The differential game with inertial players under integral constraints on controls:

Abdulla A. Azamov; Mokhisanam A. Turgunboeva

doi:10.21638/spbu10.2025.109

Authors

Abdulla A. Azamov V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan, 9, Universitetskaya ul., Tashkent, 100174, Uzbekistan https://orcid.org/0000-0003-3516-0904
Mokhisanam A. Turgunboeva Namangan State University, 316, Uychi ul., Namangan, 116019, Uzbekistan https://orcid.org/0009-0000-4408-171X

DOI:

https://doi.org/10.21638/spbu10.2025.109

Abstract

We consider pursuit-evasion differential games between inertial players (a pursuer and an evader) whose controls are subject to integral constraints. The pursuer is believed to have captured the evader when their positions coincide. The key method used to provide a win for the pursuer in the pursuit differential game is the parallel pursuit strategy (in brief, the Π-strategy). We obtain sufficient conditions for the solvability of pursuit-evasion problems. Furthermore, we investigate Isaacs’ “life-line” game in favor of the pursuer in the case of the identical initial velocities of the players. Here, the main lemma characterizing its monotonicity property provides an analytical formula for the players’ meeting domain. This paper extends and continues the works of R. Isaacs, L. A. Petrosyan, B. N. Pshenichnyi, N. Yu. Satimov, the authors of this article, and other researchers.

Keywords:

differential game, pursuer, evader, strategy, “life-line” game

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References

References

Isaacs R. Differential games. New York, John Wiley and Sons Publ., 1965, 385 p.

Pontryagin L. S. Izbrannye trudi [Selected works]. Moscow, MAKS Press, 2004, 551 p. (In Russian)

Krasovskii N. N. Teoriya upravleniya dvizheniem [Motion control theory]. Moscow, Nauka Publ., 1968, 476 p. (In Russian)

Petrosyan L. A., Dutkevich V. G. Igry s “liniei zhizni”. Sluchai l-zahvata [Games with “a survival zone”. Оccation l-catch]. Vestnik of Leningrad State University, 1969, vol. 3, no. 13, pp. 31–38. (In Russian)

Petrosyan L. A. Differential games of pursuit. Singapore, World Scientific Publ. Series on optimization, 1993, 326 p.

Petrosyan L. A. Ob odnom semeistve differencial'nyh igr na vyzhivanie v prostranstve ℝⁿ [About some of the family differential games at a survival in the space ℝⁿ]. Papers of Academy Sciences of USSR, 1965, vol. 161, no. 1, pp. 52–54. (In Russian)

Friedman A. Differential games. New York, Wiley Interscience Publ., 1971, 350 p.

Krasovskii N. N., Subbotin A. I. Game-theoretical control problems. New York, Springer, 2011, 517 p.

Bercovitz L. D. Differential game of generalized pursuit and evasion. SIAM Journal on Control and Optimization, 1986, vol. 24, no. 3, pp. 361–373. https://doi.org/10.1137/0324021

Pshenichnyi B. N. Simple pursuit by several objects. Cybernetics and System Analysis, 1976, vol. 12, no. 5, pp. 484–485. https://link.springer.com/article/10.1007/BF01070036

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. https://doi.org/10.1007/BF00934749

Chikrii A. A. Conflict-controlled processes. Dordrecht, Kluwer Academic Publ., 1997, 404 p.

Lewin J. Differential games: Theory and methods for solving game problems with singular surfaces. New York, Springer-Verlag, 1994, 242 p.

Satimov N. Yu. Metody resheniya zadachi presledovaniya v teorii differentsial’nykh igr [Methods of solving the pursuit problems in the theory of differential games]. Tashkent, National University of Uzbekistan Press, 2003, 240 p. (In Russian)

Azamov A. O zadache kachestva dlya igr prostogo presledovaniya s ogranicheniem [On the quality problem for simple pursuit games with constraint]. Serdica Bulgariacae Math., 1986, vol. 12, no. 1, pp. 38–43. (In Russian)

Azamov A. A., Samatov B. T. The П-strategy: Analogies and applications. The Fourth International Conference Game Theory and Management. St. Petersburg, 2011, vol. 4, pp. 33–47. http://mi.mathnet.ru/cgtm177

Petrov N. N. Simple group pursuit subject to phase constraints and data delay. Journal of Computer and Systems Sciences International, 2018, vol. 57, iss. 1, pp. 37–42. https://doi.org/10.1134/S1064230718010094

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. https://doi.org/10.1007/s10559-013-9563-7

Samatov B. T., Akbarov A. Kh. Differential game with a “life-line” under the Grönwall constraint on controls. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2024, vol. 20, iss. 2, pp. 265–280. https://doi.org/10.21638/spbu10.2024.211

Mamadaliev N. On a pursuit problem with integral constraints on the players controls. Siberian Mathematical Journal, 2015, vol. 55, pp. 107–124.

Levchenkov A. Y., Pashkov A. G. Differential game of optimal approach of two inertial pursuers to a noninertial evader. Journal of Optimization Theory and Applications, 1990, vol. 65, no. 3, pp. 501–518.

Ibragimov G. I., Allahabi F., Kuchkarov A. Sh. A pursuit problem in an infinite system of second-order differential equations. Ukrainian Mathematical Journal, 2014, vol. 65, no. 8, pp. 1203–1216.

Samatov B. T., Soyibboev U. B. Applications of the П-strategy when players move with acceleration. Proceedings of the IUTAM Symposium on Optimal Guidance and Control for Autonomous Systems, 2024, vol. 40, ID 165183, pp. 167–181.

Blagodatskikh V. I. Vvedenie v optimal'noe upravlenie (lineinaya teoriya) [Introduction to optimal control (linear theory)]. Moscow, Vysshaya shkola Publ., 2001, 33 p. (In Russian)