Advances in Difference Equations

, 2019:338 | Cite as

Region stability of linear stochastic discrete systems with time-delays

  • Gang LiEmail author
  • Yuxia Gao
  • Ming Chen
Open Access


In this paper, our aim is to investigate the region stability of stochastic discrete time-delay systems. Firstly, region stabilities of systems are defined by a spectrum operator. Secondly, with the aid of linear matrix inequality, some equivalent conditions and related results for region stabilities of systems are obtained. Finally, the relation between region stabilities and the convergence speed of the trajectories of systems is given.


Region-stability Time-delay Discrete stochastic system 

1 Introduction

Stability is one of essential concepts in dynamical system theory, which has been considered by the researchers in many fields, such as [1, 2, 3, 4, 5, 6, 7, 8], and [9, 10, 11, 12]. For a linear system without delays, as is well known, the stability is related to the system matrix root-clustering in subregions of the complex plane. That is, the stability of systems has a close relation with the spectrum placement of the system matrix. For example, the further left the spectrum set of the considered system is located, the faster the system response rate is.

Mean square stability has become a hot issue, which has many important applications in system analysis and designs. Some equivalent conditions for mean square stability of linear stochastic systems have been obtained in terms of generalized algebraic Riccati equation (see [13, 14, 15, 16, 17]), or linear matrix inequality (LMI) (see [18, 19, 20, 21] and [22, 23, 24, 25]), or spectra of some operators (see [26, 27, 28]). The spectrum technique is very powerful in system analysis and design, which began with [26] for a stochastic system.

Region stability considered in this paper is related with the mean square stability. Region stability problem is to check whether the spectrum set of deterministic or stochastic systems lies in a given region of the complex plane, which has become a popular research focus in the past decades (see [29, 30, 31, 32, 33], and the references therein). Compared with deterministic systems, stochastic systems can describe the complexity of practical problems more accurately. For stochastic systems, a special case of region-stability and region-stabilization, called respectively interval stability and interval stabilization, was studied, and a sufficient condition was given for the interval stabilization of general linear stochastic systems with state and control dependent noises in [29]. In [30], the authors further defined and discussed a class of relative stability and stabilization, which is a more special case of region-stability and region-stabilization of linear stochastic systems. The objective of the present work is to make a further investigation in the region stability of linear stochastic discrete systems. New results concerning the region stability of linear stochastic discrete systems are developed. By means of LMI, some necessary and sufficient conditions are presented for region-stability with generalized LMI regions. To the best of our knowledge, there were no similar results reported for such dynamical systems up to now.

This paper is organized as follows: In Sect. 2, we introduce some LMI regions and generalized LMI regions. In Sect. 3, we give some definitions of region stability and obtain some equivalent conditions and related results for the region stability of stochastic discrete systems with delays.

For convenience, we adopt the following traditional notations: \(\mathbb{S}^{n}\) is the set of all symmetric matrices, whose components may be complex; \(\mathbb{R}^{n\times m}\) is the set of all \(n\times m\) matrices, whose components are real; \(A'\) (\(\ker (A)\)) denotes the transpose (kernel space) of the matrix A; \(A\geq 0\) (\(A>0\)) is a positive semidefinite (positive definite) symmetric matrix; I is the identity matrix; \(\sigma (L)\) is the spectral set of the operator or matrix L; \(\mathbb{N}_{0}=\{0,1,2,\ldots \}\). Here ⊗ denotes Kronecker product operation of matrices (see [34] for details); \(\mathbb{R}^{n}\) is the set of all n-dimensional vectors, whose components are real. For \(x\in \mathbb{R}^{n}\), \(\Vert x \Vert =\sqrt[2]{x'x}\).

2 LMI regions

In this paper, we shall discuss the stability of time-delay systems in generalized LMI regions. First of all, we introduce some definitions and related examples in LMI regions and generalized LMI regions.

Definition 2.1


A region D in the complex plane is called an LMI region if there exist symmetric matrices \(H_{1}\) and \(H_{2}\) such that
$$\begin{aligned}& D=\bigl\{ z\in {C}\vert H_{1}+zH_{2}+ \overline{z}H_{2}'< 0\bigr\} . \end{aligned}$$

Here we give two examples of LMI region.

