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Sibirsk. Mat. Zh., 2012, Volume 53, Number 3, Pages 495–508 (Mi smj2341)  

This article is cited in 21 scientific papers (total in 21 papers)

On the asymptotic stability of solutions of nonlinear systems with delay

A. Yu. Aleksandrov, A. P. Zhabko

Saint-Petersburg State University, Saint-Petersburg

Abstract: Under study are systems of homogeneous differential equations with delay. We assume that in the absence of delay the trivial solutions to the systems under consideration are asymptotically stable. Using the direct Lyapunov method and Razumikhin's approach, we show that if the order of homogeneity of the right-hand sides is greater than 1 then asymptotic stability persists for all values of delay. We estimate the time of transitions, study the influence of perturbations on the stability of the trivial solution, and prove a theorem on the asymptotic stability of a complex system describing the interaction of two nonlinear subsystems.

Keywords: delay system, asymptotic stability, Lyapunov functions, stability with respect to nonlinear approximation, nonstationary perturbation.

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English version:
Siberian Mathematical Journal, 2012, 53:3, 393–403

Bibliographic databases:

UDC: 517.929.4
Received: 29.06.2011

Citation: A. Yu. Aleksandrov, A. P. Zhabko, “On the asymptotic stability of solutions of nonlinear systems with delay”, Sibirsk. Mat. Zh., 53:3 (2012), 495–508; Siberian Math. J., 53:3 (2012), 393–403

Citation in format AMSBIB
\by A.~Yu.~Aleksandrov, A.~P.~Zhabko
\paper On the asymptotic stability of solutions of nonlinear systems with delay
\jour Sibirsk. Mat. Zh.
\yr 2012
\vol 53
\issue 3
\pages 495--508
\jour Siberian Math. J.
\yr 2012
\vol 53
\issue 3
\pages 393--403

