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AIMS Mathematics

2020, Volume 5, Issue 6: 7467-7479. doi: 10.3934/math.2020478
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Research article

On pursuit-evasion differential game problem in a Hilbert space

  • 1.
    Department of Mathematics, Federal University, Gashu'a, Nigeria
  • 2.
    Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon (RMUTP), 1381, Pracharat 1 Road, Wongsawang, Bang Sue, Bangkok 10800, Thailand
  • 3.
    Department of Mathematical Sciences, Bayero University, Kano, 700231, Nigeria
  • Received: 12 July 2020 Accepted: 14 September 2020 Published: 22 September 2020

  • MSC : Primary: 91A23; Secondary: 49N75

  • We consider a pursuit-evasion differential game problem in which countably many pursuers chase one evader in the Hilbert space ${\ell_2}$ and for a fixed period of time. Dynamic of each of the pursuer is governed by first order differential equations and that of the evader by a second order differential equation. The control function for each of the player satisfies an integral constraint. The distance between the evader and the closest pursuer at the stoppage time of the game is the payoff of the game. The goal of the pursuers is to minimize the distance to the evader and that of the evader is the opposite. We constructed optimal strategies of the players and find value of the game.

    Citation: Jamilu Adamu, Kanikar Muangchoo, Abbas Ja'afaru Badakaya, Jewaidu Rilwan. On pursuit-evasion differential game problem in a Hilbert space[J]. AIMS Mathematics, 2020, 5(6): 7467-7479. doi: 10.3934/math.2020478

    Related Papers:

  • We consider a pursuit-evasion differential game problem in which countably many pursuers chase one evader in the Hilbert space ${\ell_2}$ and for a fixed period of time. Dynamic of each of the pursuer is governed by first order differential equations and that of the evader by a second order differential equation. The control function for each of the player satisfies an integral constraint. The distance between the evader and the closest pursuer at the stoppage time of the game is the payoff of the game. The goal of the pursuers is to minimize the distance to the evader and that of the evader is the opposite. We constructed optimal strategies of the players and find value of the game.


