Pursuit game

A two-person zero-sum differential game of pursuer (hunter) $P$ and evader (prey) $E$, whose motions are described by systems of differential equations: $$ P : \dot x = f(x,u)\,,\ \ \ E : \dot y = g(y,v) \ . $$

Here, $x,y$ are the phase vectors determining the states of the players $P$ and $E$, respectively, and $u,v$ are control parameters, chosen by the players at each moment of time from given compact sets $U,V$ in Euclidean space. The objective of $P$ can be, e.g., to approach $E$ up to a given distance, which formally means that $x$ falls in some $\ell$-neighbourhood ($\ell > 0$) of $y$. Here one distinguishes between approach with minimum time (pursuit-evasion games), up to a given time (pursuit games with prescribed duration) and up to the moment of arrival of $E$ in a certain set (games with "life-line" ). Games with complete information have been relatively well-studied; here both players know the phase state of each other at every moment of time involved. Solving a pursuit game means finding an equilibrium (cf. Saddle point in game theory).

References

[1]	L.S. Pontryagin, "On the theory of differential games" Russian Math. Surveys , 21 : 4 (1966) pp. 193–246 Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 219–274
[2]	N.N. Krasovaskii, A.I. Subbotin, "Game-theoretical control problems" , Springer (1988) (Translated from Russian)
[3]	R. Isaacs, "Differential games" , Wiley (1965)
[4]	L.A. Petrosyan, "Differential pursuit games" , Leningrad (1977) (In Russian)

Comments

Pursuit games are also called games of pursuit or games of pursuit-evasion.

Related to pursuit games are search games with (im-) mobile hider. Such games are usually stochastic, due to incomplete information.

References

[a1]	S. Gal, "Search games with mobile and immobile hider" SIAM J. Control Optim. , 17 (1979) pp. 332–349
[a2]	G.J. Olsder, G.P. Papavassilopoulos, "About when to use the searchlight" J. Math. Anal. Appl. , 136 (1988) pp. 466–478
[a3]	A. Friedman, "Differential games" , Wiley (1971)
[a4]	O. Hajek, "Pursuit games" , Acad. Press (1975)
[a5]	T. Basar, G.J. Olsder, "Dynamic noncooperative game theory" , Acad. Press (1982)