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.
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
\Bibitem{AleZha12}
\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
\mathnet{http://mi.mathnet.ru/smj2341}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2978570}
\transl
\jour Siberian Math. J.
\yr 2012
\vol 53
\issue 3
\pages 393--403
\crossref{https://doi.org/10.1134/S0037446612020218}
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Linking options:
https://www.mathnet.ru/eng/smj2341
https://www.mathnet.ru/eng/smj/v53/i3/p495
This publication is cited in the following 47 articles:
A. Aleksandrov, D. Efimov, E. Fridman, “On stability of nonlinear homogeneous systems with distributed delays having variable kernels”, Systems & Control Letters, 190 (2024), 105853
A. V. Ekimov, A. P. Zhabko, P. V. Yakovlev, “Ustoichivost differentsialno-raznostnykh sistem s lineino vozrastayuschim zapazdyvaniem. II. Cistemy s additivnoi pravoi chastyu”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 19:1 (2023), 4–9
P. A. Shamanaev, “Ob ustoichivosti nulevogo resheniya otnositelno chasti peremennykh po lineinomu priblizheniyu”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 19:3 (2023), 374–390
Alexander Aleksandrov, Denis Efimov, Emilia Fridman, “Analysis of homogeneous systems with distributed delay using averaging approach”, IFAC-PapersOnLine, 56:2 (2023), 174
Alexander Aleksandrov, Denis Efimov, Emilia Fridman, “Stability of homogeneous systems with distributed delay and time-varying perturbations”, Automatica, 153 (2023), 111058
Alexandrova V I., Zhabko A.P., “Lyapunov-Krasovskii Functionals For Homogeneous Time Delay Systems of Neutral Type”, Int. J. Robust Nonlinear Control, 32:6, SI (2022), 3251–3265
Efimov D., Aleksandrov A., “Analysis of Robustness of Homogeneous Systems With Time Delays Using Lyapunov-Krasovskii Functionals”, Int. J. Robust Nonlinear Control, 31:9, SI (2021), 3730–3746
I. V. Aleksandrova, A. P. Zhabko, “Funktsionaly Lyapunova — Krasovskogo dlya odnorodnykh sistem s neskolkimi zapazdyvaniyami”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 17:2 (2021), 183–195
Aleksandrov A., Efimov D., “Stability Analysis of Switched Homogeneous Time-Delay Systems Under Synchronous and Asynchronous Commutation”, Nonlinear Anal.-Hybrid Syst., 42 (2021), 101090
Portilla G., Alexandrova I.V., Mondie S., Zhabko A.P., “Estimates For Solutions of Homogeneous Time-Delay Systems: Comparison of Lyapunov-Krasovskii and Lyapunov-Razumikhin Techniques”, Int. J. Control, 2021
Zhabko A.P., Alexandrova V I., “Complete Type Functionals For Homogeneous Time Delay Systems”, Automatica, 125 (2021), 109456
Efimov D., Polyakov A., “Finite-Time Stability Tools For Control and Estimation”, Found. Trends Syst. Control, 9:2-3 (2021), 172+
Efimov D., Aleksandrov A.Y., “Lyapunov-Krasovskii Functional For Discretized Homogeneous Systems”, SIAM J. Control Optim., 59:4 (2021), 2546–2569
Portilla G., Alexandrova V I., Mondie S., “Estimates For Weighted Homogeneous Delay Systems: a Lyapunov-Krasovskii-Razumikhin Approach”, 2021 American Control Conference (Acc), Proceedings of the American Control Conference, IEEE, 2021, 2298–2303
Gerson Portilla, Irina V. Alexandrova, Sabine Mondie, 2021 60th IEEE Conference on Decision and Control (CDC), 2021, 4743
Efimov D., Fridman E., Perruquetti W., Richard J.-P., “Homogeneity of Neutral Systems and Accelerated Stabilization of a Double Integrator By Measurement of Its Position”, Automatica, 118 (2020), 109023
Efimov D., Aleksandrov A., “on Estimation of Rates of Convergence in Lyapunov-Razumikhin Approach”, Automatica, 116 (2020), 108928
A. V. Ekimov, O. N. Chizhova, U. P. Zaranik, “Ustoichivost odnorodnykh nestatsionarnykh sistem differentsialno-raznostnykh uravnenii s lineino vozrastayuschim zapazdyvaniem”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 15:4 (2019), 415–424
Zhabko A.P., Alexandrova I.V., “Lyapunov Direct Method For Homogeneous Time Delay Systems”, IFAC PAPERSONLINE, 52:18 (2019), 79–84
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
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