The Programmed Iteration Method in a Game Problem of Realizing Trajectories in a Function Set

We consider a differential game in which one of the players tries to keep a trajectory within a given set of vector functions on a finite time interval; the goal of the second player is opposite. To construct the set of successful solvability in this problem, which is defined by the functional target set, we apply the programmed iteration method. The essence of the method lies in a universal game problem of programmed control that depends on parameters characterizing the constraints on the initial fragments of trajectories. As admissible control procedures, we use multivalued quasistrategies (regarding a conflict-controlled system, it is assumed that the conditions of generalized uniqueness and uniform boundedness of programmed motions are satisfied).