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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:
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A two-person zero-sum [[Differential games|differential game]] of pursuer (hunter) $P$ and evader (prey) $E$, whose motions are described by systems of differential equations:
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$$
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P : \dot x = f(x,u)\,,\ \ \ E : \dot y = g(y,v) \ .
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$$
  
<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>
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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]]).
 
 
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====
 
====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>
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<table>
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<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>
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<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>
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<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>
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</table>
  
  
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====References====
 
====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>
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<table>
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<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>
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<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>
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<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>
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<TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Basar,  G.J. Olsder,  "Dynamic noncooperative game theory" , Acad. Press  (1982)</TD></TR>
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</table>
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Latest revision as of 20:17, 11 January 2017

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)
How to Cite This Entry:
Pursuit game. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pursuit_game&oldid=40166
This article was adapted from an original article by L.A. Petrosyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article