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Discrete Dynamics in Nature and Society
Volume 2016, Article ID 1386242, 8 pages
http://dx.doi.org/10.1155/2016/1386242
Research Article

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

1Institute of Mathematics, National University of Uzbekistan, 100125 Tashkent, Uzbekistan
2Department of Mathematics & Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Malaysia
3Department of Law and Economics, Mediterranea University of Reggio Calabria, 89127 Reggio Calabria, Italy
4ICRIOS-Bocconi University, 20123 Milan, Italy

Received 17 May 2016; Accepted 4 July 2016

Academic Editor: Filippo Cacace

Copyright © 2016 Atamurat Kuchkarov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.