Fixed-Time Pursuit–Evasion Differential Game with Grönwall Constraints in Hilbert Space l2

This paper studies a pursuit–evasion differential game in the Hilbert space l2 involving countably many pursuers and one evader. The players follow simple motion dynamics, while their control functions are subject to Grönwall-type constraints. The value of the game λ is defined as the infimum of the distances between the evader and the pursuers at a given terminal time θ. The pursuers aim to minimize this distance λ by the end of the game, while the evader aims to keep it as large as possible. Optimal strategies for the pursuers are constructed via the method of fictitious pursuers, and an explicit evasion strategy is derived. As a result, the optimal strategies of both sides and the value of the game are obtained.