Selected Title s i n Thi s Serie s

174 Ya-Zh e Che n an d Lan-Chen g Wu , Secon d orde r ellipti c equation s an d ellipti c systems, 199 8

173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l propertie s o f

distributions o f stochasti c functionals , 199 8

172 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finite groups . Par t 1 , 199 8

171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 199 8

170 Vikto r Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c integrals , 199 7

169 S . K . Godunov , Ordinar y differentia l equation s wit h constan t coefficient , 199 7

168 Junjir o Noguchi , Introductio n t o comple x analysis , 199 8

167 Masay a Yamaguti , Masayosh i Hata , an d Ju n Kigami , Mathematic s o f fractals , 199 7

166 Kenj i U e n o , A n introductio n t o algebrai c geometry , 199 7

165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g proble m i n

Galois theory , 199 7

164 E . I . Gordon , Nonstandar d method s i n commutativ e harmoni c analysis , 199 7

163 A . Ya . Dorogovtsev , D . S . Silvestrov , A . V . Skorokhod , an d M . I . Yadrenko ,

Probability theory : Collectio n o f problems , 199 7

162 M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d method s i n linea r

statistical models , 199 7

161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c dynamics , 199 7

160 V . G . OsmolovskiT , Linea r an d nonlinea r perturbation s o f th e operato r div , 199 7

159 S . Ya . Khavinson , Bes t approximatio n b y linea r superposition s (approximat e

nomography), 199 7

158 Hidek i Omori , Infinite-dimensiona l Li e groups , 199 7

157 V . B . Kolmanovski Y an d L . E . Shaikhet , Contro l o f system s wit h aftereffect , 199 6

156 V . N . Shevchenko , Qualitativ e topic s i n intege r linea r programming , 199 7

155. Yu . Safaro v an d D . Vassiliev , Th e asymptoti c distributio n o f eigenvalue s o f partia l

differential operators , 199 7

154 V . V . Prasolo v an d A . B . Sossinsky , Knots , links , braid s an d 3-manifolds . A n

introduction t o th e ne w invariant s i n low-dimensiona l topology , 199 7

153 S . Kh . Aranson , G . R . Belitsky , an d E . V . Zhuzhoma , Introductio n t o th e

qualitative theor y o f dynamica l system s o n surfaces , 199 6

152 R . S . Ismagilov , Representation s o f infinite-dimensiona l groups , 199 6

151 S . Yu . Slavyanov , Asymptoti c solution s o f th e one-dimensiona l Schrodinge r equation ,

1996

150 B . Ya . Levin , Lecture s o n entir e functions , 199 6

149 Takash i Sakai , Riemannia n geometry , 199 6 148 Vladimi r I . Piterbarg , Asymptoti c method s i n th e theor y o f Gaussia n processe s an d

fields, 199 6

147 S . G . Gindiki n an d L . R . Volevich , Mixe d proble m fo r partia l differentia l equation s

with quasihomogeneou s principa l part , 199 6

146 L . Ya . Adrianova , Introductio n t o linea r system s o f differentia l equations , 199 5

145 A . N . Andriano v an d V . G . Zhuravlev , Modula r form s an d Heck e operators , 199 5

144 O . V . Troshkin , Nontraditiona l method s i n mathematica l hydrodynamics , 199 5

143 V . A . Malyshe v an d R . A . Minlos , Linea r infinite-particl e operators , 199 5

142 N . V . Krylov , Introductio n t o th e theor y o f diffusio n processes , 199 5

141 A . A . Davydov , Qualitativ e theor y o f contro l systems , 199 4

140 Aizi k I . Volpert , Vital y A . Volpert , an d Vladimi r A . Volpert , Travelin g wav e solutions o f paraboli c systems , 199 4

(See th e AM S catalo g fo r earlie r titles )

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Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Serie s in Mathematic s

Number 1 8

Differential Game s

Avner Friedma n

Published fo r th e Conference Boar d of the Mathematica l Science s

by the American Mathematical Societ y

Providence, Rhod e Islan d with suppor t fro m th e

National Scienc e Foundatio n

http://dx.doi.org/10.1090/cbms/018

Expository Lecture s

from th e CBM S Regional Conferenc e

held a t th e Universit y o f Rhod e Islan d

June 4-8 , 197 3

AMS (MOS) Subjec t Classifications : Primar y 90D25 ;

Secondary 93E05 , 49A99, 49B10, 49E99

Library o f Congres s Catalogin g i n Publicatio n Dat a W ̂ M Wf

Friedman, Avner .

