doi: 10.17586/2226-1494-2017-17-5-947-951


G. I. Melnikov, V. G. Melnikov, N. A. Dudarenko, A. S. Alyshev, L. N. Ivanova

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For citation: Melnikov G.I., Melnikov V.G., Dudarenko N.A., Alyshev A.S., Ivanova L.N. Sequences of differential inequalities for Lyapunov functions in stability estimates of nonlinear dynamical systems. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2017, vol. 17, no. 5, pp. 947–951 (in Russian). doi: 10.17586/2226-1494-2017-17-5-947-951


We consider a nonlinear autonomous controlled mechanical system with several degrees of freedom with a mathematical model in the form of a polynomial system of differential equations containing homogeneous linear forms with respect to phase variables and nonlinear homogeneous forms up to the fourth power with small coefficients. By replacing variables with multipliers–exponential functions of time–the system is transformed into a system with an autonomous linear part and non-autonomous variable coefficients for homogeneous nonlinear forms with a matrix having eigenvalues ​​with negative real parts reduced in absolute value, as well as nonlinear forms with variable coefficients. A method is proposed for the formation of a sequence of linear differential inequalities for positive-definite Lyapunov functions with estimates of the system approximation to a stable equilibrium state.

Keywords: mechanical systems, dynamical systems, motion stability, Lyapunov functions, iteration method, differential inequalities sequence, transient process functional estimates

Acknowledgements. This work was supported by the RFBR Grants 16-08-00997, 17-01-00672.

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