Differential Games: Theory and Methods for Solving Game Problems with Singular Surfaces

259 Pages·1994·7.3 MB·English

by  Joseph Lewin (auth.)

Differential Games
Theory and Methods for Solving Game Problems with Singular Surfaces
Joseph Lewin
Differential Games
Theory and Methods for Solving
Game Problems with Singular Surfaces,
With 40 Figures
Springer-Verlag
London Berlin Heidelberg New York
Paris Tokyo Hong Kong
Barcelona Budapest
Joseph Lewin
Faculty of Aerospace Engineering, Technion-Israel Institute of Technology,
Haifa 32000, Israel
ISBN-13:978-1-4471-2067-4 e-ISBN-13:978-1-4471-2065-O
DOl: 10.1007/978-1-4471-2065-0
Apart from any fair dealing for the purposes of research or private study, or criticism
or review, as permitted under the Copyright, Designs and Patents Act 1988, this
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Springer-Verlag London Limited 1994
@
Softcoverreprintof the hardcover 1st edition 1994
The publisher makes no representation, express or implied, with regard to the accuracy
of the information contained in this book and cannot accept any legal responsibility or
liability for any errors or omissions that may be made.
Typesetting: Camera ready by author
Printed at the Alden Press, Oxford.
69/3830-543210 Printed on acid-free paper
To my wife, Sara,
whose love, encouragement and support
made the completion of this work possible.
List of Figures
1.1 The vectograms . . . . . . 2
1.2 The combined vectogram 3
2.1 Reference frames in the Homicidal Chauffeur game 8
2.2 The map from information into control. 17
2.3 The Obstacle Tag game ........... . 21
4.1 Semipermeable surfaces in the servo problem 36
5.1 A hodograph representation of MEl 53
5.2 A case with nonunique solution . . . 55
5.3 u* on a smooth boundary of U . . . . 59
5.4 u* on a corner on the boundary of U 60
7.1 Optimal trajectories for Example 7.3.3 84
7.2 The Lady in the Lake ...... . . . 86
7.3 Value map for the Lady in the Lake problem 89
9.1 Safe contact with a tangential junction. 124
9.2 Safe contact with a transversal junction 125
9.3 A dispersal surface 126
9.4 A universal surface .. 127
9.5 A focal surface .... 127
9.6 An equivocal surface .. 128
9.7 A switch envelope ... 129
10.1 The chatter equivalent of a singular arc with transversal trib
utaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 150
10.2 The chatter equivalent of a singular Arc with tangential
Tributaries ........................ . .. 152
11.1 The Surveillance Evasion game . . . . . . . . 158
11.2 Isochrones for the Surveillance Evasion game 163
12.1 The first field of regular optimal trajectories. 175
viii
12.2 The chatter equivalent of the safe contact motion. 175
12.3 Optimal trajectories for the Dolichbrachistochrone game 177
12.4 The Cone Surveillance Evasion problem 178
12.5 The reference frame for Example 12.5.4 ... . . . . . . 180
13.1 The reference frames for the Homicidal Chauffeur problem. 188
13.2 First field of candidate optimal trajectories ......... 192
13.3 The chatter equivalent for the motion on the Universal Surface 194
13.4 The reference frames for the Circular Wall Pursuit problem 199
13.5 The isochrones in the relative frame ..... . . . . . . . . 203
13.6 The chatter equivalent for the motion on the Focal Surface 205
14.1 Switch envelope in the Surveillance Evasion game. . . . . . 216
14.2 The chatter equivalent for the motion on the switch envelope 217
14.3 The first field of candidate optimal trajectories 222
14.4 Vectograms for a point on the equivocal surface. 223
15.1 Reference frames for the Lion and Man problem 231
Contents
Preface xv
1 A Preview Example 1
1.1 Introduction ........ . 1
1.2 A Simple Differential Game 1
1.3 Preliminary Analysis 2
1.4 A Heuristic Solution 3
1.5 Problems ..... . 4
2 The Vocabulary For Differential Games 6
2.1 Introduction ............. . 6
2.2 The State Vector and the Game-Set 7
2.3 The Equations of Motion ..... . 8
2.4 Termigation of a Differential Game. 9
2.5 Plays ... 11
2.6 Outcomes . . . . . . . . . . . . . . 12
2.7 Strategies ............. . 16
2.7.1 Decisions and Information. 16
2.7.2 Realizations of Strategies . 17
2.7.3 Strategies that Guarantee Nontermination . 17
2.7.4 Strategies that Guarantee Termination . 18
2.7.5 Admissibility of Strategies. 19
2.8 Problems .................... . 22
3 The Solution Concept 25
3.1 Introduction ........... . 25
3.2 The Solution Quintet ...... . 25
3.3 The Extended Solution Concept 28
3.4 Problems .. . . . . . . . . . 30
4 Semipermeability of Surfaces 33
4.1 Introduction.......... 33
4.2 Smooth Semipermeable Surfaces 33
x
4.3 Semipermeability of Composite Surfaces ..... . 36
4.3.1 Leaking Corners .............. . 36
4.3.2 A Modified Definition of Semipermeability. 37
