Skip to main content
SpringerLink
Log in

On Estimating the Degree of Nonconvexity of Reachable Sets of Control Systems

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

Abstract

We study the properties of reachable sets of control systems that are linear in control. For the reachable sets of a certain class of such systems, we estimate the growth of their degree of nonconvexity \(\alpha\) over time. As an auxiliary result, we establish a relationship between \(\alpha\)-sets and weakly convex sets in the sense of Vial.

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

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.

Similar content being viewed by others

References

  1. V. G. Boltyanskii and V. L. Dol’nikov, “Jung’s theorem and \(H\)-convexity,” Dokl. Math. 71 (2), 245–249 (2005) [transl. from Dokl. Akad. Nauk 401 (5), 587–591 (2005)].

    MATH  Google Scholar 

  2. A. A. Ershov and M. V. Pershakov, “On the relation between alpha-sets and other generalizations,” in Proc. VI Inf. Young Scientist School (Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 2018), pp. 143–150 [in Russian].

    Google Scholar 

  3. A. L. Garkavi, “On the Chebyshev center and convex hull of a set,” Usp. Mat. Nauk 19 (6), 139–145 (1964).

    MathSciNet  Google Scholar 

  4. P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers (North-Holland, Amsterdam, 1987).

    MATH  Google Scholar 

  5. G. E. Ivanov, Weakly Convex Sets and Functions: Theory and Applications (Fizmatlit, Moscow, 2006) [in Russian].

    MATH  Google Scholar 

  6. H. Jung, “Ueber die kleinste Kugel, die eine räumliche Figur einschliesst,” J. Reine Angew. Math. 123, 241–257 (1901).

    MathSciNet  MATH  Google Scholar 

  7. E. Michael, “Paraconvex sets,” Math. Scand. 7, 372–376 (1959).

    Article  MathSciNet  Google Scholar 

  8. H. V. Ngai and J.-P. Penot, “Paraconvex functions and paraconvex sets,” Stud. Math. 184 (1), 1–29 (2008).

    Article  MathSciNet  Google Scholar 

  9. E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis (Fizmatlit, Moscow, 2007) [in Russian].

    MATH  Google Scholar 

  10. P. V. Semenov, “Functionally paraconvex sets,” Math. Notes 54 (6), 1236–1240 (1993) [transl. from Mat. Zametki 54 (6), 74–81 (1993)].

    Article  MathSciNet  Google Scholar 

  11. V. N. Ushakov and A. A. Ershov, “Estimation of the growth of the degree of nonconvexity of reachable sets in terms of \(\alpha \)-sets,” Dokl. Math. 102 (3), 532–537 (2020) [transl. from Dokl. Ross. Akad. Nauk, Mat. Inform. Prots. Upr. 495 (1), 100–106 (2020)].

    Article  Google Scholar 

  12. V. N. Ushakov and A. A. Uspenskii, “\(\alpha \)-Sets in finite-dimensional Euclidean spaces and their properties,” Vestn. Udmurt. Univ., Mat. Mekh. Komp’yut. Nauki 26 (1), 95–120 (2016).

    Article  MathSciNet  Google Scholar 

  13. V. N. Ushakov, A. A. Uspenskii, and A. A. Ershov, “Alpha-sets in finite-dimensional Euclidean spaces and their applications in control theory,” Vestn. St.-Peterbg. Univ., Ser. 10: Prikl. Mat. Inf. Prots. Upr. 14 (3), 261–272 (2018).

    MathSciNet  Google Scholar 

  14. A. A. Uspenskii, V. N. Ushakov, and A. N. Fomin, “\(\alpha \)-Sets and their properties,” Available from VINITI, No. 543-V2004 (Inst. Math. Mech., Ural Branch Russ. Acad. Sci., Ekaterinburg, 2004).

  15. Yu. B. Zelinskii, Convexity: Selected Topics (Inst. Mat. Nats. Akad. Nauk Ukr., Kyiv, 2012) [in Russian].

    MATH  Google Scholar 

Download references

Funding

The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2021-1383).

Author information

Authors and Affiliations

  1. N. N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    V. N. Ushakov, A. A. Ershov & A. R. Matviychuk

Authors
  1. V. N. Ushakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Ershov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. R. Matviychuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. N. Ushakov.

Additional information

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 315, pp. 261–270 https://doi.org/10.4213/tm4219.

Translated by I. Nikitin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ushakov, V.N., Ershov, A.A. & Matviychuk, A.R. On Estimating the Degree of Nonconvexity of Reachable Sets of Control Systems. Proc. Steklov Inst. Math. 315, 247–256 (2021). https://doi.org/10.1134/S0081543821050199

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543821050199

Keywords

Search

Navigation