Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 1. P. 44 – 55. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2026-54-1-44-55
Research Article
Full text in English
MSC 49N79, 49N70, 91A24

Contents of this issue

Read Russian Version

On Linear Pursuit-Evasion Game Problems with GGr-Constraints on Controls of Players

B. I. Juraev^{\ast}

Andijan State University, University Str. 129, Andijan 170100, Uzbekistan

Abstract. This article investigates a differential game of pursuit-evasion for linear motion dynamics of two players, a pursuer and an evader. The pursuer’s control is subject to a geometric constraint (its Euclidean norm does not exceed a given constant \alpha), while the evader’s control satisfies a Gr¨onwall-type constraint that implies an exponential upper bound γelt on its norm. To solve the pursuit problem, a parallel approach strategy (\Pi-strategy) for the pursuer is constructed, which depends on the current evader’s control. It is shown that if \alpha > \gamma and an additional inequality involving the initial distance between the players holds, then there exists a guaranteed capture time T^{\ast}, and the \Pi-strategy ensures interception no later than this time. For the evasion problem, an admissible control function for the evader is defined, directed opposite to the initial difference vector. It is proved that if \alpha \leq \gamma, the evader can avoid capture for all time, keeping the distance strictly positive. The obtained conditions are sharp within the considered classes of constraints. These results extend classical approaches to differential games with mixed-type constraints and have potential applications in robotics and motion control.

Key words: linear differential game, pursuer, evader, strategy, pursuit, evasion, guaranteed capture time.

Received: 01.03.2026; Revised: 18.03.2026; Accepted: 20.03.2026; First online: 29.03.2026

For citation. Juraev B. I. On linear pursuit-evasion game problems with GGr-constraints on controls of players. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 1, 44-55. EDN: VEVFFW. https://doi.org/10.26117/2079-6641-2026-54-1-44-55.

Funding. Not funding

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: jbahodirjon@bk.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Juraev B. I., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

