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.

In the present paper, we investigate a differential game of pursuit for an infinite system of simple motion in a plane. The control functions of the players satisfies both geometric and integral constraints respectively. In the plane, the game is assume to be completed if the state of the pursuer xk, k = 1, 2, ... is directly coincide with that of the evader yk, k = 1, 2, ..., i.e; xk(x ) = yk(x ), k = 1,2, ..., at some time x and the evader is tries to stop the incident. In addition to that the strategy of the pursuer with respect to geometric and integral constraints will be constructed. Moreover, a numerical example will be given to illustrate the result.