Differential game with a “life line” under the Grönwall constraint on controls:

Адахамбек Хасанбаевич Акбаров; Бахром Таджиахматович Саматов

doi:10.21638/spbu10.2024.211

Авторы

Адахамбек Хасанбаевич Акбаров Андижанский государственный университет, Узбекистан, 170100, Андижан, Университетская ул., 129 https://orcid.org/0000-0001-9797-0430
Бахром Таджиахматович Саматов Наманганский государственный университет, Узбекистан, 116019, Наманган, ул. Уйчи, 316 https://orcid.org/0000-0002-0734-8507

DOI:

https://doi.org/10.21638/spbu10.2024.211

Аннотация

Изучается дифференциальная игра с «линией жизни» для одного преследователя и одного убегающего при управлениях удовлетворяющих неравенств типа Гронуолла. Убегающий считается пойманным со стороны преследователя, если состояние убегающего совпадает с состоянием преследователя. Одна из основных целей настоящей работы — построение оптимальных стратегий для игроков и определение оптимального времени поимки. Для преследователя предлагается стратегия параллельного сближения (короче, Π-стратегия) и доказывается ее оптимальность. Для решения задачи с «линией жизни» исследуется динамика области достижимости игроков методом Петросяна, т. е. найдены условия монотонности по включению относительно времени этой области достижимости. Отметим, что работа продолжает исследования Айзекса, Петросяна, Пшеничного, Азамова и др.

Ключевые слова:

дифференциальная игра, преследование, убегание, ограничение Гронуолла, стратегия, параллельное преследование, область достижимости, игра с «линией жизни», сфера Аполлония

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References

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

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

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

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

Pshenichnyi B. N. Simple pursuit by several objects. Cybernetics and System Analysis, 1976, vol. 12, no. 5, pp. 484–485.

Pshenichnyi B. N., Chikrii A. A., Rappoport J. S. Effektivnyj metod resheniya differencial'nyh igr so mnogimi presledovatelyami [An effective method for solving differential games with many pursuers]. Dokl. Akad. Nauk SSSR, 1981, vol. 256, no. 3, pp. 530–535. (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, 2010, vol. 4, pp. 33–47.

Blagodatskikh A. I., Petrov N. N. Konfliktnoe vzaimodejstvie grupp upravlyaemyh ob'ektov [Conflict interaction of groups of controlled objects]. Izhevsk, Udmurt State University Press, 2009, 266 p. (In Russian)

Chikrii A. A. Conflict-controlled processes. Dordrecht, Kluwer, 1997, 403 p.

Grigorenko N. L. Matematicheskie metody upravleniya neskol'kimi dinamicheskimi protsessami [Mathematical methods of control for several dynamic processes]. Moscow, Moscow University Press, 1990, 198 p. (In Russian)

Petrosyan L. A. Ob odnom semejstve 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)

Petrosyan L. A. Igry presledovaniya s “liniej zhizni” [Pursuit games with “a survival zone”]. Vestnik of Leningrad State University, 1967, vol. 3, no. 13, pp. 76–85. (In Russian)

Petrosyan L. A., Dutkevich V. G. Igry s “liniej zhizni”, Sluchaj 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., Rikhsiev B. B. Presledovanie na ploskosti [Pursuit on the plane]. Moscow, Nauka Publ., 1991, 96 p. (In Russian)

Petrosyan L. A., Mazalov V. V. Game theory and applications. II. New York, Nova Sci. Publ., 1996, 219 p.

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.

Samatov B. T. The pursuit-evasion problem under integral-geometric constraints on pursuer controls. Automation and Remote Control, 2013, vol. 74, no. 7, pp. 1072–1081.

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.

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.

Samatov B. T. The П-strategy in a differential game with linear control constraints. Journal of Applied Mathematics and Mechanics, 2014, vol. 78, no. 3, pp. 258–263.

Dar'in A. N., Kurzhanskii A. B. Control under indeterminacy and double constraints. Differential Equations, 2003, vol. 39, no. 11, pp. 1554–1567.

Kornev D. V., Lukoyanov N. Yu. On a minimax control problem for a positional functional under geometric and integral constraints on control actions. Proceedings of the Steklov Institute of Mathematics, 2016, vol. 293, pp. 85–100.

Satimov N. Yu. Metody resheniya zadachi presledovaniya v teorii differencial'nyh igr [Methods of solving of pursuit problem in differential games]. Tashkent, National University of Uzbekistan Press, 2003, 240 p. (In Russian)

Ibragimov G. I. A game of optimal pursuit of one object by several. Journal of Applied Mathematics and Mechanics, 1998, vol. 62, no. 2, pp. 187–192.

Ibragimov G. I. Optimal pursuit with countably many pursuers and one evader. Differential Equations, 2005, vol. 41, no. 5, pp. 627–635.

Aubin J. P., Cellina A. Differential inclusions. Set-valued maps and viability theory. Berlin, Heidelberg, New York, Tokyo, Springer-Verlag, 1984, XIII, 342 p.

Pang J. S., Stewart D. E. Differential variational inequalities. Mathematical Programming. Series A, 2008, vol. 113, no. 2, pp. 345–424.

Samatov B. T., Ibragimov G. I., Hodjibayeva I. V. Pursuit-Evasion differential games with the Grönwall type constraints on controls. Ural Mathematical Journal, 2020, vol. 6, no. 2, pp. 95–107.

Samatov B. T., Akbarov A. Kh., Zhuraev B. I. Pursuit-Evasion differential games with Gr-constraints on controls. Izvestiya Instituta matematiki i informatiki Udmurtskogo gosudarstvennogo universiteta [Processing of Institute for Mathematics and Informatics of Udmurt State University], 2022, vol. 59, pp. 67–84.

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.

Blagodatskikh V. I. Vvedenie v optimal'noe upravlenie [Introduction to optimal control]. Moscow, Vysshaya shkola Publ., 2001, 239 p. (In Russian)