# β-Stability in q-th Moment of Neutral Impulsive Stochastic Functional Differential Equations with Markovian Switching

Authors: Lassaad Mchiri; Mohamed Rhaima
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we investigate theβ-stability in q-th moment for neutral impulsive stochastic functional differential equations with Markovian switching (NISFDEwMS). Moreover, β-stability in q-th moment is studied by using the Lyapunov techniques and a new Razumikhin-type theorem to prove our result. Finally, we check the main result by a numerical example.


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## Body

## 1. Introduction

The aim of stochastic differential equations is to provide a mathematical model for a differential equation disturbed by a random noise. Consider an ordinary differential equation of the form(1)dστ=aστdτ.An ordinary differential equation is used to describe the evolution of a physical system. If we take into account the random disturbances, we add a noise term, which will be of the formgdWτ, where W denotes a Brownian motion and g, for this instant, is a constant which corresponds to the intensity of the noise. Then, we have a stochastic differential equation (SDE) of the form(2)dστ=aστdτ+gdWτ.Moreover, we have the following integral form:(3)στ=σ0+∫0τaσtdt+gWτ.We generalize equation (3) by taking g depending on the state of the system to an instant τ:(4)στ=σ0+∫0τaσtdt+∫0τgσtdWt.In view of the integral indWt, this equation will depend on the theory of stochastic integral (see [1]).One of the most important qualitative studies of SDE is the stability theory.There are a lot of papers about the stability theory of different types of SDE (see [1–3]).A special case of SDE is the neutral stochastic functional differential equations (NSFDE).In the literature, there are many research papers on the stability theory of NSFDE (see [4–6]). Recently, stability of SFDE with Markovian switching (SFDEwMS) has received a lot of attention (see [7–11]).One of the most important classes of SDE is the impulsive SDE (ISDE). In the last decades, the stability theory of ISDE takes much more attention (see [7, 11–13]).To the best of our knowledge, there are a few papers on theβ-stability or stability of SDE with general decay rate. In [14–16], the authors established the stability with general decay rate of stochastic functional differential equations with finite and infinite delay.In the literature, there is no existing paper on theβ-stability of NISFDEwMS.In this paper, we study theβ-stability in q-th moment of NISFDEwMS. In view of the other related topics on NISFDEwMS, the main contributions of this paper are to obtain a new technique to study the β-stability in a hybrid impulsive neutral stochastic system, to develop a new Razumikhin-type theorem on the Lyapunov function to prove the β-stability of the system, and to propose new theories on asymptotic properties in q-th moment of NISFDEwMS. In this sense, our results generalize the paper [16] in the case of NISFDEwMS.The paper is organized as follows. In Section2, we cite some definitions and basic notions. In Section 3, we investigate the β-stability in q-th moment of NISFDEwMS. In Section 4, one numerical example is given to show the interest of our main results.


