# New Exponential Stability Results for Neutral Stochastic Systems with Distributed Delays

Authors: Jian Wang
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper studies the stability problem of a grey neutral stochastic system with distributed delays. The main feature of a grey system is that the parameters of system are evaluated by grey numbers, and system has the strong background in engineering applications. However, the literature dealing with the stability problem for grey system seems to be scarce. The main aim of this paper is to fill the gap. Based on the Lyapunov stability theorem and Itô’s formula, especially, using the decomposition technique of the continuous matrix-covered sets of grey matrix, we propose several novel sufficient conditions, which ensure our considered grey system in mean-square exponential stability and almost surely exponential stability. Furthermore, two examples are provided to show the effectiveness of the obtained results.


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## Body

## 1. Introduction

In various physical and engineering systems, time-delay frequently causes oscillation, bifurcation, or instability. Meanwhile, the influence of stochastic phenomenon also cannot be ignored. So, time-delay stochastic system has gained many scholars’ attention and interest (see [1–9]). As pointed out in [10, 11], if the differences between nearby argument values are reduced, but the number of summands is incremental, we will find distributed delays phenomenon. In addition, we can describe many practical systems by neutral differential equations, Hence, neutral stochastic system with distributed delays has been extensively investigated by many scholars (see [12–20]). For instance, in [12], stability analysis of neutral type neural networks has been studied. Authors have proposed several delay-dependent criteria to ensure the networks to have a balance point. In [13], by the Lyapunov-Krasovkii functional, authors have studied the global exponential stability of neutral type neural networks with distributed time delays, and authors have presented new sufficient conditions of the system.It is well known that we cannot obtain parameters of systems accurately. Naturally, in order to reflect many real systems, we have to evaluate the parameters of systems. When we evaluate the parameters of system by grey numbers, the system will be described uncertainly and turn into grey system (see [21]). Recently, grey system has come to play the important role in many fields and can be widely applied to physical and engineering systems, such as population ecology, communication systems, electric power systems, and so on. Due to the strong background in engineering applications, the stability analysis for grey system has become a focused topic of theoretical as well as practical importance. But, little attention has been taken to the stability problem for grey system except some paper [21–25]. In [21], the problem of exponential stability for a grey stochastic system with distributed delays has been studied, and the authors have proposed the delay-dependent criteria, which ensure the system inp-moment exponential robust stability. In [22], the authors have investigated robust stability problem for a grey stochastic nonlinear system with distributed delays, and several exponential stability criteria have been derived in terms of LMIs.Motivated by the above discussion, using a similar method of [24], we will address the stability problem of grey system here. The main work and contribution of this paper are given as follows. (1) Compared with [24], this paper studies exponential stability problem of grey nonlinear systems with distributed delays for the first time, and the obtained results are more effective and general. (2) Different from traditional methods [10, 14, 20], for the uncertain systems, combining with concepts and conclusions of the Grey Theory (see Definitions 1 and 2 and Lemma 3), we adopt the new decomposition technique of the continuous matrix-covered sets of grey matrix. Then, several sufficient conditions are derived to ensure the systems in mean-square exponential stability and almost surely exponential stability. (3) Since the obtained criteria are presented LMIs, it can be easily solved by the Matlab LMI Control Toolbox. Meanwhile, two simple examples to illustrate the effectiveness of the obtained criteria are provided and solved.The notation in this paper is fairly standard. The superscript “T” represents the transpose and · denotes the Euclidean norm. The notation X>0, where X is real symmetric matrix, means that X is positive definite. Let Ω,F,Ftt≥0,P be a probability space with a filtration Ftt≥0 and C[-τ,0];Rn be the family of all continuous Rn-valued functions φ on [-τ,0] and LF02([-τ,0];Rn) be the family of all F0-measurable bounded C[-τ,0];Rn-valued random variables ξ=ξ(θ):-τ≤θ≤0.


