Implementation of the programmed iterations method in packages of spaces

The problem of retention studied here might be regarded (in the case of a bounded control interval) as a version of the approach problem within given constraints in the phase space and the target set given by the hyperplane of the space positions corresponding to the terminal moment of the process (the retention problem on the infinite horizon also fits to the problem statement in this paper). The essential difference of the paper from the previously considered formulations is the variability of the spaces of the system trajectories and the disturbance realizations depending on the initial moment. It is shown that the unsolvability set of the retention problem is the least element of the convexity constructed on the basis of the programmed absorption operator; under some additional consistency conditions (on the space of system trajectories and on the space of admissible disturbances corresponding to different time moments) the set of successful solvability of the retention problem is constructed as the limit of the iterative procedure in the space of sets, the elements of which are positions of the game, and the structure of resolving quasi-strategies is provided.