We study an optimal pursuit differential game problem in the Hilbert space l2r+1. The game is described by an infinite system of the first-order differential equations whose coefficients are negative. The control functions of players are subjected to integral constraints. If the state of the system coincides with the origin of the space l2r+1, then game is considered completed. We obtain an equation to find the optimal pursuit time. Moreover, we construct the optimal strategies for players.

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In the present article, we present a differential game of pursuit problem with the case of geometric constraint in the Hilbert space l 2. The game is given by system of 2-infinite systems of first order ordinary differential equations (ODEs). Geometric constraint are imposed on the control functions of players. The game is began from a given point z 0 called the initial position. It is given another point z 1 in the space l 2. The Pursuer targeting to bring the state of the system from z 0 to z 1 where an equation to find a guaranteed pursuit time is obtained while that of the Evader action is opposite. The game is assumed to be completed if z(t) = z 1 at some time t. Moreover, a control problem is studied and then extended to the differential game of pursuit where the strategy for the Pursuer is constructed explicitly.

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