Table Of ContentStatic & Dynamic Game Theory:
Foundations & Applications
Series Editor
Tamer Ba¸sar, University of Illinois, Urbana-Champaign, IL, USA
Editorial Board
Daron Acemoglu, Massachusetts Institute of Technology, Cambridge, MA, USA
Pierre Bernhard, INRIA, Sophia-Antipolis, France
Maurizio Falcone , Università degli Studi di Roma “La Sapienza”, Roma, Italy
Alexander Kurzhanski, University of California, Berkeley, CA, USA
Ariel Rubinstein, Tel Aviv University, Ramat Aviv, Israel
Yoav Shoham, Stanford University, Stanford, CA, USA
Georges Zaccour, GERAD, HEC Montréal, QC, Canada
·
Valery Y. Glizer Oleg Kelis
Singular Linear-Quadratic
Zero-Sum Differential
Games and H Control
∞
Problems
Regularization Approach
Valery Y. Glizer Oleg Kelis
The Galilee Research Center for Applied The Galilee Research Center for Applied
Mathematics Mathematics
ORT Braude College of Engineering ORT Braude College of Engineering
Karmiel, Israel Karmiel, Israel
Faculty of Mathematics
Technion-Israel Institute of Technology
Haifa, Israel
ISSN 2363-8516 ISSN 2363-8524 (electronic)
Static & Dynamic Game Theory: Foundations & Applications
ISBN 978-3-031-07050-1 ISBN 978-3-031-07051-8 (eBook)
https://doi.org/10.1007/978-3-031-07051-8
Mathematics Subject Classification: 49N70, 91A05, 91A10, 91A23, 93B36, 93C70, 93C73
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
Switzerland AG 2022
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether
the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse
of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and
transmission or information storage and retrieval, electronic adaptation, computer software, or by similar
or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, expressed or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.
This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered
company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To my family Elena, Evgeny and Svetlana
V.Y.G.
To my wife Tanya and my children Louisa,
Nadav and Gavriel
O.K.
Contents
1 Introduction ................................................... 1
References ..................................................... 4
2 Examples of Singular Extremal Problems and Some Basic
Notions ........................................................ 7
2.1 Introduction .............................................. 7
2.2 Academic Examples ....................................... 8
2.2.1 Minimization of a Function ........................... 8
2.2.1.1 The First Way of the Reformulation
of (2.1)–(2.2) ............................... 8
2.2.1.2 The Second Way of the Reformulation
of (2.1)–(2.2) ............................... 9
2.2.2 Optimal Control Problem ............................ 11
2.2.2.1 The First Way of the Reformulation
of (2.11)–(2.12), (2.13) ...................... 13
2.2.2.2 The Second Way of the Reformulation
of (2.11)–(2.12), (2.13) ...................... 14
2.2.3 Saddle Point of a Function of Two Variables ............ 18
2.2.3.1 The First Way of the Reformulation
of the Saddle-Point Problem
for the Function (2.45)–(2.46) ................ 20
2.2.3.2 The Second Way of the Reformulation
of the Saddle-Point Problem
for the Function (2.45)–(2.46) ................ 21
2.2.4 Zero-Sum Differential Game ......................... 23
2.2.4.1 The First Way of the Reformulation
of the Game (2.57)–(2.58) .................... 32
2.2.4.2 The Second Way of the Reformulation
of the Game (2.57)–(2.58) .................... 33
vii
viii Contents
2.3 Mathematical Models of Real-Life Problems .................. 37
2.3.1 Planar Pursuit-Evasion Engagement: Zero-Order
Dynamics of Participants ............................. 37
2.3.2 Planar Pursuit-Evasion Engagement: First-Order
Dynamics of Participants ............................. 39
2.3.3 Three-Dimensional Pursuit-Evasion Engagement:
Zero-Order Dynamics of Participants .................. 40
2.3.4 Infinite-Horizon Robust Vibration Isolation ............. 42
2.4 Concluding Remarks and Literature Review ................... 44
References ..................................................... 49
3 Preliminaries .................................................. 51
3.1 Introduction .............................................. 51
3.2 Solvability Conditions of Regular Problems ................... 52
3.2.1 Finite-Horizon Differential Game ..................... 52
3.2.2 Infinite-Horizon Differential Game .................... 56
3.2.3 Finite-Horizon H∞ Problem .......................... 60
3.2.4 Infinite-Horizon H∞ Problem ......................... 62
3.3 State Transformation in Linear System and Quadratic
Functional ................................................ 66
3.4 Concluding Remarks and Literature Review ................... 70
References ..................................................... 71
4 Singular Finite-Horizon Zero-Sum Differential Game ............. 73
4.1 Introduction .............................................. 73
4.2 Initial Game Formulation ................................... 74
4.3 Transformation of the Initially Formulated Game .............. 76
4.4 Regularization of the Singular Finite-Horizon Game ............ 79
4.4.1 Partial Cheap Control Finite-Horizon Game ............ 79
4.4.2 Saddle-Point Solution of the Game (4.8), (4.12) ......... 79
4.5 Asymptotic Analysis of the Game (4.8), (4.12) ................ 81
4.5.1 Transformation of the Problem (4.14) .................. 81
4.5.2 Asymptotic Solution of the Terminal-Value Problem
(4.23)–(4.25) ....................................... 83
4.5.3 Asymptotic Representation of the Value of the Game
(4.8), (4.12) ........................................ 88
4.6 Reduced Finite-Horizon Differential Game .................... 89
4.7 Saddle-Point Sequence of the SFHG ......................... 90
4.7.1 Main Assertions .................................... 90
4.7.2 Proof of Lemma 4.5 ................................. 91