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    [1] A. J. Badakaya, A Pursuit Differential Game with Different Constraints on the Control of the players, Transactions of the Nigerian Association of Mathematical Physics, 8 (2019), 17-22.
    [2] E. Bakolas, P. Tsiotras, Optimal pursuit of moving targets using dynamic Voronoi diagrams, 49th IEEE Conference on Decision and Control (CDC), (2010), 7431-7436.
    [3] D. W. Casbeer, E. Garcia, Z. E. Fuch, et al., Cooperative target defense differential game with a constrained-maneauverable defender, 2015 54th IEEE Conference on Decision and Control (CDC), (2015), 1713-1718.
    [4] H. Huang, W. Zhang, J. Ding, et al., Guaranteed decentralized pursuit-evasion in the plane with multiple pursuers, 2011 50th IEEE Conference on Decision and Control and European Control Conference, (2011), 4835-4840.
    [5] G. I. Ibragimov, N. A. Hussin, A Pursuit-Evasion Differential Game with Many Pursuers and One Evader, Malaysian Journal of Mathematical Sciences, 4 (2010), 183-194.
    [6] G. I. Ibragimov, B. B. Rikhsiev, On some Sufficient Conditions for Optimalty of the Pursuit Time in the Differential Game with Multiple Pusuers, Automation and Remote Control, 67 (2006), 529-537.
    [7] G. I. Ibragimov, M. Salimi, Pursuit-Evasion Differential Game with Many Inertial Players, Math. Probl. Eng., 2019 (2009), 1-15.
    [8] G. I. Ibragimov, A. Sh. Kuchkarov, Fixed Duration Pursuit-Evasion Differential Game with Integral Constraints, Journal of Physics: Conference Series, 435 (2013), 012017.
    [9] G. I. Ibragimov, Optimal pursuit with countably many pursuers and one evader, Differential Equations, 41 (2005), 627-635.
    [10] G. I. Ibragimov, N. Satimov, A Multiplayer Pursuit Differential Game on a Convex Set with Integral Constraints, Abstr. Appl. Anal., 2012 (2012), 1-12.
    [11] G. I. Ibragimov, A Game of Optimal Pursuit of One Object by Several, Journal of Applied Mathematics and Mechanics, 62 (1988), 187-192.
    [12] G. I. Ibragimov, On a Multiperson Pursuit Problem with Integral Constraints on the Controls of the Players, Mathematical Notes, 70 (2001), 201-212.
    [13] G. I. Ibragimov, M. Salimi, M. Amini, Evasion from many Pursuer in Simple Motion Differential Game with Integral Constraints, Eur. J. Oper. Res., 218 (2012), 505-511.
    [14] G. I. Ibragimov, A. I. Alias, U. Waziri, et al., Differential game of optimal Pursuit for an Infinite System of Differential Equations, Bull. Malays. Math. Sci. Soc., 42 (2019), 391-403.
    [15] G. I. Ibragimov, N. Abd Rashid, A. Kuchkarov, et al., Multi Pursuer Differential Game of Optimal Approach with Integral Constraints on Controls of the Players, Taiwan. J. Math., 19 (2015), 963-976.
    [16] A. B. Jaafaru, G. I. Gafurjan, On Some Pursuit and Evasion Differential Game Problems for an Infinite Numberof First-Order Differential Equations, J. Appl. Math., (2012).
    [17] A. Kuchkarov, G. I. Ibragimov, M. Ferrara, Simple Motion Pursuit and Evasion Differential Games with Many Pursuers on Manifolds with Euclidean Metric, Discrete Dyn. Nat. Soc., 2016 (2016), 1-8.
    [18] A. Yu. Levchenkov, A. G. Pashkov, Differential Game of Optimal Approach of Two Inertial Pursuers to a Noninertial Evader, J. Optimiz. Theory Appl., 65 (1990), 501-518.
    [19] V. R. Makkapati, P. Tsiotras, Optimal Evading Strategies and Task Allocation in Multi-player Pursuit-Evasion Problems, Dyn. Games Appl., 9 (2019), 116-1187.
    [20] M. S. Nikolskii, The Direct Method in Linear Differential Games with Integral Constraints in the Control System, Controlled systems, IM, IK, SO AN SSSR, 2 (1969), 49-59.
    [21] M. Pachter, E. Garcia, D. W. Casbeer, Active target defense differential game, 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton), (2014), 46-53.
    [22] M. Salimi, G. I. Ibragimov, S. Siegmund, et al., On a Fixed Duration Pursuit Differential Game with Geometric and Integral Constraints, Dyn. Games Appl., 6 (2016), 409-425.
    [23] M. Salimi, A Research Contribution on an Evasion Problem, SeMA Journal, 65 (2018), 139-144.
    [24] M. Salimi, M. Ferrara, Differential Game of Optimal Pursuit of one Evader by Many Pursuers, Int. J. Game Theory, 48 (2019), 481-490.
    [25] B. T. Samatov, Problems of Group Pursuit with Integral Constraints on Controls of the Players I, Cybernetics and System Analysis, 49 (2013), 756-767.
    [26] B. T. Samatov, Problems of Group Pursuit with Integral Constraints on Controls of the Players, II, Cybernetics and System Analysis, 49 (2013), 907-921.
    [27] N. Yu. Satimov, M. Tukhtasinov, Game Problems on a Fixed Interval in Controlled First-order Evolution Equations, Mathematical Note, 80 (2006), 578-589.
    [28] N. Yu. Satimov, M. Tukhtasinov, On Game Problems for Second-Order Evolution Equations, Russian Mathematics, 51 (2007), 49-57.
    [29] N. Siddiqova, S. Muksimova, A. Rakhmov, Research of one Problem of Pursuit with Different Constraints on the Controls, International Journal of Scientific and Research Publications, 7 (2017), 65-73.
    [30] D. A. Vagin, N. N. Petrov, A Problem of Group Pursuit with Phase Constraints, Journal of Applied Mathematics and Mechanics, 66 (2002), 225-232.
    [31] A. Von Moll, D. Casbeer, E. Garcia, et al., The multi-pursuer single-evader game, J. Intell. Robot. Syst., 96 (2019), 193-207.
    [32] Y. Wang, L. Dong, C. Sun, Cooperative control for multi-player pursuit-evasion games with reinforcement learning, Neurocomputing, 412 (2020), 101-114.
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  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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