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Stabilization of oscillations from a monoparametric family of the autonomous system

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Abstract

The problem of oscillation stabilization is solved for the autonomous system in the case when the oscillations form a monoparametric family. For each possible point type of the family the explicit formulae that solve the problem are obtained. It follows from those formulae that the dissipation is a necessary condition of the stabilization. The obtained sufficient conditions of the stabilizations coinside with the appropriate necessary conditions to within the sign of unstrict inequality.

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References

  1. Malkin, I.G., Nekotorye zadachi teorii nelineinykh kolebanii (Some Problems in the Theory of Nonlinear Oscillation), Moscow: Nauka, 1956.

    Google Scholar 

  2. Tkhai, V.N., Reversible Mechanical Systems, in Nonlinear Mechanics, Moscow: Fizmatlit, 2001, pp. 131–146.

    Google Scholar 

  3. Tkhai, V.N., Oscillations and Stability in Quasiautonomous System. I. Simple Point of the One-parameter Family of Periodic Motions, Avtom. Telemekh., 2006, no. 9, pp. 90–98.

  4. Bogolyubov, N.N., Perturbation Theory in Nonlinear Mechanics, Proc. Inst. Archit. Mech., Ukr. Acad. Sci., 1950, vol. 14, pp. 9–34.

    Google Scholar 

  5. Kapitsa, P.L., Dynamic Stability of a Pendulum with an Oscillating Suspension Point, J. Exp. Theor. Phys., 1951, vol. 21, no. 5, pp. 588–597.

    Google Scholar 

  6. Leonov, G.A. and Shumafov, M.M., Metody stabilizatsii lineinykh upravlyaemykh sistem (Stabilization Problems of Linear Control Systems), St. Petersburg: S.-Peterburg. Gos. Univ., 2002.

    Google Scholar 

  7. Lyapunov, A.M., General Problem of the Stability of Motion, London: Taylor & Francis, 1992.

    MATH  Google Scholar 

  8. Tkhai, V.N., Rotational Motions of Mechanical Systems, Prikl. Mat. Mekh., 1999, vol. 63, no. 2, pp. 179–195.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    I. N. Barabanov & V. N. Tkhai

Authors
  1. I. N. Barabanov
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  2. V. N. Tkhai
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Additional information

Original Russian Text © I.N. Barabanov, V.N. Tkhai, 2009, published in Avtomatika i Telemekhanika, 2009, No. 2, pp. 35–41.

The work was supported by the Program 22 of the Presidium of the Russian Academy of Science and by the Program “State Support of Leading Scientific Schools,” project no. N.Sh.-1676.2008.1.

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Barabanov, I.N., Tkhai, V.N. Stabilization of oscillations from a monoparametric family of the autonomous system. Autom Remote Control 70, 203–208 (2009). https://doi.org/10.1134/S0005117909020027

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  • DOI: https://doi.org/10.1134/S0005117909020027

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