L-catch and L-escape Differential Game

Bashir Mai Umar; Ali Bulama Mamman; Haruna Usman Idriss

doi:doi:10.11648/j.ijtam.20251101.12

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L-catch and L-escape Differential Game

Bashir Mai Umar^*

, Ali Bulama Mamman

, Haruna Usman Idriss

Published in International Journal of Theoretical and Applied Mathematics (Volume 11, Issue 1)

Received: 15 May 2025 Accepted: 3 June 2025 Published: 14 July 2025

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Abstract

In this work, we investigate a simple motion differential game involving one pursuer and one evader, where the dynamics of the pursuer and the evader are governed by first-order and second-order differential equations, respectively. The control functions for both players are assumed to satisfy generalized integral constraints, which limit their maneuverability over time. Our main focus is on the analysis of pursuit under the l-capture condition. Specifically, we demonstrate that pursuit is completed when the l-distance between the positions of the pursuer x(t) and the evader y(t) becomes less than or equal to a given threshold l, that is, , at some finite time t. We derive an explicit formula for the guaranteed time of pursuit, which holds for all admissible strategies of the evader. Additionally, we show that an optimal escape strategy for the evader, known as l-escape, can also be realized at this same optimal time. This simultaneous realization of pursuit and escape conditions indicates a balance in the strategies of both players, thus leading to an optimal pursuit time. Our results contribute to the broader understanding of differential games and optimal control theory.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 11, Issue 1)
DOI	10.11648/j.ijtam.20251101.12
Page(s)	18-25
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), 2025. Published by Science Publishing Group

Keywords

Optimal Pursuit Time, l-catch, l-escape, Players Control Functions

References

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[2]	Hadejia, Yunusa Aliyu, et al. “Pursuit Differential Game Problems Of L-Catch in A Hilbert Space.” (2022). https://doi.org/10.56471/slujst.v5i.282
[3]	Umar, Bashir Mai, and Aliyu Buba Aihong. “L-catch guaranteed pursuit time.” Bangmod International Journal of Mathematical and Computational Science 10 (2024): 1-9. https://doi.org/10.58715/bangmodjmcs.2024.10.1
[4]	Chikrii, A.A., andA.A.Belousov. “Onlineardifferential games with integral constraints.” Proceedings of the Steklov Institute of Mathematics 269 (2010): 69-80. https://doi.org/10.1134/S0081543810060076
[5]	Ibragimov, Gafurjan, and Nu man Satimov. “A multiplayer pursuit differential game on a closed convex set with integral constraints.” Abstract and Applied Analysis. Vol. 2012. No. 1. Hindawi Publishing Corporation, 2012. https://doi.org/10.1155/2012/460171
[6]	Umar, Bashir Mai, et al. “Pursuit and Evasion Linear Differential Game Problems with Generalized Integral Constraints.” Symmetry 16.5 (2024): 513. https://doi.org/10.3390/sym16050513
[7]	Haruna, Ahmad Yahaya, Abbas Ja’afaru Badakaya, and Jewaidu Rilwan. “Guaranteed pursuit time of a linear differential game with generalized geometric constraints on players control functions.” Bangmod International Journal of Mathematical and Computational Science 9 (2023): 63-71. https://doi.org/10.58715/bangmodjmcs.2023.9.5
[8]	Anholt, Bradley R., Donald Ludwig, and Joseph B. Rasmussen. “Optimal pursuit times: how long should predators pursue their prey?.” Theoretical Population Biology 31.3 (1987): 453-464. https://doi.org/10.1016/0040-5809(87)90015-3
[9]	Ibragimov, Gafurjan. “Optimal pursuit time for a differential game in the Hilbert space l2.” differential equations 6.8 (2013): 9. http://dx.doi.org/10.2306/scienceasia1513-1874.2013.39S.025
[10]	Ibragimov, G. I. “A problem of optimal pursuit in systems with distributed parameters.” Journal of applied mathematics and mechanics 66.5 (2002): 719-724. https://doi.org/10.1016/S0021-8928(02)90002-X
[11]	Livermore, Riley, and Tal Shima. “Deviated pure-pursuit-based optimal guidance law for imposing intercept time and angle.” Journal of Guidance, Control, and Dynamics 41.8 (2018): 1807-1814. https://doi.org/10.2514/1.G003179
[12]	Lim, Shen Hin, et al. “A time-optimal control strategy for pursuit-evasion games problems.” IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA’04. 2004. Vol. 4. IEEE, 2004. https://doi.org/10.1109/ROBOT.2004.1308889
[13]	Dall-Anese, Emiliano, and Andrea Simonetto. “Optimal power flow pursuit.” IEEE Transactions on Smart Grid 9.2 (2016): 942-952. https://doi.org/10.1109/TSG.2016.2571982
[14]	Shinar, J., and Shaul Gutman. “Three-dimensional optimal pursuit and evasion with bounded controls.” IEEE Transactions on Automatic Control 25.3 (1980): 492-496. https://doi.org/10.1109/TAC.1980.1102372
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[16]	Breakwell, JohnV.“Time-optimalpursuitinsideacircle.” Differential Games and Applications. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. 72-85. https://doi.org/10.1007/BFb0004264
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APA Style

