Some Successive Approximation Methods in Control and Oscillation Theory

249 Pages·1969·2.43 MB·English

by  Peter L. Falb and Jan L. de Jong (Eds.)

SOME SUCCESSIVE APPROXIMATION METHODS
IN CONTROL AND OSCILLATION THEORY

SOME SUCCESSIVE
APPROXIMATION METHODS
IN CONTROL AND OSCILLATION
THEORY
Peter L. Falb
Division of Applied Mathematics
Brown University
Providence, Rhode Island
Jan L. de Jong
National Aerospace Laboratory NLR
Noordoostpolder, The Netherlands
1969
ACADEMIC PRESS
New York and London
0
COPYRIGHT 1969, BY ACADEMIPCR ESSI:N C.
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PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
Successive approximation methods have been used for the solution
of two-point boundary value problems for a number of years. In this
book, we examine several of these methods. Noting that two-point
boundary value problems can be represented by operator equations,
we adopt a functional analytic viewpoint and translate results on such
operator theoretic iterative methods as Newton’s method into the
two-point boundary value problem context. Our emphasis is on
results of potential practical applicability rather than on results of the
greatest generality.
We owe a significant debt of gratitude to many of our colleagues
for their invaluable assistance in the preparation of this book. In
particular, we wish to thank Dr. W. E. Bosarge, Jr., of IBM, Professor
Elmer Gilbert of the University of Michigan, and Professor Jack
Hale of Brown University for their numerous helpful suggestions
and comments. We also gratefully acknowledge the support that we
have received from the United States Air Force under Grant No.
AFOSR 693-67 and Grant No. AFOSR 814-66 and from the National
Science Foundation under Grant No. GK-967 and Grant No.
GK-2788. Finally, we should like to express our deep appreciation
to Miss Kate Nolan for her excellent typing of the manuscript.
July I969 PETERL . FALB
JANL . DE JONG
V

CONTENTS
Preface V
CHAPTER 1 IN TRODU C TlO N
1 .I. Introduction
1.2. Control Problems and Historical Notes
1.3. Description of Contents
CHAPTER 2 OPERATOR THEORETIC ITERATIVE METHODS
2.1. Introduction 7
2.2. The Method of Contraction Mappings 7
2.3. The Modified Contraction Mapping Method 21
2.4. Newton’s Method 26
2.5. Multipoint Methods 45
CHAPTER 3 REPRESENTATION OF BOUNDARY VALUE PROBLEMS
3.1. lntrodu ction 59
3.2. Continuous Linear Two-Point Boundary Value Problems 60
3.3. Discrete Linear Two-Point Boundary Value Problems 63
3.4. Representation of Continuous Two-Point Boundary
Value Problems 67
3.5. Representation of Discrete Two-Point Boundary
Value Problem 71
3.6. A Continuous Example 75
3.7. A Discrete Example 79
3.8. Computation of Derivatives: Continuous Case 82
3.9. Computation of Derivatives: Discrete Case 88
3.10 A Lemma on Equivalence: Continuous Case 94
3.11. A Lemma on Equivalence: Discrete Case 97
Appendix. Lipschitz Norms 102
vii
viii CONTENTS
CHAPTER 4 APPLICATION TO CONTROL PROBLEMS
4.1. Introduction 104
4.2. Continuous Control Problems 104
4.3. A Continuous Example 110
4.4. Discrete Control Problems 112
4.5. Application to Continuous Problems I:
The Method of Contraction Mappings 115
4.6. Application to Continuous Problems II:
The Modified Contraction Mapping Method 134
4.7. Application to Continuous Problems 111:
Newton’s Method 137
4.8. Application to Continuous Problems IV:
Multipoint Methods 150
4.9. Application to Discrete Problems I:
The Method of Contraction Mappings 153
4.10. Application to Discrete Problems II:
The Modified Contraction Mapping Method 164
4.11. Application to Discrete Problems 111:
Newton’s Method 167
4.12. Summary 171
CHAPTER 5 APPLICATION TO OSCILLATION PROBLEMS
5.1. Introduction 17 3
5.2. Almost Linear Problems 174
5.3. Some Second-Order Examples 184
CHAPTER 6 SOME NUMERICAL EXAMPLES
6.1. Introduction 196
6.2. Constant Low-Thrust Earth-to-Mars Orbital Transfer 197
6.3. Variable Low-Thrust Earth-to-Mars Orbital Transfer 218
6.4. An Oscillation Problem 226
REFERENCES 230
AUTHOR INDEX 235
SUBJECT INDEX 237
CHAPTER 1
INTRODUCTION
.
1. l Introduction
The central theme of this book is the study of some iterative
methods for the solution of two-point boundary value problems
(TPBVP’s) of the form
or, of the form
where F,g , and h are suitable vector valued functions. and c is a
constant vector. Such TPBVP’s arise in control and oscillation
problems.
The basic approach which we use is to represent the TPBVP
by an operator equation and then apply functional analytic results on
the iterative solution of operator equations. In other words, (1.1)
or (1.2) is represented by an equivalent operator equation
on a suitable Banach space and results relating to convergence of
algorithms of the form yn,-l = v(yn) for the solution of (1.3) are
translated into convergence theorems for the iterative solution of (1.1)
or (1.2). In particular, we examine the contraction mapping method,
Newton’s method and some multipoint methods. Our goal is to obtain
convergence conditions and rates which depend upon the functions
F, g, and h and other quantities known a priori.
1