Differential games .

(Regional conferenc e serie s in mathematics , 18 )

"Expository lecture s fro m th e CBM S Regional Con -

ference hel d a t th e Universit y o f Rhod e Island ,

June 4-8 , 1973. "

Bibliography: p .

1. Differentia l games . I . Conferenc e Boar d o f

the Mathematica l Sciences . II . Title . III . Series .

QA1.R33 no . 1 8 510'.8 s [519.3 ] 73-1587 2

ISBN 0-8218-1668- 3

C o p y i n g a n d r e p r i n t i n g . Materia l i n this book ma y be reproduced b y any means fo r educationa l and scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y service s that collec t fee s for delivery o f documents an d provide d tha t th e customary acknowledgmen t o f th e source i s given. Thi s consen t doe s no t exten d t o other kind s of copying fo r genera l distribution , fo r advertising o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercia l us e o f material shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematica l Society , R O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionQams.org.

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TABLE OF CONTENTS

Introduction 1

1. Basi c definitions an d preliminar y result s 2

2. Uppe r an d lowe r value o f game s of fixe d duratio n 9

3. Existenc e o f valu e fo r game s of fixe d duratio n 1 3 4. Existenc e o f saddl e points fo r game s of fixe d duration . Exampl e 2 6

5. Existenc e theorem s fo r generalize d pursuit-evasio n game s 2 9

6. Existenc e theorem s fo r game s of surviva l 3 7

7. Compariso n theorems ; the Hamilton-Jacob i equatio n 3 9 8. Game s with restricte d phas e coordinate s 4 6

9. Capturabilit y theorem s 5 2 10. TV-perso n differentia l game s 5 4 11. Differentia l game s with partia l differentia l equation s 5 8 12. Stochasti c differentia l game s 6 0

Bibliography 6 2

Supplementary Bibliograph y 6 5

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BIBLIOGRAPHY

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tial games and control problems, SIAM J. Contro l 5 (1967), 1-24 . M R 34 #8835 .

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strategies (t o appear) .

3. E . Cockayne, Plane pursuit with curvature contraints, SIAM J. Appl . Math. 1 5 (1967), 1511-1516 . M R 36 #7422.

4. M . H. A. Davis and P . Varaiya, Dynamic programming conditions for partially

observable stochastic systems, SIA M J. Contro l 1 1 (1973), 226-261 .

5. R . J . Elliot t an d N . J. Kalton , The existence of value in differential games, Mem. Amer. Math . Soc . No. 12 6 (1972).

6. , The existence of value in differential games of pursuit and evasion, J. Differential Equation s 1 2 (1972), 504-523 .

7. , Upper values of differential games, J . Differentia l Equation s 1 4 (1973), 89-100. 8. , Cauchy problems for certain Isaacs-Bellman equations and games of sur-

vival, Trans. Amer. Math. Soc . (to appear) . 9. R . J . Elliot t an d A . Friedman , A note on generalized pursuit-evasion games, SIAM

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game theory, Ann . Math. Studies, no. 52 , Princeton Univ . Press, Princeton, N . J., 1964 , pp. 195-210 . M R 29 #6947 .

11. J.Flynn , Lion and man: the boundary constraint, SIAM J. Contro l 1 1 (1973), 397-411. 12. , Some results on max-min pursuit, SIA M J. Contro l (t o appear) .

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62

DIFFERENTIAL GAME S 63

18. A . Friedman, Differential games with restricted phase coordinates, J. Differentia l Equations 8 (1970), 135-162 . M R 42 #1534 .

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MR 45 #1613.

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Press, New York , 1972 , pp. 299-307 . 23. , Comparison theorems for differential games. I, J. Differential Equation s

12 (1972), 162-172 .

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25. , Upper and lower values of differential games, J. Differentia l Equation s 12 (1972), 462-473; Correction, ibid . 1 4 (1973).

26. , Existence of extended value for differential games of generalized pursuit-evasion, J. Differentia l Equation s 1 3 (1973), 172—181 .

27. , Remarks on differential games of survival, J . Differential Equation s 1 4 (1973), 121-128 .

28. I . I . Gihman and A. V. Skorokhod, Introduction to the theory of random pro-cesses, Nauka, Moscow, 1965 ; English transl. , Saunders , Philadelphia, Pa. , 1969 . M R 33 #6689; 40 #923 .

