Stochastic Stability Analysis of Discrete Time System Using Lyapunov Measure

TL;DR

This paper introduces a novel Lyapunov measure-based approach for analyzing the stochastic stability of discrete-time nonlinear systems, establishing theoretical connections and proposing numerical methods for finite-dimensional approximation.

Contribution

The paper develops a new Lyapunov measure framework for stochastic stability verification, linking it with Lyapunov functions and proposing numerical approximation techniques.

Findings

Lyapunov measure-based theorems verify stochastic stability.

Duality between Lyapunov functions and measures is established.

Finite-dimensional approximations enable practical stability analysis.

Abstract

In this paper, we study the stability problem of a stochastic, nonlinear, discrete-time system. We introduce a linear transfer operator-based Lyapunov measure as a new tool for stability verification of stochastic systems. Weaker set-theoretic notion of almost everywhere stochastic stability is introduced and verified, using Lyapunov measure-based stochastic stability theorems. Furthermore, connection between Lyapunov functions, a popular tool for stochastic stability verification, and Lyapunov measures is established. Using the duality property between the linear transfer Perron-Frobenius and Koopman operators, we show the Lyapunov measure and Lyapunov function used for the verification of stochastic stability are dual to each other. Set-oriented numerical methods are proposed for the finite dimensional approximation of the Perron-Frobenius operator; hence, Lyapunov measure is proposed. Stability results in finite dimensional approximation space are also presented. Finite dimensional approximation is shown to introduce further weaker notion of stability referred to as coarse stochastic stability. The results in this paper extend our earlier work on the use of Lyapunov measures for almost everywhere stability verification of deterministic dynamical systems ("Lyapunov Measure for Almost Everywhere Stability", {\it IEEE Trans. on Automatic Control}, Vol. 53, No. 1, Feb. 2008).