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    This publication is cited in the following articles:
    1. A. Yu. Aleksandrov, A. P. Zhabko, “Delay-independent stability of homogeneous systems”, Appl. Math. Lett., 34 (2014), 43–50  crossref  mathscinet  zmath  isi  elib  scopus
    2. D. Efimov, W. Perruquetti, J.-P. Richard, “Development of homogeneity concept for time-delay systems”, SIAM J. Control Optim., 52:3 (2014), 1547–1566  crossref  mathscinet  zmath  isi  elib  scopus
    3. D. Efimov, A. Polyakov, W. Perruquetti, J.-P. Richard, “Stability analysis for nonlinear time-delay systems applying homogeneity”, 2014 IEEE 53rd Annual Conference on Decision and Control (CDC), IEEE, 2014, 37–42  crossref  isi  scopus
    4. S. L. Podvalnyi, V. V. Provotorov, “Startovoe upravlenie parabolicheskoi sistemoi s raspredelennymi parametrami na grafe”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 2015, no. 3, 126–142  mathnet  elib
    5. A. Yu. Aleksandrov, A. P. Zhabko, I. A. Zhabko, “Time-delayed feedback stabilisation of nonlinear potential systems”, Int. J. Control, 88:10 (2015), 2066–2073  crossref  mathscinet  zmath  isi  elib  scopus
    6. A. Yu. Aleksandrov, E. B. Aleksandrova, A. P. Zhabko, G. Dai, “Stability analysis and estimation of the convergence rate of solutions for nonlinear time-delay systems”, 2015 7Th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), International Conference on Ultra Modern Telecommunications and Control Systems & Workshops, IEEE, 2015, 67–72  isi
    7. M. Wang, B. Chen, “Stability analysis for a class of delayed neural networks with nonlinear homogeneous activation functions”, 2015 Sixth International Conference on Intelligent Control and Information Processing (ICICIP), IEEE, 2015, 30–35  crossref  isi  scopus
    8. A. Yu. Aleksandrov, A. P. Zhabko, “Stability analysis of equilibrium positions of mechanical systems with delay via decomposition”, 2015 International Conference on Mechanics Seventh Polyakhov's Reading, ed. A. Tikhonov, IEEE, 2015  isi
    9. A. Yu. Aleksandrov, A. P. Zhabko, “On the stability of solutions of a class of nonlinear nonautonomous systems with delay”, 2015 International Conference “Stability and Control Processes” in Memory of V.i. Zubov (SCP), eds. L. Petrosyan, A. Zhabko, IEEE, 2015, 274–276  isi
    10. D. Efimov, A. Polyakov, W. Perruquetti, J.-P. Richard, “Weighted homogeneity for time-delay systems: finite-time and independent of delay stability”, IEEE Trans. Autom. Control, 61:1 (2016), 210–215  crossref  mathscinet  zmath  isi  elib  scopus
    11. A. Aleksandrov, E. Aleksandrova, A. Zhabko, “Asymptotic stability conditions and estimates of solutions for nonlinear multiconnected time-delay systems”, Circuits Syst. Signal Process., 35:10 (2016), 3531–3554  crossref  mathscinet  zmath  isi  scopus
    12. A. Yu. Aleksandrov, E. B. Aleksandrova, Y. Chen, “Partial stability analysis of nonlinear nonstationary systems via averaging”, Nonlinear Dyn., 86:1 (2016), 153–163  crossref  mathscinet  zmath  isi  scopus
    13. X. Liu, “Stability of perturbed switched homogeneous systems with delays and uncertainties”, IET Contr. Theory Appl., 10:6 (2016), 684–691  crossref  mathscinet  isi  scopus
    14. A. Yu. Aleksandrov, E. B. Aleksandrova, A. P. Zhabko, “On the asymptotic stability with respect to a part of variables of solutions of nonlinear systems with delay”, Proceedings of 2016 International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy's Conference), ed. V. Tkhai, IEEE, 2016  mathscinet  isi
    15. D. Efimov, W. Perruquetti, J.-P. Richard, “Global and local weighted homogeneity for time-delay systems”, Recent Results on Nonlinear Delay Control Systems: in Honor of Miroslav Krstic, Advances in Delays and Dynamics, 4, eds. I. Karafyllis, M. Malisoff, F. Mazenc, P. Pepe, Springer-Verlag Berlin, 2016, 163–181  crossref  mathscinet  zmath  isi  scopus
    16. K. Zimenko, D. Efimov, A. Polyakov, W. Perruquetti, “A note on delay robustness for homogeneous systems with negative degree”, Automatica, 79 (2017), 178–184  crossref  mathscinet  zmath  isi  scopus
    17. A. Aleksandrov, E. Aleksandrova, A. Zhabko, Ya. Chen, “Partial stability analysis of some classes of nonlinear systems”, Acta Math. Sci., 37:2 (2017), 329–341  crossref  mathscinet  zmath  isi  scopus
    18. V. V. Provotorov, E. N. Provotorova, “Optimal control of the linearized Navier–Stokes system in a netlike domain”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 13:4 (2017), 431–443  mathnet  crossref  elib
    19. A. Aleksandrov, E. Aleksandrova, A. Zhabko, “Stability analysis of some classes of nonlinear switched systems with time delay”, Int. J. Syst. Sci., 48:10 (2017), 2111–2119  crossref  mathscinet  zmath  isi  scopus
    20. N. V. Pertsev, B. Yu. Pichugin, A. N. Pichugina, “Primenenie M-matrits dlya issledovaniya matematicheskikh modelei zhivykh sistem”, Matem. biologiya i bioinform., 13:1 (2018), 208–237  mathnet  crossref
    21. N. V. Pertsev, B. Yu. Pichugin, A. N. Pichugina, “Application of M-matrices for the study of mathematical models of living systems”, Matem. biologiya i bioinform., 13, Suppl. (2018), 104–131  mathnet  crossref
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