# Stochastic Stability Criteria for Neutral Distributed Parameter Systems with Markovian Jump

Authors: Yanbo Li; Chao-Yang Chen; Chengqun Li
Journal: Complexity
 (2020)

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## Abstract

This paper deals with the problem of stochastic stability for a class of neutral distributed parameter systems with Markovian jump. In this model, we only need to know the absolute maximum of the state transition probability on the principal diagonal line; other transition rates can be completely unknown. Based on calculating the weak infinitesimal generator and combining Poincare inequality and Green formula, a stochastic stability criterion is given in terms of a set of linear matrix inequalities (LMIs) by the Schur complement lemma. Because of the existence of the neutral term, we need to construct Lyapunov functionals showing more complexity to handle the cross terms involving the Laplace operator. Finally, a numerical example is provided to support the validity of the mathematical results.


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## Body

## 1. Introduction

Time-delay models are popular in all kinds of fields such as demography, biology, economics, and chemistry. Neutral systems as a special type of time-delay systems are often encountered because these systems have a wider application value than the general time-delay systems in many dynamical systems such as bioengineering systems, dynamic systems of offshore platform, and dynamic economic models. Hence, there are so many investigations about time-delay systems [1–7]. As we all know, the systems inevitably receive the impact of sudden changes in the environment, abrupt failure of components, unexpected changes in system parameters, and so on. These random diversifications usually follow the law of Markov jump. These systems are called Markovian jump systems. Markovian jump systems spur investigators’ consuming interest [8, 9].The stability and performance of stochastic systems are quite different from those of deterministic systems [10–12]. More recently, neutral-type Markovian jump systems have aspirated considerable attention. All kinds of analysis methods have been used to discover the stochastic stability criteria of neutral-type Markovian jump systems such as Lyapunov–Krasovskii functional approach, reciprocally convex combination inequality method, and stochastic analysis theory in [13–16]. For less conservative results, the delay-dependent stability has been discussed in [17–20]. Other control methods have also been extensively and thoroughly studied such as robust delay-dependent H∞ control [21, 22], nonfragile control [23–25], sliding mode control [26, 27], H∞ sliding mode control [28], and the references therein.The transition rates in many references mentioned above have been supposed to be completely known. But it is very hard to acquire the accurate transfer probability, and even if the exact transfer probability can be obtained, the cost is also very huge. So the study of Markovian jump systems with general unknown transition rates [29–50] has appealed to a great many scholars. Stability, stabilization, and robust control of Markovian jump systems with partially unknown transition have been reported in [29–33]. Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities has been proposed in [34, 35]. Kao et al. and Yang et al. [36, 37] have settled the delay-dependent stability for Markovian jump systems with partially unknown transition probabilities and Markovian jump neutral stochastic systems with general unknown transition rates, respectively. Singular Markovian jump systems with general incomplete transition probabilities have been presented in [38–40]. Finite-time stochastic stability and control of Markovian jump systems with general incomplete transition probabilities have been discussed in [41–44]. Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities was probed in [45].In parallel, many researchers have extensively studied the time-delay distributed parameter systems [46–54]. Three main approaches of time-delay distributed parameter systems are semigroup theory [47], matrix norm theory [48], and linear matrix inequality theory (LMI) [49]. The semigroup method cannot guarantee the system to be of a fine dynamic character and performance index undergoing two transformations. It is not easy to apply the results to practical problems by matrix norm. The problem of exponential stability and stabilization [50], sliding mode control [51], and feedback control [52] has been proposed in terms of the linear matrix inequality approach. However, less attention has been paid to the distributed parameter systems with Markovian jump, especially neutral distributed parameter systems with Markovian jump which requires a lot of research to be performed.Based on previous discussions, we are considering the problem of stochastic stability of a class of Markovian jumping neutral distributed parameter systems in this paper. The linear matrix inequality approach together with the Lyapunov functional method is employed to develop stochastic stability criteria for the described systems. The results are given in a group of linear matrix inequalities (LMIs).


