Skip to main content
SpringerLink
Log in

The Safety Problem for Nonlinear Systems with Delay in Terms of Barrier Functions

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We study the safety problem for nonlinear nonautonomous systems described by the Cauchy problem for differential equation with delay with ordinary derivative. To analyze safety, we introduce the Razumikhin barrier function that guarantees the invariance of some safety set. Unlike traditional barrier functions used in optimization problems, our construction is not unbounded on the boundary of an admissible domain, but only changes the sign. Owing to this fact, we use the barrier function to construct a control providing the safety of the system. The results are illustrated by examples.

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. N. O. Sedova, “On the principle of reduction for the nonlinear delay systems,” Autom. Remote Control 72, No. 9, 1864–1875 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Z. Romdlony, B. Jayawardhana, “Stabilization with guaranteed safety using control Lyapunov-barrier function,” Automatica 66, 39–47 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  3. B. S. Razumikhin, “On the stability of systems with delay” [in Russian], Prikl. Mat. Mekh. 20, No. 4, 500–512 (1956).

    Google Scholar 

  4. H. K. Khalil, Nonlinear Systems, Macmillan, New York (1992).

    MATH  Google Scholar 

  5. M. Jankovic, “Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems,” IEEE Trans. Autom. Control 46, No. 7, 1048–1060 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. D. Myshkis, Linear Differential Equations with Delay Argument [in Russian], Gostekhizdat, Moscow etc. (1951).

Download references

Author information

Authors and Affiliations

  1. Ul’yanovsk State University, 42, Tolstogo St., Ul’yanovsk, 432970, Russia

    N. O. Sedova

Authors
  1. N. O. Sedova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. O. Sedova.

Additional information

Translated from Problemy Matematicheskogo Analiza 123, 2023, pp. 101-107.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sedova, N.O. The Safety Problem for Nonlinear Systems with Delay in Terms of Barrier Functions. J Math Sci 270, 733–740 (2023). https://doi.org/10.1007/s10958-023-06384-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06384-3

Search

Navigation