# A New Stability Criterion for Neutral Stochastic Delay Differential Equations with Markovian Switching

Authors: Wei Hu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this short paper, a new stability theorem for neutral stochastic delay differential equations with Markovian switching is investigated by applying stochastic analysis technique and Razumikhin stability approach. A novel criterion of thepth moment exponential stability is derived for the related systems. The feature of the criterion shows that the estimated upper bound for the diffusion operator of Lyapunov function is allowed to be indefinite, even if to be unbounded, which can loosen the constraints of the existing results. Last, an example is provided to illustrate the usefulness and significance of the theoretical results.


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## Body

## 1. Introduction

As is well-known, the stability is very hot topic in deterministic or stochastic dynamic systems (for instance, see [1–37] and references therein). Recently, a class of stochastic hybrid systems (also known as stochastic systems with Markovian switching) has been used to model many practical systems where they may experience abrupt changes in their structure and parameters. In [38], the author explained that the Markovian switching systems had been emerging as a convenient mathematical framework for the formulation of various design problems in different fields such as target tracking, fault tolerant control, and manufacturing processes, which can be seen as the motivation of wide practical use of the theoretic results for hybrid systems. Owing to their widely real applications, stochastic systems with Markovian switching have received a great deal of attention and many interesting results have been reported in the literature. For example, [39] considered the stability of stochastic differential equations with infinite Markovian switchings and [40] studied the stability for stochastic differential equations with semi-Markovian switchings. The results of stochastic hybrid systems are applied in feedback controls and neural networks; please refer to [41–45] and the reference therein.As an important class of hybrid stochastic system, neutral stochastic delay differential equations with Markovian switching (NSDDEwMS) have been applied in practice, such as traffic control, neutral networks, and chemical process. Due to its wide applications, more and more researchers focus on this system. An important question for such a model is the stability analysis. Lyapunov-Razumikhin functions method and Lyapunov-Krasovskii functionals method are two effective methods that have been exploited for the stability analysis. For example, Mao in [46, 47] applied Razumikhin approach to derive the exponential stability criteria for neutral stochastic delay differential systems with Markovian switching. In [48–53], the authors considered the exponential stability by using Lyapunov-Krasovskii functionals method. Reference [54] applied some special techniques to study the exponential stability. Besides, Zhu in [55, 56] obtained several novel exponential stability criteria for some more complex systems with impulse control and Lévy noise. In addition, [57] considered the stability in distribution of neutral stochastic differential delay equations with Markovian switching and [58, 59] studied the almost sure stability for the same system.Although the above exponential stability criteria can be used to judge the stability for many NSDDEwMS, nevertheless, these conditions in the above literature can be weakened to cover a large collection of NSDDEwMS. In other words, there are some shortcomings in the former criteria. The main is that all the above criteria for exponential stability in the related literature imposed some strict conditions on the diffusion operator of Lyapunov functions. For example, [46, 47] required the estimated upper bound for the diffusion operator of Lyapunov functions to be negative constant numbers. Reference [48] needed that the estimated upper bound λ(t) must be a negative value function. Simply speaking, the former results all need that the upper bound λ(t) for the diffusion operator of the Lyapunov function is negative definite for all t. This restriction leads to the criteria having strong conservativeness in practice due to the fact that there are a large number of time-varying systems not satisfying the above conditions. As shown by an example in Section 4, the existing results and methods cannot be applied to analyse the stability for more general time-varying systems.Motivated by the above discussion, in this short note, we focus on the exponential stability for NSDDEwMS. By using the notions such as uniformly stable function (USF), overshoot of the USF that dated from [60, 61] and then combining the stochastic analysis techniques, we obtain a novel stability criterion for NSDDEwMS. The feature of the criterion is that the coefficients of the estimated upper bound for the diffusion operator of Lyapunov functions can be allowed sign-changing as the time-varying, which can be used more widely than the existing results.The rest of the paper is organized as follows. In Section2, we introduce the model of NSDDEwMS and some preliminaries. The main result and its proof will be displayed in Section 3. An example is provided in Section 4. Finally, we conclude this paper with some general remarks in Section 5.Throughout this short paper, let(Ω,F,{Ft}t≥0,P) be a complete probability space with a natural filtration {Ft}t≥0 satisfying the usual condition (i.e., it is right continuous and F0 contains all P-null sets). |x| denotes the Euclidean norm of x. The symbol C([-τ,0],Rn) denotes the family of continuous function φ from [-τ,0] to Rn with the norm φ=sup-τ≤θ≤0|φ(θ)|. We use CFtp([-τ,0],Rn) to denote the family of all Ft-measurable, C([-τ,0],Rn) valued random variables φ satisfying sup-τ≤θ≤0E|φ(θ)|p<∞. We use E[·] to denote the expectation operator with respect to the probability measure P. Let Bt=B(t)=(B1(t),B2(t),…,Bm(t))T be an m-dimensional Brownian motion defined on a complete probability space. a∨b=max{a,b}. N denotes the set of positive integers.


