Skip to main content
SpringerLink
Log in

On certain problems of the structure of ultrafilters related to extensions of abstract control problems

  • Topical Issue
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We study the procedures for constructing ultrafilters in measurable spaces that are in a general sense based on Cartesian products. Our constructions intend to use ultrafilters as generalized elements in extension constructions for abstract reachability problems with asymptotic constraints. Constrains of this kind may arise, for instance, with sequential relaxation of boundary and intermediate conditions in control problems. In this case, families of sets of admissible regular controls in practically interesting cases represent filter bases, which makes it natural to use ultrafilters (maximal filters) that are admissible, in a certain sense, with respect to these families. We assume that counterparts of reachability sets considered in this work are sets in a topological space; the latter may be non-metrizable which occurs, for instance, in typical applications of the pointwise convergence topology. Such a topology may also be useful in impulse control problems, where one studies sheafs of motions as counterparts of reachability sets for exact and approximate satisfaction of trajectory constraints.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Warga, J., Optimal Control of Differential and Functional Equations, New York: Academic, 1972. Translated under the title Optimal’noe upravlenie differentsial’nymi i funktsional’nymi uravneniyami, Moscow: Nauka, 1979.

    MATH  Google Scholar 

  2. Gamkrelidze, R.V., Osnovy optimal’nogo upravleniya (Foundations of Optimal Control), Tbilisi: Tbilis. Univ., 1975.

    Google Scholar 

  3. Kelley, J.L., General Topology, New York: Van Nostrand, 1957. Translated under the title Obshchaya topologiya, Moscow: Nauka, 1981.

    Google Scholar 

  4. Arkhangel’skii, A.V., Compactness, in Itogi Nauki Tekhn., Ser. “Sovremen. Probl. Mat., Fundament. Napravlen., 1989, vol. 50, pp. 7–128.

    Google Scholar 

  5. Krasovskii, N.N., Teoriya upravleniya dvizheniem (Theory of Motion Control), Moscow: Nauka, 1968.

    Google Scholar 

  6. Krasovskii, N.N., Igrovye zadachi o vstreche dvizhenii (Game Problems on Meeting Motions), Moscow: Nauka, 1970.

    Google Scholar 

  7. Chentsov, A.G., Ultrafilters on Measurable Spaces as Generalized Solutions in Abstract Reachability Problems, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2011, vol. 17, no. 1, pp. 268–293.

    MathSciNet  Google Scholar 

  8. Chentsov, A.G., On One Representation of the Results of the Approximate Solutions’ Action in Problems with Asymptotic Constraints, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2011, vol. 17, no. 2, pp. 225–239.

    MathSciNet  Google Scholar 

  9. Chentsov, A.G., Some Constructions of Asymptotic Analysis Related to Stone-Čech Compactification, in Sovremennaya matematika i ee prilozheniya (Modern Mathematics and Applications), Tbilisi: Inst. of Cybernetics, Georgian Academy of Sciences, 2005, vol. 26, pp. 119–150.

    MathSciNet  Google Scholar 

  10. Chentsov, A.G., Extensions of Abstract Reachability Problems: Non-Sequential Version, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2007, vol. 13, no. 2, pp. 184–217.

    Google Scholar 

  11. Chentsov, A.G., Finitely Additive Measures and Extensions of Abstract Control Problems, J. Math. Sci., 2006, vol. 133, no. 2, pp. 1045–1206.

    Article  MATH  MathSciNet  Google Scholar 

  12. Chentsov, A.G., Ultrafilters in Constructions of Attraction Sets: The Problem of Satisfying Asymptotic Constraints, Differ. Uravn., 2011, vol. 47, no. 7, pp. 1047–1064.

    MathSciNet  Google Scholar 

  13. Duffin, R.J., Infinite Programs, in Linear Inequalities and Related Systems, Princeton: Princeton Univ. Press, 1956. Translated under the title Lineinye neravenstva i smezhnye voprosy, Moscow: Inostrannaya Literatura, 1959, pp. 263–267.

    Google Scholar 

  14. Gol’shtein, E.G., Teoriya dvoistvennosti v matematicheskom programmirovanii i ee prilozheniya (Duality Theory in Mathematical Programming and Applications), Moscow: Nauka, 1971.

    Google Scholar 

  15. Chentsov, A.G., On the Correctness of Certain Control Problems for a Material Point, Vestn. Udmurt. Univ., Mat. Mekh. Comp. Nauki, 2011, no. 3, pp. 127–141.