Example 2.1

In the complex plane, the circle region \(D(q,r)\), which has center \({(q,0)}\) and radius r, is an LMI region, where q and r satisfy \(\vert q \vert +r<1\).

Example 2.2

If \(r\in {(0,1]}\), then the fan-shaped region
$$\begin{aligned}& D(r,\theta )=\biggl\{ z=x+jy\vert x,y\in \mathbb{R},\tan \theta < - \frac{ \vert y \vert }{x}, \vert z \vert =\sqrt{x ^{2}+y^{2}}< r \biggr\} \end{aligned}$$
is an LMI region.

Different from an LMI region, we introduce another region, which is defined as follows.

Definition 2.2


For a symmetric matrix
R = ( R 1 R 2 R 2 R 3 ) , Open image in new window
the region
$$\begin{aligned}& D_{R}=\bigl\{ z\in {C}\vert R_{1}+zR_{2}+ \overline{z}R_{2}'+z\overline{z}R_{3}< 0 \bigr\} \end{aligned}$$
in the complex plane is called a generalized LMI region.

Example 2.3

The ring-shaped region \(D(0,\alpha ,\beta )\), \({0}\leq {\alpha }<{\beta }\leq {1}\) is a generalized LMI region. Indeed, the matrix in (3) of Definition 2.2 can be written as
R = ( β 2 0 0 0 0 α 2 0 0 0 0 1 0 0 0 0 1 ) . Open image in new window

Remark 2.1

In fact, let \(R_{3}=0\) in the matrix of (3). Then the region in (4) becomes the region in (1). That is, LMI region is regarded as a special case of a generalized region. The details about the LMI region and generalized region are found in [35].

3 Criteria of region-stability

In this section, we shall investigate the region stability of a kind of linear stochastic discrete systems with time-delays:
$$\begin{aligned}& \textstyle\begin{cases} x(t+1)=\sum_{j=0}^{m}[F_{j}x(t-j)+G_{j}x(t-j)\omega (t)], \\ x(k)=\varphi (k)\in {\mathbb{R}^{n}}, \quad k=0,-1,\ldots,-m, t\in \mathbb{N}_{0}, \end{cases}\displaystyle \end{aligned}$$
where \(x(t) \in \mathbb{R}^{n}\) is a column vector, \(F_{j},G_{j} \in \mathbb{R}^{n\times n}\), \(j = 0,1,\ldots, m\) are constant coefficient matrices, \(\varphi (k) \in \mathbb{R}^{n}\), \(k \in [-m, 0]\) is a deterministic initial condition, \(\{w(t), t \in \mathbb{N}_{0}\}\) is a scalar Brownian motion defined on a complete probability space \((\varOmega , \mathcal{F},\mathcal{F}_{t},\mu )\) with \(E[w(t)]=0\) and \(E[w(t)w(s)]=\delta _{s,t}\), where \(\delta _{s,t}\) is the Kronecker delta with \(\mathcal{F}_{t}=\sigma \{w(s)\vert 0\leq s \leq t\}\). For briefness, system (5) is denoted by \([\sum_{j=0}^{m}F_{j},\sum_{j=0}^{m}G_{j}]\).
To study the stability of system (5), first of all, we define two vectors \(\overline{x}(t)\) and \(\overline{x}(0)\) as follows:
$$\begin{aligned}& \overline{x}(t)=\bigl[x'(t),x'(t-1), \ldots,x'(t-m)\bigr]'\in \mathbb{R}^{n(m+1)}, \\& \overline{x}(0)=\bigl[\varphi '(0), \varphi '(-1), \ldots,\varphi '(-m)\bigr]'= \overline{x}_{0} \in \mathbb{R}^{n(m+1)}. \end{aligned}$$
So system (5) can become the following equivalent stochastic system:
$$\begin{aligned}& \textstyle\begin{cases} \overline{x}(t+1)=F\overline{x}(t)+G\overline{x}(t)\omega (t), \\ \overline{x}(0)=\overline{x}_{0}, \quad t\in \mathbb{N}_{0}, \end{cases}\displaystyle \end{aligned}$$
F = ( F 0 F 1 F m 1 F m I 0 0 0 0 0 I 0 ) , G = ( G 0 G 1 G m 1 G m 0 0 0 0 0 0 0 0 ) . Open image in new window
Let \(X(t)=E[\overline{x}(t)\overline{x}'(t)]\), and suppose \(X(t)\) satisfies
$$ X(t+1)=FX(t)F'+GX(t)G'. $$
Now we define a general Lyapunov operator as follows:
$$\begin{aligned}& L_{F,G}: X(t)\in \mathbb{S}^{n(m+1)}\rightarrow FX(t)F'+GX(t)G'\in \mathbb{S}^{n(m+1)}. \end{aligned}$$

