Cauchy problems for certain Isaacs-Bellman equations and games of survival

Elliott, Robert J.; Kalton, Nigel J.

doi:10.1090/S0002-9947-1974-0347383-8

Cauchy problems for certain Isaacs-Bellman equations and games of survival
HTML articles powered by AMS MathViewer

by Robert J. Elliott and Nigel J. Kalton PDF

Trans. Amer. Math. Soc. 198 (1974), 45-72 Request permission

Abstract:

Two person zero sum differential games of survival are considered; these terminate as soon as the trajectory enters a given closed set $F$, at which time a cost or payoff is computed. One controller, or player, chooses his control values to make the payoff as large as possible, the other player chooses his controls to make the payoff as small as possible. A strategy is a function telling a player how to choose his control variable and values of the game are introduced in connection with there being a delay before a player adopts a strategy. It is shown that various values of the differential game satisfy dynamic programming identities or inequalities and these results enable one to show that if the value functions are continuous on the boundary of $F$ then they are continuous everywhere. To discuss continuity of the values on the boundary of $F$ certain comparison theorems for the values of the game are established. In particular if there are sub- and super-solutions of a related Isaacs-Bellman equation then these provide upper and lower bounds for the appropriate value function. Thus in discussing value functions of a game of survival one is studying solutions of a Cauchy problem for the Isaacs-Bellman equation and there are interesting analogies with certain techniques of classical potential theory.

References

Robert J. Elliott and Nigel J. Kalton, Values in differential games, Bull. Amer. Math. Soc. 78 (1972), 427–431. MR 295775, DOI 10.1090/S0002-9904-1972-12929-X
Robert J. Elliott and Nigel J. Kalton, The existence of value in differential games, Memoirs of the American Mathematical Society, No. 126, American Mathematical Society, Providence, R.I., 1972. MR 0359845
Robert J. Elliott and Nigel J. Kalton, The existence of value in differential games of pursuit and evasion, J. Differential Equations 12 (1972), 504–523. MR 359846, DOI 10.1016/0022-0396(72)90022-8
Avner Friedman, Differential games, Pure and Applied Mathematics, Vol. XXV, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London, 1971. MR 0421700
Avner Friedman, Comparison theorems for differential games. I, II, J. Differential Equations 12 (1972), 162–172; ibid. 12 (1972), 396–416. MR 342198, DOI 10.1016/0022-0396(72)90011-3
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
Rufus Isaacs, Differential games. A mathematical theory with applications to warfare and pursuit, control and optimization, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0210469

Partial differential equations of second order

53

20

43

Oskar Perron, Eine neue Behandlung der ersten Randwertaufgabe für $\Delta u=0$, Math. Z. 18 (1923), no. 1, 42–54 (German). MR 1544619, DOI 10.1007/BF01192395
A. N. V. Rao, Comparison of differential games of fixed duration, SIAM J. Control 10 (1972), 393–397. MR 0303984