## 2. Preliminaries and Problem Formulation

In this section, we consider the following grey system:(1)dxt-HIxt-τ=AIxt+BIxt-τ+∫0τGsxt-sdsdt+fxt,xt-τ,tdwt,t≥0x0=ξ,ξ∈LF02-τ,0;Rn,-τ≤t≤0,where AI, BI, and HI are interval matrices, and (2)AI=La,Ua=A=aij:aij_≤aij≤aij¯,i,j=1,2,…,n,BI=Lb,Ub=B=bij:bij_≤bij≤bij¯,i,j=1,2,…,n,HI=Lh,Uh=H=hij:hij_≤hij≤hij¯,i,j=1,2,…,nare called the continuous matrix-covered sets of grey matrices A, B, and H.For grey system (1), we make the following assumptions and definitions:(H1)
f:Rn×Rn×R+→Rn×n is locally Lipschitz continuous and satisfies the linear growth condition.(H2)
There are constantsα≥0, β≥0, for arbitrary (x,y,t)∈f:Rn×Rn×R+; we have(3)TracefTx,y,tfx,y,t≤αx2+βy2.Definition 1 (see [21]).
Grey system (1) is mean square exponentially stable, if for all ξ∈LF02([-τ,0];Rn) and arbitrary A∈AI, B∈BI, H∈HI, there exist constants r>0 and C>0, such that (4)Ext;ξ2≤Ce-rtsup-τ≤θ≤0⁡Eξθ2,t≥0.Definition 2 (see [21]).
Grey system (1) is almost surely exponentially robustly stable, if for all ξ∈LF02([-τ,0];Rn) and arbitrary A∈AI, B∈BI, H∈HI, we have (5)limt→∞sup1tlnxt;ξ≤-r^2,a.s.To prove our main results, the following lemmas are very important (in particular, Lemma 3).Lemma 3 (see [21]).
IfA=(aij)n×n is a grey matrix, [aij_,aij¯] is a number-covered sets of aij; then for arbitrary matrix A∈AI, we have (1)
A=Ua+La/2+ΔA,(2)
0≤ΔA≤Ua-La/2,(3)
A≤Ua+La/2+Ua-La/2,where La=(aij_)n×n, Ua=(aij¯)n×n, ΔA=(aij¯-aij_/2rij)n×n, and rij∈[-1,1].Lemma 4 (see [26]).
For any real vectorsx,y and any matrix P>0 with appropriate dimensions, M,N∈Rn×n, ε>0, we have(6)2xTMTPNy≤εxTMTPMx+ε-1yTNTPNy.Lemma 5 (see [27]).
For a given matrixS=S11S12S12TS22 with S11=S11T, S22=S22T, the following conditions are equivalent:(1)
S<0,(2)
S22<0, S11-S12S22-1S12T<0.


