Skip to main content
SpringerLink
Log in

On the Construction of a Nonanticipating Selection of a Multivalued Mapping

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We study the properties of multivalued mappings of general form with respect to the possibility of finding their nonanticipating selections. The property of nonanticipation is formulated for an arbitrary domain by specifying some family of “test” subsets. Sufficient conditions for the existence of a nonanticipating selection of a nonanticipating multivalued mapping are proposed: the values of the mapping must be nonempty compact sets, and the family of “test” subsets must be totally ordered with respect to inclusion. We illustrate the results by showing their applicability to the pursuit–evasion differential game in the form of P. Varaya and J. Lin.

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.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. E. Roxin, “Axiomatic approach in differential games,” J. Optim. Theory Appl. 3 (3), 153–163 (1969).

  2. C. Ryll-Nardzewski, “A theory of pursuit and evasion,” in Advances in Game Theory (Princeton Univ. Press, Princeton, NJ, 1964), pp. 113–126.

  3. P. Varaiya and J. Lin, “Existence of saddle points in differential games,” SIAM J. Control 7 (1), 141–157 (1969).

  4. R. J. Elliott and N. J. Kalton, “The existence of value in differential games of pursuit and evasion,” J. Diff. Eq. 12 (3), 504–523 (1972).

  5. A. G. Chentsov, “On a game problem on the minimax of a functional,” Dokl. AN SSSR 230 (5), 1047–1050 (1976).

  6. A. G. Chentsov, Selections of Multivalued Strategies in Differential Games (Sverdlovsk, 1978), Available from VINITI, No. 3101-78 [in Russian].

  7. A. G. Chentsov, “On an alternative in a class of quasistrategies for a differential approach-evasion game,” Differential Equations 16 (10), 1167–1171 (1981).

  8. N. N. Krasovskii and A. I. Subbotin, “An alternative for the game problem of convergence,” J. Appl. Math. Mech. 34 (6), 948–965 (1970).

  9. N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].

  10. A. V. Kryazhimskii and A. G. Chentsov, On the Structure of a Game Control in Problems of Approach and Evasion (Sverdlovsk, 1979), Available from VINITI, No. 1729-80 [in Russian].

  11. A. G. Chentsov, “On a game problem of guidance,” Sov. Math. Dokl. 17 (1), 73–77 (1976).

  12. A. G. Chentsov, “On a game problem of converging at a given instant of time,” Math. USSR-Sb. 28 (3), 353–376 (1976).

  13. A. G. Chentsov, “Nonanticipating multimappings and their construction by the method of program iterations. I,” Differential Equations 37 (4), 498–509 (2001).

  14. A. G. Chentsov, “Nonanticipating selections of multivalued mappings,” Differents. Uravn. Prots. Upravl., No. 2, 1–64 (1998). http://www.math.spbu.ru/diffjournal/pdf/j019.pdf

  15. K. Kuratowski and A. Mostowski, Set Theory (North-Holland, Amsterdam, 1967; Mir, Moscow, 1970).

  16. R. Engelking, General Topology (PWN, Warsaw, 1983; Mir, Moscow, 1986).

  17. D. A. Serkov, “Unlocking of predicate: Application to constructing a non-anticipating selection,” Vestn. Udmurt. Univ., Ser. Mat. Mekh. Komp. Nauki 27 (2), 283–291 (2017).

  18. D. A. Serkov, “On the construction of the truth set of a predicate,” Izv. Inst. Mat. Inform. UdGU, No. 2(50), 37–53 (2017).

  19. D. A. Serkov, “On a condition of existence of non-anticipating selections,” IFAC-PapersOnLine 51 (32), 267–270 (2018).

  20. J. Dieudonne, Foundations of Modern Analysis (Academic, New York, 1960; Mir, Moscow, 1964).

  21. A. G. Chentsov and S. I. Morina, Extensions and Relaxations (Kluwer Acad., Dordrecht, 2002).

  22. J. Warga, Optimal Control of Differential and Functional Equations (Academic, New York, 1972; Nauka, Moscow, 1977).

  23. R. V. Gamkrelidze, Principles of Optimal Control Theory (Izd. Tbilis. Univ., Tbilisi, 1977; Plenum, New York, 1978).

  24. A. I. Subbotin and A. G. Chentsov, Guarantee Optimization in Control Problems (Nauka, Moscow, 1981)  [in Russian].

Download references

Funding

This work was supported by the Presidium of the Russian Academy of Sciences within the project “Newest Methods of Mathematical Modeling in the Study of Nonlinear Dynamic Systems.”

Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 620108, Yekaterinburg, Russia

    D. A. Serkov & A. G. Chentsov

  2. Ural Federal University, 620002, Yekaterinburg, Russia

    D. A. Serkov & A. G. Chentsov

Authors
  1. D. A. Serkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. G. Chentsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to D. A. Serkov or A. G. Chentsov.

Additional information

Translated by E. Vasil’eva

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Serkov, D.A., Chentsov, A.G. On the Construction of a Nonanticipating Selection of a Multivalued Mapping. Proc. Steklov Inst. Math. 309 (Suppl 1), S125–S138 (2020). https://doi.org/10.1134/S008154382004015X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S008154382004015X

Keywords

Search

Navigation