The paper proposes a method to set up a matrix-valued Lyapunov function for a system of differential equations with quasiperiodic coefficients. This function is used to establish asymptotic stability conditions for a class of linear systems Keywords: Lyapunov matrix-valued function, differential equations with quasiperiodic coefficients, asymptotic stability conditions 1. Introduction. Quasiperiodic motion [7] occurs in an elementary linear conservative system [9] with two degrees of freedom. Its solution has the form
The stability of linear and quasilinear systems with periodic coefficients is analyzed. The properties of stability are established in terms of matrix-valued Lyapunov functions. An algorithm is developed to set up matrix-valued Lyapunov functions for linear quasiperiodic systems. A numerical example is given to illustrate the application of the algorithm Keywords: quasiperiodic system, asymptotic stability, matrix-valued Lyapunov function 1. Introduction. The second Lyapunov method is one of the efficient methods for dynamic analysis of perturbed equations of motion [5]. The key element of this method is a Lyapunov function with special properties [7]. So far, many results in the theory of stability of linear and nonlinear systems with a finite number of degrees of freedom have been obtained in terms of Lyapunov functions with special properties. Finding appropriate Lyapunov functions for certain classes of systems of equations is still a problem of current importance. One of such classes is systems of linear differential equations with periodic coefficients. These equations are widely used in mechanics [1,6,11]. That the second Lyapunov method is universal is established by proving the inversion theorems [6,9,12]. For example, for a linear system
We establish the conditions of asymptotic stability of a linear system of matrix differential equations with quasiperiodic coefficients on the basis of constructive application of the principle of comparison with a Lyapunov matrix-valued function.
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