Under consideration is a linear differential game of several persons with integral constraints on the controls of the players. Pursuit terminates if a solution to at least one of the equations describing the differential game hits the origin at some instant of time. We establish a necessary and sufficient condition of terminating pursuit from all points of the space in the case of a single
... [Show full abstract] pursuer and a sufficient condition in the case of many pursuers. This is an English translation of the author’s article [Mat. Tr. 6, No. 2, 66-79 (2003; Zbl 1060.91032)].