Now we give some definitions of region-stability of stochastic time-delay systems by the spectrum operator \(\sigma (L_{F,G})\).

Definition 3.1

System (5) is \(D_{R}\)-stable, if \({\sigma (L_{F,G})}\subset {D_{R}}\subset {D(0,1)}\).

Definition 3.2

System (5) is \(D(q,r)\)-stable, if \({\sigma (L_{F,G})}\subset {D(q,r)}\subset {D(0,1)}\).

Definition 3.3

System (5) is \(D(r,\theta )\)-stable, if \({\sigma (L_{F,G})}\subset {D(r,\theta )}\subset {D(0,1)}\).

Definition 3.4

System (5) is \(D(0,\alpha ,\beta )\)-stable, if \({\sigma (L_{F,G})}\subset {D(0,\alpha ,\beta )}\subset {D(0,1)}\).

Now we give some equivalent conditions of these region stabilities.

Theorem 3.1

The time-delay system (5) is\(D_{R}\)-stable if and only if there exists a positive matrix\(P>0\)such that
$$\begin{aligned}& R_{1}\otimes P+R_{2}\otimes \bigl(P\varTheta \bigl(H(n,m),F,G\bigr)\bigr)+R_{2}'\otimes \bigl( \varTheta \bigl(H(n,m),F,G\bigr)'P\bigr) \\& \quad {}+R_{3}\otimes (\varTheta \bigl(H(n,m),F,G \bigr)'P\bigl(\varTheta \bigl(H(n,m),F,G\bigr)\bigr)< 0, \end{aligned}$$
where\(\varTheta (H(n,m),F,G) =[H(n,m)'H(n,m)]^{-1}H(n,m)'[{F}\otimes {F}+{G}\otimes {G}]H(n,m)\).


Let \(X(t)=E[\overline{x}(t)\overline{x}'(t)]\). Then system (5) can be rewritten as a stochastic system
$$\begin{aligned}& X(t+1)=FX(t)F'+GX(t)G'. \end{aligned}$$
With the knowledge of Kronecker matrix product, system (8) becomes
$$\begin{aligned}& \overrightarrow{X}(t+1)=\widehat{F}\overrightarrow{X}(t), \end{aligned}$$
where \(\overrightarrow{X}(t)=[X_{1,1}(t),\ldots,X_{1,n}(t),\ldots,X _{1,n(m+1)}(t),\ldots,X_{n(m+1),n(m+1)}(t)]'\in \mathbb{R}^{n^{2}(m+1)^{2}}\) and \(\widehat{F}=F\otimes F+G\otimes G\). By the theory of H representation method in [36], there exists a full column rank matrix \(H(n,m)\in \mathbb{R}^{n^{2}(m+1)^{2}\times \frac{n(m+1)[n(m+1)+1]}{2}}\) such that \(\overrightarrow{X}=H(n,m) \widetilde{X}\), \(\widetilde{X}\in \mathbb{R}^{ \frac{n(m+1)[n(m+1)+1]}{2}}\), so system (9) is equivalent to the following discrete system:
$$ \widetilde{X}(t+1)=\varTheta \bigl(H(n,m),F,G\bigr)\widetilde{X}(t), $$
where \(\varTheta (H(n,m),F,G) =[H(n,m)'H(n,m)]^{-1}H(n,m)'[{F}\otimes {F}+{G}\otimes {G}]H(n,m)\). By Theorem 1 in [33], we immediately obtain the result. □

From Eqs. (8) and (9) in Theorem 3.1 and Theorem 1 in [27], the relation between the spectrum of operator \(L_{F,G}\) and matrix \(\varTheta (H(n,m),F,G)\) is obtained as follows.

