Quasistrategies in an abstract guidance control problem

We consider an abstract guidance control problem with elements of uncertainty, where the control procedures are identified with set-valued quasistrategies. The goal of the control consists in the guidance to the goal set under phase constraints. We construct the solvability set by the method of programmed iterations which is well known in the theory of differential games. We prove that the limit of the iterative procedure has the sense of the set of positional absorption (the stable bridge) introduced by N. N. Krasovskii. This limit coincides with the solvability set in the class of quasistrategies.