The elements of the operator convexity in the construction of the programmed iteration method

The problem of retention studied here can be regarded (in the case of bounded control interval) as a variant of the approach problem within the 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 the problem stated in the work). The main difference of the problem from the previously considered formulation is the possibility of variation of the spaces of system trajectories and disturbance realizations depending on the initial moment of control. It is shown that the unsolvability set of the retention problem is the operator convex hull of the empty set constructed on the base of programmed absorption operator. Under some additional coherence conditions (on the spaces of system trajectories and disturbance realizations corresponding to different initial moments) the set of successful solvability is constructed as the limit of the iterative procedure in the space of sets, elements of which are positions of the game; in this case the structure of resolving quasistrategy is also given.