Evasion differential game from many pursuers of one evader whose control set is a sector

From mathematical point of view, game involves a number of players, a set of strategies for each player, and analysis of the game outcome which conclude either victory or defeat for each player involved. A common type of game is often called the pursuit-evasion game. Pursuit-evasion game is one of the widely studied game theory where it involves players of two opposite sides, which are pursuer and evader. The pursuer’s goal is to capture the evader while oppositely, the evader is to avoid being captured. Strategy to be constructed for players depend on the purpose of the game. It can be solved as an evasion problem for the evader. In this case, strategy for evader will be constructed and behavior of the pursuer is assumed to be any. On the other hand, the game could also be a pursuit problem and thus construction of the strategy is for the pursuer, with assumption that the evader can move freely. In this thesis, we study an evasion differential game of many pursuers x₁; : : : ;xm against one evader y in the plane R2. Movements of the players are described by simple differential equations. Control functions of players are subjected to geometric constraints where maximum speed of each pursuer is equal to 1, and maximum speed of the evader is a > 1. Control set of the evader is a sector S with radius a. We say that evasion is possible if xi(t) 6= y(t) for all t ≥ 0 and i = 1; : : : ;m. In other words, the evasion problem is solved when it is proved that the position of the evader never coincides with the position of each pursuer at all time. To achieve the solution, conditions of evasion that guarantee the evasion from any initial positions of players are to be found. We examine game with one pursuer by constructing the evader’s strategy, checking the admissibility, and estimating distances between the evader and pursuer for the possibility of evasion. Then we show that evasion game is solvable for the case of k pursuers. The motivation behind the study is to construct a new admissible strategy for the evasion to be possible in the evasion differential game of one evader versus many pursuers, which were studied in many works before.

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