## 2. Problem Formulation and Preliminaries

Consider the neutral distributed parameter systems with Markovian jump of the following form:(1)∂∂tWx,t−CrtWx,t−σ=DrtΔWx,t+ArtWx,t+A1rtWx,t−τ,where x,t∈Ω×R+, Ω=x,x<l<+∞⊂Rm is the bounded domain with smooth boundary ∂Ω, and mesΩ>0. Also,(2)∇Wx,t=col∇w1x,t,∇w2x,t,…,∇wnx,t,where ∇=∂/∂x1,∂/∂x2,…,∂/∂xm is the gradient operator. Wx,t=colw1x,t,w2x,t,…,wnx,t∈Rn is the state function, and Δ=∑k=1m∂2/∂xk2 is the Laplace operator on Ω.The initial value and boundary value conditions satisfy(3)Wx,t=0,x,t∈∂Ω×−γ,+∞,Wx,t=ψx,t,x,t∈Ω×−γ,0,∂Wx,t∂n=0,x,t∈∂Ω×−γ,+∞,where γ=maxσ,τ and σ>0 and τ>0 are constants. n is the unit outward normal vector of ∂Ω, and ψx,t is the smooth function. τ>0,σ>0, and Drt>0 are constants; Art, Crt, and A1rt are constant matrices.Letrt,t≥0 be a right-continuous Markov process and take values in a finite set F=1,2,…,N with transition probability matrix Π=πij; the mode transition probabilities are defined as follows:(4)Prrt+h=jrh=i=πijh+oh,i≠j,1+πiih+oh,i=j,where h>0 and limδ⟶0oh/h=0. πij≥0,i≠j donates the transition rate from mode i to mode j in the time interval h and πii=−∑j=1,j≠iNπij. For each rt=i∈F, let Art=Ai, A1rt=A1i, Drt=Di, and Crt=Ci. Then, we can represent system (1) in the following form:(5)∂∂tWx,t−CiWx,t−σ=DiΔWx,t+AiWx,t+A1iWx,t−τ.Lemma 1 (see [53] (Green formula)).
LetΩ⊂Rn be the bounded domain with smooth boundary ∂Ω, n is the unit outward normal vector of ∂Ω, and G⊂Ω is the smooth subdomain. If u,v∈C2G¯, then(6)∫GuΔvdx=∫∂Ωu∂v∂nds−∫Ω∇u∇vdx,where ∇ is the Hamilton operator and ds is the area element over the boundary region.Lemma 2 (see [54] (Friedrichs’s inequality)).
Letw∈C01Ω and Ω be included in the closed region Ω1:0≤xi≤li=1,2,…,n. Then,(7)∫Ωw2xdx≤∫Ω∑i=1n∂w∂x2dx=c∫Ω∇w2dx,where c=l2/n.Lemma 3 (see [55]).
LetV1,V2,andV3 be the real matrices and V3=V3T>0; then, for an arbitrary given scalar α>0, the following inequality holds:(8)V2TV1+V1TV2≤α−1V1TV3−1V1+αV2TV3V2.


