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Asymptotic stability of solutions of a class of systems of nonlinear differential equations with delay

Abstract

We study systems of differential equations with delay whose right-hand sides are represented as sums of potential and gyroscopic components of vector fields. We assume that in the absence of a delay zero solutions of considered systems are asymptotically stable. By the Lyapunov direct method, using the Razumikhin approach, we prove that in the case of essentially nonlinear equations the asymptotic stability of zero solutions is preserved for any value of the delay.

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References

  1. 1.

    L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments (Nauka, Moscow, 1971; Academic Press, New York, 1973).

    Google Scholar 

  2. 2.

    J. K. Hale, Theory of Functional Differential Equations (Springer, Berlin, 1977; Mir, Moscow, 1984).

    MATH  Book  Google Scholar 

  3. 3.

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

    Google Scholar 

  4. 4.

    N. N. Krasovskii, Some Problems of Stability of Motions (Fizmatgiz, Moscow, 1959) [in Russian].

    Google Scholar 

  5. 5.

    V. L. Kharitonov, “Lyapunov Functionals with a Given Time Derivative. I. Complete-Type Functionals,” Vestnik St.-Peterb. Univ., Ser. 10, No. 1, 110–117 (2005).

  6. 6.

    B. S. Razumikhin, “Stability of Systems with a Delay,” Prikl. Matem. iMekhan. 20(4), 500–512 (1956).

    Google Scholar 

  7. 7.

    V. I. Zubov, Stability Problem of Control Processes (Sudpromgiz, Leningrad, 1980) [in Russian].

    MATH  Google Scholar 

  8. 8.

    V. I. Zubov, Stability of Motion (Vysshaya Shkola, Moscow, 1973) [in Russian].

    Google Scholar 

  9. 9.

    A. Yu. Aleksandrov, “On Asymptotic Stability of Solutions to Systems of Nonstationary Differential Equations with Homogeneous Right-Hand Sides,” Phys. Dokl. 349(3), 295–296 (1996).

    Google Scholar 

  10. 10.

    A. Yu. Aleksandrov, “On aMethod for Construction of Lyapunov’s Functions for Nonlinear Nonautonomous Systems,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 1, 3–10 (1998) [Russian Mathematics (Iz. VUZ) 42 (1), 1–8 (1998)].

  11. 11.

    R. E. Vinograd, “On a Criterion of Instability in the Sense of Lyapunov of the Solutions to a Linear System of Ordinary Differential Equations,” Sov. Phys. Dokl. 84(2), 201–204 (1952).

    MathSciNet  MATH  Google Scholar 

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Correspondence to A. Yu. Aleksandrov.

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Original Russian Text © A.Yu. Aleksandrov and A.P. Zhabko, 2012, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2012, No. 5, pp. 3–12.

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Aleksandrov, A.Y., Zhabko, A.P. Asymptotic stability of solutions of a class of systems of nonlinear differential equations with delay. Russ Math. 56, 1–8 (2012). https://doi.org/10.3103/S1066369X12050015

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Keywords and phrases

  • delay systems
  • asymptotic stability
  • Lyapunov functions
  • Razumikhin condition
  • nonstationary perturbations