## 2. Preliminaries

Let{r(t),t≥0} be a right-continuous Markov chain on the probability space (Ω,F,P) taking values in a finite state space S={1,2,…,N} with generator Q=(qij)N×N given by (1)Prt+Δt=j∣rt=i=qijΔt+oΔt,ifi≠j1+qiiΔt+oΔt,ifi=jwhere Δt>0. Here qij≥0 is the transition rate from i to j if i≠j while qii=-∑j≠i‍qij=-qi.In this short paper, we will consider the following neutral stochastic delay differential system with Markovian switching:(2)dxt-ut,xt-τ,rt=ft,xt,xt-τ,rtdt+gt,xt,xt-τ,rtdBtwith the initial data x0=ϕ={ϕ(θ),-τ≤θ≤0}∈CF0p([-τ,0],Rn). We assume that f, g satisfy the local Lipschitz condition and the linear growth condition. Additionally, we impose the following condition:(3)ut,x,i-ut,y,i≤κx-y,κ∈0,1,i∈Son u(·) to guarantee the existence and uniqueness of solution x(t,ϕ) for system (2) for all t≥0. We also assume that f(t,0,0,i)=0, g(t,0,0,i)=0, and u(t,0,i)=0, which implies that the trivial solution of system (2) exists. In order to use Lyapunov’s method to study the pth moment exponential stability, we need the following definitions.Definition 1.
The trivial solution of (2) is said to be pth moment exponentially stable if (4)limsupt→∞lnExt,ϕpt<0for all ϕ∈CF0p([-τ,0],Rn). When p=2, 2th moment exponential stability is usually called mean square exponentially stable.Definition 2.
The functionV(t,x,i):R+×Rn×S→R+ belongs to class Ψ if it is a continuously twice differentiable with respect to x and once differentiable with respect to t.Next, we need an operatorLV from R+×Rn×Rn×S to R by (5)LVt,x,y,i=Vtt,x-ut,y,i,i+Vxt,x-ut,y,i,ift,x,y,i+12tracegTt,x,y,iVxxt,x-ut,y,i,igt,x,y,i+∑j=1NqijVt,x-ut,y,i,j,where (6)Vxt,x,i=∂Vt,x,i∂x1,∂Vt,x,i∂x2,…,∂Vt,x,i∂xn,Vxxt,x,i=∂2Vt,x,i∂xi∂xjn×n,Vtt,x,i=∂Vt,x,i∂t.In order to overcome the difficult caused by the neutral item, we need the following inequality.Lemma 3.
Letp≥1. Then (7)1-κp-1xp-κ1-κp-1yp≤x-ut,y,ip≤1+κp-1xp+κypholds for any (t,x,y,i)∈R+×Rn×Rn×S.The following lemma, which is important in the proof of the main result, can be found in Lemma 1 of [62].Lemma 4.
LetT∗ be a constant. Let a function w:[-T∗,∞)→[0,∞) admit a sequence {vi} and positive constants v¯a and v¯b, such that v0=t0, vi+1-vi∈[v¯a,v¯b] for all i≥0. w is continuous on [vi,vi+1) for all i≥0 and the left limit limt→vi-w(t) exist. Assume that there exists a constant ρ∈(0,1) such that w(t)≤ρsups∈[t-T∗,t]w(s) holds for all t≥t0. Then (8)wt≤elnρ/T∗t-t0sups∈t0-T∗,t0wsholds for all t≥t0.The following definitions, which are dated from [60], are important in the proof of our main result.Definition 5.
A piecewise continuous functionμ is said to be a USF if the following linear time-varying equation is globally uniformly asymptotically stable: (9)dyt=μtytdt,∀t≥0Definition 6.
Letμ(t) be a USF. Then the set (10)Ωμ=T>0:supt≥0∫tt+Tμsds<0is said to be the uniform convergence set of μ(t). For any T≥0, the overshoot φμ(T)≔φμ of μ(t) is defined as follows: (11)φμ=supt≥0maxθ∈0,T∫tt+θμsds.


