Skip to main content
SpringerLink
Log in

The invariance principle for nonautonomous differential equations with discontinuous right-hand side

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We study limit differential inclusions for nonautonomous differential equations with discontinuous right-hand side and Filippov solutions. Using Lyapunov functions with derivatives of constant sign, we establish an analog of LaSalle’s invariance principle. We study differential equations with either measurable or piecewise continuous right-hand side.

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. Rouche N., Habets P., and Laloy M., Stability Theory by Lyapunov’s Direct Method, Springer-Verlag, New York, Heidelberg, and Berlin (1977).

    Book  MATH  Google Scholar 

  2. Barbashin E. A., The Lyapunov Functions [in Russian], Nauka, Moscow (1970).

    MATH  Google Scholar 

  3. Sell G. R., “Nonautonomous differential equations and topological dynamics. 1 and 2,” Trans. Amer. Math. Soc., 22, 241–283 (1967).

    MathSciNet  MATH  Google Scholar 

  4. Artstein Z., “Topological dynamics of an ordinary differential equation,” J. Differential Equations, 23, 216–223 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  5. Artstein Z., “Topological dynamics of an ordinary differential equation,” J. Differential Equations, 23, 224–243 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  6. Artstein Z., “The limiting equations of nonautonomous ordinary differential equations,” J. Differential Equations, 25, 184–202 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  7. Artstein Z., “Uniform asymptotic stability via limiting equations,” J. Differential Equations, 27, 172–189 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  8. Andreev A. S., “The Lyapunov functionals method in stability problems for functional differential equations,” Automation and Remote Control, 70, No. 9, 1438–1486 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  9. Finogenko I. A., “Limit differential inclusions and the invariance principle for nonautonomous systems,” Sib. Math. J., 55, No. 2, 372–386 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  10. Finogenko I. A., “Limit functional-differential inclusions and the invariance principle for nonautonomous systems with delay,” Dokl. Math., 89, No. 2, 239–241 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  11. Finogenko I. A., “The invariance principle for nonautonomous functional differential inclusions,” Trudy Inst. Mat. i Mekh. UrO RAN, 20, No. 1, 271–284 (2014).

    MathSciNet  MATH  Google Scholar 

  12. Filippov A. F., Differential Equations with Discontinuous Right-Hand Side [in Russian], Nauka, Moscow (1985).

    MATH  Google Scholar 

  13. Borisovich Yu. G., Gelman B. D., Myshkis A. D., and Obukhovskii V. V., Introduction to the Theory of Multivalued Maps and Differential Inclusions [in Russian], KomKniga, Moscow (2005).

    MATH  Google Scholar 

  14. Kuratowski K., Topology. Vol. 1 [Russian translation], Mir, Moscow (1966).

  15. Davy J. L., “Properties of solution set of a generalized differential equation,” Bull. Austral. Math. Soc., 6, 379–398 (1972).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Systems Dynamics and Control Theory, Irkutsk, Russia

    I. A. Finogenko

Authors
  1. I. A. Finogenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. A. Finogenko.

Additional information

Original Russian Text Copyright © 2016 Finogenko I. A.

The author was supported by the Russian Foundation for Basic Research (Grant 13–01–00287).

Irkutsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 4, pp. 913–927, July–August, 2016; DOI: 10.17377/smzh.2016.57.413. Original article submitted May 30, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Finogenko, I.A. The invariance principle for nonautonomous differential equations with discontinuous right-hand side. Sib Math J 57, 715–725 (2016). https://doi.org/10.1134/S0037446616040133

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446616040133

Keywords

Search

Navigation