References

  1. Isaacs R. Differential games. New York: John Wiley and Sons, 1965.
  2. Pontryagin L. S. Selected Works. Moscow: MAKS Press, 2004 (In Russian https://cs.msu.ru/en/node/1221).
  3. Krasovskii N. N. Control of a Dynamical System. Moscow: Nauka, 1985 (In Russian https://cs.msu.ru/en/node/1221).
  4. Friedman A. Differential games. Mineola, New York s: Dover Publications, Inc., 2006.
  5. Fleming W. H. The convergence problem for differential games. II, Advances in Game Theory, Ann. Math. Studies, 1964. vol. 52, pp. 195–210.
  6. Berkovitz L. D. Characterizations of the differential games, J. Math. Anal. Appl., 1988. vol. 129, no. 2, pp. 493–504 DOI: 10.1007/BF01448365.
  7. Petrosjan L. A. Differential Games of Pursuit. Singapore: World Scientific Publishing, 1993.
  8. Petrosyan L. A. Pursuit Games with “a Survival Zone”, Vestnik Leningard State Univ., 1967. vol. 13, pp. 76–85 (In Russian).
  9. Chikrii A. A. Conflict-Controlled Processes. Dordrecht: Kluwer Academic Publishers, 1997 DOI: 10.1007/978-94-017-1135-7.
  10. Pshenichnii B. N. Simple pursuit by several objects, Cybernetics and System Analysis, 1976 DOI: 10.1007/BF01070036. vol. 12, no. 4, pp. 484–485.
  11. Chernous’ko F. L., Melikyan A. A. Game Problems of Control and Search. Moscow: Nauka, 1978 (In Russian).
  12. Mishchenko E. F.On the problem of evading the encounter in differential games, SIAM J. Control, 1974. vol. 12, pp. 300–310 DOI: 10.1137/0312023.
  13. Subbotin A. I. Generalization of the main equation of differential game theory, Journal of Optimization Theory and Applications, 1984. vol. 43, no. 1, pp. 103–133 DOI: 10.1007/BF00934749.
  14. Chentsov A. G. Guidance–Evasion Differential Game: Alternative Solvability and Relaxations of the Guidance Problem, Proceedings of the Steklov Institute of Mathematics, 2021. vol. 315, no. 1, pp. 270–289 DOI: 10.1134/S0081543821050217.
  15. Osipov Yu. S.On the theory of differential games of systems with aftereffect, J. Appl. Maths and Mechs, 1971. vol. 35, pp. 262–272 DOI: 10.1016/0021-8928(71)90032-3.
  16. Kriazhimskii A. V.On stable position control in differential games, J. Appl. Maths and Mechs, 1978. vol. 42, no. 6, pp. 963–968 DOI: 10.1016/0021-8928(78)90054-0.
  17. Kurzhanskii A. B. Differential games of observation, Dokl. Akad. Nauk SSSR, 1972. vol. 207, no. 3, pp. 527–530 (In Russian).
  18. Nikol’skiiM. S. Some topical problems in the theory of differential games, Differential Equations, 1997. vol. 33, no. 11, pp. 1563–1565.
  19. Dar’in A. N., Kurzhanskii A. B. Control Under Indeterminacy and Double Constraints, Differential Equations, 2003. vol. 39, no. 11, pp. 1554–1567 DOI: 10.1023/B:DIEQ.0000019347.24930.a3.
  20. Gomoyunov M. I., Lukoyanov N.Yu.On the numerical solution of differential games for neutraltype linear systems, Trudy Inst. Mat. i Mekh. UrO RAN, 2017. vol. 23, no. 1, pp. 75–87 DOI: 10.21538/0134-4889-2017-23-1-75-87.
  21. Garcia E., Casbeer D. W., Pachter M. Optimal strategies of the differential game in a circular region, IEEE Control Systems Letters, 2019. vol. 4, no. 2, pp. 492–497 DOI: 10.1109/LCSYS.2019.2963173.
  22. Von Moll A., Casbeer D. W., Garcia E., Milutinović D., Pachter M. The Multi-pursuer Single-Evader Game, Journal of Intelligent and Robotic Systems, 2019. vol. 96, pp. 193–207 DOI: 10.1007/s10846-018-0963-9.
  23. Weintraub I. E., Pachter M., Garcia E. An Introduction to Pursuit–evasion Differential Games, American Control Conference (ACC), 2020, pp. 1049–1066 DOI: 10.23919/ACC45564.2020.9147205.
  24. Satimov N.Yu. Methods of solving pursuit problems in the theory of differential games. Tashkent: Izd-vo NBRUz, 2019 (In Russian).
  25. Azamov A. A.On the quality problem for the games pursuit with the restriction, Serdica Bulgarian Math., 1986. vol. 12, no. 1, pp. 38–43.
  26. Ibragimov G. I. Evasion Differential Game of One Evader and Many Slow Pursuers, Dynamic Games and Applications, 2024. vol. 14, no. 3, pp. 665–685 DOI: 10.1007/s13235-023-00501-2.
