ПЕРШИЙ ПРЯМИЙ МЕТОД ПОНТРЯГІНА ДЛЯ ДИФЕРЕНЦІАЛЬНИХ ВКЛЮЧЕНЬ

• Дата: 2020
• Автор: Iskanadjiev, I.M.
• Формат: Стаття
• Мова: English
• Опубліковано: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Теми: диференціальна гра; диференціальне включення; стратегія; переслідувач; допустиме управління; інтеграл; багатозначне відображення; апроксимація; наближені формули
• Онлайн доступ: https://jais.net.ua/index.php/files/article/view/447
• Назва журналу: Problems of Control and Informatics

Pontryagin’s direct methods played a large role in the development t of the theory of differential games and its application to specific applied problems. It turned out to be useful in control theory under conditions of uncertainty, also in solving the problem of control synthesis. Since direct methods have proved themselves as an effective means to solve the problem of pursuit and control, a lot of research have been devoted to the development of the corresponding theory. Pontryagin’s direct methods are based on the consideration of integrals, which has a number of significant differences from the classical integral. One of the differences is the use of the multi-valued mapping integral in their definitions. In this connection, some difficulties arise in calculating these integrals. In this paper, we consider a differential game described by differential inclusions of the form z E - F (t, v) , where F is a continuous compact valued map. The first direct method is described with respect to such class of games. In particular, the class of stroboscopic strategies of the pursuer, the trajectory of the system, is determined. For these class of games, it is proved that if the starting point belongs to the first integral (the integral of the multi valued (compact-valued) mapping that is present in the definition of the first direct method, then this is necessary and sufficient to complete the game at a fixed point in time in the class of stroboscopic strategies. Schemes are proposed for the approximate calculation of the integral of the first direct method. The approximative properties of this integral are studied. The semi-stability of such integrals with respect to the initial data of the differential game is proved. The first integral is stable under unilateral perturbations, as it were, not profitable for the pursuer of the initial data of the differential game.