## 2. Preliminaries and Definitions

The complete probability space is denoted byΩ,F,Fττ≥0,ℙ, where Fττ≥0 is a filtration satisfying the usual conditions. Wτ is an m-dimensional Brownian motion defined on the probability space. Let l>0 and C−l,0;ℝi be the family of continuous functions ψ from −l,0 to ℝi with the norm ψ=sup−l≤b≤0ψb, where z=zTz for any z∈ℝi. If B is a matrix, its trace norm is denoted by B=traceBTB, where its norm is given by B=supz=1Bz.LetPC−l,0;ℝi=ϕ:−l,0⟶ℝi|ϕt+=ϕt, where ϕt− exists and ϕt−=ϕt for all but at most a finite number of points t, with the uniform norm ϕ=sup−l≤b≤0ϕb, where ϕt+ and ϕt− denote the right-hand and left-hand limits of ϕt at t, respectively. For q≥1, let Lq−∞,0,ℝi be the family of all function μ:−∞,0⟶ℝi such that ∫−∞0μτqdτ<∞.Letmτ,τ≥τ0 be a right-continuous Markov chain on the probability space taking values in a finite state space S=1,2,3,…,N with Γ=γjkℕ×ℕ as a generator given by(5)Pms+ϖ=k|ms=j=γjkϖ+oϖ,ifj≠k,1+γjjϖ+oϖ,ifj=k,while ϖ>0. Here, γjk≥0 is the transition rate from j to k, if j≠k, while(6)γjj=−∑j≠kγjk.We consider the following NISFDEwMS:(7)dστ−Gστ=f1στ,τ,mτdτ+f2στ,τ,mτdwτ,τ≥0,Δστp=Ipστp,στp,τp,mτp,p=1,2,…,στ=χ∈PC−l,0;ℝi,τ∈−l,0,where στ is defined by στθ=στ+θ,−l≤θ≤0, which is a PC−l,0;ℝi-valued stochastic process, τp≥0 are impulsive moments satisfying τp<τp+1 and limτp⟶+∞τp=+∞, and Δστp=στp+−στp represents the jump in the state σ at τp with Ip determining the size of the jump. Furthermore, we assume that(8)f1:PC−l,0;ℝi×0,+∞×S⟶ℝi,f2:PC−l,0;ℝi×0,+∞×S⟶ℝn×m,G:PC−l,0;ℝi⟶ℝi,Ip:ℝi×PC−l,0;ℝi×0,+∞×S⟶ℝi.LetC1,2ℝi×−l,0×S,ℝ+ be the family of all nonnegative functions Vσ,τ,j on ℝi×−l,0×S, which are twice continuously differentiable with respect to σ and once continuously differentiable with respect to τ.Define the operatorℒV:ℝi×PC−l,0;ℝi×ℝ+×S⟶ℝ by (see [9])(9)ℒVσ,ψ,τ,j=Vτσ−Gψ,τ,j+Vσσ−Gψ,τ,jf1ψ,τ,j+12tracef2Tψ,τ,jVσσσ−Gψ,τ,jf2ψ,τ,j+∑k=1NγjkVσ−Gψ,τ,k,where(10)Vτσ,τ,j=∂Vσ,τ,j∂τ,Vσσ,τ,j=∂Vσ,τ,j∂σ1,…,∂Vσ,τ,j∂σi,Vσσσ,τ,j=∂2Vσ,τ,j∂σm∂σni×i.Throughout this paper, we assume thatf1, f2, G, and Ip are smooth enough to ensure the global existence and uniqueness of solutions for all τ≥0. For any χ∈LF0q−l,0;ℝi, there is a unique solution satisfying (7), which is continuous on the left-hand side and have a finite limit on the right-hand side. We assume that f0,τ,j=0, g0,τ,i=0, and Ip0,0,τ,j=0, where ∀τ≥0, j∈S, and p=1,2,…, which implies that 0 is an equilibrium solution.Definition 1.
The functionβ:ℝ+⟶0,∞ is said to be β-type function if it satisfies the following conditions:(i)
It is continuous and nondecreasing inℝ+ and continuously differentiable in ℝ+(ii)
β0=1, limτ⟶∞βτ=∞, and Mβ=supt>0β′τ/βτ<∞(iii)
For anys, τ≥0, βτ≤βsβτ−sDefinition 2.
System (7) is said to be β-stable in q-th moment (β-s.q.m), if there exists a function β satisfying Definition 1 and positive constants b1 and η such that, for any χ∈PC−l,0;ℝi∩Lq−∞,0,ℝi,(11)Eστq≤b1Eχqβ−ητ,τ≥0,which is equivalent to(12)lim supτ⟶∞lnEστqlnβτ≤−η,when q=2, it is said to be β-stable in mean square.