  27. Mamadaliev N. A. The pursuit problem for linear games with integral constraints on player’s controls, Russian Mathematics, 2020. vol. 64, no. 3, pp. 9–24 DOI: 10.3103/S1066369X20030020.
  28. Mamadaliev N. A. Pursuit problems in linear differential games with delay, Russian Mathematics, 2010. vol. 54, no. 6, pp. 13–18 DOI: 10.3103/S1066369X10060022.
  29. Samatov B. T. Problems of group pursuit with integral constraints on controls of the players I, Cybernetics and Systems Analysis, 2013. vol. 49, no. 5, pp. 756–767 DOI: 10.1007/s10559-013-9563-7.
  30. Samatov B. T. Problems of group pursuit with integral constraints on controls of the players II, Cybernetics and Systems Analysis, 2013. vol. 49, no. 6, pp. 907–921 DOI: 10.1007/s10559-013-9581-5.
  31. Ibragimov G. I., Rikhsiev B. B.On some sufficient conditions for optimality of the pursuit time in the differential game with multiple pursuers, Automation and Remote Control, 2006. vol. 67, no. 4, pp. 529–537 DOI: 10.1134/S0005117906040023.
  32. Shiyuan J., Zhihua Q. Pursuit-evasion games with multi-pursuer vs. One fast evader, Proceedings of the 8th World Congress on Intelligent Control and Automation, 2010, pp. 3184–3189 DOI: 10.1109/WCICA.2010.5553770.
  33. Grigorenko N. L. / Mathematical Methods of Control for Several Dynamic Processes. Moscow, Izdat. Gos. Univ., 1990 (In Russian).
  34. Blagodatskikh A. I., Petrov N. N. Conflict interaction of groups of controlled objects. Izhevsk: Udmurt State University, 2009 (In Russian).
  35. Petrov N. N. Multiple capture of a given number of evaders in the problem of simple pursuit with phase restrictions on timescales, Dynamic Games and Applications, 2022. vol. 12, no. 2, pp. 632–642 DOI: 10.1007/s13235-021-00387-y.
  36. Petrov N. N., Mozhegova E. S. Simple pursuit problem with phase constraints of two coordinated evaders on time scales, Doklady Mathematics, 2023. vol. 108, no. 1, pp. 86–91 DOI: 10.1134/S1064562423600720.
  37. Azamov A. A., Kuchkarov A. Sh., Samatov B. T. The relation between problems of pursuit, controllability and stability in the large in linear systems with different types of constraints, J. Appl. Maths and Mechs, 2007. vol. 71, no. 2, pp. 229–233 DOI: 10.1016/j.jappmathmech.2007.06.006.
  38. Azamov A. A., Samatov B. T. The Π-Strategy: Analogies and Applications / The Fourth International Conference Game Theory and Management, St. Petersburg, 4, 2011, pp. 33–47.
  39. Samatov B. T.On a pursuit-evasion problem under a linear change of the pursuer resource, Siberian Advances in Mathematics, 2013. vol. 23, no. 10, pp. 294–302 DOI: 10.3103/S1055134413040056.
  40. Samatov B. T., Ibragimov G. I., Khodjibayeva V. I. Pursuit-evasion differential games with Grönwall type constraints on controls, Ural Mathematical Journal, 2019. vol. 6, no. 2, pp. 95–107 DOI: 10.15826/umj.2020.2.010.
  41. Samatov B. T., Umaraliyeva N. T., Uralova S. I. Differential Games with the Langenhop Type Constraints on Controls, Lobachevskii Journal of Mathematics, 2020. vol. 6, no. 2, pp. 95–107 DOI: 10.1134/S1995080221120295.
  42. Samatov B. T., Horilov M. A., Akbarov A. Kh. Differential Game: “Life Line” for Non-Stationary Geometric Constraints On Controls, Lobachevskii Journal of Mathematics, 2022. vol. 43, no. 1, pp. 237–248 DOI: 10.1134/S1995080222040187.
  43. Samatov B. T., Soyibboev U. B. Differential game with a lifeline for the inertial movements of players, Ural Mathematical Journal, 2021. vol. 7, no. 2, pp. 94–109 https://doi.org/10.15826/umj.2021.2.007.
  44. Samatov B. T., Soyibboev U. B. Applications of the Π-Strategy When Players Move with Acceleration,
    Proceedings of the IUTAM Symposium on Optimal Guidance and Control for Autonomous Systems, 2024. vol. 40, pp. 165–183 DOI: 10.1007/978-3-031-39303-7−10.
  45. Grönwall T. H. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations., Annals of Mathematics Second Series, 1919. vol. 20, no. 4, pp. 292–296.

Information about the author

Juraev Bahodirjon Inomjon ugli – PhD (Phys. & Math.), Head of the Department of Applied Mathematics, Andijan State University, Andijan,
Uzbekistan, ORCID 0000-0002-6920-4314.

Read articleDownload article