View source for Pursuit game

A two-person zero-sum differential game (cf. [[Differential games|Differential games]]) of pursuer (hunter) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p0759201.png" /> and evader (prey) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p0759202.png" />, whose motions are described by systems of differential equations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p0759203.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p0759204.png" /> are the phase vectors determining the states of the players <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p0759205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p0759206.png" />, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p0759207.png" /> are control parameters, chosen by the players at each moment of time from given compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p0759208.png" /> in Euclidean space. The objective of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p0759209.png" /> can be, e.g., to approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p07592010.png" /> up to a given distance, which formally means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p07592011.png" /> falls in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p07592012.png" />-neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p07592013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p07592014.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075920/p07592015.png" /> 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|Saddle point in game theory]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Krasovaskii,   A.I. Subbotin,   "Game-theoretical control problems" , Springer  (1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Isaacs,   "Differential games" , Wiley  (1965)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.A. Petrosyan,   "Differential pursuit games" , Leningrad  (1977)  (In Russian)</TD></TR></table>



====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Gal,   "Search games with mobile and immobile hider"  ''SIAM J. Control Optim.'' , '''17'''  (1979)  pp. 332–349</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.J. Olsder,   G.P. Papavassilopoulos,   "About when to use the searchlight"  ''J. Math. Anal. Appl.'' , '''136'''  (1988)  pp. 466–478</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Friedman,   "Differential games" , Wiley  (1971)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O. Hajek,   "Pursuit games" , Acad. Press  (1975)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Basar,   G.J. Olsder,   "Dynamic noncooperative game theory" , Acad. Press  (1982)</TD></TR></table>