MSC: 49N70
DOI: 10.21538/0134-4889-2022-28-3-285-295
Полный текст статьи (Full text)
In the paper, we investigate pursuit-evasion problems in a simple motion differential game with two players, termed a pursuer and an evader. We put different kinds of non-stationary integral constraints, which restrict the energy consumption rate of the players. On the other hand, it is assumed that at each time the players have some additional amount of control resource. The integral constraint on the control of the pursuer is given under certain conditions, which include a non-stationary integral constraint. Firstly, the reachable set of each player is determined. We put forward the parallel approach strategy, which is known as a Π-strategy, for the pursuer, and as a result, we get necessary and sufficient conditions of capture. To solve the evasion problem, a specific admissible strategy is provided for the evader and a sufficient condition is obtained. Furthermore, in the pursuit problem, an optimal capture time is found through the strategy of the evader. In order to illustrate the obtained results, several examples are given, where guaranteed capture times are proposed for the pursuit problems and lower bounds for the distances between the players are obtained for the evasion problem. This work extends the results and methods from the works of R. Isaacs, L.A. Petrosjan, N.N. Krasovskii, A.A. Chikrii, A.A. Azamov, and other authors.
Keywords: pursuit-evasion differential games, simple motion, non-stationary integral constraint, pursuer, evader, strategy, guaranteed capture time
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Received May 10, 2022
Revised August 11, 2022
Accepted August 15, 2022
Bahrom Tadjiahmatovich Samatov, Doctor of Physical and Mathematical Sciences, Professor, Department of Mathematical Analysis, Namangan State University, Uychi str., 316, Namangan, 116019, Uzbekistan, e-mail: samatov57@inbox.ru
Bakhodirjon Inomjon ugli Juraev, PhD student, Department of Mathematics, Andijan State University, University str., 129, Andijan, 170100, Uzbekistan, e-mail: jbahodirjon@bk.ru
Cite this article as: B.T. Samatov, B.I. Juraev. Pursuit-evasion problems under nonlinear increase of the pursuer’s resource. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 285–295.
Русский
Б.Т. Саматов, Б.И. Жураев. Задачи преследования-уклонения при нелинейном увеличении ресурса преследователя
Исследуются задачи преследования-уклонения в дифференциальной игре с простым движением и двумя игроками, называемыми преследователем и убегающим. На управления игроков накладываются различные типы нестационарных интегральных ограничений, связанных со скоростью расходования энергии. Интегральное ограничение на управление преследователя задано при определенных условиях и включает в себя нестационарное интегральное ограничение. Управление убегающего подчиняется геометрическому ограничению. Во-первых, найдено множество достижимости каждого игрока. При использовании преследователем стратегии параллельной сходимости, известной как Π-стратегия, получены необходимые и достаточные условия поимки. Для решения задачи уклонения указана конкретная допустимая стратегия убегающего и получено достаточное условие уклонения. Далее, с помощью стратегии убегающего в задаче преследования найдено оптимальное время поимки. Для иллюстрации полученных результатов приводится несколько примеров с численными решениями и рисунками. Настоящая работа дополняет результаты и методы работ Р. Айзекса, Л. А. Петросяна, Н. Н. Красовского, А. А. Чикрия, А. А. Азамова и других ученых, включая авторов данной статьи.
Ключевые слова: дифференциальные игры, преследователь, убегающий, интегральное ограничение, стратегия, преследование, уклонение, гарантированное время поимки, оптимальное время поимки
