# Exponential Stability of Impulsive Stochastic Delay System Based on Razumikhin Method and Its Application to Chaos Control

Authors: Shiguo Huang; Yujun Niu; Yajing Xu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the exponential stability of a stochastic delay system with impulsive signal is considered, and stability theorem of this system is proposed based on the Lyapunov–Razumikhin method; the convergence rate is also given, which gives theoretical foundation to chaos control and synchronization using the impulsive method. Finally, the classic delay chaos system with white noise and impulsive signal is employed to verify the feasibility and effectiveness of our theorem.


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## Body

## 1. Introduction

The control of dynamic systems, including but not limited to nonstationary control [1], chaos control [2], etc., has always been an important topic. The impulsive method is an important control scheme, especially for chaos control. It can obtain stability and synchronization using only small signal, which is very important when the system cannot tolerate continuous disturbance or the control signal is expensive. Many scholars studied impulsive phenomenon [3–13]. Yang [7] and Sun and Zhang [9] made constructive contribution in the determinate impulsive system. Wu [14] studied the P-moment stability of a jumping system. Cao et al. [15, 16] investigated the Pth moment exponential stability for stochastic functional differential equations under different conditions. Li et al., Niu et al., and some other scholars [17–21] investigated the stability of impulsive stochastic differential equations and its application to chaos control and synchronization using the impulsive method. In addition, delay is a common phenomenon in nonlinear science, which can induce complex dynamic action and has aroused the strong interest of many scholars. Yang et al. [22, 23] studied the mean stability and exponential P-stability of impulsive stochastic differential equations with delays. We know that the Lyapunov–Razumikhin method is an important and effective tool to study the stability of delay differential equations. Yan and Shen [24] and Wang and Liu [25] investigated the impulsive stabilization of a class of delay differential equations, and numerical examples were given to verify the method mentioned in their work. Unfortunately, for the impulsive stochastic delay differential system, there is no theory about the stability using the Razumikhin method, which gives theoretical guarantee to impulsive control and synchronization in the stochastic delay system. In this paper, a judgement theorem about impulsive stochastic delay differential equations based on the Lyapunov–Razumikhin method is deduced, and chaos control of stochastic Lorenz system using the impulsive method is employed to verify the effectiveness of our theorem.


## 2. Preliminary Notes and Definitions

LetRn be an n-dimensional Euclidean space with the norm ⋅. Rn×m is the set of all n×m matrices, R+=x:x∈R,x≥0, Rt0=t:t∈R+,t≥t0, and Sρ0=X∈Rn|X<ρ0. The family of all nonnegative functions from Rt0×Rn to R+ is denoted by C1,2Rt0×Rn,R+, of which these functions are first-order continuously differentiable with respect to the first variable and second-order differentiable with respect to the second variable. ψτ=sup−τ≤s≤0Eψs, for ψ∈PC−τ,0,Rn,τ>0, where EX means the expectation of X and PC−τ,0,Rn denotes the family of all piecewise right continuous functions.Consider the following impulsive stochastic delay system:(1)dXt=Ft,Xtdt+Gt,XtdWt,t≠tk,ΔXt=Iktk,Xtk−,k∈N,Xt0=X0,where F,G,Ik:Rt0×PC−τ,0,Rn⟶Rn, Wt is a normal m-dimensional Wiener process, X0∈PC−τ,0,Rn, 0≤t0<t1<t2<⋯<tk<⋯, limk⟶∞tk=∞, tk is the moment when impulse happens, ΔXt=Xt−Xt−, Xi∈PC−τ,0,Rn, and Xis=Xi+s, i=t,t−, s∈−τ,0. We suppose there is a solution Xt=Xt,t0,X0 satisfying (1), which is left continuous: Xτi−=limt⟶τi−0Xt=Xτi and has right limit: Xτi+=limt⟶τi+0Xt.Definition 1.
FunctionV: Rt0×Rn⟶R+ belongs to class V0 if(1)
V is continuous in each of sets τk−1,τk×Rn, for all t≥0, Vt,0=0, and(2)limt,y⟶τk+,xVt,y=Vτk+,x.(2)
V is locally Lipschitzian in x∈Rn.Definition 2.
For eachVt,X∈C1,2Rt0×Rn,R+, define an operator L from Rt0×Rn to R as LVt,X=Vtt,X+VXt,Xft,X+1/2tracegTt,XVXXgt,X, where Vtt,X=∂Vt,X/∂t, VXXt,X=∂2Vt,X/∂xi∂xjn×n, and(3)VXt,X=∂Vt,X∂x1,∂Vt,X∂x2,…,∂Vt,X∂xn.Definition 3.
Trivial solution of (1) is global exponentially stable if there is α>0, M≥1 such that(4)EXt,t0,X0≤MX0τe−αt−t0holds for Xt0=X0 and t,X0∈PC−τ,0,Rn.