## 3. Main Results

Theorem 1.
Given matricesAi,A1i,andCi, time-delay constants τ>0 and σ>0 and constant Di>0, the neutral distributed parameter systems with Markovian jump (5) is stochastically stable. If there exist positive symmetric matrices Pi, Qi, M, N, R, and Z and positive scalars αi, such that for any Markovian jump mode i∈F, the following linear matrix inequalities (LMIs) hold:(9)αi−2DiIDiCiDiCiT−αiI<0,Θ=Π10−AiTCi0000−Pi−A1iTCi000∗∗−Qi000000Π2A1iTDiCi−CiTAi000∗−M−CiTA1i000∗∗−N<0,where β=maxπii,i∈F, Π1=AiT+Ai+Pi+Qi+βτR+βσZ, Π2=M+N+Ai+AiT−2DiI,(10)Pi<R,(11)Qi<Z.Proof.
For system (5), we construct the following stochastic Lyapunov functional:(12)Vit,Wx,t=∑n=19Vin,where(13)V1i=∫ΩYTx,tYx,tdx,whereYx,t=Wx,t−CiWx,t−σ,Vi2=∫Ω∫t−τtWTx,θPiWx,θdθdx,Vi3=∫Ω∫t−σtWTx,θQiWx,θdθdx,Vi4=∫Ω∫t−τt∇WTx,θM∇WTx,θTdθdx,Vi5=∫Ω∫t−σt∇WTx,θN∇WTx,θTdθdx,Vi6=∫Ω∇YTx,θ∇YTx,θTdx,Vi7=αi∫Ω∫t−σtΔWx,θTΔWx,θdθdx,Vi8=βi∫Ω∫−τ0∫t+stWTx,θRWx,θdθdsdx,Vi9=βi∫Ω∫−σ0∫t+stWTx,θZWx,θdθdsdx.
LetL be the weak infinitesimal generator; then, we calculate(14)LVit,Wx,t=∑n=19LVin,where(15)LV1i=2∫ΩYTx,t∂Yx,t∂tdx=2Di∫ΩWTx,tΔWx,tdx+∫ΩWTx,tAiT+AiWx,tdx+2∫ΩWTx,tA1iWx,t−τdx−2Di∫ΩWTx,t−σCiTΔWx,tdx−2∫ΩWTx,t−σCiTAiWx,tdx−2∫ΩWTx,t−σCiTA1iWx,t−τdx.
Applying Lemma1 and Lemma 2,(16)2Di∫ΩWTx,tΔWx,tdx=∑k=1n∑l=1n∫∂Ωwkx,t∂wlx,t∂nds−∫Ω∇wkx,t∇wlx,tdx=−∑k=1n∑l=1n∫Ω∇wkx,t∇wlx,tdx=−2Di∫Ω∇WTx,t∇WTx,tTdx,(17)−2Di∫ΩWTx,t−σCiTΔWx,tdx=−2Di∫Ω∑l=1n∑m=1nwmx,t−σcml∇⋅∇wlx,tdx=2Di∑l=1n∑m=1n∫Ω∇wmx,t−σcml⋅∇wlx,tdx=2Di∑l=1n∑m=1n∑k=1m∫Ω∂wmx,t−σ∂xkcml∂wlx,t∂xkdx=2Di∫Ω∇WTx,t−σCi∇WTx,tTdx.
Substituting (16) and (17) into (15), we have(18)LV1i=−2Di∫Ω∇WTx,t∇WTx,tTdx+∫ΩWTx,tAiT+AiWx,tdx+2∫ΩWTx,tA1iWx,t−τdx+2Di∫Ω∇WTx,t−σCiT∇WTx,tTdx−2∫ΩWTx,t−σCiTAiWx,tdx−2∫ΩWTx,t−σCiTA1iWx,t−τdx,(19)LVi2t,Wx,t=∫ΩWTx,tPiWx,tdx−∫ΩWTx,t−τPiWx,t−τdx+∫Ω∫t−τtWTx,θ∑j=1NπijPjWx,θdθdx.
Noticingπij>0i≠j and combining Pj<R and β=maxπii,i∈F, we derive the following inequality:(20)∫Ω∫t−τtWTx,θ∑j=1NπijPjWx,θdθdx≤∫Ω∫t−τtWTx,θ∑j=1,i≠jNπijPjWx,θdθdx=−πii∫Ω∫t−τtWTx,θPjWx,θdθdx≤β∫Ω∫t−τtWTx,θPjWx,θdθdx≤β∫Ω∫t−τtWTx,θRWx,θdθdx.
Then,(21)LVi2t,Wx,t≤∫ΩWTx,tPiWx,tdx−∫ΩWTx,t−τPiWx,t−τdx+β∫Ω∫t−τtWTx,θRWx,θdθdx.
For the same reason, we can also obtain(22)∫Ω∫t−σtWTx,θ∑j=1NπijQjWx,θdθdx≤β∫Ω∫t−σtWTx,θQjWx,θdθdx≤β∫Ω∫t−σtWTx,θZWx,θdθdx.
So(23)LVi3t,Wx,t=∫ΩWTx,tQiWx,tdx−∫ΩWTx,t−σQiWx,t−σdx+∫Ω∫t−σtWTx,θ∑j=1NπijQjWx,θdθdx≤∫ΩWTx,tQiWx,tdx−∫ΩWTx,t−σQiWx,t−σdx+β∫Ω∫t−σtWTx,θZWx,θdθdx,(24)LVi4=∫Ω∇WTx,tM∇WTx,tTdx−∫Ω∇WTx,t−τM∇WTx,t−τTdx,(25)LVi5=∫Ω∇WTx,tN∇WTx,tTdx−∫Ω∇WTx,t−σN∇WTx,t−σTdx,(26)LVi6=2∫Ω∇∂YTx,t∂tT∇YTx,tTdx=−2∫Ω∂YTx,t∂tΔYTx,tTdx=−2Di∫ΩΔWx,tTΔWx,tdx+2Di∫ΩΔWx,tTCiΔWx,t−σdx−2∫ΩWTx,tAiTΔWx,tdx−2∫ΩWTx,t−τA1iTΔWx,tdx+2∫ΩWTx,tAiTCiΔWx,t−σdx+2∫ΩWTx,t−τA1iTCiΔWx,t−σdx.
Through applying Lemma3, the following inequality holds:(27)2Di∫ΩΔWx,tTCiΔWx,t−σdx≤αi−1∫ΩΔWx,tTDi2CiCiTΔWx,tdx+αi∫ΩΔWx,t−σTΔWx,t−σdx.
Taking advantage of Lemma1 and Lemma 2, we obtain(28)−2∫ΩWTx,tAiTΔWx,tdx=2∫Ω∇WTx,tAi∇WTx,tTdx,(29)2∫ΩWTx,tAiTCiΔWx,t−σdx=−2∫Ω∇WTx,tCiTAi∇WTx,t−σTdx,(30)−2∫ΩWTx,t−τA1iTΔWx,tdx=2∫Ω∇WTx,t−τA1i∇WTx,tTdx,(31)2∫ΩWTx,t−τA1iTCiΔWx,t−σdx=−2∫Ω∇WTx,t−τCiTA1i∇WTx,t−σTdx.
Combining (26)–(31) together yields(32)LVi6≤∫ΩΔWx,tTαi−1Di2CiCiT−2DiIΔWx,tdx+αi∫ΩΔWx,t−σTΔWx,t−σdx+2∫Ω∇WTx,tAi∇WTx,tTdx+2∫Ω∇WTx,t−τA1i∇WTx,tTdx−2∫Ω∇WTx,tCiTAi∇WTx,t−σTdx−2∫Ω∇WTx,t−τCiTA1i∇WTx,t−σTdx,(33)LVi7=αi∫ΩΔWx,tTΔWx,tdx−αi∫ΩΔWx,t−σTΔWx,t−σdx,(34)LVi8=βτ∫ΩWTx,tRWx,tdx−β∫Ω∫t−τtWTx,θRWx,θdθdx,(35)LVi9=βσ∫ΩWTx,tZWx,tdx−β∫Ω∫t−σtWTx,θZWx,θdθdx.
Synthesizing (18), (21), (23)–(25), and (32)–(35), the following inequality holds:(36)LVt,Wx,t≤∫ΩΔWx,tTαiI+αi−1Di2CiCiT−2DiIΔWx,tdx+∫ΩWTx,tΠ1Wx,tdx+∫Ω∇WTx,tΠ2∇WTx,tTdx−∫ΩWTx,t−τPiWt−τdx−∫ΩWTx,t−σQiWt−σdx−∫Ω∇WTx,t−τM∇WTx,t−τTdx−∫Ω∇WTx,t−σN∇WTx,t−σTdx+2∫Ω∇WTx,t−σDiCiT−AiTCi∇WTx,tTdx+2∫Ω∇WTx,t−τA1i∇WTx,tTdx−2∫Ω∇WTx,t−τCiTA1i∇WTx,t−σTdx−2∫ΩWTx,t−σCiTAiWx,tdx−2∫ΩWTx,t−σCiTA1iWx,t−τdx.
Select appropriateαi>0 such that αiI+αi−1Di2CiCiT−2DiI<0, then we transform it into (9) by the Schur complement lemma.
Set(37)Xt=Wx,tWx,t−τWx,t−σ∇WTx,tT∇WTx,t−τT∇WTx,t−σT.
Then,(38)LVx,Wx,t≤∫ΩXTtΘXtdx<0.
The proof is concluded.Remark 1.
Because of the simultaneous existence of neutral term and Markov jump, the stochastic stability of neutral distributed parameter systems with jump is given by finding the maximum value of absolute value on the main diagonal of the state transition probability matrix. Compared with the general time-delay system, the derivation process of the system is complex, but the conclusion is simple.


