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## Scientific Bulletin. Physical and Mathematical Research

#### Abstract

A scientific school on differential games was founded by N.Y.Satimov in Uzbekistan, and is being continued by A.Azamov now. The representatives of this school have achieved significant success in this area, have created remarkable ways in the theory of pursuit with a geometric constraint on the parameters of control. It is noteworthy that the theorem of A.Azamov  on this problem practically solved the famous problem created by Pontryagin. Significant results were also obtained with an integral constraint on the player control parameter. Influential world journals published the results obtained in this direction by N.Y.Satimov, B.B.Rikhsiev, G.I.Ibragimov, A.A.Khamdamov, and others [4-6]. Thanks to in-depth scientific research, A.Z.Fozilov and M.S.Nikolskiy achieved the transfer of the second Pontryagin method to a position with an integral constraint of control parameters . We study a pursuit differential game of m pursuers and one evader Ewhich is described by the following differential equations (1) where — control parameter of the pursuer , and E of evader— . Assume controls of the pursuers and evader are defined as the measurable function , , subjected to contraints: (2) (3) (4) (5) where – positive numbers. Defenition 1.A function , Is referred to as the strategy of the pursuer iffor an arbitrary admissible controlof the evader , the initial value problem has a unique solution and along this solution the following coordinate-wise integral contraints are satisfied. Assume.There are sets , satisfying condition , for set and , let it satisfy . Theorem 1.If assume is performed, then ingame(1), (2)-(5) d multiple pursuit comes from any starting position . Theorem 2.If the following inequalities satisfy longd multiple pursuit occur. Theorem 3. If one-time pursuit occur.

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