## 3. Main Results and Proofs

Several novel stability conditions are proposed to guarantee grey system (1) in the mean-square exponential stability and almost surely exponential stability. The main results are given in the following theorems and corollaries.Theorem 6.
System (1) is mean square exponentially stable, if k=Uh+Lh/2+Uh-Lh/2<1 and there exist symmetric matrices P>0, Q>0, R>0 and scalars ε1>0, ε2>0, ε3>0, ε4>0, such that(7)λminR≥ε2-1+ε4-1τsup0≤s≤τGs2,(8)Δ1PUb+Lb2MUbT+LbT2PΔ20MT0-J<0,where (9)Δ1=Q+τR+PUa+La2+UaT+LaT2P+λmaxP×2Ua-La2+ε2+αIn+ε3-1Ua+La2+Ua-La22In+ε1P2,Δ2=-Q+ε1-1Ub-Lb22+ε3Uh+Lh2+Uh-Lh22+ε4Uh+Lh2+Uh-Lh22+2Uh+Lh2·Ub+Lb2+Uh+Lh2·Ub-Lb2+Uh-Lh2·Ub+Lb2+Uh-Lh2·Ub-Lb2+λmaxP×βIn,M=PPPP,J=diag⁡ε1Inε2Inε3Inε4In.Then, for all ξ∈LF02([-τ,0];Rn), we have(10)Ext,ξ2≤C01-k1-kerτe-rtsup-τ≤θ≤0Eξθ2;here, (11)C0=λmaxP1+k22+rτerτ+rτ2erτ+τλmaxQ+rτ3erτ+τ2λmaxR,and r is the unique positive solution of the following equation(12)rλmaxP1+k21+erτ+λmaxQ+τλmaxRrτerτ+λmaxΨ=0,(13)withΨ≔Δ1+∑i=14εi-1P2PUb+Lb2UbT+LbT2PΔ2<0,and 1+rτ>0, 1-kerτ>0.Proof.
Using Lemma5, (14)Δ1PUb+Lb2MUbT+LbT2PΔ20MT0-J<0implies that (15)ψ≔Δ1+∑i=14εi-1P2PUb+Lb2UbT+LbT2PΔ2<0.For convenience, let (16)xt,ξ=xt,yt=xt-τ,zt=∫0τGsxt-sds,ηt=xt-HIxt-τ.Firstly, arbitrarily fix ξ∈LF02([-τ,0];Rn), and A∈AI, B∈BI, H∈HI; we introduce the following Lyapunov-Krasovskii functional:(17)Vxt,t=ηTtPηt+∫t-τtxTsQxsds+∫-τ0∫t+θtxTsRxsdsdθ.Using Itô’s differential formula, we have(18)LVxt,t=xTtQ+τRxt-yTtQyt-∫t-τtxTsRxsds+2xTtPAxt+2xTtPByt+2xTtPzt-2yTtHTAxt-2yTtHTByt-2yTtHTzt+TracefTxt,yt,tPfxt,yt,t.By applying Lemmas 3 and 4, it follows that (19)2xTtPAxt≤xTtPUa+La2+UaT+LaT2Pxt+λmaxP×2Ua-La2xTtxt,(20)2xTtPByt≤xTtPUb+Lb2yt+yTtUbT+LbT2Pxt+ε1xTtP2xt+ε1-1Ub-Lb22yTtyt,(21)2xTtPzt≤λmaxP×ε2xTtxt+ε2-1zTtzt,(22)-2yTtHTAxt≤ε3yTtHTHyt+ε3-1xTtATAxt≤ε3Uh+Lh2+Uh-Lh22yTtyt+ε3-1Ua+La2+Ua-La22xTtxt,(23)-2yTtHTByt≤2yTtLhTLbyt+2yTtLhTΔByt+2yTtΔTHLbyt+2yTtΔTHΔByt≤2Uh+Lh2·Ub+Lb2+Uh+Lh2·Ub-Lb2+Uh-Lh2·Ub+Lb2+Uh-Lh2·Ub-Lb2yTtyt,(24)-2yTtHTzt≤ε4yTtHTHyt+ε4-1zTtzt≤ε4Uh+Lh2+Uh-Lh22yTtyt+ε4-1zTtzt,(25)zTtzt≤τ∫0τGsxt-s2ds≤τsup0≤s≤τGs2∫t-τtxTsxsds.From assumptions (H2), we can get(26)TracefTxt,yt,tPfxt,yt,t≤λmaxPαxTtxt+βyTtyt.(27)Obviously,-∫t-τtxTsRxsds≤-λminR∫t-τtxTsxsds.Then, combining (18)–(27) and noting (7), we have(28)LVxt,t≤xTt,yTtΨxtyt≤λmaxΨxt2+yt2≤λmaxΨxt2,where Ψ is defined in (13).
Now, letΦ=ertV(x(t),t), and by (28), we can get (29)LΦxt,t≤rertλmaxP1+k2xt2+yt2+λmaxQ∫t-τtxs2ds+τλmaxR∫t-τtxs2ds+ertλmaxΨxt2.Thus, it can be seen that(30)∫0tELΦxs,sds≤rλmaxP1+k21+erτ+rτerτλmaxQ+τλmaxR+λmaxΨ∫0tersExs2ds+rτλmaxP1+k2erτ+rτ2erτλmaxQ+τλmaxRsup-τ≤θ≤0Eξθ2.Set f(r)=rλmax(P)(1+k2)(1+erτ)+rτerτ[λmax(Q)+τλmax(R)]+λmax(Ψ).
We havef′(r)=λmax(P)(1+k2)+τλmax(Q)+τ2λmax(R)(1+rτ)erτ.
Sincef′(r)>0, f(0)=λmax(Ψ)<0 and f(+∞)=+∞, then (12) has a unique solution r>0.