## 3. Main Results

Theorem 1.
Assume thatV∈C1,2ℝi×−l,∞×S,ℝ+, constants q>1, ε1>0, ε2>0, aj,p≥0, a¯j,p≥0, aj,p2+a¯j,p2≠0, γ1>0, γ2>0, αj≥0, j∈S, and p=1,2,…, and a function β satisfying Definition 1 such that(i)
ε1σq≤Vσ,τ,j≤ε2σq and ∀σ,τ,j∈ℝi×−l,+∞×S.(ii)
∀τ∈τp−1,τp and j∈S satisfying(13)EℒVψ0,ψ,τ,j≤αjβ′τβτEVψ0−Gψ,τ,j,where∀τ≥0 and those ψ∈PC−l,0;ℝi∩Lq−∞,0,ℝi satisfying(14)Emin1≤j≤NVψθ,τ+θ,j≤sβγ2lEmax1≤j≤NVψ0−Gψ,τ,j.(iii)
∀j∈S satisfying(15)EVψ0−Gψ+Ipψ0,ψθ,τp,j,τk+,j≤aj,pEVψ0−Gψ,τp,j+a¯j,psup−l≤θ≤0EVψθ−Gψ,τp,j,.(iv)
sup1≤p<∞τp−τp−1≤γ1.(v)
For anyj∈S, γ2+αj≤lns/γ1.Then, the solution of system (7) is β-stable in p-th moment, where 2q−1kqβγ2l<1 and s=1/maxj∈S,1≤p<∞aj,p+a¯j,pβγ2l>1.Proof.
For anyχ∈PC−l,0;ℝi∩Lq−∞,0,ℝi, letστ=στ,χ be the solution of (7) and mτ=m0=m0, where ∀τ∈−l,0. Let δ>0 be sufficiently small such that τ+δ∈τp−1,τp. By using Itô formula, we obtain(16)EVστ+δ−Gστ+δ,τ+δ,mτ+δ=EVστ−Gστ,τ,mτ+∫ττ+δEℒVσt,t,mtdt.
Letδ⟶0, we can derive that, for τ∈τp−1,τp,(17)D+EVστ−Gστ,τ,mτ=EℒVστ,στ,τ,mτ,where(18)D+gτ=limsuph⟶0+gτ+h−gτh.
LetSτ=βγ2τEVστ−Gστ,τ,j, then we have for τ∈τp−1,τp,(19)D+Sτ=γ2β′τβτβγ2τEVστ−Gστ,τ,j+βγ2τEℒVστ,στ,τ,j.
By (iii), we obtain(20)Sτp+=βγ2τpVστp+−Gστp+,τp+,j≤aj,pSτp+a¯j,pβγ2lsup−l≤θ≤0Sτp+θ.
LetC>0 such that(21)sup−l≤θ≤0Sθ<Cs.
We can claim that, forτ≥−l,(22)Sτ<C.
By (19), it is clear to see that Sτ<C for τ∈−l,0. We will prove that(23)Sτ<C,τ∈0,τ1.
Otherwise, there existsτ¯∈0,τ1 satisfying Sτ¯=C and Sτ<C, for τ∈−l,τ¯. By the continuity of Sτ in 0,τ1, there exists τ˜∈0,τ¯ satisfying(24)Sτ˜=Cs,Sτ>Cs,τ∈τ˜,τ¯.
Forτ∈τ˜,τ¯and θ∈−l,0, we obtain(25)Sτ>Sτ+θ.
Thus, we have(26)sβγ2lEmax1≤j≤NVψ0−Gψ,τ,j≥sβ−γ2τβγ2lSτ>β−γ2τβγ2lSτ+θ≥Emin1≤j≤NVστ+θ−Gστ+θ,τ+θ,j.
Combining (26) with (19) and (ii), for τ∈τ˜,τ¯, we have(27)D+Sτ≤γ2Mββγ2τEVστ−Gστ,τ,j+αjMββγ2τEVστ−Gστ,τ,j=Mββγ2τγ2+αjEVστ−Gστ,τ,j=Mβγ2+αjSτ.
Then,(28)C=Sτ¯≤Sτ˜eMβγ2+αjτ¯−τ˜=CseMβγ2+αjτ¯−τ˜<CseMβγ2+αjτ1≤C.
This is a contradiction. Therefore, (23) holds. By (iii), we can derive(29)Sτ1+≤aj,1Sτ1+a¯j,1βγ2lsup−l≤θ≤0Sθ+τ1<Cs<C.
Now, we will prove that(30)Sτ<C,τ∈τ1,τ2.
Suppose that (30) does not hold, there exists τ¯1∈τ1,τ2 such that(31)Sτ¯1=C,Sτ¯1<C,τ∈−l,τ¯1.
By the continuity ofSτ in τ1,τ2, there exists τ˜1∈τ1,τ¯1 satisfying(32)Sτ˜1=Cs,Sτ>Cs,τ∈τ˜1,τ¯1.
Forτ∈τ˜1,τ¯1 and θ∈−l,0, we have(33)sSτ>Sτ+θ.
Using (19) and (ii), for τ∈τ˜1,τ¯1, we obtain(34)D+Sτ≤Mβγ2+αjSτ.
