On stability of linear dynamical systems with small Markov perturbations

Abstract

The mean-square stability analysis of linear equations with coefficients perturbed by small bounded functions of Markov processes is studied. The stability is determined by the sign of the mean-square Lyapunov exponent. This Lyapunov index depends on a small parameter epsilon related to the strength of the perturbations. A method is proposed for calculating the Lyapunov index from the largest real eigenvalue of a specially constructed matrix expanded as, power series in epsilon. An algorithms presented for calculating the terms in the expansion as well as the number of terms needed to be included in the expansion for the purpose of determining the system stability. The foundation of the algorithm is based on well-known results of the Kato perturbation theory for closed operators. For sufficiently small epsilon, the methodology gives the necessary and sufficient conditions for mean-square stability The method and the algorithm are illustrated by analyzing the stability of MDOF linear-systems. Examples include the parametric resonance of an oscillator with low damping, as well as the stability of a bridge deck subject to wind excitation