Abstract
Based on perturbation theory methods, criteria for the Lyapunov stability of Lurie systems with weakly oscillating parameters are proposed. The main attention is paid to obtaining the first approximation formulas for perturbations of multiple definite and indefinite multipliers of linear Hamiltonian systems and their applications to stability analysis. The formulas proposed lead to new criteria for the Lyapunov stability of Lurie systems in critical cases. Applications to the problem of a parametric resonance in fundamental resonances are considered. The results obtained are stated in terms of the original equations and brought to the stage of design formulas and algorithms. The efficiency of the formulas is illustrated by the example of the problem on the parametric resonance in a system of coupled oscillators.
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REFERENCES
Leonov, G.A., Teoriya upravleniya (Control Theory), St. Petersburg: Izd. S.-Peterburg. Univ., 2006.
Voronov, A.A., Vvedenie v dinamiku slozhnykh upravlyaemykh sistem (Introduction to the Dynamics of Complex Controlled Systems), Moscow: Nauka, 1985.
Krasnosel’skii, A.M. and Rachinskii, D.I., On Hamiltonian nature of Lurie systems, Autom. Remote Control, 2000, vol. 61, no. 8, pp. 1259 – 1262.
Bryuno, A.D., On types of stability in Hamiltonian systems, Preprint of Keldysh Inst. Appl. Math., Moscow, 2020, no. 021. https://doi.org/10.20948/prepr-2020-2
Bryuno, A.D., Normal form of a Hamiltonian system with a periodic perturbation, Preprint of Keldysh Inst. Appl. Math., Moscow, 2019, no. 057. https://doi.org/10.20948/prepr-2019-57
Zhuravlev, V.F., Petrov, F.G., and Shunderyuk, M.M., Izbrannye zadachi gamil’tonovoi mekhaniki (Selected Problems of Hamiltonian Mechanics), Moscow: Lenand, 2015.
Markeev, A.P., Lineinye gamil’tonovy sistemy i nekotorye zadachi ob ustoichivosti dvizheniya sputnika otnositel’no tsentra mass (Linear Hamiltonian Systems and Some Problems of Stability of Satellite Motion with Respect to the Center of Mass), Moscow–Izhevsk: Inst. Komp’yut. Issled., 2009.
Yakubovich, V.A. and Starzhinskii, V.M., Parametricheskii rezonans v lineinykh sistemakh (Parametric Resonance in Linear Systems), Moscow: Nauka, 1987.
Meyer, K.R. and Hall, G.R., Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, New York: Springer, 2009.
Seyranian, A.P. and Mailybaev, A.A., Multiparameter Stability Theory with Mechanical Applications, New Jersey: World Sci., 2003.
Yakubovich, V.A. and Starzhinskii, V.M., Lineinye differentsial’nye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya (Linear Differential Equations with Periodic Coefficients and Applications), Moscow: Nauka, 1972.
Landa, P.S., Nelineinye kolebaniya i volny (Nonlinear Oscillations and Waves), Moscow: Librokom, 2015.
Lanchares, V., On the stability of Hamiltonian dynamical systems, in Monografias Matematicas Garca de Galdeano, 2014, pp. 155–166.
Yumagulov, M.G., Ibragimova, L.S., Muzafarov, S.M., and Nurov, I.D., The Andronov–Hopf bifurcation with weakly oscillating parameters, Autom. Remote Control, 2008, vol. 69, no. 1, pp. 36–41.
Shil’nikov, L.P., Shil’nikov, A.L., Turaev, D.V., and Chua, L., Metody kachestvennoi teorii v nelineinoi dinamike. Chast’ 2 (Methods of Qualitative Theory in Nonlinear Dynamics. Part 2), Moscow–Izhevsk: Regulyarnaya Khaoticheskaya Din., Inst. Komp’yut. Issled., 2009.
Yumagulov, M.G., Ibragimova, L.S., and Belova, A.S., Methods for studying the stability of linear periodic systems depending on a small parameter, Itogi Nauki Tekh. Ser. Sovrem. Mat. Pril., 2019, vol. 163, pp. 113–126.
Polyak, B.T. and Kvinto, Ya.I., Stability and synchronization of oscillators: new Lyapunov functions, Autom. Remote Control, 2017, vol. 78, no. 7, pp. 1234–1242.
Funding
The research by A.S. Belova was carried out within the framework of the state order from the Ministry of Science and Higher Education of the Russian Federation, scientific topic code no. FZWU-2020-0027.
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Translated by V. Potapchouck
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Yumagulov, M.G., Ibragimova, L.S. & Belova, A.S. Investigation of the Problem on a Parametric Resonance in Lurie Systems with Weakly Oscillating Coefficients. Autom Remote Control 83, 252–263 (2022). https://doi.org/10.1134/S0005117922020072
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DOI: https://doi.org/10.1134/S0005117922020072