29. R . Isaacs , Differential games, Wiley, New York, 1965 . M R 35 #1362 .

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32. D . L. Lukes and D. R. Russel , The quadratic criterion for distributed systems, SIAM J. Contro l 7 (1969), 101-121 . M R 40 #3403 .

33. O . A. Oleihik an d S. N. Kruzkov, Quasi-linear parabolic equations of the second order with many independent variables, Uspeh i Mat . Nauk 1 6 (1961), no. 5 (101), 115-15 5 = Russian Math . Surveys 1 6 (1961), no. 5 , 105-146 . M R 25 #5289 .

34. L . S . Pontrjagin, V . G. Boltjanskif , R . V. Gamkrelidze an d E . F . MiScenko, The mathematical theory of optimal processes, Fizmatgiz, Moscow, 1961 ; English transl., Wiley, New York, 1962 ; Macmillan, New York, 1964 . M R 2 9 #3316a , b .

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36. B . N. P3eni£nyi and M . I. Sagaidak , Differential games with fixed time, Kibernetik a (Kiev) 1970 , no. 2, 54-63 = Cybernetic s 6 (1970) , 74-83. M R 44 #5126 .

37. G . T. Rublein , On pursuit with curvature constraints, SIAM J. Contro l 1 0 (1972), 37-39.

38. R . C . Scalzo, Differential games with restricted phase coordinates, SIAM J. Con-trol (to appear) .

64 AVNER FRIEDMA N

39. R . C . Scalzo, N-person linear quadratic differential games with constraints, SIAM

J. Contro l (t o appear) .

40. L . C. Stecher, Pursuit with general curvature constraints. I , I I (to appear) .

41. P . Varaiya, N-person non-zero sum differential games with linear dynamics, SIA M

J. Control 8 (1970), 441-449. M R 43 #4489 .

42. P . Varaiya an d J . Lin , Existence of saddle points in differential games, SIAM J .

Control 7 (1969), 142-157 .

43. J . Flynn , Lion and man: the general case, SIAM J. Contro l (t o appear) .

SUPPLEMENTARY BIBLIOGRAPH Y

1. A Bensoussan, Points de Nash dans le cas de fonctionnelles quadratiques et jeux

differentiels lineaires a N personnes, SIAM J. Contro l (t o appear) . 2. R . Chattopadhyay , Completion of linear differential games, J. Optimizatio n Theor y

Appl. 3 (1969), 220-226. M R 40 #3912 . 3. M . D. Ciletti, New results in the theory of differential games with information

time lag, J . Optimizatio n Theor y Appl . 8 (1971), 287-315 . 4. Ju . G. Dutkevi t an d L . A. Petrosjan, Games with a ''life-line. " The case of l-cap-

ture, Vestnik Leningrad . Univ . 24 (1969), no. 13 , 31-38; English transl. , SIAM J. Contro l 10 (1972), 40-47. M R 41 #6585 .

5. R . J . Elliot t an d N . J. Kalton , Boundary value problems for non-linear partial

differential operators, J. Math . Anal . Appl . (to appear) . 6. R . J. Elliott , N. J. Kalto n an d L . Markus, Saddle points for linear differential

games, SIAM J. Contro l 1 1 (1973), 100-112 . 7. W . H. Fleming, The Cauchy problem for degenerate parabolic equations, J . Math .

Mech. 1 3 (1964), 987-1008. M R 31 #3721 . 8. A . Friedman, The Cauchy problem for first order partial differential equations,

Indiana Univ . Math. J . 2 3 (1973), 27-40 . 9. V . M. Gorlov, The existence of the value of a game in pursuit problems, Z . Vy£isl.

Mat. i Mat. Fiz . 12 (1972) , 78-88 = USS R Comput . Math , an d Math . Phys. 1 2 (1972), no. 1,93-108 . M R 45 #4841 .

10. V . A. Jakubovifc, On the synthesis of optimal controls in a linear differential game

on a finite time interval with quadratic payoff functional, Dokl . Akad . Nauk SSS R 20 0 (1971), 548-551 = Sovie t Math . Dokl. 1 2 (1971), 1454-1458 . M R 44 #6354 .

11. N . N. Krasovskii, Programmed absorption in differential games, Dokl. Akad . Nau k SSSR 20 1 (1971) , 270-272 = Sovie t Math . Dokl . 1 2 (1971), 1642-1645 . M R 45 #6465.

