Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics

Hassan Abdullahi; Salisu Saleh; Bilyaminu Muhammad

doi:doi:10.11648/j.ajce.20261405.19

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Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics

Hassan Abdullahi^*, Salisu Saleh, Bilyaminu Muhammad

Published in American Journal of Applied Mathematics (Volume 14, Issue 5)

Received: 8 September 2026 Accepted: 8 September 2026 Published: 8 October 2026

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Abstract

Pursuit-evasion differential games are mathematical models used to study competitive interactions between pursuers and an evader, with applications in control theory, optimization, and dynamical systems. Although several pursuit-evasion models have been studied in finite-dimensional spaces and under first- or higher-order dynamics, relatively limited attention has been given to infinite-dimensional settings involving players with different orders of dynamics. This study investigates a pursuit-evasion differential game in the Hilbert space ℓ₂, where each pursuer is governed by an nth-order differential equation, while the evader is governed by an mth-order differential equation, with n < m. The controls of both the pursuers and the evader are subject to integral constraints. The main objective is to establish sufficient conditions under which the pursuers can guarantee the capture of the evader in finite time and to construct an effective pursuit strategy. To achieve this objective, the problem is formulated within the framework of differential games in ℓ₂, and the properties of the players' higher-order dynamics, control constraints, and geometric relationships are analyzed. Sufficient conditions for guaranteed capture are derived, and a constructive, non-singular pursuit strategy is established. The results demonstrate that the pursuers can achieve capture despite the unequal orders of the players' dynamics and the infinite-dimensional nature of the state space. Consequently, the study extends existing results on first-order and higher-order pursuit-evasion differential games to infinite-dimensional Hilbert spaces with unequal dynamic orders and provides a framework for analyzing pursuit problems under integral control constraints.

Published in	American Journal of Applied Mathematics (Volume 14, Issue 5)
DOI	10.11648/j.ajce.20261405.19
Page(s)	359-363
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Dynamic Equations, Integral Constraint, Guaranteed Capture Time, Hilbert Space

References

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[2]	Krasovskii, N.N., Subbotin, A.I. Game-Theoretical Control Problems. Springer, New York, 1988.
[3]	Petrosyan, L.A. Differential Games of Pursuit. World Scientific, Singapore, 1993.
[4]	Ibragimov, G.I., Mat Hasim, R. Pursuit and Evasion Differential Games in Hilbert Space. International Game Theory Review. 2010, 12(3), 239-251. https://doi.org/10.1142/S0219198910002647
[5]	Ja'afar, A.B., Ibragimov, G. On Some Pursuit and Evasion Differential Game Problems for an Infinite Number of First-Order Differential Equations. Journal of Applied Mathematics. 2012, 2012, 717124. https://doi.org/10.1155/2012/717124
[6]	Badakaya, A.J., Halliru, A.S., Adamu, J. A Game Problem of Pursuit-Evasion with nth Order Differential Equations Describing Players' Dynamics. Journal of Control and Decision. 2022, 9(2), 193-201. https://doi.org/10.1080/23307706.2021.1941336
[7]	Adamu, J., Halliru, A.S., Abdulhamid, B.M. On Some Pursuit Differential Game Problem in a Hilbert Space. Journal of the Nigerian Society of Physical Sciences. 2022, 4(1), 83-87. https://doi.org/10.46481/jnsps.2022.379
[8]	Muhammad, B., Abdullahi, H. Differential Game with Integral Constraints in a Hilbert Space. International Journal of Research and Innovation in Applied Science. 2025, 10(5), 201-206. https://doi.org/10.51584/IJRIAS.2025.100500020
[9]	Umar, B.M., Abdullahi, H., Haruna, A.Y., Tsoho, S.M. Many Pursuers and One Evader Game. International Journal of Research and Innovation in Applied Science. 2025, 10(4), 712-722. https://doi.org/10.51584/IJRIAS.2025.10040060
[10]	Abdullahi, H., Umar, B.M., Muhammad, B., Tsoho, S.M. Pursuit Differential Game with High-Order Dynamic. Proceedings of the Nigerian Society of Physical Sciences. 2026, 3, 315. https://doi.org/10.61298/pnspsc.2026.3.315
[11]	Ibragimov, G., Egamberganova, O., Alias, I.A., Luckraz, S. On Some New Results in a Pursuit Differential Game with Many Pursuers and One Evader. AIMS Mathematics. 2023, 8(3), 6581-6589. https://doi.org/10.3934/math.2023332
[12]	Rilwan, J., Ferrara, M., Badakaya, A.J., Pansera, B.A. On Pursuit and Evasion Game Problems with Gr"onwall-Type Constraints. Quality & Quantity. 2023, 57, 5551-5562. https://doi.org/10.1007/s11135-023-01616-9
[13]	Ibragimov, G., Ruziboev, M., Zaynabiddinov, I., Pansera, B.A. Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in ell₂. Games. 2023, 14(4), 52. https://doi.org/10.3390/g14040052
[14]	Badakaya, A.J., Tsoho, S.M., Salimi, M. On Some ell-Catch Pursuit Differential Games with Different Players' Dynamic Equations. Differential Equations and Dynamical Systems. 2024, 33, 1175-1187. https://doi.org/10.1007/s12591-023-00673-8
[15]	Umar, B.M., Rilwan, J., Aphane, M., Muangchoo, K. Pursuit and Evasion Linear Differential Game Problems with Generalized Integral Constraints. Symmetry. 2024, 16(5), 513. https://doi.org/10.3390/sym16050513
[16]	Kazimirova, R., Ibragimov, G., Pansera, B.A., Ibragimov, A. Multi-Pursuer and One-Evader Evasion Differential Game with Integral Constraints for an Infinite System of Binary Differential Equations. Mathematics. 2024, 12(8), 1183. https://doi.org/10.3390/math12081183
[17]	Odiliobi, C.S., Mat Hasim, R., Ibragimov, G. Coordinate-Wise Integral Constraint Pursuit Game of State-Transition Modelled by Infinite System of Two-Coupled Differential Equations. BAREKENG: Jurnal Ilmu Matematika dan Terapan. 2026, 20(4), 2949-2966. https://doi.org/10.30598/barekengvol20iss4pp2949-2966

