Optimal Pursuit Time in Linear Differential Game

We consider a differential game of many pursuers and one evader. The motion of the players is described by a linear differential equations. Control functions of the players are subject to generalized integral constraints. The position of the pursuit and evader at some time t is described by x(t) and y(t) respectively, z(t) represents the state of the game express as z(t) = y(t) - x(t). Game is said to be completed if z(t) = 0, that is if the position of pursuer and evader is the same. Pursuer tries to complete the game and evader pursues the opposite goal. We construct a formula for guaranteed pursuit time and prove that it an optimal pursuit time. To this end, we construct the optimal strategies of the players. Lastly, we demonstrate our results with a numerical example.