Thus, we obtain(35)C=Sτ¯1≤Sτ˜1eMβγ2+αjτ¯1−τ˜1=CseMβγ2+αjτ¯1−τ˜1<CseMβγ2+αjτ2−τ1≤C.
This is a contradiction. Then, (30) holds. By induction, we can show that, for p=1,2,…,(36)Sτ<C,τ∈τp−1,τp.
Hence, we can derive forτ≥−l,(37)Sτ<C.
By (i) and (37), we obtain(38)ε1Eστ−Gστq≤EVστ−Gστ,τ,j≤Cβ−γ2τ,which implies that(39)βγ2τEσσ−Gστq≤Cϵ1.
On the other hand, for anyT>0, we have(40)sup0≤τ≤Tβγ2τEστq≤2q−1sup0≤τ≤Tβγ2τEστ−Gστq+2q−1sup0≤τ≤Tβγ2τEGστq≤2q−1sup0≤τ≤Tβγ2τEστ−Gστq+2q−1kqsup0≤τ≤Tβγ2τEστq≤2q−1sup0≤τ≤Tβγ2τEστ−Gστq+2q−1kqsup−l≤τ≤Tβγ2τ−θEστq≤2q−1Cϵ1+2q−1kqsup−l≤τ≤0βγ2τβγ2−θEστq+2q−1kqsup0≤τ≤Tβγ2τβγ2−θEστq≤2q−1Cϵ1+2q−1kqβγ2lβγ20Eχq+2q−1kqβγ2lsup0≤τ≤Tβγ2τEστq.
Then,(41)sup0≤τ≤Tβγ2τEστq≤11−2q−1kqβγ2l×2q−1Cε1+kqβγ2lEχq.
LettingT⟶+∞, we have(42)sup0≤τ≤+∞βγ2τEστq≤C1,where(43)C1=11−2q−1kqβγ2l×2q−1Cϵ1+kqβγ2lEχq.
This implies that, for allτ≥0,(44)Eστq≤C1β−γ2τ.
Therefore,(45)limsupτ⟶∞lnEστqlnβτ≤−γ2,as desired.Theorem 2.
Assume thatV∈C1,2ℝi×−l,∞×S,ℝ+, constants q>1, ε1>0, ε2>0, aj,p≥0, a¯j,p≥0, aj,p2+a¯j,p2≠0, γ>0, λ>0, μj>0, j∈S, andp=1,2,…, and a function β satisfying Definition 1 such that(i)
ε1σq≤Vσ,τ,j≤ε2σq and ∀σ,τ,j∈ℝi×−l,+∞×S.(ii)
∀τ∈τp−1,τp and j∈S satisfying(46)EℒVψ0,ψ,τ,j≤−μjβ′τβτEVψ0−Gψ,τ,j,where∀τ≥0 and those ψ∈PC−l,0;ℝi∩Lq−∞,0,ℝi satisfying(47)Emin1≤j≤NVψθ,τ+θ,j≤sβγlEmax1≤j≤NVψ0−Gψ,τ,j.(iii)
∀j∈S satisfying(48)EVψ0−Gψ+Ipψ0,ψθ,τp,j,τp+,j≤aj,pEVψ0−Gψ,τp,j+a¯j,psup−l≤θ≤0EVψθ−Gψθ,τp,j,.(iv)
inf1≤p<∞τp−τp−1≥λ.(v)
For anyj∈S, μj−γ>lns/λ.Then, the solution of system (7) is β-stable in q-th moment, where 2q−1kqβγl<1 and s=1/maxj∈S,1≤p<∞aj,p+a¯j,pβγl>1.Proof.
For anyχ∈PC−l,0;ℝi∩Lq−∞,0,ℝi, let στ=στ,χ be the solution of (7) and mτ=m0=m0, where ∀τ∈−l,0. Since (v) is satisfied, we can choose δ>0 sufficiently small such that(49)μj−γ≥lns+δλ,∀j∈S.
Let(50)Sτ=βγτEVστ−Gστ,τ,j.
Forτ∈τp−1,τp, we obtain(51)D+Sτ=γβ′τβτβγτEVστ−Gστ,τ,j+βγτEℒVστ,στ,τ,j.
By (iii), we can derive(52)Sτp+=βγτpEVστp+−Gστp+,τp+,j≤aj,pSτp+a¯j,pβγlsup−l≤θ≤0Sτp+θ.
LetC>0 such that(53)sup−l≤θ≤0Sθ<Cs+δ.
We will prove that, forτ≥−l,(54)Sτ<C.
Using (51), it is obvious to see that(55)Sτ<C,forτ∈−l,0.
Now, we will prove that(56)Sτ<C,forτ∈0,τ1.
Otherwise, there existsτ¯∈0,τ1 satisfying Sτ¯=C and Sτ<C, for τ∈−l,τ¯. By the continuity of Sτ in 0,τ1, there exists τ˜∈0,τ¯ satisfying(57)Sτ˜=Cs+δ,Sτ>Cs+δ,τ∈τ˜,τ¯.
Forτ∈τ˜δ,τ¯ and θ∈−l,0, we obtain(58)s+δSτ>Sτ+θ.
