Numerical Approximation of Exact Controls for Waves

Cover......Page 1
Numerical Approximation \rof Exact Controls for Waves......Page 4
Preface......Page 6
Acknowledgments......Page 8
Contents......Page 10
Introduction......Page 14
1.1.1 An Abstract Functional Setting......Page 19
1.1.2 Contents of Chap.1......Page 22
1.2.1 An ``Algorithm\'\' in an Infinite-Dimensional Setting......Page 24
1.2.2 The Continuous Approach......Page 27
1.2.3 The Discrete Approach......Page 29
1.3.1 Classical Convergence Results......Page 31
1.3.2 Convergence Rates in Xs......Page 32
1.4 The Continuous Approach......Page 34
1.4.1 Proof of Theorem 1.2......Page 35
1.5 Improved Convergence Rates: The Discrete Approach......Page 36
1.5.1 Proof of Theorem 1.4......Page 37
1.6.1 The Number of Iterations......Page 38
1.6.2 Controlling Non-smooth Data......Page 39
1.6.3 Other Minimization Algorithms......Page 41
1.7.1.1 The Continuous Case......Page 42
1.7.1.2 The Continuous Approach......Page 45
1.7.1.3 The Discrete Approach......Page 52
1.7.2 Distributed Control......Page 58
1.7.2.1 The Continuous Setting......Page 59
1.7.2.2 The Continuous Approach......Page 61
1.8.1 The Setting......Page 62
1.8.2 Numerical Approximation Methods......Page 64
1.8.2.1 The Continuous Approach......Page 65
1.8.2.2 The Discrete Approach......Page 66
2.1 Objectives......Page 67
2.2 Spectral Decomposition of the Discrete Laplacian......Page 68
2.3.1 The Multiplier Identity......Page 69
2.3.2 Proof of the Uniform Hidden Regularity Result......Page 70
2.4.1 Equipartition of the Energy......Page 71
2.4.2 The Multiplier Identity Revisited......Page 72
2.4.3.1 Estimates on  Yh(t)......Page 73
2.4.3.2 Estimates on Xh(t)......Page 74
2.4.4 Proof of Theorem 2.3......Page 76
3.2 Extension Operators......Page 77
3.2.2 Other Extension Operators......Page 78
3.3 Orders of Convergence for Smooth Initial Data......Page 82
3.4.1 Strongly Convergent Initial Data......Page 88
3.4.2 Smooth Initial Data......Page 89
3.4.3 General Initial Data......Page 91
3.4.4 Convergence Rates in Weaker Norms......Page 92
3.5 Numerics......Page 93
4.1 The Setting......Page 97
4.2.1 Natural Functional Spaces......Page 99
4.2.2 Stronger Norms......Page 104
4.2.3 Numerical Results......Page 106
4.3 Uniform Bounds on yh......Page 107
4.3.1 Estimates in C([0,T]; L2(0,1))......Page 108
4.3.2 Estimates on t yh......Page 111
4.4.1 Main Convergence Result......Page 116
4.4.2.1 Computation of 01 y(T) T......Page 118
4.4.2.2 Computation of 01 yh(T) T-.4......Page 119
4.4.2.3  Estimating the Difference 01 y(T) T- 01 yh(T) T-.4......Page 121
4.4.3 Convergence of t yh......Page 122
4.4.4 More Regular Data......Page 124
4.5 Further Convergence Results......Page 128
4.6 Numerical Results......Page 129
5.1 Discrete Versus Continuous Approaches......Page 133
5.2 Comparison with Russell\'s Approach......Page 134
5.4 Optimal Control Theory......Page 135
5.5 Fully Discrete Approximations......Page 136
References......Page 137