHAPTER 7. EQUATIONS WITH BOUNDE
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80 CHAPTER 8. EQUATIONS WITH UNBOUN
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100 CHAPTER 9. CONTROL OF A SEMILIN
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110 CHAPTER 10. ALGORITHMS FOR SOLV
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120 BIBLIOGRAPHY [16] I. Lasiecka,
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