Stability analysis for a class of mechanical systems with piecewise constant coefficients

Abstract

A nonlinear mechanical system described by a vector equation of the Lienard type is considered. It is assumed that the forces acting on the mechanical system are given by functions that are piecewise constant with respect to time. With the aid of a special variable substitution, the equation is reduced to a system that can be considered as an impulsive switched system with infinite numbers of operating modes. The stability problem for the trivial equilibrium position of the obtained system is studied. The stability analysis is carried out using a specially constructed discontinuous Lyapunov function. This function can be understood as a multiple Lyapunov function, consisting of an infinite number of partial Lyapunov functions, each of which is used for a certain time interval. Differentiating the chosen Lyapunov function with respect to solutions of the given system on the corresponding time intervals and applying the theory of differential inequalities, sufficient conditions of the asymptotic stability of the equilibrium position are established.