phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2021 - 2 issue » pp. 226-240

Archive of Issues

Russia Saint Petersburg
Year
2021
Volume
31
Issue
2
Pages
226-240
<<
>>
Section Mathematics
Title Stability analysis for the Lienard equation with discontinuous coefficients
Author(-s) Platonov A.V.a
Affiliations Saint Petersburg State Universitya
Abstract A nonlinear mechanical system, whose dynamics is described by a vector ordinary differential equation of the Lienard type, is considered. It is assumed that the coefficients of the equation can switch from one set of constant values to another, and the total number of these sets is, in general, infinite. Thus, piecewise constant functions with infinite number of break points on the entire time axis, are used to set the coefficients of the equation. A method for constructing a discontinuous Lyapunov function is proposed, which is applied to obtain sufficient conditions of the asymptotic stability of the zero equilibrium position of the equation studied. The results found are generalized to the case of a nonstationary Lienard equation with discontinuous coefficients of a more general form. As an auxiliary result of the work, some methods for analyzing the question of sign-definiteness and approaches to obtaining estimates for algebraic expressions, that represent the sum of power-type terms with non-stationary coefficients, are developed. The key feature of the study is the absence of assumptions about the boundedness of these non-stationary coefficients or their separateness from zero. Some examples are given to illustrate the established results.
Keywords nonlinear mechanical systems, discontinuous coefficients, asymptotic stability, Lyapunov functions
UDC 517.9
MSC 34A38, 34D20
DOI 10.35634/vm210205
Received 30 December 2020
Language Russian
Citation Platonov A.V. Stability analysis for the Lienard equation with discontinuous coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 226-240.
References
  1. Matrosov V.M. Metod vektornykh funktsii Lyapunova: analiz dinamicheskikh svoistv nelineinykh sistem (The method of vector Lyapunov functions: the analysis of a dynamical properties of nonlinear systems), Moscow: Fizmatlit, 2001.
  2. Vulfson I.I. Description of nonlinear dissipative forces under limited starting information, The Theory of Mechanisms and Machines, 2003, vol. 1, issue 1, pp. 70-77 (in Russian). https://www.elibrary.ru/item.asp?id=17092689
  3. Kozlov V.V. On the stability of equilibrium positions in non-stationary force fields, Journal of Applied Mathematics and Mechanics, 1991, vol. 55, issue 1, pp. 14-19. https://doi.org/10.1016/0021-8928(91)90054-X
  4. Hatvani L. The effect of damping on the stability properties of equilibria of non-autonomous systems, Journal of Applied Mathematics and Mechanics, 2001, vol. 65, issue 4, pp. 707-713. https://doi.org/10.1016/S0021-8928(01)00076-4
  5. Sun J., Wang Q.-G., Zhong Q.-C. A less conservative stability test for second-order linear time-varying vector differential equations, International Journal of Control, 2007, vol. 80, issue 4, pp. 523-526. https://doi.org/10.1080/00207170601028899
  6. Aleksandrov A.Yu. The stability of the equilibrium positions of non-linear non-autonomous mechanical systems, Journal of Applied Mathematics and Mechanics, 2007, vol. 71, issue 3, pp. 324-338. https://doi.org/10.1016/j.jappmathmech.2007.07.015
  7. Andreyev A.S. The stability of the equilibrium position of a non-autonomous mechanical system, Journal of Applied Mathematics and Mechanics, 1996, vol. 60, issue 3, pp. 381-389. https://doi.org/10.1016/S0021-8928(96)00048-2
  8. Liberzon D. Switching in systems and control, Boston: Birkhäuser, 2003. https://doi.org/10.1007/978-1-4612-0017-8
  9. Shorten R., Wirth F., Mason O., Wulf K., King G. Stability criteria for switched and hybrid systems, SIAM Review, 2007, vol. 49, issue 4, pp. 545-592. https://doi.org/10.1137/05063516X
  10. Zhai G., Hu B., Yasuda K., Michel A.N. Disturbance attention properties of time-controlled switched systems, Journal of the Franklin Institute, 2001, vol. 338, issue 7, pp. 765-779. https://doi.org/10.1016/S0016-0032(01)00030-8
  11. Ding X., Liu X. On stabilizability of switched positive linear systems under state-dependent switching, Applied Mathematics and Computation, 2017, vol. 307, pp. 92-101. https://doi.org/10.1016/j.amc.2017.03.007
  12. Xiang W., Xiao J. Stabilization of switched continuous-time systems with all modes unstable via dwell time switching, Automatica, 2014, vol. 50, issue 3, pp. 940-945. https://doi.org/10.1016/j.automatica.2013.12.028
  13. Yang H., Jiang B., Cocquempot V. A survey of results and perspectives on stabilization of switched nonlinear systems with unstable modes, Nonlinear Analysis: Hybrid Systems, 2014, vol. 13, pp. 45-60. https://doi.org/10.1016/j.nahs.2013.12.005
  14. Larina Ya.Yu., Rodina L.I. Asymptotically stable sets of control systems with impulse actions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 4, pp. 490-502 (in Russian). https://doi.org/10.20537/vm160404
  15. Cruz-Zavala E., Moreno J.A. Homogeneous high order sliding mode design: a Lyapunov approach, Automatica, 2017, vol. 80, pp. 232-238. https://doi.org/10.1016/j.automatica.2017.02.039
  16. Wang R., Xing J., Xiang Z. Finite-time stability and stabilization of switched nonlinear systems with asynchronous switching, Applied Mathematics and Computation, 2018, vol. 316, pp. 229-244. https://doi.org/10.1016/j.amc.2017.08.017
  17. Wang F., Zhang X., Chen B., Lin C., Li X., Zhang J. Adaptive finite-time tracking control of switched nonlinear systems, Information Sciences, 2017, vol. 421, pp. 126-135. https://doi.org/10.1016/j.ins.2017.08.095
  18. Shen Y., Huang Y., Gu J. Global finite-time observers for lipschitz nonlinear systems, IEEE Transactions on Automatic Control, 2011, vol. 56, issue 2, pp. 418-424. https://doi.org/10.1109/TAC.2010.2088610
  19. Zhang B. On finite-time stability of switched systems with hybrid homogeneous degrees, Mathematical Problems in Engineering, 2018, vol. 2018, pp. 1-7. https://doi.org/10.1155/2018/3096986
  20. Platonov A.V. On the asymptotic stability of nonlinear time-varying switched systems, Journal of Computer and Systems Sciences International, 2018, vol. 57, issue 6, pp. 854-863. https://doi.org/10.1134/S1064230718060084
  21. Platonov A.V. Stability analysis for nonstationary switched systems, Russian Mathematics, 2020, vol. 64, issue 2, pp. 56-65. https://doi.org/10.3103/S1066369X20020061
  22. Zubov V.I. Methods of A.M. Lyapunov and their applications, Groningen: Noordhoff, 1964.
  23. Aleksandrov A.Yu., Aleksandrova E.B., Lakrisenko P.A., Platonov A.V., Chen Y. Asymptotic stability conditions for some classes of mechanical systems with switched nonlinear force fields, Nonlinear Dynamics and Systems Theory, 2015, vol. 15, issue 2, pp. 127-140. https://zbmath.org/?q=an:1335.34087
Full text
/doc/issues-2021/21-02-05.pdf
<< Previous article
Next article >>