Multi pursuer differential game of optimal approach with integral constraints on player controls

A game contains three important elements, which is players, set of strategies, and payoff that quantitatively identified either win or lose for each players in term of the amount called value game. Pursuit-evasion game is a common type of game that describes how to guide one or a group of pursuers to catch one or a group of moving evader in a given environment. This thesis studied a pursuit-evasion differential game of optimal approach of finite or countable pursuers with one evader in the Hilbert space. The movements of players are described by the ordinary differential equation of first and second order. The control functions of players are subject to integral constraints, such constraints arise in modelling the constraints on energy. Important point to note is resource energy for the control of each pursuer need not to be greater than that of evader. Duration of the game q is fixed. The payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is finish. The pursuers try to minimize the functional and, the evader tries to maximize it. The formula to calculate the value of the game is given and construction of optimal strategies of the players. To solve the main theorem of the problem in this thesis relies on the solutions of auxiliary differential game in half space, and some properties of balls in half space. In the first part of proof of the theorem, the method of counterfeit or fictitious pursuers is used.The thesis shows the sufficient condition for the pursuers to catch the evader explicitly, and also prove the admissibility of the strategy. It is shows that the strategies of players ensure the value of the game