4.4 Problems ...................... . 38
5 Necessary Conditions 39
5.1 Introduction............... 39
5.2 Properties of the Target Set . . . . . . 40
5.2.1 Partitioning of the Target Set. 40
5.2.2 The Relation Between J(x) and G(x) 42
5.3 Semipermeability of the Boundary of the Escape Set :F . 43
5.4 Properties of Optimal Trajectories . . . . 44
5.4.1 Principle of Optimality (weak) . . 44
5.4.2 Continuity Of The Value Function 46
5.4.3 j(x) Along Optimal Trajectories 47
5.4.4 The Hamiltonian on Optimal Trajectories 48
5.5 The Isaacs Equations. . . . . . . . . . . . . . . 48
5.5.1 Semi-Local Deviations From Optimality 48
5.5.2 The Isaacs Main Equations . . . . . . . 50
5.5.3 The hodograph representations of MEl 52
5.5.4 The Viscosity Form of Isaacs Equations 55
5.6 The Adjoint Equations. . . . . . . . . . . . . . 57
5.6.1 The Retro Time Form of the Adjoint Equations. 61
5.7 Problems .. . . . . . . . . . . . . . . . . . . . . . . . . 61
6 Sufficient Conditions 65
6.1 Introduction........ 65
6.2 The Sufficiency Theorem. . 65
6.3 Validity of Partial Solutions 67
6.4 Estimatioms of the Value Function 67
6.5 Problems .............. 69
7 Regular Construction 70
7.1 Introduction............. 70
7.2 The Regular Procedure ...... 71
7.2.1 Partitioning the Target-Set 71
7.2.2 Candidate Optimal Control Laws. 71
7.2.3 Retro-Integration of the Adjoint Equations 72
7.2.4 Properties of the Manifolds of Candidate Optimal
Trajectories . . . . 73
7.3 Examples . . . . . . . . . 76
7.4 Linear Quadratic Games . 89
7.4.1 Introduction ... 89
7.4.2 LQG with Fixed Duration and Unbounded Controls 90
7.4.3 Infinite Horizon Linear Quadratic Games . . . . .. 94
xi
7.4.4 LQG and Controller Design 102
7.5 Problems .............. 106
8 Construction of SPS 113
8.1 Introduction................ 113
8.2 Construction of Semipermeable Surfaces 114
8.2.1 The Regular Construction. . . . 114
8.2.2 Semipermeability of the Constructed Manifold 116
8.3 Examples 120
8.4 Problems ......................... 122
9 A Topography of the Value Map 123
9.1 Introduction........ 123
9.2 Barriers and Safe Contact 123
9.2.1 Barriers..... 123
9.2.2 State Costraints 124
9.2.3 Safe Contact .. 124
9.2.4 The Tributaries. 124
9.3 Switch Surfaces . . . . . 125
9.4 Dispersal Surfaces ... 125
9.5 Universal and Focal Surfaces 126
9.5.1 General characterization. 126
9.5.2 Universal Surfaces 126
9.5.3 Focal Surfaces ...... 126
9.6 Corner Surfaces. . . . . . . . . . 128
9.6.1 General characterization. 128
9.6.2 Equivocal Surfaces . . . . 128
9.6.3 Switch Envelopes. . . . . 128
10 Necessary Conditions (Singular) 130
10.1 Introduction. . . . . . . 130
10.2 The Projection Lemma. . . . . . 131
10.3 Open Barriers. .......... 132
10.4 Isaacs Equations for Singular Arcs 134
10.4.1 The Hamiltonian on Singular Surfaces 134
10.4.2 Isaacs Theorems for Singular Arcs 135
10.4.3 Hamiltonians on Seams . . . 136
10.5 Junctions to Singular Arcs. . . . . . 137
10.5.1 Controls Along Singular Arcs 137
10.5.2 The Junction Conditions .. 141
10.6 Adjoint Equations for Singular Arcs 143
10.7 Properties of Regular Switch Surfaces 147
10.8 The Chatter Equivalent of Singular Arcs. 148
10.8.1 Introduction ............ 148
10.8.2 Singular Arcs with Tributaries Joining Transversely 148
xii
10.8.3 Singular Arcs with Tributaries Joining Tangentially 151
10.9 Sufficient conditions 153
10.10Problems ........................... . 153
11 Dispersal Surfaces 155
11.1 introduction .............. . 155
11.2 Region of Multiple Choices ..... . 155
11.3 Characterization of Dispersal Surfaces 155
11.4 Examples 157
11.5 Problems ............... . 164
12 Singular Arcs of Safe Contact 167
12.1 Introduction ......... . 167
12.2 Characterization of Safe Contact . 167
12.3 Construction of Safe Contact Arcs 168
12.3.1 Introduction ....... . 168
12.3.2 Safe Contact with Tangential Junctions 169
12.3.3 Safe Contact with Transversal Junctions. 170
12.4 Examples 170
12.5 Problems ..................... . 177
13 Universal and Focal Surfaces 182
13.1 Introduction ........ . 182
13.2 Characterization of Universal Surfaces 182
13.3 Examples ............. . 188
13.3.1 The Chatter Equivalent .. 193
13.4 Characterization of Focal Surfaces 194
13.5 Construction of Focal Surfaces 197
13.6 An Example of a Focal Surface 198
13.6.1 The Chatter Equivalent 205
13.7 Problems ........... . 206
14 Corner Surfaces 208
14.1 Introduction. 208
14.2 Characterization of Corner Surfaces 208
14.3 The Switch Envelope ... 213
14.4 Chatter Equivalent of SE 216
14.5 The Equivocal Surface .. 217
14.6 Chatter Equivalent of ES 218
14.7 Problems ..... . 225
15 The Envelope Barrier 227
15.1 Introduction ..... 227
15.2 The Envelope Barrier ... 227
15.2.1 Dominated Surfaces 227