Quadratic Lyapunov function for stochastic mechanical systems with switching impacts

Abstract

A general class of stochastic mechanical systems (including closed-loop control ones) developing under impacts generating both jumps and switchings is considered. We establish existence of a quadratic Lyapunov function for a system of this class: provided that the corresponding switching systems of ordinary differential equations have a common quadratic Lyapunov function and stochastic actions supplemented to them are moderate, this ordinary Lyapunov function is also a Lyapunov function for the system. Parameters of stochastic actions must satisfy the following constraints: disturbances in the force field are wiener processes with constant matrix coefficients; for jumps a certain combination of their distributions and of impacts arrival intensities is bounded relative to the ordinary common Lyapunov function. Systems of the class may have discontinuous coefficients and impacts arrival intensities; both are allowed to be time and state dependent.