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On parametric instability of singularly perturbed systems

  • Nonlinear Systems
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Abstract

Consideration is given to the class of singularly perturbed systems that admit the separation into a slow subsystem and a boundary layer. On the basis of some suppositions as to behavior of stated subsystems, sufficient conditions of instability of a moving equilibrium state are assigned for the system considered under parametric perturbations.

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References

  1. Thomson, J.M.T., Instability and Catastrophes in Science and Technology, New York: Wiley, 1982. Translated under the title Neustoichivost’ i katastrofy v nauke i tekhnike, Moscow: Mir, 1985.

    Google Scholar 

  2. Lyapunov, A.M., Obshchaya zadacha ob ustoichivosti dvizheniya (The Common Problem on Stability of Motion), Moscow: GITTL, 1950.

    Google Scholar 

  3. Mitropol’skii, Yu.A., Metody nelineinoi mekhaniki (Methods of Nonlinear Mechanics), Kiev: Naukova Dumka, 2005.

    Google Scholar 

  4. Chetaev, N.G., Ustoichivost’ dvizheniya (Stability of Motion), Moscow: Nauka, 1965, 3rd ed.

    Google Scholar 

  5. Andronov, A.A., Vitt, A.A., and Khaikin, S.E., Teoriya kolebanii (Theory of Oscillations), Moscow: Nauka, 1981.

    Google Scholar 

  6. Martynyuk, A.A. and Khoroshun, A.S., To the Problem of Motion Stabilization of the Parametric Set of Nonlinearly Perturbed Systems, Nelineinye Kolebaniya, 2011, vol. 14, no. 2, pp. 238–254.

    Google Scholar 

  7. Khoroshun, A.S. and Martynyuk, A.A., Parametric Stability of Singularly Perturbed Nonlinear Uncertain Systems, Int. Appl. Mech., 2010, vol. 46, no. 10, pp. 1177–1190.

    MathSciNet  Google Scholar 

  8. Martynyuk, A.A. and Khorishun, A.S., To the Theory of Parametric Stabilization of Inexact Singularly Perturbed Systems, Nelineinye Kolebaniya, 2012, vol. 15, no. 3, pp. 367–380.

    Google Scholar 

  9. Binning, H.S. and Goodall, D.P., Output Feedback Controls for a Class of Singularly Perturbed Uncertain Nonlinear Systems, Autom. Remote Control, 1997, vol. 58, no. 7, part 1, pp. 1132–1145.

    MathSciNet  MATH  Google Scholar 

  10. Kolpakova, E.A. and Subbotina, N.N., Defining the Asymptotics for One Class of Singularly Perturbed Problems of Vibrational Mechanics, Autom. Remote Control, 2007, vol. 68, no. 11, pp. 2029–2041.

    Article  MathSciNet  MATH  Google Scholar 

  11. Kremlev, A.G., Asymptotic Properties of a Set of Trajectories of a Singularly Perturbed System in an Optimal Control Problem, Autom. Remote Control, 1993, vol. 54, no. 9, part 1, pp. 1353–1367.

    MathSciNet  Google Scholar 

  12. Barbashin, E.A., Vvedenie v teoriyu ustoichivosti (Introduction to the Stability Theory), Moscow: Nauka, 1967.

    Google Scholar 

  13. Demidovich, B.P., Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the Mathematical Stability Theory), Moscow: Nauka, 1967.

    Google Scholar 

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Authors and Affiliations

  1. Timoshenko Institute of Mechanics, Academy of Sciences of Ukraine, Kiev, Ukraine

    A. A. Martynyuk & A. S. Khoroshun

Authors
  1. A. A. Martynyuk
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  2. A. S. Khoroshun
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Original Russian Text © A.A. Martynyuk, A.S. Khoroshun, 2013, published in Avtomatika i Telemekhanika, 2013, No. 1, pp. 59–78.

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Martynyuk, A.A., Khoroshun, A.S. On parametric instability of singularly perturbed systems. Autom Remote Control 74, 46–61 (2013). https://doi.org/10.1134/S0005117913010050

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  • DOI: https://doi.org/10.1134/S0005117913010050

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