    Google Scholar 

  16. Chentsov, A.G., Finitely Additive Measures and Relaxations of Extremal Problems, New York: Plenum, 1996.

    MATH  Google Scholar 

  17. Chentsov, A.G., Asymptotic Attainability, Dordrecht: Kluwer, 1997.

    Book  MATH  Google Scholar 

  18. Chentsov, A.G. and Morina, S.I., Extensions and Relaxations, Dordrecht: Kluwer, 2002.

    Book  MATH  Google Scholar 

  19. Arkhangel’skii, A.V., Topologicheskie prostranstva funktsii (Topological Spaces of Functions), Moscow: Mosk. Gos. Univ., 1989.

    Google Scholar 

  20. Skvortsova, A.V. and Chentsov, A.G., On the Construction of an Asymptotic Counterpart of the Sheaf of Trajectories for a Linear System with Single-Impulse Control, Differ. Uravn., 2004, vol. 40, no. 12, pp. 1645–1657.

    MathSciNet  Google Scholar 

  21. Edwards, R., Functional Analysis: Theory and Applications, New York: Dovers, 2011. Translated under the title Funktsional’nyi analiz. Teoriya i prilozheniya, Moscow: Mir, 1969.

    Google Scholar 

  22. Chentsov, A.G., On the Problem of Attraction Set Structure in a Topological Space, Izv. Inst. Mat. Informat. Udmurt. Gos. Univ., 2012, no. 1(39), pp. 147–150.

    Google Scholar 

  23. Mostowski, K. and Kuratowski, A., Set Theory, New York: North-Holland, 1967. Translated under the title Teoriya mnozhestv, Moscow: Mir, 1970.

    Google Scholar 

  24. Bulinskii, A.V. and Shiryaev, A.N., Teoriya sluchainykh protsessov (Theory of Random Processes), Moscow: Fizmatlit, 2005.

    Google Scholar 

  25. Bourbaki, N., Topologie general. Les structures fondamentales de l’analyse, Paris: Hermann, 1966. Translated under the title Obshchaya topologiya. Osnovnye struktury, Moscow: Nauka, 1968.

    Google Scholar 

  26. Chentsov, A.G., On One Example of Representing the Space of Ultrafilters for an Algebra of Sets, Tr. Inst. Mat. Mekh., 2011, vol. 17, no. 4, pp. 293–311.

    Google Scholar 

  27. Chentsov, A.G., Filters and Ultrafilters in Constructions of Attraction Sets, Vestn. Udmurt. Univ., Mat. Mekh. Comp. Nauki, 2011, no. 1, pp. 113–142.

    Google Scholar 

  28. Engelking, R., General Topology, Warszawa: PWN, 1985. Translated under the title Obshchaya topologiya, Moscow: Mir, 1986.

    Google Scholar 

  29. Chentsov, A.G., Elementy konechno-additivnoi teorii mery. I (Elements of the Theory of Finitely Additive Measures. I), Ekaterinburg: UGTU-UPI, 2008.

    Google Scholar 

  30. Neveu, J., Mathematical Foundations of the Calculus of Probability, San Francisco: Holden-Day, 1965. Translated under the title Matematicheskie osnovy teorii veroyatnostei, Moscow: Mir, 1969.

    MATH  Google Scholar 

  31. Chentsov, A.G., On the Problem of Representing Attraction Elements in Abstract Reachability Problems with Asymptotic Constraints, Izv. Vyssh. Uchebn. Zaved, Mat., 2012, no. 10, pp. 45–59.

    Google Scholar 

  32. Chentsov, A.G., Elementy konechno-additivnoi teorii mery. II (Elements of the Theory of Finitely Additive Measures. II), Ekaterinburg: UGTU-UPI, 2010.

    Google Scholar 

  33. Bhaskara, Rao K.P.S. and Bhaskara, Rao M., Theory of Charges. A Study of Finitely Additive Measures, New York: Academic, 1983.

    MATH  Google Scholar 

  34. Krasovskii, N.N. and Subbotin, A.I., Pozitsionnye differentsial’nye igry (Positional Differential Games), Moscow: Nauka, 1974.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, Russia

    A. G. Chentsov

Authors
  1. A. G. Chentsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © A.G. Chentsov, 2013, published in Avtomatika i Telemekhanika, 2013, No. 12, pp. 119–139.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chentsov, A.G. On certain problems of the structure of ultrafilters related to extensions of abstract control problems. Autom Remote Control 74, 2020–2036 (2013). https://doi.org/10.1134/S0005117913120060

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117913120060

Keywords

Search

Navigation