Splines, both interpolatory and smoothing, have a long and rich history that has largely been application driven. This book unifies these constructions in a comprehensive and accessible way, drawing from the latest methods and applications to show how they arise naturally in the theory of linear control systems. Magnus Egerstedt and Clyde Martin are leading innovators in the use of control theoretic splines to bring together many diverse applications within a common framework. In this book, they begin with a series of problems ranging from path planning to statistics to approximation. Using the tools of optimization over vector spaces, Egerstedt and Martin demonstrate how all of these problems are part of the same general mathematical framework, and how they are all, to a certain degree, a consequence of the optimization problem of finding the shortest distance from a point to an affine subspace in a Hilbert space. They cover periodic splines, monotone splines, and splines with inequality constraints, and explain how any finite number of linear constraints can be added. This book reveals how the many natural connections between control theory, numerical analysis, and statistics can be used to generate powerful mathematical and analytical tools. This book is an excellent resource for students and professionals in control theory, robotics, engineering, computer graphics, econometrics, and any area that requires the construction of curves based on sets of raw data.

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