Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Relaxation of reachable sets and extension constructions

  • 16 Accesses

  • 1 Citations

Abstract

Regularization of the reachability set is performed by relaxing the constraints on the system. We use compactification constructions in the solution space from the class of finitely additive measures and derive conditions of asymptotic robustness to various disturbances.

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

References

  1. 1.

    N. N. Krasovskii, Game-Theoretical Problems on Meeting of Motions [in Russian], Nauka, Moscow (1970).

  2. 2.

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

  3. 3.

    A. I. Panasyuk and V. I. Panasyuk, Asymptotic Turnpike Optimization of Controlled Systems [in Russian], Nauka i Tekhnika, Minsk (1986).

  4. 4.

    A. Dantchev, Optimal Control Systems. Perturbations, Approximations, and Sensitivity Analysis [Russian translation], Mir, Moscow (1987).

  5. 5.

    A. G. Chentsov, "Vector finitely additive measures and extensions in one class of nonlinear extremal problems," IMM UrO Akad. Nauk SSSR, Sverdlovsk (1983). Unpublished manuscript, VINITI No. 8191-V88.

  6. 6.

    J. L. Kelley, General Topology [Russian translation], Nauka, Moscow (1981).

  7. 7.

    R. Engelking, General Topology [Russian translation], Mir, Moscow (1986).

  8. 8.

    J. Warga, Optimal Control of Differential and Functional Equations [Russian translation], Nauka, Moscow (1974).

  9. 9.

    L. Young, Lectures on Calculus of Variations and Optimal Control Theory [Russian translation], Mir, Moscow (1974).

  10. 10.

    R. V. Gamkrelidze, Foundations of Optimal Control [in Russian], Izd. Tbiliss. Univ., Tbilisi (1977).

  11. 11.

    A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).

  12. 12.

    I. Ekeland and R. Temam, Convex Analysis and Variational Problems [Russian translation], Mir, Moscow (1979).

  13. 13.

    R. J. Duffin, "Infinite programs," in: Linear Inequalities and Related Topics [Russian translations], Inostr. Lit., Moscow (1959), pp. 263–267.

  14. 14.

    E. G. Gol'shtein, Duality Theory in Mathematical Programming [in Russian], Nauka, Moscow (1971).

  15. 15.

    A. G. Chentsov, "Optimization under fuzzy constraints," Preprint, UrO AN SSSR, Inst. Matem. i Mekhan., Sverdlovsk (1986).

  16. 16.

    A. G. Chentsov, "Finitely additive measures and minimum problems," Kibernetika, No. 3, 67–70 (1988).

  17. 17.

    A. G. Chentsov, "Two-valued measures on the semialgebra of sets and some applications to infinite-dimensional mathematical programming problems," Kibernetika, No. 6, 72–76, 89 (1988).

  18. 18.

    A. G. Chentsov, "On universal integrability of bounded functions," Mat. Sb.,131, No. 1, 73–93 (1986).

  19. 19.

    A. G. Chentsov, Applications of Measure Theory to Control Problems [in Russian], Sred.-Ural. Knizhnoe Izd., Sverdlovsk (1985).

  20. 20.

    J. Neveu, Mathematical Foundations of the Calculus of Probability [Russian translation], Mir, Moscow (1969).

  21. 21.

    N. Dunford and J. T. Schwartz, Linear Operators. General Theory, [Russian translation], Vol. 1, Inostr. Lit., Moscow (1962).

  22. 22.

    H. Sheffer, Topological Vector Spaces [Russian translation], Mir, Moscow (1971).

  23. 23.

    H. Maynard, "A Radon—Nikodym theorem for finitely additive bounded measures," Pacific J. Math.,83, No. 2, 401–413 (1979).

  24. 24.

    A. G. Chentsov, "On some representations of finitely additive measures approximated by indefinite integrals," UPI im. S. M. Kirova, Sverdlovsk (1987). Unpublished manuscript, VINITI No. 8511-V87.

  25. 25.

    A. G. Chentsov, "Extension of extremal problems in the class of finitely additive measures," UPI im. S. M. Kirova, Sverdlovsk (1989). Unpublished manuscript, VINITI No. 4201-V88.

  26. 26.

    R. A. Aleksandryan and E. A. Mirzakhanyan, General Topology [in Russian], Vysshaya Shkola, Moscow (1979).

  27. 27.

    N. Bourbaki, General Topology. Basic Structures [Russian translation], Nauka, Moscow (1968).

Download references

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 78–87, July–August, 1992.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chentsov, A.G. Relaxation of reachable sets and extension constructions. Cybern Syst Anal 28, 554–561 (1992). https://doi.org/10.1007/BF01124991

Download citation

Keywords

  • Operating System
  • Artificial Intelligence
  • System Theory
  • Solution Space
  • Additive Measure