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APA Style

Abdullahi, H., Saleh, S., Muhammad, B. (2026). Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics. American Journal of Applied Mathematics, 14(5), 359-363. https://doi.org/10.11648/j.ajce.20261405.19

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ACS Style

Abdullahi, H.; Saleh, S.; Muhammad, B. Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics. Am. J. Appl. Math. 2026, 14(5), 359-363. doi: 10.11648/j.ajce.20261405.19

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AMA Style

Abdullahi H, Saleh S, Muhammad B. Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics. Am J Appl Math. 2026;14(5):359-363. doi: 10.11648/j.ajce.20261405.19

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@article{10.11648/j.ajce.20261405.19,
  author = {Hassan Abdullahi and Salisu Saleh and Bilyaminu Muhammad},
  title = {Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {5},
  pages = {359-363},
  doi = {10.11648/j.ajce.20261405.19},
  url = {https://doi.org/10.11648/j.ajce.20261405.19},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajce.20261405.19},
  abstract = {Pursuit-evasion differential games are mathematical models used to study competitive interactions between pursuers and an evader, with applications in control theory, optimization, and dynamical systems. Although several pursuit-evasion models have been studied in finite-dimensional spaces and under first- or higher-order dynamics, relatively limited attention has been given to infinite-dimensional settings involving players with different orders of dynamics. This study investigates a pursuit-evasion differential game in the Hilbert space ℓ2, where each pursuer is governed by an nth-order differential equation, while the evader is governed by an mth-order differential equation, with n < m. The controls of both the pursuers and the evader are subject to integral constraints. The main objective is to establish sufficient conditions under which the pursuers can guarantee the capture of the evader in finite time and to construct an effective pursuit strategy. To achieve this objective, the problem is formulated within the framework of differential games in ℓ2, and the properties of the players' higher-order dynamics, control constraints, and geometric relationships are analyzed. Sufficient conditions for guaranteed capture are derived, and a constructive, non-singular pursuit strategy is established. The results demonstrate that the pursuers can achieve capture despite the unequal orders of the players' dynamics and the infinite-dimensional nature of the state space. Consequently, the study extends existing results on first-order and higher-order pursuit-evasion differential games to infinite-dimensional Hilbert spaces with unequal dynamic orders and provides a framework for analyzing pursuit problems under integral control constraints.},
 year = {2026}
}

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TY  - JOUR
T1  - Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics
AU  - Hassan Abdullahi
AU  - Salisu Saleh
AU  - Bilyaminu Muhammad
Y1  - 2026/10/08
PY  - 2026
N1  - https://doi.org/10.11648/j.ajce.20261405.19
DO  - 10.11648/j.ajce.20261405.19
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 359
EP  - 363
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajce.20261405.19
AB  - Pursuit-evasion differential games are mathematical models used to study competitive interactions between pursuers and an evader, with applications in control theory, optimization, and dynamical systems. Although several pursuit-evasion models have been studied in finite-dimensional spaces and under first- or higher-order dynamics, relatively limited attention has been given to infinite-dimensional settings involving players with different orders of dynamics. This study investigates a pursuit-evasion differential game in the Hilbert space ℓ2, where each pursuer is governed by an nth-order differential equation, while the evader is governed by an mth-order differential equation, with n < m. The controls of both the pursuers and the evader are subject to integral constraints. The main objective is to establish sufficient conditions under which the pursuers can guarantee the capture of the evader in finite time and to construct an effective pursuit strategy. To achieve this objective, the problem is formulated within the framework of differential games in ℓ2, and the properties of the players' higher-order dynamics, control constraints, and geometric relationships are analyzed. Sufficient conditions for guaranteed capture are derived, and a constructive, non-singular pursuit strategy is established. The results demonstrate that the pursuers can achieve capture despite the unequal orders of the players' dynamics and the infinite-dimensional nature of the state space. Consequently, the study extends existing results on first-order and higher-order pursuit-evasion differential games to infinite-dimensional Hilbert spaces with unequal dynamic orders and provides a framework for analyzing pursuit problems under integral control constraints.
VL  - 14
IS  - 5
ER  -