By (30) and if (12) holds, we can obtain (31)∫0tELΦxs,sds≤rτλmaxP1+k2erτ+rτ2erτλmaxQ+τλmaxRsup-τ≤θ≤0Eξθ2.On the other hand, it is easy to see(32)EΦx0,0≤λmaxP2+2k2+τλmaxQ+τ2λmaxRsup-τ≤θ≤0Eξθ2.Then, it follows from (31) and (32) that(33)EΦxt,t=EΦx0,0+∫0tELΦxs,sds≤λmaxP1+k22+rτerτ+rτ2erτ+τλmaxQsup-τ≤θ≤0Eξθ2+rτ3erτ+τ2λmaxRsup-τ≤θ≤0Eξθ2.Note that (34)EertVxt,t≥ert1-kExt2-1k-1k2Eyt2,H≤k=Uh+Lh2+Uh-Lh2<1.Then, we can derive(35)ertExt,t2≤11-kEertVxt,t+kertsup-τ≤θ≤0Ext+θ2.By (33), and note that definition of C0, it is easy to obtain(36)ertExt,t2≤C01-ksup-τ≤θ≤0Eξθ2+kerτsup-τ≤t≤TertExt2.So, we also have(37)sup-τ≤t≤TertExt,t2≤C01-ksup-τ≤θ≤0Eξθ2+kerτsup-τ≤t≤TertExt2.By (37) and noting 1-kerτ>0, we can get(38)sup-τ≤t≤TertExt,t2≤C01-k1-kerτsup-τ≤θ≤0Eξθ.That is, (10) holds. This completes the proof of Theorem 6.Corollary 7.
System (1) is mean square exponentially stable, if k=Uh+Lh/2+Uh-Lh/2<1 and there exist symmetric matrices Q>0 and scalars m>0, ε1>0, ε2>0, ε3>0, ε4>0, such that (39)Σ1mUb+Lb2NmUbT+LbT2Σ20NT0-J<0,where (40)Σ1=Q+τ2ε2-1+ε4-1sup0≤s≤τGs2In+mUa+La2+UaT+LaT2+2Ua-La2+ε2+αmIn+ε3-1Ua+La2+Ua-La22In+ε1m2In,Σ2=-Q+ε1-1Ub-Lb22+ε3Ug+Lg2+Ug-Lg22+ε4Uh+Lh2+Uh-Lh22+2Uh+Lh2·Ub+Lb2+Uh+Lh2·Ub-Lb2+Uh-Lh2·Ub+Lb2+Uh-Lh2·Ub-Lb2+mβIn,N=mInmInmInmIn,J=diag⁡ε1Inε2Inε3Inε4In.Then, for all ξ∈LF02([-τ,0];Rn), we have(41)Ext,ξ2≤C11-k1-kerτe-rtsup-τ≤θ≤0Eξθ2;here, (42)C1=m1+k22+rτerτ+rτ2erτ+τλmaxQ+rτ4erτ+τ3ε2-1+ε4-1sup0≤s≤τGs2,and r is the unique positive solution of the following equation(43)rm1+k21+erτ+rτerτλmaxQ+τ2ε2-1+ε4-1sup0≤s≤τGs2+λmaxΦ=0,withΦ≔Σ1+∑i=14εi-1m2InmUb+Lb2mUbT+LbT2Σ2<0,and 1+rτ>0, 1-kerτ>0.Proof.
LetP=mIn, R=[(ε2-1+ε4-1)τsup0≤s≤τ⁡G(s)2]In; similar to the method used in Theorem 6, the proof of Corollary 7 is completed.Remark 8.
If the interval matricesAI, BI, and HI are replaced by deterministic matrices A^, B^, and H^, then grey system (1) will become the deterministic stochastic system: (44)dxt-H^xt-τ=A^xt+B^xt-τ+∫0τGsxt-sdsdt+fxt,xt-τ,tdwt,t≥0,x0=ξ,ξ∈LF02-τ,0;Rn,-τ≤t≤0.Let La=Ua=A^, Lb=Ub=B^, and Lh=Uh=H^, and similar to the method used in Corollary 7, the criterion of the deterministic system (44) can be obtained.Corollary 9.
System (44) is mean square exponentially stable, if p=H<1 and there exist symmetric matrices Q>0 and scalars m>0, ε1>0, ε2>0, ε3>0, ε4>0, such that (45)Ω1mB^NmB^TΩ20NT0-J<0,where (46)Ω1=Q+τ2ε2-1+ε4-1sup0≤s≤τGs2In+mA^+A^T+ε2+αmIn+ε3-1A^2In+ε1m2In,Ω2=-Q+ε3H^2+ε4H^2+2H^·B^+mβIn,N=mInmInmInmIn,J=diag⁡ε1Inε2Inε3Inε4In.Then, for all ξ∈LF02([-τ,0];Rn), we have(47)Ext,ξ2≤C21-p1-perτe-rtsup-τ≤θ≤0Eξθ2;here, (48)C2=m1+p22+rτerτ+rτ2erτ+τλmaxQ+rτ4erτ+τ3ε2-1+ε4-1sup0≤s≤τGs2,and r is the unique positive solution of the following equation(49)rm1+p21+erτ+rτerτλmaxQ+τ2ε2-1+ε4-1sup0≤s≤τGs2+λmaxΠ=0,withΠ≔Ω1+∑i=14εi-1m2InmB^mB^TΩ2<0,and 1+rτ>0, 1-kerτ>0.Theorem 10.
Under the conditions of Theorem6, grey system (1) is almost surely exponential stability. That is, for all ξ∈LF02([-τ,0];Rn), we have (50)limt→∞sup1tlnxt;ξ≤-r^2,a.s.,where r^=min{r,τ-1,ln⁡k-1}, k=Uh+Lh/2+Uh-Lh/2<1, and r satisfies (12).Proof.
Under the conditions of Theorem6, system (1) is mean square exponentially stable, so we have(51)Ext;ξ2≤Ksup-τ≤θ≤0Eξθ2e-rt.Using Doob’s martingale inequality, Cauchy inequality, from (51), we have(52)Pω:sup0≤s≤τxkτ+s-G⊗^xkτ+s2>e-r-δkτ≤K1+k21+e-rτe-δkτsup-τ≤θ≤0Eξθ2.Then, using Borel-Cantelli lemma, and similar to the method used in Theorem 2 (see [23, 24]), we complete the proof of Theorem 10.