Corollary 3.1

\(\sigma (L_{F,G})=\sigma (\varTheta (H(n,m),F,G))\).

Remark 3.1

From Theorem 3.1, we know that the time-delay system (5) is \(D_{R}\)-stable if and only if a certain LMI is feasible.

Similar to Theorem 3.1, it is easy to draw some results as follows.

Theorem 3.2

System (5) is\(D(q,r)\)-stable if and only if there exists a positive matrix\(P>0\)such that the following inequality holds:
( r q q r ) P + ( 0 1 0 0 ) ( P Θ ( H ( n , m ) , F , G ) ) + ( 0 1 0 0 ) ( Θ ( H ( n , m ) , F , G ) P ) < 0 . Open image in new window

Theorem 3.3

System (5) is\(D(r,\theta )\)-stable if and only if there exists a positive matrix\(P>0\)satisfying the inequality in Theorem 3.2and the following inequality:
( M N N M ) < 0 , Open image in new window
$$\begin{aligned}& M=\sin \theta \bigl(\varTheta \bigl(H(n,m),F,G\bigr)P+P\varTheta \bigl(H(n,m),F,G\bigr)'\bigr), \\& N=\cos \theta \bigl(\varTheta \bigl(H(n,m),F,G\bigr)P-P\varTheta \bigl(H(n,m),F,G\bigr)'\bigr). \end{aligned}$$

Theorem 3.4

System (5) is\(D(0,\alpha ,\beta )\)-stable if and only if there exists a positive matrix\(P>0\)satisfying
( β 2 0 0 α 2 ) P + ( 1 0 0 1 ) ( Θ ( H ( n , m ) , F , G ) ) P Θ ( H ( n , m ) , F , G ) < 0 . Open image in new window

Using MATLAB, it is quick to check the above criterion of \(D_{R}\)-stability with LMI. Now we give an example as follows.

Example 3.1

In the time-delay system \([\sum_{j=0}^{m}F _{j},\sum_{j=0}^{m}G_{j}]\), let
$$ F_{0}=-\frac{1}{4}, \quad\quad F_{1}= \frac{1}{8}, \quad\quad G_{0}=\frac{1}{16}, \quad \quad G_{1}=\frac{1}{32}. $$
F = ( 1 4 1 8 1 0 ) , G = ( 1 16 1 32 0 0 ) . Open image in new window
Through a simple calculation, we can obtain that
H ( 1 , 1 ) = ( 1 0 0 0 1 0 0 1 0 0 0 1 ) , Θ ( H ( 1 , 1 ) , F , G ) = ( 0.0664 0.0664 0.0166 0.2500 0.1250 0 1.0000 0 0 ) . Open image in new window
Given \(q=\frac{1}{4}\), \(r=\frac{1}{6}\), using MATLAB, we cannot find a feasible solution of the inequality in Theorem 3.2. That is, system (5) is not \(D(\frac{1}{4},\frac{1}{6})\)-stable.
Choosing \(q=\frac{1}{4}\), \(r=\frac{1}{2}\) and using MATLAB, we can obtain that there exists a positive region matrix
P = ( 29.7165 0.8568 0.0985 0.8568 16.4863 2.7158 0.0985 2.7158 3.2825 ) > 0 Open image in new window
such that
( 1 6 P 1 4 P + P Θ ( H ( 1 , 1 ) , F , G ) 1 4 P + Θ ( H ( 1 , 1 ) , F , G ) P 1 6 P ) < 0 . Open image in new window
By Theorem 3.2, the system is \(D(\frac{1}{4},\frac{1}{2})\)-stable.

