Filters and ultrafilters in the constructions of attraction sets

Ченцов, А. Г.

doi:10.20537/vm110112

Cited by 16 publications

(10 citation statements)

References 4 publications

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“…But, now the analogous employment of filters and ultrafilters is more appropriate. The equivalence of the representations with employment of nets and ultrafilters was noted (in particular) in §3 of [7]. Now, we discuss the representation realized in class of ultrafilters.…”

Section: Elements Of Topologymentioning

confidence: 72%

“…Now, we discuss the representation realized in class of ultrafilters. So, by definition of §3 of [7] (…”

Section: Elements Of Topologymentioning

confidence: 99%

“…We consider the sets (4.1) and (4.2) as AS corresponding to given nonempty family of subsets of E (we recall that, under employment of nets in E, we obtain the equivalent representation of AS; in this connection, see constructions of §3 in [7]). By (4.1) the following singularity is realized: ultrafilters play the role of approximate solutions of J.Warga.…”

Section: Elements Of Topologymentioning

confidence: 99%

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Constraints of Asymptotic Character and Attainability Problems

Ченцов¹

2017

IIM

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The attainability problem with "asymptotic constraints" is considered. Concrete variants of this problem arise in control theory. Namely, we can consider the problem about construction and investigation of attainability domain under perturbation of traditional constraints (boundary and immediate conditions; phase constraints). The natural asymptotic analog of the usual attainability domain is attraction set, for representation of which, the Warga generalized controls can be applied. More exactly, for this, attainability domain in the class of generalized controls is constructed. This approach is similar to methods for optimal control theory (we keep in mind approximate and generalized controls of J. Warga). But, in the case of attainability problem, essential difficulties arise. Namely, here it should be constructed whole set of limits corresponding to different variants of all more precise realization of usual solutions in the sense of constraints validity. Moreover, typically, the abovementioned control problems are infinite-dimensional. Real possibility for investigation of the arising limit sets is connected with extension of control space. For control problems with geometric constraints on the choice of programmed controls, procedure of this extensions was realized (for extremal problems) by J. Warga. More complicated situation arises in theory of impulse control. It is useful to note that, for investigation of the problem about constraints validity, it is natural to apply asymptotic approach realized in part of perturbation of standard constraints. And what is more, we can essentially generalize self notion of constraints: namely, we can consider arbitrary systems of conditions defined in terms of nonempty families of sets in the space of usual controls. Thus, constraints of asymptotic character arise.

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Section: Elements Of Topologymentioning

confidence: 72%

“…Now, we discuss the representation realized in class of ultrafilters. So, by definition of §3 of [7] (…”

Section: Elements Of Topologymentioning

confidence: 99%

Section: Elements Of Topologymentioning

confidence: 99%

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Constraints of Asymptotic Character and Attainability Problems

Ченцов¹

2017

IIM

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“…We recall that ultrafilters were used as generalized elements in problems connected with attainability under constraints of asymptotic character (see, for example, [6][7][8]). Now, we seek to explore spaces which are comprehending for ultrafilters.…”

Section: Introductionmentioning

confidence: 99%

Some Representations Connected With Ultrafilters and Maximal Linked Systems

Ченцов

2017

UMJ

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Ultrafilters and maximal linked systems (MLS) of a lattice of sets are considered. Two following variants of topological equipment are investigated: the Stone and Wallman topologies. These two variants are used both in the case of ultrafilters and for space of MLS. Under Wallman equipment, an analog of superextension is realized. Namely, the space of MLS with topology of the Wallman type is supercompact topological space. By two above-mentioned equipments a bitopological space is realized.

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“…In this connection, let us recall that the Stone-Čech compactification (with certain insignificant modifications) was applied in [7] to extend an abstract reachability problem. This direction of research was developed in a number of publications; let us mention papers [8][9][10], where ultrafilters of broadly understood measurable spaces were used as generalized elements. A general approach to the construction of extensions in the class of ultrafilters was outlined in [8].…”

Section: Introductionmentioning

confidence: 99%