The Ulam Problem of Optimal Motion of Line Segments, 1985, 128 pp., ISBN 0-911575-04-9, $58.00
THIS TITLE IS OUT OF PRINT BUT THE PUBLISHER CAN PREPARE A COPY
Investigates the general problem of motion of line segments
by the integral maximum principle. Includes many new results of potential
application to optimization problems with state constraints. "In this remarkable
book, Dr. Dubovitskij has succeeded in solving in closed form, using the
Dubovitskij-Milyutin theory, a generalization of a problem of Stanislaw
Ulam ..." -- from the foreword by M.R. Hestenes.
TABLE OF CONTENTS
Preface ..... xi
CHAPTER 1. GENERAL VARIATIONAL STUDY
OF THE ULAM PROBLEM ..... 1
1. Regularity of the Lumped Constraints ..... 3
2. The Maximum Principle ..... 4
3. Analysis of the Local Maximum Principle ..... 8
4. Stationary Sets ..... 17
5. Operation of Parallel Displacement of Vectors on Surfaces and Equations of a Stationary Set ..... 20
6. Standard Regimes ..... 22
CHAPTER 2. OPTIMAL MOTION OF SEGMENTS IN THE n-DIMENSIONAL
SPACE AND ON THE n-DIMENSIONAL SPHERE ..... 29
1. The Sliding Regime ..... 30
2. The Rotating Regime ..... 33
3. The Degenerate Regime ..... 36
CHAPTER 3. OPTIMAL MOTION OF SEGMENTS
ON THE CIRCULAR CYLINDER ..... 47
1. The Sliding Regime ..... 48
2. The Rotating Regime ..... 53
3. The Degenerate Regime ..... 59
4. Description of Extremals ..... 61
CHAPTER 4. OPTIMAL MOTION OF SEGMENTS
ON THE ANISOTROPIC PLANE ..... 65
CHAPTER 5. SYNTHESIS OF OPTIMAL MOTIONS OF SEGMENTS
IN THE SPACE AND ON THE PLANE ..... 73
1. Geometric Properties of Extremals ..... 74
2. Optimal Motion of Segments in the Space E3 ..... 79
3. Atlas of Optimal Motions of Segments on the Plane .....
83
Appendix. The Maximum Principle in Optimal Control Problems with Regular Constraints ..... 105
Bibliography ..... 111
Index ..... 113