Thus,(59)s+δβγlEmax1≤j≤NVψ0−Gψ,τ,j≥s+δβ−γτβγlSτ>β−γτβγlSτ+θ≥Emin1≤j≤NVστ+θ−Gστ+θ,τ+θ,j.
Combining (59) with (51) and (ii), for τ∈τ˜,τ¯, we obtain(60)D+Sτ≤γMββγτEVστ−Gστ,τ,j−μjMββγτEVστ−Gστ,τ,j=Mβγ−μjβγτEVστ−Gστ,τ,j=Mβγ−μjSτ≤0.
Then,(61)C=Sτ¯≤Sτ˜=Cs+δ<C,which is a contradiction. Hence, (56) holds. Now, we will prove that(62)Sτ1≤Cs+δ.
If it is not satisfied, we haveSτ1>C/s+δ. By the continuity of Sτ in 0,τ1, there exists τ∗∈0,τ1 satisfying(63)Sτδ∗=Cs+δ,Sτ>Cs+δ,τ∈τ∗,τ1.
Forτ∈τ∗,τ1 and θ∈−l,0, we obtain(64)s+δSτ>Sτ+θ.
By (51) and (ii), for τ∈τ∗,τ1 and θ∈−l,0, we have(65)D+Sτ≤Mβγ−μjSτ≤0.
Then,(66)Sτ1≤Sτ∗=Cs+δ.
This is a contradiction. Using (52), (56), and (62), it follows that(67)Sτ1+≤aj,1Sτ1+a¯j,1βγlsup−l≤θ≤0Sτ1+θ≤aj,1+a¯j,1βγls+δC<C.
Moreover, we can show that(68)Sτ<C,∀τ∈τ1,τ2.
Indeed, there isτ¯1∈τ1,τ2 satisfying(69)Sτ¯1=C,Sτ<C,τ∈−l,τ¯1.
There existsSτ>C/s+δ for τ∈τ1,τ¯1. Thus, s+δSτ>Sτ+θ for τ∈τ1,τ¯1 and θ∈−l,0. Then, by (51) and (ii), for τ∈τ1,τ¯1, we obtain(70)D+Sτ≤Mβγ−μjSτ≤0,which implies that(71)Sτ¯1≤Sτ1+<C,which is a contradiction. If there is τ¯¯1∈τ1,τ¯1 satisfying Sτ¯¯1≤C/s+δ. By the continuity of Sτ in τ1,τ2, there exists τ1∗¯¯∈τ¯¯1,τ¯1 satisfying(72)Sτ1∗¯¯=Cs+δ,Sτ>Cs+δ,τ∈τ1∗¯¯,τ¯1.
Thus, forτ∈τ1∗¯¯,τ¯1 and θ∈−l,0,(73)s+δSτ>Sτ+θ.
Then, for(74)D+Sτ≤Mβγ−μjSτ≤0,we have(75)Sτ¯1≤Sτ1∗¯¯=Cs+δ<C.
This is a contradiction. Hence, we can see thatSτ2≤C/s+δ. If this is false, we have Sτ2>C/s+δ. To prove this claim, there are two cases to be presented:Case 1: forτ∈τ1,τ2, Sτ>C/s+δ. By (53), (56), and (69), we have for τ∈τ1,τ2 and θ∈−l,0,(76)s+δSτ>Sτ+θ.Thus, by (51) and (ii), for τ∈τ1,τ2,(77)D+Sτ≤Mβγ−μjSτ.Then,(78)Sτ2≤Sτ1+eMβγ−μjτ2−τ1<CeMβγ−μjτ2−τ1≤Cs+δ,which implies a contradiction.Case 2: there isτ1∗˜∈τ1,τ2 satisfying Sτ1∗˜≤C/s+δ. Since Sτ2>C/s+δ and by the continuity of Sτ in τ1,τ2, there is τ1∗˜˜∈τ1∗˜,τ2 satisfying(79)Sτ1∗˜˜=Cs+δ,Sτ>Cs+δ,τ∈τ1∗˜˜,τ2.
Forτ∈τ1∗˜˜,τ2 and θ∈−l,0, we obtain(80)s+δSτ>Sτ+θ.
By (51) and (ii), for τ∈τ1∗˜˜,τ2, we have(81)D+Sτ≤Mβγ−μjSτ≤0.
Then,(82)Sτ2≤Sτ1∗˜˜=Cs,which is a contradiction. By induction, we can show that, for p=1,2,…,(83)Sτ<C,τ∈τp−1,τp.
Finally, we can derive that, forτ≥−l, Sτ<C.
Proceeding as the proof of Theorem1, we have ∀τ≥0 and (84)Eστq≤C1β−γτ,where(85)C1=11−2q−1kqβγl×2q−1Cε+kqβγlEχq.
Therefore,(86)limsupτ⟶∞lnEστqlnβτ≤−γ,as desired.


