Compactifiers in extension constructions for reachability problems with constraints of asymptotic nature

  • A. G. ChentsovEmail author


A reachability problem with constraints of asymptotic nature is considered in a topological space. The properties of a rather general procedure that defines an extension of the problem are studied. In particular, we specify a rule that transforms an arbitrary extension scheme (a compactifier) into a similar scheme with the property that the continuous extension of the objective operator of the reachability problem is homeomorphic. We show how to use this rule in the case when the extension is realized in the ultrafilter space of a broadly understood measurable space. This version is then made more specific for the case of an objective operator defined on a nondegenerate interval of the real line.


attraction set topological space ultrafilter quotient space 


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  1. 1.
    J. Warga, Optimal Control of Differential and Functional Equations (Academic, New York, 1972; Nauka, Moscow, 1977).zbMATHGoogle Scholar
  2. 2.
    R. V. Gamkrelidze, Principles of Optimal Control Theory (Izd. Tbilis. Univ., Tbilisi, 1977; Plenum, New York, 1978).CrossRefzbMATHGoogle Scholar
  3. 3.
    N. N. Krasovskii, Theory of Motion Control (Nauka, Moscow, 1968) [in Russian].Google Scholar
  4. 4.
    N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].zbMATHGoogle Scholar
  5. 5.
    N. N. Krasovskii and A. I. Subbotin, “An alternative for the game problem of convergence,” J. Appl. Math. Mech. 34 (6), 948–965 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. V. Kryazhimskii, “On the theory of positional differential games of approach–evasion,” Dokl. Akad. Nauk SSSR 239 (4), 779–782 (1978).MathSciNetGoogle Scholar
  7. 7.
    A. V. Bulinskii and A. N. Shiryaev, The Theory of Random Processes (Fizmatlit, Moscow, 2005) [in Russian].Google Scholar
  8. 8.
    R. Engelking, General Topology (Heldermann, Berlin, 1989; Mir, Moscow, 1986).zbMATHGoogle Scholar
  9. 9.
    A. G. Chentsov, “Attraction sets in abstract attainability problems: Equivalent representations and basic properties,” Russ. Math. 57 (11), 28–44 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A. G. Chentsov, “Some properties of ultrafilters related to extension constructions,” Vestn. Udmurt. Univ., Ser. Mat. Mekh. Komp. Nauki, No. 1, 87–101 (2014).CrossRefzbMATHGoogle Scholar
  11. 11.
    N. Bourbaki, General Topology (Hermann, Paris, 1940; Nauka, Moscow, 1968).Google Scholar
  12. 12.
    A. G. Chentsov, “Filters and ultrafilters in constructions of attraction sets,” Vestn. Udmurt. Univ., Ser. Mat. Mekh. Komp. Nauki, No. 1, 113–142 (2011).CrossRefzbMATHGoogle Scholar
  13. 13.
    A. G. Chentsov and S. I. Morina, Extensions and Relaxations (Kluwer Acad., Dordrecht, 2002).CrossRefzbMATHGoogle Scholar
  14. 14.
    J. L. Kelley, General Topology (Van Nostrand, Princeton, NJ, 1955; Nauka, Moscow, 1968).zbMATHGoogle Scholar
  15. 15.
    P. S. Aleksandrov, Introduction to the Theory of Sets and General Topology (Editorial URSS, Moscow, 2004) [in Russian].Google Scholar
  16. 16.
    A. G. Chentsov, “Tier mappings and ultrafilter-based transformations,” Trudy Inst. Mat. Mekh. UrO RAN 18 (4), 298–314 (2012).Google Scholar
  17. 17.
    A. G. Chentsov, “An abstract reachability problem: “Purely asymptotic” version,” Trudy Inst. Mat. Mekh. UrO RAN 21 (2), 289–305 (2015).Google Scholar
  18. 18.
    A. G. Chentsov, “On the question of validity of constraints in a class of generalized elements,” Vestn. Udmurt. Univ., Ser. Mat. Mekh. Komp. Nauki, No. 3, 90–109 (2014).CrossRefzbMATHGoogle Scholar
  19. 19.
    A. G. Chentsov, “On the question of the structure of attraction sets in a topological space,” Izv. Inst. Mat. Informat. Udmurt. Univ., No. 1, 147–150 (2012).zbMATHGoogle Scholar
  20. 20.
    A. G. Chentsov, “On one example of representing the ultrafilter space for an algebra of sets,” Trudy Inst. Mat. Mekh. 17 (4), 293–311 (2011).Google Scholar
  21. 21.
    A. G. Chentsov, Asymptotic Attainability (Kluwer Acad., Boston, 1997).CrossRefzbMATHGoogle Scholar
  22. 22.
    N. Dunford and J. Schwartz, Linear Operators: General Theory (Interscience, New York, 1958; Inostrannaya Lit., Moscow, 1962).zbMATHGoogle Scholar
  23. 23.
    A. G. Chentsov, A. P. Baklanov, and I. I. Savenkov, “An attainability problem with constraints of asymptotic nature,” Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., No. 1 (47), 54–118 (2016).MathSciNetGoogle Scholar
  24. 24.
    A. G. Chentsov and A. P. Baklanov, “A problem related to asymptotic attainability in the mean,” Dokl. Math. 90 (3), 762–765 (2014).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Ural Federal UniversityYekaterinburgRussia

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