An Approach for Solving Discrete Game Problems with Total Constraints on Controls

Abstract

We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Controls of the players satisfy total constraints. Terminal set $M$ is a subset of ${ℝ}^n$ and it is assumed to have nonempty interior. Game is said to be completed if $y\left(k\right)-x\left(k\right)\in M$ at some step $k$. To construct the control of the pursuer, at each step $i$, we use positions of the players from step 1 to step $i$ and the value of the control parameter of the evader at the step $i$. We give sufficient conditions of completion of pursuit and construct the control for the pursuer in explicit form. This control forces the evader to expend some amount of his resources on a period consisting of finite steps. As a result, after several such periods the evader exhausted his energy and then pursuit will be completed.