Spectral Perspective on the Stability of Discrete-Time Markov Jump Systems with Multiplicative Noise

Since the pioneering research of [1], Markov jump systems have received an increasing attention from the control community. The main impetus for studying this kind of systems arises from the widespread dynamics that are subject to abrupt structural changes, such as random component failures or repairs, sudden variation in operating points of nonlinear plants, the switching between the economic scenarios, and temporary loss of communication signal. By now, the study of Markov jump systems has become one of the most active areas in the control theory and great progress has been made in the relevant analysis and synthesis. Among many contributions, we would like to mention that structural properties and quadratic optimal control of continuous-time Markov jump linear systems have been considered in [2]. Stability and robust control design have been performed in [3] for discrete-time Markov jump linear systems. Moreover, robustly asymptotic stabilization for a class of stochastically nonlinear singular jump systemshas been tackled in [4] based on the Lyapunov-Krasovskii functional. Besides, stability analyses for Markovian jumping neural networks with delays have also been elaborately addressed; see [5–10] and the references therein. Recently, many researchers are attracted to the study of stochastic linear systems subject to both multiplicative noise perturbations and Markov jump, which are commonly viewed as a powerful mathematical tool to investigate the financial phenomena and engineering problems. For instance, the indefinite stochastic linear quadratic (LQ) optimal control was tackled in [11] for discrete-time multiplicative noise systems with Markov jump parameters. Robust H ∞ filtering has been reported in [12] for stochastic nonlinear Itô systems with Markov jumps. In addition, by means of four coupled matrix recursions, mixed H 2 /H ∞


Introduction
Since the pioneering research of [1], Markov jump systems have received an increasing attention from the control community.The main impetus for studying this kind of systems arises from the widespread dynamics that are subject to abrupt structural changes, such as random component failures or repairs, sudden variation in operating points of nonlinear plants, the switching between the economic scenarios, and temporary loss of communication signal.By now, the study of Markov jump systems has become one of the most active areas in the control theory and great progress has been made in the relevant analysis and synthesis.Among many contributions, we would like to mention that structural properties and quadratic optimal control of continuous-time Markov jump linear systems have been considered in [2].Stability and robust control design have been performed in [3] for discrete-time Markov jump linear systems.Moreover, robustly asymptotic stabilization for a class of stochastically nonlinear singular jump systems has been tackled in [4] based on the Lyapunov-Krasovskii functional.Besides, stability analyses for Markovian jumping neural networks with delays have also been elaborately addressed; see [5][6][7][8][9][10] and the references therein.Recently, many researchers are attracted to the study of stochastic linear systems subject to both multiplicative noise perturbations and Markov jump, which are commonly viewed as a powerful mathematical tool to investigate the financial phenomena and engineering problems.For instance, the indefinite stochastic linear quadratic (LQ) optimal control was tackled in [11] for discrete-time multiplicative noise systems with Markov jump parameters.Robust  ∞ filtering has been reported in [12] for stochastic nonlinear Itô systems with Markov jumps.In addition, by means of four coupled matrix recursions, mixed  2 / ∞ control problem has been settled in [13] for a class of discretetime Markov jump systems with multiplicative noise.For more details of recent developments, interested readers are referred to [14][15][16] and the references cited therein.
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 [17][18][19], asymptotic stability in probability [20], finite-time stability [21], global asymptotical stability [22,23], and absolute stability [24] have been studied extensively.It can be remarked that almost all existing works about the stability analysis of stochastic systems are based on the Lyapunov functional method.However, our main purpose is to provide a spectral perspective about the stability of discrete-time Markov jump systems with state-multiplicative noise.Concretely speaking, we will employ the spectral analysis technique to classify three different kinds of stabilities: asymptotic mean square stability, critical stability, and 2 Mathematical Problems in Engineering essential instability.For them, spectral criteria are derived according to a generalized Lyapunov operator generated from the system coefficients and transition probability matrix of Markov jump parameter.
The rest of this paper is organized as follows.In Section 2, we introduce a generalized Lyapunov operator which serves as the fundamental tool in the stability analysis.In Section 3, necessary and sufficient conditions, including spectral criteria, are supplied for three kinds of stochastic stabilities, respectively.Section 4 ends this paper with a brief concluding remark.

Lyapunov-Type Operator and Its Spectra
On a given probability space (Ω, F, P), we consider the following linear system with Markovian jumps and multiplicative noise:  ( + 1) =  (  )  () +  (  )  ()  () , where () ∈ R  represents the state of the system.() is a Markov chain taking values in  and the transition probability matrix is In the sequel, system (1) is also written as [,  | P] for short.First of all, we show a useful representation which characterizes the state evolution of system (1).
Given  = ( 1 ,  2 , . . .,   ) ∈ S   , we introduce the following Lyapunov operator: According to the formulation of   () ( ∈ ) derived in Theorem 1, the spectra of L , are defined as follows.] . When In the case of   = 2, By ( 5) and noting the symmetry of , we get where * is the ellipsis of the symmetric terms.The eigenvalue equation in ( 6) can be equivalently expressed as Further, we can derive It is easy to obtain that the spectral set of L , is where    (,  = 1, 2, . . ., ) is the element of   .That is, ⃗ (  ) is a column vector consisting of all elements of   , while φ(  ) is the column vector generated by all upper diagonal elements of   .The relation between them can be dominated by the unique matrix   2 ,((+1)/2) that is of full column rank: In fact, if we take that where then the expression of   turns out to be By Definition 2, the operator L , is a linear operator defined on the Hilbert space S   with the inner product ⟨, ⟩ = ∑  =1 Tr(    ), where ,  ∈ S   .The adjoint operator of L , induced by the definition of inner production and the property of trace is given as follows: We can show that L * , and L , have the same spectral radius.Particularly, when the coefficients of [A, C | P] are all real matrices, we have (L , ) = (L * , ).

Stability Analysis
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 (1).

Asymptotic Mean Square
Stability.Above all, we recall the well-known asymptotic mean square stability of stochastic Markov jump systems.
From Definition Let There exists satisfying the following LMIs: It thus follows by Theorem 6 that the system denoted by this example is asymptotically mean square stable.

Critical Stability.
In this subsection, we are concerned with the critical stability, which was first proposed in [27]  Next, we give a spectral criterion for essential instability of system (1).The detailed proof can be given by following the similar procedure of Theorem 1 [25].

Conclusions
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.

Definition 2 .
For [A, C | P], let L , be the linear operator from S   to S   .Then, the spectral set of L , is  (L , ) := { ∈  : L , () = ,  ∈ S   ,  ̸ = 0} .(6) Let us see an illustrative example which shows how to compute all the spectra of L , .Example 3. In [A, C | P], the jump parameter   takes values in  = {1, 2} and the transition probability matrix is given by