## 3. Main Results

In this section, we will use stochastic analysis theory, Lyapunov’s method, and Lemmas3 and 4 to obtain a criterion for pth moment exponential stability of system (2). Our main result is the following.Theorem 7.
Letp,c1,c2,κ,τ be all positive numbers, and μ(t) is a USF. If there exist a function V∈Ψ and a constant T∈Ωμ such that the following conditions hold for all t≥0,
(1) for allx∈Rn, i∈S, (12)c1xp≤Vt,x,i≤c2xp,
(2) the following Razumikhin-type condition holds:(13)ELVt,xt,xt-τ,rt≤μtEVt,x~t,rtifmini∈SEVt+θ,xt+θ,i≤qmaxi∈SEVt,x~t,i∀θ∈-τ,0,whereq>c2c1eφμ1-κ-p.
Then, system (2) is pth moment exponentially stable; here x~(t)≜x(t)-u(t,x(t-τ),r(t)).Proof.
First of all, we will prove the following fact: ifq-1EV(t+θ,x(t+θ),r(t))≤EV(t,x~(t),r(t)) holds for all θ∈[-τ,0] which implies ELV(t,x(t),x(t-τ),r(t))≤μ(t)EV(t,x~(t),r(t)), then, for any t>T,(14)EVt,x~t,rt≤maxEVt-T,x~t-T,rt-Te∫t-Ttμsds,q-1sups∈t-T,tsupθ∈-τ,0EVs+θ,xs+θ,rs+θeφμ.In order to prove inequality (14), we consider the following two cases:
(A):supθ∈[-τ,0]EV(s+θ,x(s+θ),r(s+θ))≤qEV(s,x~(s),r(s)) holds for all s∈[t-T,t].
(B):supθ∈[-τ,0]EV(s+θ,x(s+θ),r(s+θ))≤qEV(s,x~(s),r(s)) does not hold for some s∈[t-T,t].
For (A), we know thatELV(s,x(s),x(s-τ),r(s))≤μ(s)EV(s,x~(s),r(s)) holds for any s∈[t-T,t]. By using Ito’s formula and the standard stopping times technique, we obtain that (15)Ee-∫t-TtμsdsVt,x~t,rt=EVt-T,x~t-T,rt-T+E∫t-Tt-μse-∫t-TsμvdvVs,x~s,rs+e-∫t-TsμvdvLVs,xs,xs-τ,rsds≤EVt-T,x~t-T,rt-T,which implies that EV(t,x~(t),r(t))≤EV(t-T,x~(t-T),r(t-T))e∫t-Tt‍μ(s)ds.
For (B), definet∗=sup{s∈[t-T,t]:supθ∈[-τ,0]EV(s+θ,x(s+θ),r(s+θ))>qEV(s,x~(s),r(s))}. Then t∗<t or t∗=t. If t∗<t, then, for all s∈[t∗,t], supθ∈[-τ,0]EV(s+θ,x(s+θ),r(s+θ))≤qEV(s,x~(s),r(s)). So according to the result of (A), we obtain that (16)EVt,x~t,rt≤e∫t∗tμsdsEVt∗,x~t∗,rt∗=e∫t∗tμsdsq-1supθ∈-τ,0EVt∗+θ,xt∗+θ,rt∗+θ≤e∫t∗tμsdsq-1sups∈t-T,tsupθ∈-τ,0EVs+θ,xs+θ,rs+θ≤eφμq-1sups∈t-T,tsupθ∈-τ,0EVs+θ,xs+θ,rs+θ.
Ift∗=t, from the definition of t∗, we have (17)EVt,x~t,rt=EVt∗,x~t∗,rt∗≤q-1supθ∈-τ,0EVt∗+θ,xt∗+θ,rt∗+θ≤eφμq-1sups∈t-T,tsupθ∈-τ,0EVs+θ,xs+θ,rs+θ.Combining the above three inequalities, we can see that inequality (14) holds.
Assume thatq-1EV(t+θ,x(t+θ),r(t))≤EV(t,x~(t),r(t)) holds for all θ∈[-τ,0], then q-1mini∈S{EV(t+θ,x(t+θ),i)}≤q-1EV(t+θ,x(t+θ),r(t))≤EV(t,x~(t),r(t))≤maxi∈S{EV(t,x~(t),i)} holds for all θ∈[-τ,0]. According to condition (2), we know that ELV(t,x(t),x(t-τ),r(t))≤μ(t)EV(t,x~(t),r(t)). Thus, from inequality (14), condition (1), and Lemma 3, we conclude that, for any t>T, (18)EVt,x~t,rt≤maxEVt-T,x~t-T,rt-Te∫t-Ttμsds,q-1sups∈t-T,tsupθ∈-τ,0EVs+θ,xs+θ,rs+θeφμ≤maxc2Ex~t-Tpe∫t-Ttμsds,q-1sups∈t-T,tsupθ∈-τ,0EVs+θ,xs+θ,rs+θeφμ≤maxc21+κp-1Ext-Tp+κExt-T-τpe∫t-Ttμsds,q-1sups∈t-T,tsupθ∈-τ,0EVs+θ,xs+θ,rs+θeφμ≤maxc2c11+κpe∫t-Ttμsds,q-1eφμsups∈t-T,tsupθ∈-τ,0EVs+θ,xs+θ,rs+θ.By