Optimal strategies of players in linear differential games.

A game involves a number of players, says N, a set of strategies for each player, and a pay of that quantitatively describes the outcome of each play of the game in terms of the amount that each player wins or loses. A common type of game is often called the pursuit-evasion game. Pursuit-evasion game is about how to guide one or a group of pursuers to catch one or a group of moving evaders. In the general definition of a pursuit-evasion game, there will typically be N players with opposing goals, each of them conflicts the other. Each player tries to fulfill his or her goals, and it is assumed that all players always do their best to fulfill their goals. These goals are formally expressed in terms of minimizing or maximizing a pay of functional. In this thesis, we study a pursuit-evasion differential game of countably many players in Hilbert space. Motions of the players are described by the ordinary differential equations of first and second order. The control functions of players are subject to geometric and integral constraints. Resource for the control of each pursuer is greater than that of the evader. Duration of the game is ¯xed. The payo® functional is the greatest lower bound of the distances between the pursuers and evader when the game is terminated. The pursuers try to minimize the payo® functional, and the evader tries to maximize it. We give a formula to calculate the value of the game and construct optimal strategies of the players. To solve the ¯rst part of the problem, the pursuit game, we use the method of fictitious pursuers. In addition, we consider an evasion di®erential game of several pursuers and one evader with simple motions and integral constraints on control functions of players. We ¯nd the su±cient condition for the evader to escape from all pursuers. We present explicit strategy for the evader and show that the proposed escape is possible, no matter what control is adapted by the pursuers. We prove the admissibility of our strategy as well. Finally, an application of pursuit-evasion game in a missile guidance system is introduced by constructing optimal strategy of pursuer missile which guarantees capturing of the evader missile