We use the LMI toolbox of Matlab and commands for obtaining P as follows:

H=[1 0 0;0 1 0;0 1 0;0 0 1];

F=[-1/4 1/8;1 0];

G=[-1/16 1/32;0 0];




E=[1 0 0;0 1 0;0 0 1];







lmiterm([1 1 1 P],B1,1)

lmiterm([1 1 2 P],1,A2)

lmiterm([1 2 2 P],B1,1)

lmiterm([-2 1 1 P],1,1)





Remark 3.2

Theorems 3.13.4 give a kind of characterization for region stability. But the relation between region stabilities and the convergence speed of the trajectories of a system need to be further investigated, which is an interesting and practical problem.

Theorem 3.5

System (5) is\(D(0,\alpha ,\beta )\)-stable if and only if system (1) is such for arbitrary sufficiently small real number\(\varepsilon >0\), the convergence speed of the system\([\sum_{j=0}^{m}F_{j},\sum_{j=0}^{m}G_{j}]\)is slower than\(O((\alpha +\varepsilon )^{t})\), but is faster than\(O((\beta -\varepsilon )^{t})\). That is, there exist two constants\(C_{1}>0\), \(C_{2}>0\)such that
$$ C_{2}{ \Vert \overline{x}_{0} \Vert }^{2}(\alpha +\varepsilon )^{t} \leq E{ \bigl\Vert x(t) \bigr\Vert }^{2}\leq {C_{1}{ \Vert \overline{x}_{0} \Vert }^{2} (\beta -\varepsilon )^{t}}. $$


