Moment stability of stochastic processes with applications to control systems

TL;DR

This paper develops new conditions for bounding moments of discrete-time stochastic processes with state-dependent noise, providing insights into their stability and ergodicity, with applications in control and dynamical systems.

Contribution

It introduces novel moment bounds under weak negative drift and state-dependent jump restrictions, applicable to multiplicative-noise processes, and establishes ergodicity for Markovian systems.

Findings

Derived uniform moment bounds for stochastic processes

Proved ergodicity under Markovian assumptions

Demonstrated applicability through illustrative examples

Abstract

We establish new conditions for obtaining uniform bounds on the moments of discrete-time stochastic processes. Our results require a weak negative drift criterion along with a state-dependent restriction on the sizes of the one-step jumps of the processes. The state-dependent feature of the results make them suitable for a large class of multiplicative-noise processes. Under the additional assumption of Markovian property, new result on ergodicity has also been proved. There are several applications to iterative systems, control systems, and other dynamical systems with state-dependent multiplicative noise, and we include illustrative examples to demonstrate applicability of our results.