Game value for a pursuit-evasion differential game problem in a Hilbert space

We consider a pursuit-evasion differential game problem with countable number pursuers and one evader in the Hilbert space \begin{document}$ l_{2}. $\end{document} Players' dynamic equations described by certain \begin{document}$ n^{th} $\end{document} order ordinary differential equations. Control functions of the players subject to integral constraints. The goal of the pursuers is to minimize the distance to the evader and that of the evader is the opposite. The stoppage time of the game is fixed and the game payoff is the distance between evader and closest pursuer when the game is stopped. We study this game problem and find the value of the game. In addition to this, we construct players' optimal strategies.