Linear pursuit-evasion differential games with integral constraints on control functions.

Recently use of decision-making in modern life has extensively increased. This lead to review subject of Pursuit-Evasion (PE) di®erential games. A di®erential game models a situation where two or more players operate in a same environment with con°icting goals. In this work, we attempt to solve general linear PE games in time-varying systems with continuous time. Most studies related to PE games in the current literature concentrate on two-player games with a single Pursuer and a single Evader and the results for general multi-player PE games are still largely sparse. The purpose of this study is to provide a theoretical foundation for linear PE games with integral constraints under the theory of the di®erential game and optimal control theory. The results of this study contain of four parts, in the ¯rst part, the linear pursuit-evasion game by using optimal control techniques which is based on structured controls of the players has been solved. We obtain a formula for the optimal pursuit time and construct the optimal strategies for the players when the control resource of the Pursuer is greater than the Evader. In addition, a new method for solving of the evasion problem is proposed where the control resources of the Pursuer are less than or equal to the Evader. Secondly, the more general linear pursuit-evasion game in the case where the ter-minal set closed and convex has been solved. For this case, we construct the set of attainability which is an ellipsoid. Some conditions on capturability are also discussed. The construction of the optimal pursuit time and optimal strategies for the players are the main objectives of this part. The third part deals with the study of di®erential game of optimal approach with many Pursuers and one Evader, which can be considered as the generalized case of a pursuit-evasion game with one Pursuer and one Evader. This part is devoted to the problem of capture of one Evader by many Pursuers. The case of integral constrains is considered and the strategies for the players are constructed. Con- ditions are obtained for the existence of solutions for a multi-Pursuer game. In order to estimate the value of the game, we obtain several lemmas and theorems. In the fourth part, the optimal control is obtained by using the method of the maximum principle of Pontryagin, where only a special case is studied. The result shows an applications of Pontryagin's maximum principle in a linear quadratic di®erential game (LQDG) with integral constraints.

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