## 4. Example

Consider the following NISFDEwMS:(87)dστ−Gστ−1=f1στ−1,τ,mτdτ+f2στ−1,τ,mτdwτ,τ≠τp,Δστp=−ϵpστp+ϵpστp−1,andτp=0.5p,p=1,2,…,στ=χ,τ∈−1,0.where(88)f1στ−1,τ,1=−α11+τστ+α2τ1+τ2Gστ−1,f1στ−1,τ,2=12f1στ−1,τ,1,f2στ−1,τ,1=α31+τστ−Gστ−1,f2στ−1,τ,2=12f2στ−1,τ,1,where ε>0 is arbitrary and l=1. We assume that α1≥α2>0 and α3∈ℝ such that α1>α32/2, wτ is standard one-dimensional Brownian Motion, and G:PC−1,0,ℝi⟶ℝi is Borel measurable such that G0=0, and there exists k∈0,1 such that(89)Gψ1−Gψ2≤ksup−1≤θ≤0ψ1θ−ψ2θ,where ∀ψ1,ψ2∈PC−1,0ℝi and(90)Γ=−111−1.Let(91)Vσ,τ,1=Vσ,τ,2=σ2,then(92)ℒVστ,στ−1,τ,1≤2στ−Gστ−1T−α11+τστ+α2τ1+τ2Gστ−1+α321+τ2στ−Gστ−12≤−2α11+τστ−Gστ−1Tστ−Gστ−1+α321+τ2στ−Gστ−12≤−11+τ2α1−α32Vστ−Gστ−1,τ,1.Then,(93)EℒVστ,στ−1,τ,1≤−11+τ2α1−α32EVστ−Gστ−1,τ,1.Thus,μ1=2α1−α32 and βτ=1+τ.By the same computation, we have(94)EℒVστ,στ−1,τ,2≤−121+τ2α1−α32EVστ−Gστ−1,τ,2.Thus,μ2=1/22α1−α32 and βτ=1+τ. For j=1,2,(95)EVψ0−Gψ+Ipψ0,ψθ,τp,j,τp+,j=Eστp−Gστp−1+εpστp−1−εpστp2=E1−εpστp−Gστp−1+εpστp−1−Gστp−12≤21−εp2Eστp−Gστp−12+2ε2psup−1≤θ≤0Eστp+θ−Gστp+θ2=aj,pEVστp−Gστp−1,τp,j+a¯j,psup−1≤θ≤0EVστp+θ−Gστp+θ,τp+θ,j,where a1,p=a2,p=21−εp2 and a¯1,p=a¯2,p=2ε2p. Hence, we can choose λ, γ, and ϵ>0 such that max1≤p<∞1−εp2<1/4, max1≤p<∞ε2p2γ<1/4, λ≤0.5, μj−γ>lns/λ, and(96)s=1maxj∈1,2,1≤p<∞aj,p+a¯j,p2γ>1.Then, system (87) is polynomially stable in mean square.The response curve of the Markovian switching withm0=1 is given in Figure 1. Taking Gx=1/2sinx and choosing α1=2, α2=1, α3=1, and ε=0.8, we obtain the response curve of solution of system (87) in Figure 2. In Figure 3, it can be seen that continuous dynamics of system (87) with initial condition χτ=τ2−1 can be successfully stabilized by delayed impulses.Figure 1
Response curve of Markovian switching inmτ in system (87).Figure 2
Response curve of system (87) without impulses with initial condition χτ=τ2−1.Figure 3
Response curve of system (87) with impulses with initial condition χτ=τ2−1.From Figures2 and 3, although neutral SFDEwMS without impulsive effect may be polynomially unstable in mean square, adding impulsive effect may become polynomially stable in mean square, which implies that impulsive effect may change the stable behavior of a system.


## 5. Conclusion

In this paper,β-stability in q-th moment for NISFDEwMS is obtained. For future work, we will try to obtain the β-stability of random impulsive stochastic pantograph differential equation.



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*Source: 9975897-2021-05-08.xml*