About an example of the attraction set construction with employment of Stone space

A. G. Chentsov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2012), pp. 108-124

Résumé

The extension construction of the abstract problem of attainability realized with employment of the Stone compactum (the space of ultrafilters in the traditional equipment) is considered. The questions connected with the structure of attraction sets are investigated; these attraction sets define possibilities for attainability of desired states in topological space under employment of asymptotic analogs of usual solutions. Constraints of asymptotic character are given. This constraints can be arising under the weakening of standard constraints used in control theory (the natural prototype of the investigated abstract problem is the problem about the construction of the asymptotic analog of the attainability domain for the control system under vanishingly small weakening of some constraints on the choice of the programmed control). Using the natural modification of Warga approach, we can introduce (along with precise solutions) so-called approximate solutions in the form of sequences of usual solutions satisfying the conditions (realizing in the totality “asymptotic constraints”) “with reinforcing exactness”. Sometimes, the employment of only such (sequential) approximate solutions can be insufficient. Nets or filters are required. The last objects are used as the basic type of (asymptotic in essence) solutions in this investigation under construction of attraction sets in the attainability problem with constraints of asymptotic character. And what is more, in these constructions, we can confine ourselves to the employment of ultrafilters. For a particular case, on this basis, the concrete structure of attraction set is established.

Détail

Keywords: attraction set, constraints of asymptotic character, ultrafilter.