Skip to main content
SpringerLink
Log in

Asymptotic stability and instability with respect to part of variables for solutions to impulsive systems

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We obtain sufficient conditions for asymptotic stability with respect to part of variables for the zero solution to an impulsive system with the fixed moments of impulse effects.

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. Rumyantsev V. V. and Oziraner A. S., Motion Stability and Stabilization with Respect to Part of Variables [in Russian], Nauka, Moscow (1987).

    MATH  Google Scholar 

  2. Vorotnikov V. I., Stability of Dynamical Systems with Respect to Part of Variables [in Russian], Nauka, Moscow (1991).

    MATH  Google Scholar 

  3. Savchenko A. Ya. and Ignat’ev A. O., Some Problems of the Stability of Nonautonomous Dynamical Systems [in Russian], Naukova Dumka, Kiev (1989).

    MATH  Google Scholar 

  4. Vorotnikov V. I. and Rumyantsev V. V., Stability and Control with Respect to Part of Coordinates for the Phase Vector in Dynamical Systems: Theory, Methods, and Applications [in Russian], Nauchnyi Mir, Moscow (2001).

    Google Scholar 

  5. Ignatyev A. O., “On the partial equiasymptotic stability in functional differential equations,” J. Math. Anal. Appl., 268, No. 2, 615–628 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  6. Andreev A. S. and Pavlikov S. V., “The partial stability for a nonautonomous functional differential equation,” Prikl. Mat. Mekh., 63, No. 1, 3–12 (1999).

    MATH  MathSciNet  Google Scholar 

  7. Bernfeld S., Corduneanu C., and Ignatyev A. O., “On the stability of invariant sets of functional differential equations,” Nonlinear Anal., 55, No. 6, 641–656 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  8. Corduneanu C. and Ignatyev A. O., “Stability of invariant sets of functional differential equations with delay,” Nonlinear Funct. Anal. Appl., 10, No. 1, 11–24 (2005).

    MATH  MathSciNet  Google Scholar 

  9. Milman V. D. and Myshkis A. D., “On the stability of motion in the presence of impulses,” Sibirsk. Mat. Zh., 1, No. 2, 233–237 (1960).

    Google Scholar 

  10. Myshkis A. D. and Samoilenko A. M., “Systems with impulses at given times,” Mat. Sb., 74, No. 2, 202–208 (1967).

    MathSciNet  Google Scholar 

  11. Samoilenko A. M. and Perestyuk N. A., Impulsive Differential Equations, World Scientific, Singapore (1995).

    MATH  Google Scholar 

  12. Borysenko S. D., Iovane G., and Giordano P., “Investigations of the properties motion for essential nonlinear systems perturbed by impulses on some hypersurfaces,” Nonlinear Anal., 62, No. 2, 345–363 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  13. Boichuk A. A., Perestyuk N. A., and Samoilenko A. M., “Periodic solutions of impulse differential systems in critical cases,” Differential Equations, 27, No. 9, 1070–1073 (1991).

    MathSciNet  Google Scholar 

  14. Gladilina R. I. and Ignat’ev A. O., “Stability of periodic impulsive systems,” Math. Notes, 76, No. 1, 41–47 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  15. Ignat’ev A. O., “Method of Lyapunov functions in problems of stability of solutions of systems of impulsive differential equations,” Sb.: Math., 194, No. 10, 1543–1558 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  16. Perestyuk N. A. and Chernikova O. S., “Stability of solutions of impulse systems,” Ukrain. Mat. Zh., 49, No. 1, 98–111 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  17. Kenzhebaev K. K. and Stanzhitskii A. N., “Invariant sets of impulsive systems and their stability,” Nonlinear Oscillations, 7, No. 1, 78–82 (2004).

    Article  MathSciNet  Google Scholar 

  18. Simeonov P. S. and Bainov D. D., “Stability with respect to part of the variables in systems with impulse effect,” J. Math. Anal. Appl., 117, No. 1, 247–263 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  19. Gladilina R. I., “The partial stability in impulsive systems,” Trudy IPMM NAN Ukraine, 8, 7–18 (2003).

    MATH  MathSciNet  Google Scholar 

  20. Gladilina R. I., “Lyapunov’s direct method in the partial stability problems for systems with impulse effect,” Trudy Inst. Prikl. Mat. Mekh., 9, 46–52 (2004).

    MATH  MathSciNet  Google Scholar 

  21. Marachkov V. P., “A theorem on stability,” Izv. Fiz.-Mat. Obshch. i Nauchno-Issled. Inst. Mat. i Mekh. pri Kazan. Univ. Ser. 3., 12, 171–174 (1940).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of the Ukraine, Donetsk, the Ukraine

    A. O. Ignat’ev

Authors
  1. A. O. Ignat’ev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. O. Ignat’ev.

Additional information

Original Russian Text Copyright © 2008 Ignat’ev A. O.

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 1, pp. 125–133, January–February, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ignat’ev, A.O. Asymptotic stability and instability with respect to part of variables for solutions to impulsive systems. Sib Math J 49, 102–108 (2008). https://doi.org/10.1007/s11202-008-0009-9

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-008-0009-9

Keywords

Search

Navigation