Differential Games

by Avner Friedman

• ISBN13: 9780486449951
• ISBN10: 0486449955
• Format: Paperback
• Copyright: 2006-05-12
• Publisher: Dover Publications

Summary

This volume lays the mathematical foundations for the theory of differential games, developing a rigorous mathematical framework with existence theorems. Topics include games of fixed duration, games of pursuit and evasion, the computation of saddle points, games of survival, and games with restricted phase coordinates. 1971 edition.

Table of Contents

Introduction
1. Definition of a Differential Game
1.1. Results on ordinary differential equations	4	(2)
1.2. Differential equations with control functions	6	(3)
1.3. Upper and lower δ-strategies and δ-games	9	(3)
1.4. Properties of upper and lower δ-values	12	(7)
1.5. Definition of a differential game and of its value	19	(2)
1.6. Strategies and payoff sets	21	(6)
1.7. Examples	27	(5)
2. Games of Fixed Duration
2.1. Generalization of the concept of a differential game	32	(2)
2.2. Existence of upper and lower values	34	(4)
2.3. Existence of value	38	(5)
2.4. Compactness of the space of trajectories	43	(7)
2.5. Existence of a saddle point	50	(2)
2.6. Continuity of the value	52	(6)
2.7. Construction of a saddle point	58	(6)
2.8. Computation of a saddle point for a class of games	64	(14)
3. Games of Pursuit and Evasion
3.1. Existence of extended value	78	(3)
3.2. Pursuer's upper δ-strategies near the terminal set	81	(6)
3.3. Pursuer's lower δ-strategies near the terminal set	87	(2)
3.4. Existence of value	89	(5)
3.5. Existence of a saddle point	94	(3)
3.6. Continuity of the value	97	(4)
3.7. Construction of a saddle point	101	(6)
3.8. The dolichobrachistochrone game	107	(9)
3.9. Capturability	116	(6)
4. Computation of Saddle Points
4.1. Rademacher's theorem	122	(4)
4.2. Derivation of the Isaacs equation	126	(7)
4.3. The max-min principle for synthesized saddle points	133	(15)
4.4. Solution of the Hamilton-Jacobi equation	148	(9)
4.5. a-saddle points	157	(7)
4.6. Construction of η-saddle points for games of fixed duration	164	(6)
4.7. Construction of η-saddle points for games of pursuit-evasion	170	(3)
4.8. The isotropic rocket game (special case)	173	(12)
4.9. The isotropic rocket game (general case)	185	(7)
5. Games of Survival
5.1. Reduction to games with terminal payoff	192	(2)
5.2. Existence of value	194	(3)
5.3. Existence of a saddle point	197	(2)
5.4. Continuity of the value	199	(5)
5.5. Computation of saddle points	204	(3)
6. Games with Restricted Phase Coordinates
6.1. Definition of a game with restricted phase coordinates	207	(3)
6.2. Existence of value	210	(13)
6.3. Existence of value (continued)	223	(7)
6.4. Examples	230	(4)
6.5. Existence of an X-saddle point	234	(2)
6.6. Continuity of the value	236	(2)
6.7. Computation of X-saddle points	238	(3)
6.8. Players satisfying separate equations	241	(2)
6.9. Games of survival	243	(3)
7. Selected Topics
7.1. Games with integral bounds on the controls	246	(9)
7.2. Capturability	255	(12)
7.3. Games with delayed information	267	(4)
7.4. Games with partial differential equations	271	(7)
7.5. Fleming's definition of a value of a differential game	278	(4)
7.6. The definition of Varaiya-Lin of a differential game	282	(3)
8. N-Person Games
8.1. Definition of N-person games	285	(6)
8.2. The Hamilton-Jacobi equations	291	(4)
8.3. Construction of an equilibrium point	295	(2)
8.4. Games with integral bounds on the controls	297	(10)
8.5. Linear-quadratic games: Synthesis of the open-loop equilibrium strategy	307	(11)
8.6. Linear-quadratic games: Closed loop equilibrium strategy	318	(4)
8.7. Linear-quadratic games: Existence of stable equilibrium value	322	(8)
8.8. Other solutions for N-person games	330	(5)
Bibliographical Remarks	335	(4)
Bibliography	339	(8)
Index	347	(3)
Index of Conditions	350