In a finite-dimensional Euclidean space, a pursuit problem is considered in which a group of pursuers chases a group of evaders, described by the system
$$
\dot z_{ij} = \alpha z_{ij} + u_i - v, \quad u_i, v\in V.
$$
It is assumed that the evaders use the same control and remain within a convex polyhedral set. The pursuers employ quasi-strategies. The set of admissible controls $V$ is the unit ball centered at the origin, and the target sets are the origin. The objective of the group of pursuers is to capture at least one evader by two pursuers. In terms of the initial positions and the game parameters, sufficient conditions for capture are obtained. The method of resolving functions is used as the basic tool for studying the pursuit problem.
Keywords: differential game, group pursuit, evader, pursuer, double capture
Received November 27, 2025
Revised December 26, 2025
Accepted January 19, 2026
Funding Agency:This research was funded by the Ministry of Science and Higher Education of the Russian Federation in the framework of state assignment, project FEWS-2024-0009.
Nikolai Nikandrovich Petrov, Dr. Phys.-Math. Sci., Prof., Udmurt. State University, Izhevsk, 426034, Russia, e-mail: kma3@list.ru
REFERENCES
1. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. NY, Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8 . Original Russian text was published in Krasovskii N. N., Subbotin A. I. Pozitsionnye differentsial’nye igry, Moscow, Nauka Publ., 1974, 456 p.
2. Pontryagin L.S. Izbrannye nauchnye trudy, II [Selected scientific works, II], Moscow, Nauka Publ., 1988, 575 p. ISBN: 5-02-14410-X .
3. Chikrii A.A. Conflict-controlled processes. Dordrecht, Springer, 1997, 404 p. Original Russian text published in Konfliktno upravlyaemye processy, Kiev, Naukova Dumka Publ., 1992, 384 p.
4. Grigorenko N.L. Matematicheskie metody upravleniya neskol’kimi dinamicheskimi protsessami [Mathematical methods of control a few dynamic processes], Moscow, Moscow State Univ. Publ., 1990, 198 p. ISBN: 5-211-00954-1 .
5. Bopardikar S.D., Suri S. k-Capture in multiagent pursuit evasion, or the lion and the gyenas. Theor. Comput. Sci., 2014, vol. 522, pp. 13–23. https://doi.org/10.1016/j.tcs.2013.12.001
6. Petrov N.N. Multiple capture in Pontryagin’s example with phase constraints. Prikl. Mat. Mekh., 1997, vol. 61, no. 5, pp. 747–754 (in Russian).
7. Satimov N., Mamatov M.S. On problems of pursuit and evasion away from meeting in differential games between the group of pursuers and evaders. Dokl. Akad. Nauk Uzb. SSR, 1983, vol. 4, pp. 3–6 (in Russian).
8. Vagin D.A., Petrov N.N. A problem of group pursuit with phase constraints. J. Appl. Math. Mech., 2002, vol. 66, no. 2, pp. 225–232. https://doi.org/10.1016/S0021-8928(02)00027-8
9. Blagodatskikh A.I. Synchronous implementation of simultaneous multiple captures of evaders. Izv. IMI UdGU, 2023, vol. 61, pp. 3–26 (in Russian). https://doi.org/10.35634/2226-3594-2023-61-01
10. Blagodatskikh A.I., Bannikov A.S. Simultaneous multiple capture in the presence of evader’s defenders. Izv. IMI UdGU, 2023, vol. 62, pp. 10–29. https://doi.org/10.35634/2226-3594-2023-62-02
11. Petrov N.N. Double capture of coordinated evaders in recurrent differential games. Izv. IMI UdGU, 2024, vol. 63, pp. 49–60. https://doi.org/10.35634/2226-3594-2024-63-04
12. Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya [Guarantee optimization in control problems]. Moscow, Nauka Publ., 1981, 288 p.
13. Petrov N.N. About controllability of autonomous systems. Differ. Uravn., 1968, vol. 4, no. 4, pp. 606–617 (in Russian).
Cite this article as: N.N. Petrov. Double capture of coordinated evaders in linear group pursuit problem with a simple matrix and phase constraints. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 2, pp. 174–185.