(Necessity) Assume that system (5) is \(D(0,\alpha ,\beta )\)-stable. For \({\lambda _{i}}\in {\sigma (L_{F,G})}\), \(i=1,2,\ldots, \frac{n(m+1)[n(m+1)+1]}{2}\), without loss of generality, let
$$ \max \vert \lambda _{i} \vert = \vert \lambda _{1} \vert < \beta , \quad\quad \min \vert \lambda _{i} \vert = \vert \lambda _{2} \vert >\alpha . $$
And let \(X(t)=E[\overline{x}(t)\overline{x}'(t)]\), where \(\overline{x}(t)=[x'(t),x'(t-1),\ldots,x'(t-m)]'\); \(x(t)\) is the orbit of system (5) satisfying the initial condition \(x(k)=\varphi (k)\), \(k=0,-1,\ldots,-m\) (i.e., \(\overline{x}_{0}=[\varphi '(0),\varphi '(-1),\ldots,\varphi '(-m)]'\)). Using the same proof process as that of Theorem 2.1 in [27], system (5) is equivalent to
$$\begin{aligned}& X(t+1)=L_{F,G}\bigl(X(t)\bigr). \end{aligned}$$
By the theory of H representation method in [36], the initial problem of system (10) is equivalent to the initial problem of the following discrete system:
$$\begin{aligned}& \textstyle\begin{cases} \widetilde{X}(t+1)=\varTheta (H(n,m),F,G)\widetilde{X}(t), \\ \widetilde{X}(0)=[H(n,m)'H(n,m)]^{-1}H(n,m)'\overrightarrow{X}(0). \end{cases}\displaystyle \end{aligned}$$
By the stability theory of linear deterministic discrete system, there exist two positive real constants \(c_{0}\) and \(c^{0}\) such that the following inequality holds:
$$\begin{aligned}& { \bigl\Vert \widetilde{X}(t) \bigr\Vert }\leq {c_{0}{ \bigl\Vert \widetilde{X}(0) \bigr\Vert }\Biggl(\sum_{i=1} ^{r_{1}}t^{i-1}\Biggr) \vert \lambda _{1} \vert ^{t}}, \\& { \bigl\Vert \widetilde{X}(t) \bigr\Vert }\geq {c^{0}{ \bigl\Vert \widetilde{X}(0) \bigr\Vert }\Biggl(\sum_{i=1} ^{r_{2}}t^{i-1}\Biggr) \vert \lambda _{2} \vert ^{t}}, \end{aligned}$$
here \(r_{1}\) and \(r_{2}\) are respectively the multiplicities of the characteristic roots \(\lambda _{1}\) and \(\lambda _{2}\). For \(\vert \lambda _{1} \vert <\beta \), \(\vert \lambda _{2} \vert >\alpha \), by the denseness of real numbers, there exists a sufficiently small number \(\varepsilon >0\), such that \(\vert \lambda _{1} \vert -\beta +\varepsilon <0\), \(\vert \lambda _{2} \vert -\alpha -\varepsilon >0\), and so, by the results in [29], we can obtain the following inequality:
$$ E{ \Vert \overline{x} \Vert }^{2}\leq {\sqrt{n(m+1)} { \Vert \widetilde{X} \Vert }}, \quad \quad { \Vert \widetilde{X} \Vert }\leq {\sqrt{ \frac{n(m+1)+1}{2}}E{ \Vert \overline{x} \Vert }^{2}}. $$
$$ E{ \bigl\Vert \overline{x}(t) \bigr\Vert }^{2}\leq {c_{1}{ \Vert \overline{x}_{0} \Vert }^{2}( \beta -\varepsilon )^{t}}, \qquad E{ \bigl\Vert \overline{x}(t) \bigr\Vert }^{2}\geq {c_{2}{ \Vert \overline{x}_{0} \Vert }^{2}( \alpha +\varepsilon )^{t}}, $$
$$\begin{aligned}& c_{1}=c_{0}\sqrt{\frac{n(m+1)[n(m+1)+1]}{2}}\sup _{t\in [0,\infty )}\Biggl( \sum_{i=1}^{r_{1}}{t^{i-1}} \Biggr) \biggl(\frac{ \vert \lambda _{1} \vert }{\beta -\varepsilon }\biggr)^{t}< \infty , \\& c_{2}=c^{0}\sqrt{\frac{2}{n(m+1)[n(m+1)+1]}}. \end{aligned}$$
Hence, from the above inequality, we can obtain that there exist two constants \(C_{1}>0\), \(C_{2}>0\) such that
$$ E{ \bigl\Vert x(t) \bigr\Vert }^{2}\leq {C_{1}{ \Vert \overline{x}_{0} \Vert }^{2}{(\beta - \varepsilon )^{t}}}, \qquad E{ \bigl\Vert x(t) \bigr\Vert }^{2}\geq {C_{2}{ \Vert \overline{x}_{0} \Vert }^{2}(\alpha +\varepsilon )^{t}}. $$
(Sufficiency) Suppose that system (5) is \((\alpha ,\beta )\) region stable. For arbitrary real number \(\varepsilon >0\), the convergence speed of the system \([\sum_{j=0}^{m}F_{j},\sum_{j=0}^{m}G _{j}]\) is slower than \(O((\alpha +\varepsilon )^{t})\), but is faster than \(O((\beta -\varepsilon )^{t})\). That is, there exist two constants \(C_{1}>0\), \(C_{2}>0\) such that
$$ C_{2}{ \Vert \overline{x}_{0} \Vert }^{2}(\alpha +\varepsilon )^{t} \leq E{ \bigl\Vert x(t) \bigr\Vert }^{2}\leq {C_{1}{ \Vert \overline{x}_{0} \Vert }^{2} (\beta -\varepsilon )^{t}}. $$
Note that \(\overline{x}(t)=[x'(t),x'(t-1),\ldots,x'(t-m)]'\), and it is easy to obtain that there exist two numbers \(c_{1}\), \(c_{2}\) satisfying
$$\begin{aligned}& c_{2}{ \Vert \overline{x}_{0} \Vert }^{2}(\alpha +\varepsilon )^{t} \leq E{ \bigl\Vert \overline{x}(t) \bigr\Vert }^{2}\leq {c_{1}{ \Vert \overline{x}_{0} \Vert }^{2} (\beta - \varepsilon )^{t}}. \end{aligned}$$
Note that \(\overline{x}(t)\) is the trajectory of system (6), so from (12), we can obtain \(\sigma (L_{F,G})\subset {D(0,\alpha ,\beta )} \subset D(0,1)\). The proof is completed. □

Corollary 3.2

If system (5) is\(D_{R}\)-stable then there exists a constant\(C>0\)such that\(\lim_{t \rightarrow +\infty }E { \Vert x(t) \Vert }^{2}=0\).

