Simple Motion Pursuit and Evasion Differential Games with Many Pursuers on Manifolds with Euclidean Metric

Abstract

We consider pursuit and evasion differential games of a group of pursuers and one evader on manifolds with Euclidean metric. The motions of all players are simple, and maximal speeds of all players are equal. If the state of a pursuer coincides with that of the evader at some time, we say that pursuit is completed. We establish that each of the differential games (pursuit or evasion) is equivalent to a differential game of groups of countably many pursuers and one group of countably many evaders in Euclidean space. All the players in any of these groups are controlled by one controlled parameter. We find a condition under which pursuit can be completed, and if this condition is not satisfied, then evasion is possible. We construct strategies for the pursuers in pursuit game which ensure completion the game for a finite time and give a formula for this time. In the case of evasion game, we construct a strategy for the evader. Hindawi Publishing Corporation

On a Linear Differential Game in the Hilbert Space ℓ2

“…Fix T > ϑ. We define u(t, v) = v − e −tA * • W −1 (ϑ 1 )x 0 (20) Let v(•) be any admissible control of the evader. We show that (20) is admissible.…”

“…Differential games for infinite dimensional systems are also well studied, for example, when the evolution of the system is governed by parabolic equations pursuit-evasion problems are considered in [14][15][16], where the problem for the partial differential equations is reduced to an infinite system of ordinary differential equations. Pursuit and evasion games with many players considered in [17][18][19][20].…”

Pursuit differential game problem with integral and geometric constraints.

“…pursuer and evader) are explicitly stated and are considered to be differential equations of the same order. For example, in the papers [3,8,10,13,20,24], motion of each of the player is considered to obey first order differential equation. In other studies such as [5,6,17,22], players' motions are described by second order differential equations.…”

Linear evasion differential game of one evader and several pursuers with integral constraints

“…In Kuchkarov et al (2012) the game problem of many pursuers and one evader was studied on a cylinder. In the recent work of Kuchkarov et al (2016), the results of Pshenichnii (1976) were extended to differential games on manifolds with Euclidean metric. In Blagodatskikh and Petrov (2009) Blagodatskikh and Petrov obtained necessary and sufficient condition of evasion in a simple motion differential game of a group of pursuers and a group of evaders in R n where all evaders use the same control.…”