On game value of a differential game problem with Grönwall-type constraints on players control functions

The game value of a pursuit-evasion differential game is an estimation of the game's payoff at the instant when all the players employ their optimal strategies. In this paper, we estimate this value for a fixed duration differential game problem of countably many pursuers and one evader with the Gronwall-type constraints, a generalization of the well known geometric constraints, imposed on all the players' control functions. The players' dynamics are governed by a generalized dynamic equations. We construct the attainability domain and a Gronwall-type optimal strategies for the players. The constructed strategies are then employed in establishing that the estimated game value is guaranteed for the players.