Corollary 3.3

If system (5) is\(D(q,r)\)-stable then there exist two constants\(C_{1}>0\), \(C_{2}>0\)such that\(C_{2}{ \Vert \overline{x} _{0} \Vert }^{2}( \Vert q \Vert +\varepsilon )^{t} \leq E{ \Vert x(t) \Vert }^{2}\leq {C_{1} { \Vert \overline{x}_{0} \Vert }^{2} (( \Vert q \Vert +r)-\varepsilon )^{t}}\).

Corollary 3.4

If system (5) is\(D(r,\theta )\)-stable then there exists a constant\(C>0\)such that\(E{ \Vert x(t) \Vert }^{2}\leq {C{ \Vert \overline{x}_{0} \Vert }^{2} (r-\varepsilon )^{t}}\).

Remark 3.3

From Theorem 3.5 and Corollaries 3.23.4, region stabilities defined in this paper imply mean square stability of system (5).

4 Conclusion

In this paper, we investigated the region stability of discrete stochastic time-delay systems. First of all, we defined the notions of region-stability of discrete stochastic time-delay systems. We then gave some equivalent conditions for the region stabilization of linear stochastic discrete time-delay systems. We only considered the stability of discrete stochastic time-delay systems on some LMI regions. But the stability of discrete stochastic time-delay systems on general regions is still a challenging problem.



The authors wish to thank editor and anonymous reviewer for his/her valuable suggestions to this paper.

Availability of data and materials

We don’t apply new software, databases and relevant raw data.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final paper.


This work was supported by the National Natural Science Foundation of China under Grants 61503224, National Science Foundation of Shandong Province ZR2017MF054, Qingdao Postdoctoral Applied Research Project No. 2015188 and SDUST Research Fund No. 2015TDJH105.

Competing interests

All the authors declare that there is no conflict of interest regarding the publication of this paper.