condition (1), we can see that (19)Ex~tp≤c2c1maxc2c11+κpe∫t-Ttμsds,q-1eφμsups∈t-T,tsupθ∈-τ,0Exs+θp≔ηsups∈t-T,tsupθ∈-τ,0Exs+θp.Using Lemma 3 again, we obtain that (20)Extp≤Ex~tp+κ1-κp-1Ext-τp1-κ1-p=1-κ1-pEx~tp+κExt-τp≤1-κ1-pη+κsups∈t-T,tsupθ∈-τ,0Exs+θp.In order to use Lemma 4, we need to justify that (1-κ)1-pη+κ<1. In fact, on one hand, due to q>c2/c1eφμ(1-κ)-p, there exists a constant ρ∈(0,1) such that c2/c1q-1eφμ(1-κ)1-p≤ρ-κ. On the other hand, due to μ(t) being a USF and T∈Ωμ, there exists a constant ρ1∈(0,1) such that e∫t-Tt‍μ(s)ds<(c1/c2)2((1-κ)/(1+κ))p(ρ1-κ)/(1-κ)<1. Thus (21)Extp≤1-κ1-pη+κsups∈t-T,tsupθ∈-τ,0Exs+θp=1-κ1-pc2c1maxc2c11+κpe∫t-Ttμsds,q-1eφμ+κsups∈t-T,tsupθ∈-τ,0Exs+θp≤maxρ1-κ,ρ-κ+κsups∈t-T,tsupθ∈-τ,0Exs+θp≤ρ∗sups∈t-T,tsupθ∈-τ,0Exs+θp≤ρ∗sups∈t-T-τ,tExspwhere ρ∗=ρ∨ρ1∈(0,1). By Lemma 4, we derive that (22)Extp≤elnρ∗/T+τt-Tsups∈-τ,TExsp,which implies that (23)limsupt→∞lnExtpt=lnρ∗T+τ<0.In other words, we have proved that system (2) is pth moment exponentially stable.Remark 8.
Here, by using stochastic analysis theory, some notions as uniformly stable function (USF), overshoot of the USF that dated from [60, 61], and some inequality techniques, we obtain a novel exponential stability criterion with respect to the related systems. The method we use here is rather different from the traditional Razumikhin approach. Thus, we provide a new method to investigate the stability analysis for neutral stochastic delayed differential equations with Markovian switchings. In the future, we can develop this approach to study the stability for impulsive neutral stochastic delayed differential equations.Remark 9.
If system (2) has no Markovian switching and μ(t)≡μ<0, then eφμ=; the condition of Theorem 7q>c2/c1eφμ(1-κ)-p becomes q>c2/c1(1-κ)-p, which is the same as the conditions in [46, 47]. If the model we considered has no Markovian switching, stochastic disturbance, and neutral item, then we can obtain the same result as Theorem 1 in [60]. Thus, our result is the generalization of Theorem 1 in [60] and the main results in [46, 47].Remark 10.
In Theorem 3.1 of [53], the condition (3.3) ensured that the coefficient of the estimated upper bound for the diffusion operator of Lyapunov function is a negative value, and the condition of Theorem 3.1 in [48] required that the coefficient function of the estimated upper bound for the diffusion operator of Lyapunov function is a negative definite function, which are very conservative in the practice if the system is time-varying system. But our conditions allow this function to take values on R and even allow it to be unbounded. Thus, our result can be used more widely and is less conservative than the existing results. See Example 1 for more details.


