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Home » Issues » Archive of Issues » 2015 - 2 issue » pp. 212-229

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Russia Yekaterinburg
Year
2015
Volume
25
Issue
2
Pages
212-229
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Section Mathematics
Title To question about realization of attraction elements in abstract attainability problems
Author(-s) Chentsov A.G.a
Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa
Abstract An abstract attainability problem under constraints of asymptotic character is considered; the corresponding solution is identified with an attraction set in the class of ultrafilters of the space of ordinary solutions. The remainder of the above-mentioned set with respect to closuring the set of results supplied by precise solutions is investigated (the given notion of a precise solution conceptually corresponds to Warga scheme although it is applied to the case of more general constraints). To represent the above-mentioned (basic) attraction set, the corresponding analog (of the last set) realized in the space of generalized elements is used. For thus obtained auxiliary attraction set, the remainder is analyzed; its connection with the remainder of the basic attraction set is investigated. Conditions of identifying the remainders for basic and auxiliary attraction sets are obtained. General statements are detailed for the case when generalized elements are defined in the form of ultrafilters of widely interpreted measurable spaces where free ultrafilters are responsible for the realization of remainders. It is established that, under existence of a remainder, the set of generalized admissible elements does not coincide with closuring a set of ordinary solutions (this set does not admit standard realization).
Keywords remainder, attraction set, ultrafilter
UDC 519.6
MSC 28A33
DOI 10.20537/vm150206
Received 15 April 2015
Language Russian
Citation Chentsov A.G. To question about realization of attraction elements in abstract attainability problems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 2, pp. 212-229.
References
  1. Chentsov A.G. To the validity of constraints in the class of generalized elements, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2014, no. 3, pp. 90-109 (in Russian).
  2. Chentsov A.G. Filters and ultrafilters in the constructions of attraction sets, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 1, pp. 113-142 (in Russian).
  3. Warga J. Optimal'noe upravlenie differentsial'nymi i funktsional'nymi uravneniyami (Optimal control of differential and functional equations), Moscow: Nauka, 1977, 624 p.
  4. Gamkrelidze R.V. Osnovy optimal'nogo upravleniya (Foundations of optimal control), Tbilisi: Izd. Tbilis. Univ., 1975, 230 p.
  5. Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p.
  6. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi (Control of dynamic system), Moscow: Nauka, 1985, 518 p.
  7. Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya (Optimization of guarantee in control problems), Moscow: Nauka, 1981, 288 p.
  8. Gryzlov A.A., Bastrykov E.S., Golovastov R.A. About points of compactification of $N$, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 3, pp. 10-17 (in Russian).
  9. Gryzlov A.A., Golovastov R.A. The Stone spaces of Boolean algebras, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 1, pp. 11-16 (in Russian).
  10. Flachsmeyer J., Terpe F. Some applications of the theory of compactifications of topological spaces and measure theory, Russian Mathematical Surveys, 1977, vol. 32, no. 5, pp. 133-171. DOI: 10.1070/RM1977v032n05ABEH003866
  11. Kuratovskii K., Mostovskii A. Teoriya mnozhestv (Theory of sets), Moscow: Mir, 1970, 416 p.
  12. Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov (Theory of stochastic processes), Мoscow: Fizmatlit, 2005, 402 p.
  13. Engelking R. Obshchaya topologiya (General topology), Moscow: Mir, 1986, 751 p.
  14. Burbaki N. Obshchaya topologiya (General topology), Moscow: Nauka, 1968, 272 p.
  15. Chentsov A.G. Some ultrafilter properties connected with extension constructions, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2014, no. 1, pp. 87-101 (in Russian).
  16. Chentsov A.G., Morina S.I. Extensions and relaxations, Dordrecht-Boston-London: Kluwer Academic Publishers, 2002, 408 p.
  17. Chentsov A.G. Representation of attraction elements in abstract attainability problems with asymptotic constraints, Russian Mathematics, 2012, vol. 56, issue 10, pp. 38-49.
  18. Chentsov A.G. Attraction sets in abstract attainability problems: Equivalent representations and basic properties, Russian Mathematics, 2013, vol. 57, issue 11, pp. 28-44.
  19. Chentsov A.G. Tier mappings and ultrafilter-based transformations, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2012, vol. 18, no. 4, pp. 298-314 (in Russian).
  20. Chentsov A.G. The nonsequential approximate solutions in problems of asymptotic analysis, Soochow Journal of Mathematics, 2006, vol. 32, no. 3, pp. 441-475.
  21. Chentsov A.G. Elementy konechno-additivnoi teorii mery, I (The elements of finitely additive measures theory, I), Yekaterinburg: USTU-UPI, 2008, 388 p.
  22. Chentsov A.G. Elementy konechno-additivnoi teorii mery, II (The elements of finitely additive measures theory, II), Yekaterinburg: USTU-UPI, 2010, 541 p.
  23. Chentsov A.G. On one example of representing the ultrafilter space for an algebra of sets, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2011, vol. 17, no. 4, pp. 293-311 (in Russian).
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