Skip to main content
SpringerLink
Log in

Stability of Systems with Aftereffect

  • Chapter
  • First Online:
Qualitative Analysis of Set-Valued Differential Equations

  • 588 Accesses

Abstract

In this chapter we consider a set of equations with aftereffect and uncertain parameters. As a result of regularization of the family of equations according to the scheme adopted in the book, a set of equations with aftereffect are obtained, for which the solution existence conditions are established, an estimate of the distance between the extreme solution sets is obtained, and stability conditions for the set of stationary solutions on a finite time interval are found as well as the attenuation conditions for the set of trajectories.

This chapter deals with the families of equations with aftereffect and uncertain parameter values. The families of equations with aftereffect are obtained as a result of regularization of equations according to the scheme adopted in the book. For the obtained families of equations, the solution existence conditions are established, the estimates of the distance between extreme sets of solutions are found, and the stability conditions for the set of stationary solutions on a finite time interval are obtained.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Azbelev, N.V., Simonov, P.M.: Stability of Differential Equations with Aftereffect. Taylor and Francis, London (2002)

    MATH  Google Scholar 

  2. Boltzmann, L.: Zur Theorie der elastischen Nachwirkungen. In: Wissenschaftliche Abhandlung von Ludwig Boltzmann, I. Band (1865–1874), pp. 616–644. Verlag Johann Ambrosius Barth, Leipzig (1909)

    Google Scholar 

  3. Burton, T.A.: Stability and Periodic Solutions for Ordinary and Functional Differential Equations. Academic, New York (1985)

    MATH  Google Scholar 

  4. El’sgol’ts, L.E.: Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day, San Francisco (1966)

    MATH  Google Scholar 

  5. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

    Book  Google Scholar 

  6. Krasovskii, N.N.: Stability on Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford (1963)

    MATH  Google Scholar 

  7. Lakshmikantham, V., Martynyuk, A.A.: Development of the direct Lyapunov’s method for systems with delay. Int. Appl. Mech. 29(2), 3–14 (1993)

    Article  Google Scholar 

  8. Lakshmikantham, V., Rama Mohana Rao, M.: Theory of Integro-Differential Equations. Gordon and Breach Science Publishers, Amsterdam (1995)

    Google Scholar 

  9. Lakshmikantham, V., Leela, S., Sivasundaram, S.: Lyapunov functions on product spaces and stability theory of delay differential equations. J. Math. Anal. Appl. 154(2), 391–402 (1991)

    Article  MathSciNet  Google Scholar 

  10. Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability Analysis of Nonlinear Systems, 2nd edn. Springer, Basel (2015)

    Book  Google Scholar 

  11. Martynyuk, A.A.: Stability of Motion. The Role of Multicomponent Liapunov’s Functions. Cambridge Scientific Publishers, Cambridge (2007)

    Google Scholar 

  12. Milman, V.D., Myshkis, A.D.: On motion stability with shocks. Sibirsk. Mat. Zh. I(2), 233–237 (1960)

    Google Scholar 

  13. Minorsky, N.: Self-excited oscillations in dynamical systems possessing retarded actions. J. Appl. Mech. 9(1), 65–71 (1942)

    Google Scholar 

  14. Razumikhin, B.S.: On the stability of systems with a delay. J. Appl. Math. Mech. 22, 215–227 (1958)

    Article  MathSciNet  Google Scholar 

  15. Schoen, G.M.: Stability and stabilization of time-delay systems. Zurich, Diss. ETH No. 11166 (1995)

    Google Scholar 

  16. Stamova, I.M.: Stability Analysis of Impulsive Functional Differential Equations. Walter De Gruyter Inc., New York (2009)

    Book  Google Scholar 

  17. Vasundhara Devi, J., Vatsala, A.S.: A study of set differential equations with delay, dynamics of continuous, discrete and impulsive systems. Math. Anal. A 11, 287–300 (2004)

    MATH  Google Scholar 

  18. Yoshizawa, T.: Stability Theory by Liapunov’s Second Method. Mathematics Society Japan, Tokyo (1966)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine

    Anatoly A. Martynyuk

Authors
  1. Anatoly A. Martynyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Martynyuk, A.A. (2019). Stability of Systems with Aftereffect. In: Qualitative Analysis of Set-Valued Differential Equations. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-07644-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation