Many Pursuers and One Evader Game

Many Pursuers and One Evader Game

Bashir Mai Umar^1*, Hassan Abdullahi², Ahmad Yahaya Haruna³ and Sani Musa Tsoho⁴

¹Department of Mathematics Federal University, Gashua, M.B 1005, Yobe, Nigeria

²Department of Mathematics Zamfara State University, Talata Mafara, Zamfara Nigeria

³Mathematics Unit, Department of General Studies Federal Polytechnic Kabo, Kano Nigeria

^1,2,3,4.Differential Game Research Group Nigeria (DGRGN). Bayero University, Kano Chapter.

*Corresponding Author

DOI: https://doi.org/10.51584/IJRIAS.2025.10040060

Received: 03 April 2025; Accepted: 07 April 2025; Published: 14 May 2025

ABSTRACT

In this paper we study a pursuit differential game in space Rⁿ. The dynamic equation of finite number of pursuers and a single evader are describe by certain first order and second order differential equations respectively. The control functions (of the players) are subject to generalized coordinate-wise integral constraint. Pursuers are focused to catch the evader when the geometric position of at least one pursuer coincides with that of the evader. The pursuers’ strategies are describe in phases. Moreover, we provide conditions that ensure completion of pursuit.

Keywords: Pursuit game; Pursuer; Evader; generalized Coordinate-wise integral constraint

INTRODUCTION

Differential games is a research area with diverse literature, as a result of different types of games, number of players, nature of the players’ dynamic equations, the constraints on the players’ state or on the control functions of the players etc. The works [1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] are all concerned with study of differential games.

In the following papers [20], [24], [28], [29], and [31] the authors studied game problem with geometric constraints. Examples of studies in which the integral constraints are considered includes [1], [2], [6], [8], [9], [10], [11], [13], [14], [15], [19], [25], [26], [27], [32], [33] and [34].

The integral constraints are of different forms. The conventional integral constraints was considered in the works [6], [11], [13], [14], [15], [25], [26], [27], [33] and some references therein that is

In the work by Ahmed et. al [1] a pursuit game with a more general integral constraints on control functions of the player was considered and obtained sufficient conditions for completion of pursuit, also in the work of Umar at. al [32], pursuit-evasion differential game was studied with generalized integral constraints

conditions for completion of pursuit were provided and conditions that guarantee evasion were also found. There is also coordinate-wise integral constraint which was considered in many works such as [2],[8], [9], [10], [34], [12].

In terms of players’ dynamic equation, some authors used a first order differential equation to represent motion of the player. In the work of Umar and Jibo [33] a guaranteed pursuit time was investigated whereby the evader’s motion is described by a second order differential equation.

where $P$ and $E$ represent pursuer and evader respectively.

We are motivated by the following papers [2],[4],[32] and [33]. For the paper [2], the authors used coordinate- wise integral constraints on the control function of the players, in [4] the authors studied a pursuit problem with many pursuers and an evader in a closed convex subset of a Euclidian space. Control functions of each of the players are subject to the coordinate-wise integral constraints. Also in [32] the authors presented their with a more general integral constraint, while in [33] evader’s motion was described second order differential equation. In this paper we will investigate a pursuit differential game in which the motion of pursuer’s and evader is described by first order and second order differential equations with generalized coordinate-wise integral constraints.

PRELIMINARY

THE MAIN RESULT

NUMERICAL EXAMPLE

CONCLUSION

We have studied a pursuit problem with many pursuers and an evader. Control functions of each of the players are subject to the generalized coordinate-wise integral constraints. We proved that pursuers can be completed. The strategy of pursuer is constructed in phases. Each phase of the game last for some time τ_i. The time is defined by a given formula and depends on the current positions of of the players and the time span of the immediate previous phase of the game as well as the control used by the evader in that phase.

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