Automation and Remote Control

, Volume 80, Issue 6, pp 1016–1025 | Cite as

Sufficient Conditions for the Existence of Asymptotic Quiescent Position for One Class of Differential-Difference Systems

  • S. E. KuptsovaEmail author
  • N. A. Stepenko
  • S. Yu. Kuptsov
Nonlinear Systems


The time-delay systems are considered and the limiting behavior of their solutions is investigated. The case in which the solutions have the trivial equilibrium that may not be an invariant set of the system is studied. The notion of an asymptotic quiescent position for the trajectories of delayed systems is introduced. Its stability is analyzed by the method of Lyapunov functions using the Razumikhin approach. Sufficient conditions for the existence of an asymptotic quiescent position for one class of the systems of differential-difference equations are established. Some illustrative examples of nonlinear differential equations with delay that have an asymptotic quiescent position are given and the sufficient conditions are applied to them.


Lyapunov stability nonlinear time-delay systems asymptotic quiescent position Lyapunov function Razumikhin approach 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Some Problems in the Theory of Motion Stability), Moscow: Fizmatlit, 1959.Google Scholar
  2. 2.
    Bellman, R. and Cooke, K.L., Differential-Difference Equations, New York: Academic, 1963. Translated under the title Differentsial’no-raznostnye uravneniya, Moscow: Mir, 1967.zbMATHGoogle Scholar
  3. 3.
    Hale, J.K., Theory of Functional Differential Equations, New York: Springer Verlag, 1977. Translated under the title Teoriya funktsional’no-differentsial’nykh uravnenii, Moscow: Mir, 1984.CrossRefzbMATHGoogle Scholar
  4. 4.
    Zubov, V.I., Lektsii po teorii upravleniya (Lectures on Control Theory), Moscow: Nauka, 1975.zbMATHGoogle Scholar
  5. 5.
    Zubov, V.I., To the Theory of Linear Time-Invariant Systems with Delayed Argument, Izv. Vyssh. Uchebn. Zaved., Mat., 1958, no. 6, pp. 86–95.Google Scholar
  6. 6.
    Zubov, V.I., Kolebaniya i volny (Oscillations and Waves), Leningrad: Leningr. Gos. Univ., 1989.zbMATHGoogle Scholar
  7. 7.
    Kuptsova, S.E., Asymptotically Invariant Sets, Tr. 37 Mezhd. Konf. “Protsessy upravleniya i ustoichivost” (Proc. 37th Int. Conf. “Control Processes and Stability”), 2006, pp. 50–56.Google Scholar
  8. 8.
    Kuptsova, S.E., On Asymptotic Behavior of the Solutions of Systems of Nonlinear Time-Varying Differential Equations, Tr. Srednevolzh. Mat. Obshch., 2006, vol. 8, @@no. 1, pp. 235–243.Google Scholar
  9. 9.
    Zhabko, A.P., Tikhomirov, O.G., and Chizhova, O.N., Stability of Asymptotic Equilibrium of Perturbed Homogeneous Time-Varying Systems, Zh. Srednevolzh. Mat. Obshch., 2018, vol. 20, @@no. 1, pp. 13–22.zbMATHGoogle Scholar
  10. 10.
    Ekimov, A.V. and Svirkin, M.V., Analysis of Asymptotic Equilibrium State of Differential Systems Using Lyapunov Function Method, 2015 Int. Conf. on Stability and Control Processes in Memory of V.I. Zubov (SCP 2015), IEEE, 2015, pp. 45–47.Google Scholar
  11. 11.
    Kuptsova, S.E., Asymptotic Equilibria of Systems of Difference Equations, Sist. Upravlen. Inform. Tekh., 2014, vol. 56, @@no. 2, pp. 67–71.Google Scholar
  12. 12.
    Kuptsov, S.Yu., Kuptsova, S.E., and Zaranik, U.P., On Asymptotic Quiescent Position of Nonlinear Difference Systems with Perturbations, 2015 Int. Conf. on Stability and Control Processes in Memory of V.I. Zubov (SCP 2015), IEEE, 2015, pp. 20–22.Google Scholar
  13. 13.
    Razumikhin, B.S., On Stability of Delayed Systems, Prikl. Mat. Mekh., 1956, vol. 56, @@no. 2, pp. 500–512.MathSciNetGoogle Scholar
  14. 14.
    Razumikhin, B.S., The Application of Lyapunov’s Method to Problems in the Stability of Systems with Delay, Autom. Remote Control, 1960, vol. 21, @@no. 6, pp. 740–748.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kuptsova, S.E., Kuptsov, S.Yu., and Stepenko, N.A., On the Limiting Behavior of the Solutions of Systems of Differential Equations with Delayed Argument, Vestn. St. Petersburg Univ., Prikl. Mat., Informat., Prots. Upravlen., 2018, vol. 14, @@no. 2, pp. 173–182.Google Scholar
  16. 16.
    Zaranik, U.P., Kuptsova, S.E., and Stepenko, N.A., Sufficient Conditions for the Stability of Asymptotic Equilibrium of Delayed Systems, Zh. Srednevolzh. Mat. Obshch., 2018, vol. 20, @@no. 2, pp. 175–186.Google Scholar
  17. 17.
    Kudryavtsev, L.D., Kurs matematicheskogo analiza (Course on Mathematical Analysis), Moscow: Vysshaya Shkola, 1988, 2nd ed. This paper was recommended for publication by L.B. Rapoport, a member of the Editorial Board zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  • S. E. Kuptsova
    • 1
    Email author
  • N. A. Stepenko
    • 2
  • S. Yu. Kuptsov
    • 1
  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.OOO OGS RUSSIASt. PetersburgRussia

Personalised recommendations