12. , Extremal control in a nonlinear positional differential game, Dokl . Akad .

Nauk SSS R 20 3 (1972) , 520-52 3 = Sovie t Math . Dokl . 1 3 (1972), 411-415. M R 45 #9711. 13. N . N. Krasovskn an d Ju . S . Osipov, Linear differential-difference games, Dokl.

Akad. Nauk SSS R 19 7 (1971), 777-780 = Sovie t Math . Dokl . 1 2 (1971), 554-558 . MR 43 #8395.

14. V . N. Lagunov, Asymptotic properties of nonlinear pursuit games, Dokl. Akad . Nauk SSS R 19 7 (1971) , 271-274 = Sovie t Math . Dokl . 1 2 (1971), 436^440 . M R 43 #4486.

65

66 AVNER FRIEDMA N

15. V . N. Lagunov, A nonlinear differential game of evasion, Dokl. Akad. Nau k SSSR 202 (1972), 522-52 5 = Sovie t Math . Dokl. 1 3 (1972), 131-135 . M R 45 #4843.

16. V . Lakshmikantha m an d A . N. V. Rao , Differential inequalities and differential

games, J. Math , an d Appl . 41 (1973) , 490-496 . 17. A . A. Melikjan an d F . L . Cernousko, On differential games with changing condi-

tions on information, Dokl . Akad. Nauk SSS R 20 3 (1972) , 46-49 = Sovie t Phys . Dokl . 1 7 (1972/73), 185-188 .

18. M . S. Nikol'skff, Linear differential games of pursuit with delays, Dokl. Akad . Nauk SSS R 19 7 (1971) , 1018-102 1 = Sovie t Math . 1 2 (1971), 614-618. M R 44 #6356 .

19. , Linearizable objects and their application in differential pursuit games,

Dokl. Akad. Nauk SSS R 20 5 (1972) , 786-790 = Sovie t Math . Dokl. 1 3 (1972), 1026-1031 . 20. Ju . S . Osipov, Differential games of systems with after effect, Dokl . Akad. Nauk

SSSR 19 6 (1971), 779-782 = Sovie t Math . Dokl . 1 2 (1971), 262-266. M R 43 #1658 . 21. , Minimax absorption in difference-differential games, Dokl. Akad . Nauk

SSSR 20 3 (1972), 32-35 = Soviet Math . Dokl . 1 3 (1972), 337-341 . M R 45 #9716 . 22. L . A. Petrosyan, Topological games and their applications to pursuit problems. I ,

Vestnik Leningrad . Univ . 24 (1969), no . 19 , 55-63; Englis h transl. , SIAM J. Contro l 1 0 (1972), 194-202 . M R 41 #9594 .

23. A . N. V. Rao, Comparison of differential games of fixed duration, SIA M J. Con-trol 1 0 (1972), 393-397 .

24. R . J . Stern , Asymptotic behavior of the value of differential games, SIAM J. Con -trol 1 0 (1972), 747-753 .

25. , Open-loop approximation of differential games, SIAM J. Contro l 1 1 (1973), 20-31 .

26. , Open loop equilibrium of N-person non-zero sum linear-quadratic differ-

ential games with magnitude restraints, J. Math . Anal . Appl. (to appear) . 27 ? Differential games of survival with space-like terminal set, SIA M J . Con-

trol (t o appear) . 28. A . I. Subbotin , Extremal strategies in differential games with perfect memory,

Dokl. Akad. Nauk SSS R 20 6 (1972), 552-555 = Sovie t Math . Dokl. 1 3 (1972), 1263-1267 . 29. M . I. Zelikin , On a differential game with incomplete information, Dokl . Akad .

Nauk SSS R 20 2 (1972), 998-1000 = Soviet Math . Dokl . 1 3 (1972), 228-231 .

30. D . Zonnevend, On a pursuit method, Dok l Akad . Nauk SSS R 20 4 (1972) , 1296 -1299 = Sovie t Math . Dokl . 1 3 (1972), 816-820 .

31. , On a type of player superiority, Dokl. Akad. Nauk SSS R 20 8 (1973), 520-523 = Soviet Math . Dokl . 14(1973) , 106-110 .

32. V . I. Zubov , Construction of controls in pursuit problems, Dokl . Akad. Nau k

SSSR 20 6 (1972) , 1288-128 9 = Sovie t Math . Dokl . 1 3 (1972), 1400-1402 .