4.7.3 Proof of Lemma 4.6 ................................. 96
4.7.3.1 Stage 1: Regularization of (4.85)–(4.86) ........ 97
4.7.3.2 Stage 2: Asymptotic Analysis
of the Problem (4.85), (4.90) ................. 97
4.7.3.3 Stage 3: Deriving the Expression
for the Optimal Value of the Functional
in the Problem (4.85)–(4.86) .................. 100
Contents ix
4.8 Examples ................................................ 101
4.8.1 Example 1 ......................................... 101
4.8.2 Example 2: Solution of Planar Pursuit-Evasion
Game with Zero-Order Dynamics of Players ............ 102
4.8.3 Example 3: Solution of Three-Dimensional
Pursuit-Evasion Game with Zero-Order Dynamics
of Players .......................................... 104
4.8.3.1 Pursuit-Evasion Game with One “Regular”
and One “Singular” Coordinates
of the Pursuer’s Control ...................... 105
4.8.3.2 Pursuit-Evasion Game with Both
“Singular” Coordinates of the Pursuer’s
Control .................................... 107
4.9 Concluding Remarks and Literature Review ................... 109
References ..................................................... 110
5 Singular Infinite-Horizon Zero-Sum Differential Game ............ 113
5.1 Introduction .............................................. 113
5.2 Initial Game Formulation ................................... 114
5.3 Transformation of the Initially Formulated Game .............. 116
5.4 Auxiliary Lemma ......................................... 118
5.5 Regularization of the Singular Infinite-Horizon Game .......... 120
5.5.1 Partial Cheap Control Infinite-Horizon Game ........... 120
5.5.2 Saddle-Point Solution of the Game (5.9), (5.29) ......... 121
5.6 Asymptotic Analysis of the Solution to the Game (5.9), (5.29) ... 122
5.6.1 Transformation of the Eq. (5.33) ...................... 122
5.6.2 Asymptotic Solution of the Set (5.44)–(5.46) ............ 124
5.6.3 Asymptotic Representation of the Value of the Game
(5.9), (5.29) ........................................ 129
5.7 Reduced Infinite-Horizon Differential Game .................. 129
5.8 Saddle-Point Sequence of the SIHG .......................... 131
5.8.1 Main Assertions .................................... 131
5.8.2 Proof of Lemma 5.6 ................................. 133
5.8.3 Proof of Lemma 5.7 ................................. 135
5.8.3.1 Stage 1: Regularization of (5.95)–(5.96) ........ 136
5.8.3.2 Stage 2: Asymptotic Analysis
of the Problem (5.95), (5.100) ................ 137
5.8.3.3 Stage 3: Deriving the Expression
for the Optimal Value of the Functional
in the Problem (5.95)–(5.96) .................. 139
5.9 Examples ................................................ 140
5.9.1 Example 1 ......................................... 140
5.9.2 Example 2: Solution of Infinite-Horizon Vibration
Isolation Game ..................................... 141
5.10 Concluding Remarks and Literature Review ................... 145
References ..................................................... 147
x Contents
6 Singular Finite-Horizon H∞ Problem ............................ 149
6.1 Introduction .............................................. 149
6.2 Initial Problem Formulation ................................. 150
6.3 Transformation of the Initially Formulated H∞ Problem ........ 151
6.4 Regularization of the Singular Finite-Horizon H∞ Problem ...... 153
6.4.1 Partial Cheap Control Finite-Horizon H∞ Problem ...... 153
6.4.2 Solution of the H∞ Problem (6.9), (6.11) ............... 154
6.5 Asymptotic Analysis of the H∞ Problem (6.9), (6.11) .......... 154
6.5.1 Transformation of the Problem (6.13) .................. 155
6.5.2 Asymptotic Solution of the Problem (6.20)–(6.22) ....... 156
6.6 Reduced Finite-Horizon H∞ Problem ........................ 159
6.7 Controller for the SFHP .................................... 160
6.7.1 Formal Design of the Controller ....................... 160
6.7.2 Properties of the Simplified Controller (6.39) ........... 162
6.7.3 Proof of Theorem 6.1 ................................ 163
6.7.4 Proof of Theorem 6.2 ................................ 168
6.7.4.1 Auxiliary Proposition ........................ 168
6.7.4.2 Main Part of the Proof ....................... 169
6.8 Example ................................................. 171
6.9 Concluding Remarks and Literature Review ................... 172
References ..................................................... 173
7 Singular Infinite-Horizon H∞ Problem ........................... 175
7.1 Introduction .............................................. 175
7.2 Initial Problem Formulation ................................. 176
7.3 Transformation of the Initially Formulated H∞ Problem ........ 177
7.4 Regularization of the Singular Infinite-Horizon H∞ Problem .... 179
7.4.1 Partial Cheap Control Infinite-Horizon H∞ Problem ..... 179
7.4.2 Solution of the H∞ Problem (7.9), (7.15) ............... 179
7.5 Asymptotic Analysis of the H∞ Problem (7.9), (7.15) .......... 180
7.5.1 Transformation of Eq. (7.17) ......................... 180
7.5.2 Asymptotic Solution of the Set (7.24) .................. 181
7.6 Reduced Infinite-Horizon H∞ Problem ....................... 184
7.7 Controller for the SIHP ..................................... 185
7.7.1 Formal Design of the Controller ....................... 185
7.7.2 Properties of the Simplified Controller (7.45) ........... 187
7.7.3 Proof of Theorem 7.1 ................................ 188
7.7.4 Proof of Theorem 7.2 ................................ 191
7.8 Examples ................................................ 195
7.8.1 Example 1 ......................................... 195
7.8.2 Example 2: H∞ Control in Infinite-Horizon
Vibration Isolation Problem .......................... 196
7.9 Concluding Remarks and Literature Review ................... 199
References ..................................................... 200
Index ............................................................. 201