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Author Information

Hassan Abdullahi

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

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Salisu Saleh

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

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Bilyaminu Muhammad

Department of Mathematics, Federal College of Education (Technical), Gusau, Nigeria

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APA Style

Abdullahi, H., Saleh, S., Muhammad, B. (2026). Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics. American Journal of Applied Mathematics, 14(5), 359-363. https://doi.org/10.11648/j.ajce.20261405.19

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ACS Style

Abdullahi, H.; Saleh, S.; Muhammad, B. Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics. Am. J. Appl. Math. 2026, 14(5), 359-363. doi: 10.11648/j.ajce.20261405.19

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AMA Style

Abdullahi H, Saleh S, Muhammad B. Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics. Am J Appl Math. 2026;14(5):359-363. doi: 10.11648/j.ajce.20261405.19

Copy | Download

@article{10.11648/j.ajce.20261405.19,
  author = {Hassan Abdullahi and Salisu Saleh and Bilyaminu Muhammad},
  title = {Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {5},
  pages = {359-363},
  doi = {10.11648/j.ajce.20261405.19},
  url = {https://doi.org/10.11648/j.ajce.20261405.19},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajce.20261405.19},
  abstract = {Pursuit-evasion differential games are mathematical models used to study competitive interactions between pursuers and an evader, with applications in control theory, optimization, and dynamical systems. Although several pursuit-evasion models have been studied in finite-dimensional spaces and under first- or higher-order dynamics, relatively limited attention has been given to infinite-dimensional settings involving players with different orders of dynamics. This study investigates a pursuit-evasion differential game in the Hilbert space ℓ2, where each pursuer is governed by an nth-order differential equation, while the evader is governed by an mth-order differential equation, with n < m. The controls of both the pursuers and the evader are subject to integral constraints. The main objective is to establish sufficient conditions under which the pursuers can guarantee the capture of the evader in finite time and to construct an effective pursuit strategy. To achieve this objective, the problem is formulated within the framework of differential games in ℓ2, and the properties of the players' higher-order dynamics, control constraints, and geometric relationships are analyzed. Sufficient conditions for guaranteed capture are derived, and a constructive, non-singular pursuit strategy is established. The results demonstrate that the pursuers can achieve capture despite the unequal orders of the players' dynamics and the infinite-dimensional nature of the state space. Consequently, the study extends existing results on first-order and higher-order pursuit-evasion differential games to infinite-dimensional Hilbert spaces with unequal dynamic orders and provides a framework for analyzing pursuit problems under integral control constraints.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Multiple-Pursuer Evasion Differential Games with Higher-Order Dynamics
AU  - Hassan Abdullahi
AU  - Salisu Saleh
AU  - Bilyaminu Muhammad
Y1  - 2026/10/08
PY  - 2026
N1  - https://doi.org/10.11648/j.ajce.20261405.19
DO  - 10.11648/j.ajce.20261405.19
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 359
EP  - 363
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajce.20261405.19
AB  - Pursuit-evasion differential games are mathematical models used to study competitive interactions between pursuers and an evader, with applications in control theory, optimization, and dynamical systems. Although several pursuit-evasion models have been studied in finite-dimensional spaces and under first- or higher-order dynamics, relatively limited attention has been given to infinite-dimensional settings involving players with different orders of dynamics. This study investigates a pursuit-evasion differential game in the Hilbert space ℓ2, where each pursuer is governed by an nth-order differential equation, while the evader is governed by an mth-order differential equation, with n < m. The controls of both the pursuers and the evader are subject to integral constraints. The main objective is to establish sufficient conditions under which the pursuers can guarantee the capture of the evader in finite time and to construct an effective pursuit strategy. To achieve this objective, the problem is formulated within the framework of differential games in ℓ2, and the properties of the players' higher-order dynamics, control constraints, and geometric relationships are analyzed. Sufficient conditions for guaranteed capture are derived, and a constructive, non-singular pursuit strategy is established. The results demonstrate that the pursuers can achieve capture despite the unequal orders of the players' dynamics and the infinite-dimensional nature of the state space. Consequently, the study extends existing results on first-order and higher-order pursuit-evasion differential games to infinite-dimensional Hilbert spaces with unequal dynamic orders and provides a framework for analyzing pursuit problems under integral control constraints.
VL  - 14
IS  - 5
ER  -

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