## 4. Numerical Examples

Example 1.
Consider a grey neutral systems with distributed delays:(53)dxt-HIxt-τ=AIxt+BIxt-τ+∫00.5Gsxt-sdsdt+fxt,xt-τ,tdwt,t≥0,x0=ξ,ξ∈LF02-0.5,0;Rn,-0.5≤t≤0.Here, the lower bound and upper bound matrices of AI, BI, and HI are(54)La=-3.75-0.520.53-3.35,Ua=-3.63-0.480.58-3.25,Lb=-0.650.200.13-0.66,Ub=-0.620.220.18-0.60,Lh=-0.350.130.15-0.28,Uh=-0.300.190.21-0.25.Moreover, (55)fxt,xt-0.5,t=12x1tsin⁡x2t-0.512x2tsin⁡x1t-0.5,Gs=e-s+3/600e-s+2/3.Then, we have(56)TracefTxt,xt-0.5,tfxt,xt-0.5,t≤0.25x2t,sup0≤s≤0.5Gs2=e-1.By the calculation procedure programmed in [24], we can obtain and optimize scalars ε1, ε2, ε3, and ε4, which satisfy the criteria conditions in Theorem 6, and make the Lyapunov exponent much smaller. Now, the computed allowable values of constants and the Lyapunov exponent are shown in Table 1.
According to Theorem6, and from Table 1, it is obvious that grey system (53) is exponentially stable in mean square, although the given system is described uncertainly. Moreover, it should be pointed out that [24] merely studies the exponential stability of a linear grey stochastic system. But, in this example, we can see that our work is suited to more general nonlinear grey stochastic systems. In the meantime, Theorem 6 only needs four positive scalars.Table 1
Calculation values ofε1, ε2, ε3, ε4, and r in Example 1.
Constants The Lyapunov exponent ε 1 ε 2 ε 3 ε 4 r Values 0.5281 0.7163 0.8210 0.8516 1.1025Example 2.
Consider the stochastic system (44) with(57)A^=-3.70-0.550.49-3.26,B^=-0.600.280.16-0.72,H^=-0.320.170.20-0.26.For this deterministic system, if Corollary 9 in this paper is used, and by the Matlab LMI Control Toolbox, we can know that the Lyapunov exponent r of this example is 1.0296.
For a comparison with the results of other methods and ours, the Lyapunov exponentr is shown in Table 2.
It can be seen from Table2 that,the Lyapunov exponent r obtained by Corollary 9 is larger. That is, the result obtained by Corollary 9 is less conservative than that in [7, 15] and is an improvement of those results. Therefore, the criterion in Corollary 9 is valid. In addition, it is worth noting that since the parameters of given system are deterministic, the criterion in [7, 15] cannot be applied to Example 1. But Corollary 9 can be generalized, and we can obtain Corollary 7 to solve Example 1. Meanwhile, in this example, Corollary 9 only seeks a positive define matrix.Table 2
Comparison of the Lyapunov exponentr for different methods.
Methods The Lyapunov exponent Sun and Liu [7] 0.8985 Song et al. [15] 0.9962 Corollary9 1.0296


## 5. Conclusion

This paper has focused on the exponential stability problem for a grey neutral stochastic system with distributed delays. By the Lyapunov stability theorem, Itô’s formula, and decomposition technique of the continuous matrix-covered sets of grey matrix, some novel stability criteria have been obtained, which ensure our grey system in the mean-square exponential stability and almost surely exponential stability. Finally, we provide two examples to show the effectiveness of the proposed method.



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