Transfinite Sequences in the Programmed Iteration Method

Abstract

We consider the problem of retaining the motions of an abstract dynamic system in a given constraint set. Constructions from the programmed iteration method are extended to problems whose dynamics, in general, does not possess any topological properties. The weaker requirements are compensated by introducing transfinite iterations of the programmed absorption operator. The technique of fixed points of mappings in chain-complete partially ordered sets is used in the proofs. The proposed procedure produces a set where the retention problem is solved in the class of quasistrategies. The control interval is not assumed to be bounded.