Linear discrete pursuit game with phase constraints

We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Position of the evader satisfies phase constraints: , where is a subset of . We considered two cases: (1) controls of the players satisfy geometric constraints, and (2) controls of the players satisfy total constraints. Terminal set is a subset of and it is assumed to have a nonempty interior. Game is said to be completed if at some step ; thus, the evader has not the right to leave set . To construct the control of the pursuer, at each step , we use the value of the control parameter of the evader at the step . We obtain sufficient conditions of completion of pursuit from certain initial positions of the players in finite time interval and construct a control for the pursuer in explicit form