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Russia Saint Petersburg
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.
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