Some Topological Structures of Extensions of Abstract Reachability Problems / Chentsov A. G.,Pytkeev E. G. // PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS. - 2016. - V. 292, l. 1. - P. S36-S54.

ISSN/EISSN:

0081-5438 / 1531-8605

Type:

Article

Abstract:

We consider the problem of the reachability of states that are elements of a topological space under constraints of asymptotic nature on the choice of an argument of a given objective mapping. We study constructions that have the sense of extensions of the original space and are implemented with the use of methods that are natural for applied mathematics but employ elements of extensions used in general topology. The study is oriented towards the application in the problem on the construction and investigation of properties of reachability sets for control systems. Constructions involving an approximate observation of constraints in control problems, as well as various generalized regimes, were widely used by N.N. Krasovskii and his students. In particular, this approach was applied in the proof of N.N. Krasovskii and A.I. Subbotin's fundamental theorem of the alternative, which made it possible to establish the existence of a saddle point in a nonlinear differential game. In the investigation of impulse control problems, Krasovskii used techniques from the theory of generalized functions, which formed the basis for many studies in this direction. A number of A.B. Kurzhanski's papers are devoted to the solution of control problems related in one way or another to the construction of reachability sets. Control problems with incomplete information, duality issues for control and observation problems, and team control problems constitute a far from exhaustive list of research areas where Kurzhanskii obtained profound results. These studies are characterized by the use of a wide range of tools and methods from applied mathematics and various constructions as well as by the combination of theoretical investigations and procedures related to the possibility of computer modeling. The research direction developed in the present paper mainly concerns the problem of constraint observation (including ``asymptotic{''} constraints) and involves other issues. Nevertheless, the idea of constructing generalized elements of various nature (in particular, generalized controls) seems to be useful for the purpose of asymptotic analysis of control problems that do not possess stability as well as problems on the comparison of different tendencies in the choice of control in the form of dependences on a complex of factors inherent in the original real-life problem. The use of such tools as the Stone Cech compactification and Wallman's extension is, of course, oriented toward the study of qualitative issues. In the authors' opinion, the combined application of the approaches to the construction of extensions used in control theory and in general topology holds promise from the point of view of both pure and applied mathematics. Apparently, the present paper can be considered as a certain step in this direction.

Author keywords:

attraction set; topological space; ultrafilter ULTRAFILTERS; SPACES

DOI:

10.1134/S0081543816020048

Web of Science ID:

ISI:000376272600004

Соавторы в МНС:

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Другие поля

Поле	Значение
Month	APR
Publisher	MAIK NAUKA/INTERPERIODICA/SPRINGER
Address	233 SPRING ST, NEW YORK, NY 10013-1578 USA
Language	English
EISSN	1531-8605
Keywords-Plus	ULTRAFILTERS; SPACES
Research-Areas	Mathematics
Web-of-Science-Categories	Mathematics, Applied; Mathematics
Author-Email	chentsov@imm.uran.ru pyt@imm.uran.ru
Funding-Acknowledgement	Presidium of the Russian Academy of Sciences {[}12-P-1-1012, 12-P-1-1019]; Russian Foundation for Basic Research {[}12-01-00537 a, 13-01-00304 a, 13-01-90414 ukr\_f\_a]
Funding-Text	This work was supported by the Presidium of the Russian Academy of Sciences (project nos. 12-P-1-1012 and 12-P-1-1019) and by the Russian Foundation for Basic Research (project nos. 12-01-00537 a, 13-01-00304 a, and 13-01-90414 ukr\_f\_a).
Number-of-Cited-References	22
Usage-Count-Last-180-days	1
Usage-Count-Since-2013	3
Journal-ISO	Proc. Steklov Inst. Math.
Doc-Delivery-Number	DM3UQ