Differential Games

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.

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

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