UDC 518.9

DOI: 10.15507/0236-2910.027.201702.239-249

Cooperative option of pursuit game with two pursuers and one evader. A strong stability of division variety

Viktor D. Shiryayev
Professor of Chair of Fundamental Informatics, Faculty of Mathematics and Information Technology, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Physics and Mathematics), docent, ORCID: http://orcid.org/0000-0003-0497-3769, This email address is being protected from spambots. You need JavaScript enabled to view it.

Ravilya R. Bikmurzina
Associated Professor of Chair of Fundamental Informatics, Faculty of Mathematics and Information Technology, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Pedagogy), ORCID: http://orcid.org/0000-0002-7651-6340, This email address is being protected from spambots. You need JavaScript enabled to view it.

Introduction: The article deals with a simple differential game on the plane of pursuit with two consistently active players and one evader E; the game is considered in the form of the characteristic function.
Materials and Methods: The geometric constructions and methods are used for solving the problem. The security zone of the escapee is bounded by the Apollonius circle, the pursuit team uses a strategy of parallel approach.
Results: A method of finding the optimal players strategies and the optimal players’ trajectory is proposed. The way of forming the characteristic function is provided. All the variety of division is considered as a solution. However, the use of the results of cooperative theory of differential games is impossible without solving the problems associated with the specifics of differential equations of motion. Foremost, it is the problem of dynamic stability of optimality principles. The article introduces an auxiliary function of making the redistribution of winnings in time, keeping his total winnings throughout the game. The dynamic stability of the cooperative solution is determined with the help of this function. Strong dynamic stability of the entire set of solutions is shown.
Discussion and Conclusions: The obtained results are consistent with similar research of other authors. Further research in this field can be used in the development of methods for “regularization” of optimality principles, for which the condition of dynamic stability is always fulfilled.

Keywords: simple movement, Apollonius circle, coalition, characteristic function, division, weak stability of the solution, strong stability of the solution

For citation: Shiryayev V. D., Bikmurzina R. R. Cooperative option of pursuit game with two pursuers and one evader. A strong stability of division variety. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2017; 27(2):239-249. DOI: 10.15507/0236-2910.027.201702.239-249

Contribution of the co-authors: V. Shiryayev: formulation of the problem, scientific supervision, writing the draft with subsequent revision; R. Bikmurzina: reviewing of literature, computer processing of data.