  1. 1.
    Mao, X.: Stochastic Differential Equations and Their Applications. Horwood, Chichester (1997) zbMATHGoogle Scholar
  2. 2.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955) zbMATHGoogle Scholar
  3. 3.
    Has’minskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen (1980) CrossRefGoogle Scholar
  4. 4.
    Kushner, H.J.: Stochastic Stability and Control. Academic Press, New York (1967) zbMATHGoogle Scholar
  5. 5.
    Chen, M., Xia, D., Wang, D., Han, J., Liu, Z.: An analytical method for reducing metal artifacts in X-ray CT images. Math. Probl. Eng. 2019, Article ID 2351878 (2019) MathSciNetGoogle Scholar
  6. 6.
    Li, G., Chen, M.: Intertwined phenomenon of a kind of dynamical system. Adv. Differ. Equ. 2013(1), Article ID 265 (2013) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, G., Ding, C., Chen, M.: Intertwined basins of attraction of dynamical systems. Appl. Math. Comput. 213(1), 272–274 (2009) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Zhang, T., Meng, X., Zhang, T.: Global analysis for a delayed SIV model with direct and environmental transmissions. J. Appl. Anal. Comput. 6(2), 479–491 (2016) MathSciNetGoogle Scholar
  9. 9.
    Baleanu, D., Wu, G., Bai, Y., Chen, F.: Stability analysis of Caputo-like discrete fractional systems. Commun. Nonlinear Sci. Numer. Simul. 48, 520–530 (2017) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wu, G., Baleanu, D., Zeng, S.: Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion. Commun. Nonlinear Sci. Numer. Simul. 57, 299–308 (2018) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wu, G., Baleanu, D., Zeng, S.: Several fractional differences and their applications to discrete maps. J. Appl. Nonlinear Dyn. 4, 339–348 (2015) CrossRefGoogle Scholar
  12. 12.
    Li, G., Gao, Y., Chen, M.: Attractors of dynamical systems in locally compact spaces. Open Math. 17, 465–471 (2019) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li, G., Chen, M.: On uniqueness of strong solution of stochastic systems. Abstr. Appl. Anal. 2014, Article ID 890925 (2014) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gao, M., Sheng, L., Zhang, W.: Stochastic \(H_{2}/H_{\infty }\) control of nonlinear systems with time-delay and state-dependent noise. Appl. Math. Comput. 266, 429–440 (2015) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Liu, X., Li, Y., Zhang, W.: Stochastic linear quadratic optimal control with constraint for discrete-time systems. Appl. Math. Comput. 228, 264–270 (2014) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Zhang, W., Zhang, H., Chen, B.S.: Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion. IEEE Trans. Autom. Control 53, 1630–1642 (2008) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Li, G., Zhang, W., Chen, M.: Robust \(H_{2}/H_{\infty }\) control for periodic stochastic difference systems with multiplicative noise. IET Control Theory Appl. 9(16), 2451–2457 (2015) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Yan, Z., Zhang, G., Wang, J., Zhang, W.: State and output feedback finite-time guaranteed cost control of linear ito stochastic systems. J. Syst. Sci. Complex. 28, 813–829 (2015) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) CrossRefGoogle Scholar
  20. 20.
    Ait Rami, M., Zhou, X.Y.: Linear matrix inequalities, Riccati equations and indefinite stochastic linear quadratic control. IEEE Trans. Autom. Control 45, 1131–1142 (2000) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Boyd, S., Ghaoui, E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) CrossRefGoogle Scholar
  22. 22.
    Mobayen, S., Baleanu, D.: Linear matrix inequalities design approach for robust stabilization of uncertain nonlinear systems with perturbation based on optimally-tuned global sliding mode control. J. Vib. Control 23, 1285–1295 (2017) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Faieghi, M.R., Kuntanapreeda, S., Delavari, H., Baleanu, D.: Robust stabilization of fractional-order chaotic systems with linear controllers: LMI-based sufficient conditions. J. Vib. Control 20, 1042–1051 (2014) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mobayen, S., Baleanu, D., Tchier, F.: Second-order fast terminal sliding mode control design based on LMI for a class of non-linear uncertain systems and its application to chaotic systems. J. Vib. Control 23, 2912–2925 (2016) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Li, G., Chen, M.: Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems. Adv. Differ. Equ. 2015, Article ID 14 (2015) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhang, W., Chen, B.S.: On stabilizability and exact observability of stochastic systems with their applications. Automatica 40, 87–94 (2004) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Li, G., Chen, M.: The stability and stabilization of stochastic delay-time systems. Math. Probl. Eng. 2014, Article ID 272745 (2014) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Hou, T., Ma, H., Zhang, W.: Spectral tests for observability and detectability of periodic Markov jump systems with nonhomogeneous Markov chain. Automatica 63, 175–181 (2016) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zhang, W., Xie, L.: Interval stability and stabilization of linear stochastic systems. IEEE Trans. Autom. Control 54, 810–815 (2009) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhang, W.: General D-stability and D-stabilization for linear stochastic systems: continuous-time case. In: 2010 8th IEEE International Conference on Control and Automation (2010) Google Scholar
  31. 31.
    Fragoso, M.D., Costa, O.L.V., de Souza, C.E.: A new approach to linearly perturbed Riccati equations arising in stochastic control. Appl. Math. Optim. 37, 99–126 (1998) MathSciNetCrossRefGoogle Scholar
  32. 32.
    El Harraki, I., El Alami, A., Boutoulout, A., Serhani, M.: Regional stabilization of semi-linear parabolic systems. IMA J. Math. Control Inf. 34, 961–971 (2017) MathSciNetzbMATHGoogle Scholar
  33. 33.
    Peaucelle, D., Arzelier, D., Bachelier, O., Bernussou, J.R.K.: A new robust D-stability condition for real convex polytypic uncertainty. Syst. Control Lett. 40, 21–30 (2000) CrossRefGoogle Scholar
  34. 34.
    Ortega, J.M.: Matrix Theory. Plenum, New York (1987) CrossRefGoogle Scholar
  35. 35.
    Chilali, M., Gahinet, P.: \(H_{\infty }\) design with pole placement constraints: an LMI approach. IEEE Trans. Autom. Control 41, 358–367 (1996) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zhang, W., Chen, B.S.: H-representation and applications to generalized Lyapunov equations and linear stochastic systems. IEEE Trans. Autom. Control 57, 3009–3022 (2012) MathSciNetCrossRefGoogle Scholar

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoP.R. China

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