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Mat. Zametki, 2004, Volume 76, Issue 1, Pages 44–51 (Mi mz89)  

This article is cited in 12 scientific papers (total in 12 papers)

On the Stability of Periodic Impulsive Systems

R. I. Gladilina, A. O. Ignatyev

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

Abstract: In this paper, we consider periodic systems of ordinary differential equations with impulse perturbations at fixed points of time. It is assumed that the system possesses the trivial solution. We show that if the trivial solution of the system is stable or asymptotically stable, then it is uniformly stable or uniformly asymptotically stable, respectively. By using the method of Lyapunov functions, we establish criteria of uniform asymptotical stability and instability.

DOI: https://doi.org/10.4213/mzm89

Full text: PDF file (190 kB)
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English version:
Mathematical Notes, 2004, 76:1, 41–47

Bibliographic databases:

UDC: 517.925.3
Received: 21.05.2002
Revised: 25.06.2003

Citation: R. I. Gladilina, A. O. Ignatyev, “On the Stability of Periodic Impulsive Systems”, Mat. Zametki, 76:1 (2004), 44–51; Math. Notes, 76:1 (2004), 41–47

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. O. Ignatyev, O. A. Ignatyev, A. A. Soliman, “Asymptotic stability and instability of the solutions of systems with impulse action”, Math. Notes, 80:4 (2006), 491–499  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. R. I. Gladilina, A. O. Ignatyev, “On retention of impulsive system stability under perturbations”, Autom. Remote Control, 68:8 (2007), 1364–1371  mathnet  crossref  mathscinet  zmath
    3. A. O. Ignatyev, “Asymptotic stability and instability with respect to part of variables for solutions to impulsive systems”, Siberian Math. J., 49:1 (2008), 102–108  mathnet  crossref  mathscinet  zmath  isi  elib
    4. Ignat'ev A.O., “On equiasymptotic stability of solutions of doubly-periodic impulsive systems”, Ukrainian Math. J., 60:10 (2008), 1528–1539  crossref  mathscinet  zmath  isi  scopus
    5. Perestyuk M.O., Chernikova O.S., “Some modern aspects of the theory of impulsive differential equations”, Ukrainian Math. J., 60:1 (2008), 91–107  crossref  mathscinet  zmath  isi  scopus
    6. Gladilina, RI, “Necessary and sufficient stability conditions for invariant sets of nonlinear impulsive systems”, International Applied Mechanics, 44:2 (2008), 228  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. Ignatyev O.A., “Homogeneous polynomials as Lyapunov functions in the stability research of solutions of difference equations”, Applied Mathematics and Computation, 216:2 (2010), 388–394  crossref  mathscinet  zmath  isi  scopus  scopus
    8. Ignatyev A.O., Ignatyev O., “Quadratic Forms as Lyapunov Functions in the Study of Stability of Solutions to Difference Equations”, Electron. J. Differ. Equ., 2011, 19  mathscinet  zmath  isi
    9. A. I. Dvirnyi, V. I. Slynko, “Analog kriticheskogo sluchaya Kamenkova dlya sistem differentsialnykh uravnenii s impulsnym vozdeistviem”, Sib. zhurn. industr. matem., 15:1 (2012), 22–33  mathnet  mathscinet
    10. A. I. Dvirnyj, V. I. Slyn'ko, “Application of Lyapunov's Direct Method to the Study of the Stability of Solutions to Systems of Impulsive Differential Equations”, Math. Notes, 96:1 (2014), 26–37  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. Ya. Yu. Larina, L. I. Rodina, “Asimptoticheski ustoichivye mnozhestva upravlyaemykh sistem s impulsnym vozdeistviem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 26:4 (2016), 490–502  mathnet  crossref  mathscinet  elib
    12. Shahinyan S., “About Stability of Dynamical Systems With Integrally Small Perturbations”, AIP Conference Proceedings, 2046, ed. Sivasundaram S., Amer Inst Physics, 2018, UNSP 020086  crossref  isi
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