A Multiplayer Pursuit Differential Game on a Closed Convex Set with Integral Constraints

Abstract

We study a simple motion pursuit differential game of many pursuers and many evaders on a nonempty convex subset of ${ℝ}^n$. In process of the game, all players must not leave the given set. Control functions of players are subjected to integral constraints. Pursuit is said to be completed if the position of each evader $y_j$, $j\in \{1,2,\mathrm{...}k\}$, coincides with the position of a pursuer $x_i$, $i\in \{1,\mathrm{...},m\}$, at some time $t_j$, that is, $x_i(t_j)=y_j(t_j)$. We show that if the total resource of the pursuers is greater than that of the evaders, then pursuit can be completed. Moreover, we construct strategies for the pursuers. According to these strategies, we define a finite number of time intervals $[{\theta }_{i-1},{\theta }_i]$ and on each interval only one of the pursuers pursues an evader, and other pursuers do not move. We derive inequalities for the resources of these pursuer and evader and, moreover, show that the total resource of the pursuers remains greater than that of the evaders.