## 4. An Example

In this section, we will give an example to illustrate the validity and significance of our result.Example 1.
Now we will consider the following neutral-type stochastic delay differential system with Markovian switching:(24)dxt-0.2xt-0.5=ft,xt,xt-0.5,rtdt+gt,xt,xt-0.5,rtdBt,where(25)
f
t
,
x
t
,
x
t
-
0.5
,
i
=
1
16
t
c
o
s
t
2
-
3
2
x
t
-
0.5
+
1
16
t
c
o
s
t
2
x
t
,
i
=
1
,
1
8
t
c
o
s
t
2
-
1
x
t
-
0.5
,
i
=
2
,and(26)gt,xt,xt-0.5,i=tanhtxt-0.5,i=1,e-tx2t-0.5xt-0.5,i=2.r(t) is a right-continuous Markov chain on the state space S={1,2} with the generator (27)Q=-112-2.
Theorem 3.1 in [48] is void in determining the exponential stability for such a system. In fact, taking V(t,x,i)=|x|2 and c1=c2=1, then, by Ito’s formula, (28)ELVt,xt,xt-0.5,1≤2Ext-0.2xt-0.5116tcost2-32xt-0.5+116tcost2xt+Ext-0.52≤740tcost2-34Ext2+340tcost2-120Ext-0.52and (29)ELVt,xt,xt-0.5,2≤2Ext-0.2xt-0.518tcost2-1xt-0.5+Ext-0.52≤18tcost2-1Ext2+110tcost2-25Ext-0.52.Obviously, the conditions in Theorem 3.1 [48] do not hold. But we can judge the exponential stability for such system by using our result. Take V(t,x,i)=|x|2 and c1=c2=1. By Ito’s formula, (30)ELVt,xt,xt-0.5,1≤18tcost2-3Ex~txt-0.5+18tcost2Extx~t+Ext-0.52≤18tcost2-3q+12+qEx~t2+18tcost2Extx~t≤18tcost2-3q+12+qEx~t2+18q+12tcost2Ex~t2=18tcost2q+1-3q+12+qEx~t2,and(31)ELVt,xt,xt-0.5,2≤18tcost2-1Ex~t2+18tcost2-1Ext-0.52+Ext-0.52≤14tcost2-2q+12+qEx~t2=18tcost2q+1+qEx~t2.Choosing q=3, then ELV(t,x(t),x(t-0.5),r(t))≤μ(t)EV(t,x~(t),r(t)), where μ(t)=1/2tcost2-1. Here, μ(t) is an unbounded function, taking values in R, so we cannot use any former results to judge the mean-square exponential stability for such system. But, from our result, we can determine its stability. In fact, taking T=1, then ∫tt+1‍μ(s)ds=-1/2<0 holds for all t≥0. Thus T∈Ωμ. Additionally, eφμ≤e0.5, and q-1eφμ(1-k)-p≤3-1×e0.5×0.8-2<1. By Theorem 7, the system is mean square exponentially stable. See Figure 1.Figure 1
The state response of Example1.


## 5. Conclusion

In this paper, we investigate the stability criterion for neutral stochastic delay differential equations with Markovian switching. By using stochastic analysis technique and Razumikhin approach, we overcome the difficulty caused by the neutral item for the related system. Finally, we derive a new and novel Razumikhin exponential stability criterion. The feature of the result is that the estimated upper bound for the diffusion operator of Lyapunov function is allowed to take values onR, even if it is allowed to be unbounded. The criterion can reduce some restrictiveness of the related results that are existing in the previous literature. An example is provided to show the superiority of the new exponential stability criterion.



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