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Mat. Zametki, 2006, Volume 80, Issue 4, Pages 516–525 (Mi mz2844)  

This article is cited in 20 scientific papers (total in 20 papers)

Asymptotic stability and instability of the solutions of systems with impulse action

A. O. Ignatyeva, O. A. Ignatyevb, A. A. Solimanc

a Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
b Kent State University
c Benha University

Abstract: In this paper, we study the stability of the zero solution of a system of ordinary differential equations subject to impulse action. Using the method of Lyapunov functions, we obtain tests for asymptotic stability or instability of the system. Illustrative examples are given.

DOI: https://doi.org/10.4213/mzm2844

Full text: PDF file (488 kB)
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English version:
Mathematical Notes, 2006, 80:4, 491–499

Bibliographic databases:

UDC: 517.925
Received: 26.07.2005
Revised: 08.02.2006

Citation: A. O. Ignatyev, O. A. Ignatyev, A. A. Soliman, “Asymptotic stability and instability of the solutions of systems with impulse action”, Mat. Zametki, 80:4 (2006), 516–525; Math. Notes, 80:4 (2006), 491–499

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Ignatyev A. O., “On the stability of invariant sets of systems with impulse effect”, Nonlinear Anal., 69:1 (2008), 53–72  crossref  mathscinet  zmath  isi  elib  scopus
    2. 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
    3. Borysenko S.D., Toscano S., “Impulsive differential systems: the problem of stability and practical stability”, Nonlinear Anal., 71:12 (2009), e1843–e1849  crossref  mathscinet  zmath  isi  scopus
    4. Ignatyev O.A., “Homogeneous polynomials as Lyapunov functions in the stability research of solutions of difference equations”, Appl. Math. Comput., 216:2 (2010), 388–394  crossref  mathscinet  zmath  isi  scopus
    5. Li X., Rakkiyappan R., Balasubramaniam P., “Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations”, J. Franklin Inst., 348:2 (2011), 135–155  crossref  mathscinet  zmath  isi  scopus
    6. Dvirnyi A.I., Slyn'ko V.I., “On the Stability of Abstract Monotone Impulsive Differential Equations in Terms of Two Measures”, Ukr. Math. J., 63:7 (2011), 1042–1064  crossref  mathscinet  zmath  isi  scopus
    7. Ignat'ev A.O., “On the Existence of a Lyapunov Function as a Quadratic Form for Impulsive Systems of Linear Differential Equations”, Ukr. Math. J., 62:11 (2011), 1680–1689  crossref  mathscinet  isi  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. Lin D., Li X., O'Regan D., “Stability Analysis of Generalized Impulsive Functional Differential Equations”, Math. Comput. Model., 55:5-6 (2012), 1682–1690  crossref  mathscinet  zmath  isi  scopus
    11. Lin D., Li X., O'Regan D., “Mu-Stability of Infinite Delay Functional Differential Systems with Impulsive Effects”, Appl. Anal., 92:1 (2013), 15–26  crossref  mathscinet  zmath  isi  scopus
    12. A. I. Dvirnyj, V. I. Slyn'ko, “Stability in terms of two measures for a class of semilinear impulsive parabolic equations”, Sb. Math., 204:4 (2013), 485–507  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Li X., Akca H., Fu X., “Uniform Stability of Impulsive Infinite Delay Differential Equations with Applications to Systems with Integral Impulsive Conditions”, Appl. Math. Comput., 219:14 (2013), 7329–7337  crossref  mathscinet  zmath  isi  scopus
    14. 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
    15. Slyn'ko V.I., Denysenko V.S., “The Stability Analysis of Abstract Takagi-Sugeno Fuzzy Impulsive System”, Fuzzy Sets Syst., 254 (2014), 67–82  crossref  mathscinet  zmath  isi  scopus
    16. Li X., Song Sh., “Research on Synchronization of Chaotic Delayed Neural Networks With Stochastic Perturbation Using Impulsive Control Method”, Commun. Nonlinear Sci. Numer. Simul., 19:10 (2014), 3892–3900  crossref  mathscinet  adsnasa  isi  scopus
    17. A. I. Dvirnyi, V. I. Slyn'ko, “A counterpart of A. M. Molchanov's critical case for impulse systems”, Autom. Remote Control, 76:6 (2015), 945–956  mathnet  crossref  isi  elib  elib
    18. Slyn'ko V.I. Tunc C., “Global Asymptotic Stability of Nonlinear Periodic Impulsive Equations”, Miskolc Math. Notes, 19:1 (2018), 595–610  crossref  isi
    19. V. I. Slyn'ko, “The stability conditions of linear periodic systems of ordinary differential equations”, St. Petersburg Math. J., 30:5 (2019), 885–900  mathnet  crossref  isi  elib
    20. S. V. Kravchuk, V. I. Slyn'ko, “Robust stability of linear periodic systems”, Autom. Remote Control, 80:12 (2019), 2108–2125  mathnet  crossref  crossref  isi  elib
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