^{1}

^{2}

^{1}

^{1}

^{2}

We apply the spectrum analysis approach to address the stability of discrete-time Markov jump systems with state-multiplicative noise. In terms of the spectral distribution of a generalized Lyapunov operator, spectral criteria are presented to testify three different kinds of stochastic stabilities: asymptotic mean square stability, critical stability, and essential instability.

Since the pioneering research of [

It is well known that stability is one of the most fundamental notions in the modern control theory. It is the primal factor to be taken into account in the controller design. For stochastic Itô systems and Markov jump systems, various stability concepts such as stochastic stability, exponential stability [

The rest of this paper is organized as follows. In Section

Throughout this paper, the following notations are adopted:

On a given probability space

First of all, we show a useful representation which characterizes the state evolution of system (

For

Since

Given

For

Let us see an illustrative example which shows how to compute all the spectra of

In

The matrix

Similar to [

By Definition

Stability is one of the core concepts in the modern control theory. It is a common sense that stability is a necessary prerequisite for the system to behave well. Next, we will make use of the spectrum technique to discuss the stability of system (

Above all, we recall the well-known asymptotic mean square stability of stochastic Markov jump systems.

The system described by the state equation (

In terms of the spectra of the operator

The following assertions are equivalent to AMSS of

for any given

for any given

there exists

there exists

We only show the validity of (i). The rigorous arguments of (ii)–(v) can be found in [

In

In this subsection, we are concerned with the critical stability, which was first proposed in [

The following result presents useful criteria for verifying the critical stability of

for any given

for any given

for any given

for any given

for any given

Essential instability for stochastic Itô systems was first proposed in [

Next, we give a spectral criterion for essential instability of system (

Let

In this paper, we have applied the technique of operator spectrum to demonstrate some new aspects on the stability of discrete-time Markov jump systems with multiplicative noise. With the help of the spectra of a generalized Lyapunov operator, we have distinguished three kinds of stochastic stabilities and obtained their spectral criteria which are easy to test. As one of our future works, we will focus on how to use the proposed stability theories to devise the stochastic LQ optimal control with convergence rate constraints, which still remains open to date.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (nos. 61304074 and 61174078), the Research Award Fund for Outstanding Young Scientists of Shandong Province (no. BS2013DX009), the Research Fund for the Taishan Scholar Project of Shandong Province of China, the SDUST Research Fund (no. 2011KYTD105), and the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (Grant no. LAPS13018).