Control Theoretic Splines

Preface ix

Chapter 1: INTRODUCTION 1

1.1 From Interpolation to Smoothing 1

1.2 Background 2

1.3 The Introduction of Control Theory 4

1.4 Applications 7

1.5 Topical Outline of the Book 8

Chapter 2: CONTROL SYSTEMS AND MINIMUM NORM PROBLEMS 11

2.1 Linear Control Systems 11

2.2 Hilbert Spaces 14

2.3 The Projection Theorem 15

2.4 Optimization and Gateaux Derivatives 18

2.5 The Point-to-Point Transfer Problem 21

Chapter 3: EIGHT FUNDAMENTAL PROBLEMS 25

3.1 The Basic Set-Up 26

3.2 Interpolating Splines 29

3.3 Interpolating Splines with Constraints 31

3.4 Smoothing Splines 35

3.5 Smoothing Splines with Constraints 38

3.6 Dynamic Time Warping 45

3.7 Trajectory Planning 48

Chapter 4: SMOOTHING SPLINES AND GENERALIZATIONS 53

4.1 The Basic Smoothing Problem 56

4.2 The Basic Algorithm 60

4.3 Interpolating Splines with Initial Data 62

4.4 Problems with Additional Constraints 63

Chapter 5: APPROXIMATIONS AND LIMITING CONCEPTS 73

5.1 Basic Assumptions 73

5.2 Convergence of the Smoothing Spline 75

5.3 Quadrature Schemes 80

5.4 Rate of Convergence 82

5.5 Cubic Spline Convergence Bounds 83

Chapter 6: SMOOTHING SPLINES WITH CONTINUOUS DATA 87

6.1 Continuous Data 89

6.2 The Continuous Smoothing Problem 89

6.3 The Basic Two-Point Boundary Value Problem 91

6.4 The General Two-Point Boundary Value Problem 95

6.5 Multipoint Problems 99

6.6 Recursive Splines 101

Chapter 7: MONOTONE SMOOTHING SPLINES 113

7.1 The Monotone Smoothing Problem 113

7.2 Properties of the Solution 115

7.3 Dynamic Programming 118

7.4 Monotone Cubic Splines 120

7.5 Probability Densities 126

Chapter 8: SMOOTHING SPLINES AS INTEGRAL FILTERS 133

8.1 Smoothing Concepts 133

8.2 Splines from Statistical Data 136

8.3 The Optimal Control Problem 141

8.4 The Cubic Smoothing Spline 146

Chapter 9: OPTIMAL TRANSFER BETWEEN AFFINE VARIETIES 155

9.1 Point-to-Point Transfer 155

9.2 Transfer between Affine Varieties 156

9.3 Transfer through Dynamic Programming 158

9.4 A Multi-Agent Problem 164

Chapter 10: PATH PLANNING AND TELEMETRY 169

10.1 The Telemetry Problem 169

10.2 Splines on Spheres 171

10.3 Splines and Bezier Curves 176

10.4 Conflict Resolution for Autonomous Vehicles 185

Chapter 11: NODE SELECTION 193

11.1 Background 193

11.2 Sampling for Interpolation and Smoothing 194

11.3 Optimal Timing Control 195

11.4 Applications to Smoothing Splines 199

Bibliography 205

Index 215