# Stability of Systems with Aftereffect

Chapter

First Online:

## 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.

## References

- 9.Azbelev, N.V., Simonov, P.M.: Stability of Differential Equations with Aftereffect. Taylor and Francis, London (2002)zbMATHGoogle Scholar
- 14.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
- 16.Burton, T.A.: Stability and Periodic Solutions for Ordinary and Functional Differential Equations. Academic, New York (1985)zbMATHGoogle Scholar
- 26.El’sgol’ts, L.E.: Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day, San Francisco (1966)zbMATHGoogle Scholar
- 32.Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)CrossRefGoogle Scholar
- 40.Krasovskii, N.N.: Stability on Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford (1963)zbMATHGoogle Scholar
- 43.Lakshmikantham, V., Martynyuk, A.A.: Development of the direct Lyapunov’s method for systems with delay. Int. Appl. Mech.
**29**(2), 3–14 (1993)CrossRefGoogle Scholar - 44.Lakshmikantham, V., Rama Mohana Rao, M.: Theory of Integro-Differential Equations. Gordon and Breach Science Publishers, Amsterdam (1995)Google Scholar
- 48.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)MathSciNetCrossRefGoogle Scholar - 51.Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability Analysis of Nonlinear Systems, 2nd edn. Springer, Basel (2015)CrossRefGoogle Scholar
- 69.Martynyuk, A.A.: Stability of Motion. The Role of Multicomponent Liapunov’s Functions. Cambridge Scientific Publishers, Cambridge (2007)Google Scholar
- 94.Milman, V.D., Myshkis, A.D.: On motion stability with shocks. Sibirsk. Mat. Zh.
**I**(2), 233–237 (1960)Google Scholar - 95.Minorsky, N.: Self-excited oscillations in dynamical systems possessing retarded actions. J. Appl. Mech.
**9**(1), 65–71 (1942)Google Scholar - 104.Razumikhin, B.S.: On the stability of systems with a delay. J. Appl. Math. Mech.
**22**, 215–227 (1958)MathSciNetCrossRefGoogle Scholar - 107.Schoen, G.M.: Stability and stabilization of time-delay systems. Zurich, Diss. ETH No. 11166 (1995)Google Scholar
- 110.Stamova, I.M.: Stability Analysis of Impulsive Functional Differential Equations. Walter De Gruyter Inc., New York (2009)CrossRefGoogle Scholar
- 118.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)zbMATHGoogle Scholar - 122.Yoshizawa, T.: Stability Theory by Liapunov’s Second Method. Mathematics Society Japan, Tokyo (1966)Google Scholar

## Copyright information

© Springer Nature Switzerland AG 2019