Umar, B. M., Mamman, A. B., Idriss, H. U. (2025). L-catch and L-escape Differential Game. International Journal of Theoretical and Applied Mathematics, 11(1), 18-25. https://doi.org/10.11648/j.ijtam.20251101.12

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ACS Style

Umar, B. M.; Mamman, A. B.; Idriss, H. U. L-catch and L-escape Differential Game. Int. J. Theor. Appl. Math. 2025, 11(1), 18-25. doi: 10.11648/j.ijtam.20251101.12

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AMA Style

Umar BM, Mamman AB, Idriss HU. L-catch and L-escape Differential Game. Int J Theor Appl Math. 2025;11(1):18-25. doi: 10.11648/j.ijtam.20251101.12

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@article{10.11648/j.ijtam.20251101.12,
  author = {Bashir Mai Umar and Ali Bulama Mamman and Haruna Usman Idriss},
  title = {L-catch and L-escape Differential Game
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {11},
  number = {1},
  pages = {18-25},
  doi = {10.11648/j.ijtam.20251101.12},
  url = {https://doi.org/10.11648/j.ijtam.20251101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20251101.12},
  abstract = {In this work, we investigate a simple motion differential game involving one pursuer and one evader, where the dynamics of the pursuer and the evader are governed by first-order and second-order differential equations, respectively. The control functions for both players are assumed to satisfy generalized integral constraints, which limit their maneuverability over time. Our main focus is on the analysis of pursuit under the l-capture condition. Specifically, we demonstrate that pursuit is completed when the l-distance between the positions of the pursuer x(t) and the evader y(t) becomes less than or equal to a given threshold l, that is, , at some finite time t. We derive an explicit formula for the guaranteed time of pursuit, which holds for all admissible strategies of the evader. Additionally, we show that an optimal escape strategy for the evader, known as l-escape, can also be realized at this same optimal time. This simultaneous realization of pursuit and escape conditions indicates a balance in the strategies of both players, thus leading to an optimal pursuit time. Our results contribute to the broader understanding of differential games and optimal control theory.
},
 year = {2025}
}

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TY  - JOUR
T1  - L-catch and L-escape Differential Game

AU  - Bashir Mai Umar
AU  - Ali Bulama Mamman
AU  - Haruna Usman Idriss
Y1  - 2025/07/14
PY  - 2025
N1  - https://doi.org/10.11648/j.ijtam.20251101.12
DO  - 10.11648/j.ijtam.20251101.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 18
EP  - 25
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20251101.12
AB  - In this work, we investigate a simple motion differential game involving one pursuer and one evader, where the dynamics of the pursuer and the evader are governed by first-order and second-order differential equations, respectively. The control functions for both players are assumed to satisfy generalized integral constraints, which limit their maneuverability over time. Our main focus is on the analysis of pursuit under the l-capture condition. Specifically, we demonstrate that pursuit is completed when the l-distance between the positions of the pursuer x(t) and the evader y(t) becomes less than or equal to a given threshold l, that is, , at some finite time t. We derive an explicit formula for the guaranteed time of pursuit, which holds for all admissible strategies of the evader. Additionally, we show that an optimal escape strategy for the evader, known as l-escape, can also be realized at this same optimal time. This simultaneous realization of pursuit and escape conditions indicates a balance in the strategies of both players, thus leading to an optimal pursuit time. Our results contribute to the broader understanding of differential games and optimal control theory.

VL  - 11
IS  - 1
ER  -

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Author Information

Bashir Mai Umar

Department of Mathematics Federal University, Gashua, Nigeria

Contact Email

http://orcid.org/0009-0007-0862-540X
Ali Bulama Mamman

Department of Mathematics Federal University, Gashua, Nigeria

Contact Email

http://orcid.org/0009-0009-1455-3590
Haruna Usman Idriss

Department of Mathematics Federal University, Gashua, Nigeria

Contact Email

http://orcid.org/0009-0009-5267-3921

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Plain Text BibTeX RIS

APA Style

Umar, B. M., Mamman, A. B., Idriss, H. U. (2025). L-catch and L-escape Differential Game. International Journal of Theoretical and Applied Mathematics, 11(1), 18-25. https://doi.org/10.11648/j.ijtam.20251101.12

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ACS Style

Umar, B. M.; Mamman, A. B.; Idriss, H. U. L-catch and L-escape Differential Game. Int. J. Theor. Appl. Math. 2025, 11(1), 18-25. doi: 10.11648/j.ijtam.20251101.12

Copy | Download

AMA Style

Umar BM, Mamman AB, Idriss HU. L-catch and L-escape Differential Game. Int J Theor Appl Math. 2025;11(1):18-25. doi: 10.11648/j.ijtam.20251101.12

Copy | Download

@article{10.11648/j.ijtam.20251101.12,
  author = {Bashir Mai Umar and Ali Bulama Mamman and Haruna Usman Idriss},
  title = {L-catch and L-escape Differential Game
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {11},
  number = {1},
  pages = {18-25},
  doi = {10.11648/j.ijtam.20251101.12},
  url = {https://doi.org/10.11648/j.ijtam.20251101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20251101.12},
  abstract = {In this work, we investigate a simple motion differential game involving one pursuer and one evader, where the dynamics of the pursuer and the evader are governed by first-order and second-order differential equations, respectively. The control functions for both players are assumed to satisfy generalized integral constraints, which limit their maneuverability over time. Our main focus is on the analysis of pursuit under the l-capture condition. Specifically, we demonstrate that pursuit is completed when the l-distance between the positions of the pursuer x(t) and the evader y(t) becomes less than or equal to a given threshold l, that is, , at some finite time t. We derive an explicit formula for the guaranteed time of pursuit, which holds for all admissible strategies of the evader. Additionally, we show that an optimal escape strategy for the evader, known as l-escape, can also be realized at this same optimal time. This simultaneous realization of pursuit and escape conditions indicates a balance in the strategies of both players, thus leading to an optimal pursuit time. Our results contribute to the broader understanding of differential games and optimal control theory.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - L-catch and L-escape Differential Game

AU  - Bashir Mai Umar
AU  - Ali Bulama Mamman
AU  - Haruna Usman Idriss
Y1  - 2025/07/14
PY  - 2025
N1  - https://doi.org/10.11648/j.ijtam.20251101.12
DO  - 10.11648/j.ijtam.20251101.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 18
EP  - 25
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20251101.12
AB  - In this work, we investigate a simple motion differential game involving one pursuer and one evader, where the dynamics of the pursuer and the evader are governed by first-order and second-order differential equations, respectively. The control functions for both players are assumed to satisfy generalized integral constraints, which limit their maneuverability over time. Our main focus is on the analysis of pursuit under the l-capture condition. Specifically, we demonstrate that pursuit is completed when the l-distance between the positions of the pursuer x(t) and the evader y(t) becomes less than or equal to a given threshold l, that is, , at some finite time t. We derive an explicit formula for the guaranteed time of pursuit, which holds for all admissible strategies of the evader. Additionally, we show that an optimal escape strategy for the evader, known as l-escape, can also be realized at this same optimal time. This simultaneous realization of pursuit and escape conditions indicates a balance in the strategies of both players, thus leading to an optimal pursuit time. Our results contribute to the broader understanding of differential games and optimal control theory.

VL  - 11
IS  - 1
ER  -

Copy | Download