## 4. Examples

Consider a neutral distributed parameter system (5) with two Markovian jump modes and the parameters as follows:(39)A1=−100−0.9,A2=−100−0.1,A11=10.1−0.51,A12=−120−1,C1=−0.30.50−0.2,C2=−0.20.3−0.1−1.6,D1=D2=1,τ=0.1,σ=0.2.The transition rate matrix is defined by(40)Π=−110.5−0.5.Obviously, the absolute maximum of the principle diagonal line is 1. We get feasible solutions by solving linear matrix inequalities (9)–(12) in Theorem 1 and obtain the following parameters:(41)M=2.41830.31140.31140.4770,N=0.64910.00220.00221.1412,P1=17.32371.31971.319721.9271,P2=35.049315.369715.369780.9490,Q1=0.0358−0.0829−0.08290.2264,Q2=0.0844−0.1372−0.13720.5661,R=5.26930.25870.25873.0645,Z=3.67310.21790.21791.4507.Obviously matricesP1,P2,Q1,Q2,M,N,R,andZ are positive. The effectiveness of the method of Theorem 1 is illustrated.


## 5. Conclusion

First, we choose a set of appropriate Lyapunov stochastic functionals; some of them show more complexity in dealing with the neutral cross terms containing the Laplace operator. Then, by taking advantage of boundary conditions, Green formula, and Schur complement lemma, we obtain a sufficient condition for stochastic stability of the studied model in this paper via linear matrix inequalities. At last, a numerical example is given to prove the effectiveness of the proposed method.



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