Pursuit game

Pursuit game - an antagonistic differential game of the pursuer (catching up) $P$ ${\ displaystyle P}$ $P$ and chased (running away) $E$ ${\ displaystyle E}$ $E$ whose movements are described by systems of differential equations:

$P:{\dot {x}}=f(x,u),$ ${\ displaystyle P: {\ dot {x}} = f (x, u),}$ ${\ displaystyle P: {\ dot {x}} = f (x, u),}$ $E:{\dot {y}}=g(y,v),$ ${\ displaystyle E: {\ dot {y}} = g (y, v),}$ ${\ displaystyle E: {\ dot {y}} = g (y, v),}$

Where $x, y$ ${\ displaystyle x, y}$ $x, y$ - phase vectors that determine the state of the players $P$ ${\ displaystyle P}$ $P$ and $E$ ${\ displaystyle E}$ $E$ respectively; $u, v$ ${\ displaystyle u, v}$ $u, v$ - control parameters selected by the players at each moment of time from the given compact sets $U, V$ ${\ displaystyle U, V}$ ${\ displaystyle U, V}$ Euclidean spaces. Purpose $P$ ${\ displaystyle P}$ $P$ maybe, for example, rapprochement with $E$ ${\ displaystyle E}$ $E$ at a given distance, which formally means hitting $x$ ${\ displaystyle x}$ $x$ at $\ell$ ${\ displaystyle \ ell}$ $\ ell$ neighborhood $y$ ${\ displaystyle y}$ $y$ ( $\ell >0$ ${\ displaystyle \ ell> 0}$ ${\ displaystyle \ ell> 0}$ ) In this case, cases of rapprochement for the minimum time (pursuit game for speed), to a given point in time (pursuit game with the prescribed duration) and until the player reaches $E$ ${\ displaystyle E}$ $E$ some set (a game with a "life line"). Games with full information are relatively well studied, when both players know each other's phase states at each current point in time. The decision of a game of pursuit is understood as finding a situation of equilibrium.

The game began to be studied with the advent of guided torpedoes and missiles : what should be the tactics of a rocket to bring down a fighter? A fighter to get away from a rocket? At the same time, the missile is much faster than a fighter, but is limited in maneuvers and does not live long.

Literature

Pontryagin L. S. , “Successes Mat. Sciences ", 1966, t. 21, c. 4, p. 219-74
Krasovsky H. N., Subbotin A. I., Positional differential games, M., 1974
Isaacs R., Differential Games, trans. from English., M., 1967
Petrosyan L.A. , Differential games of pursuit, L., 1977

Pursuit game

Literature

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