## 3. Exponential Stability for Impulsive Stochastic Delay Differential Equations Based on Lyapunov–Razumikhin Method

In this section, the Lyapunov–Razumikhin method is employed to study the exponential stability of impulsive stochastic delay differential equations, and judgement theorem of this problem will be deduced too.Theorem 1.
If there exists functionV∈V0 and p,c,c1,c2>0,α>τ,λ>c such that(I)
c1EXp≤EVt,X≤c2EXp for t∈R+andX∈Rn;(II)
ELVt,X≤cEVt,X holds for all t∈tk−1,tk,k∈N, q≥e2λα, whenever qEVt,X≥EVt+s,Xts for s∈−τ,0;(III)
EVtk+,Xtk+Iktk,Xtk≤dkEVtk,Xtk, where dk>0,k∈N;(IV)
τ≤tk−tk−1≤α,lndk+λα<−λtk+1−tk,then the trivial solution of system (1) is exponentially stable, and the convergence rate is λ/p.Proof.
ChooseM≥1 such that(5)c2EX0τp<MEX0τpe−λt1−t0−αc<MEX0τpe−λt1−t0≤qc2EX0τp.
LetXt=Xt,t0,X0 be the solution of system (1), Xt0=X0, and Vt=Vt,X. Then, we will prove that the following inequality holds:(6)EVt≤MEX0τpe−λtk−t0,t∈tk−1,tk,k∈N.
First, we will show that (6) holds for k = 1, i.e.,(7)EVt≤MEX0τpe−λt1−t0,t∈t0,t1.
From condition (I) and equation (5), for t∈t0−τ,t0, we have(8)EVt≤c2EXp≤c2EXτp<MEXτpe−λt1−t0−αc<MEXτpe−λt1−t0.
If (7) is not true, there must be some t−∈t0,t1 such that(9)EVt−>MEX0τpe−λt1−t0>MEX0τpe−λt1−t0−αc>c2EX0τp≥Vt0+s,s∈−τ,0.
AsVt is continuous in t0,t−, there must exist t∗∈t0,t− such that(10)EVt∗=MEX0τpe−λt1−t0,and when t∈t0−τ,t∗,(11)EVt≤MEX0τpe−λt1−t0.
There existst∗∗∈t0,t− such that(12)EVt∗∗=c2EX0τp.
Whent∈t∗∗,t∗, we have(13)EVt≥c2EX0τp.
So, for allt∈t∗∗,t∗,(14)EVt+s≤MEX0τpe−λt1−t0≤qc2EX0τp≤qEVt,s∈−τ,0.
From condition (II),LVt≤cVt holds for t∈t∗∗,t∗; then, we have(15)EVt∗∗≥EVt∗e−αc,i.e.c2EX0τp≥MEX0τpe−λt1−t0−αc,which is in conflict with (5). In other words, (6) holds for k = 1.
Suppose that (6) holds for k=m, m∈N, i.e.,(16)EVt≤MEX0τpe−λtk−t0,t∈tk−1,tk,holds for k=1,2,…,m. From condition (III) and (16), we know that(17)EVtm≤dmEVtm−<e−λαe−λtm+1−tmMEX0τpe−λtm−t0<MEX0τpe−λtm+1−t0.
In the following, we will prove that (16) holds for k=m+1, i.e.,(18)EVt≤MEX0τpe−λtm+1−t0,t∈tm,tm+1.
If (18) is not true, let(19)t−=inft∈tm,tm+1|EVt>MEX0τpe−λtm+1−t0.
From (17), we know that t−≠tm. As Vt is continuous in tm,tm+1, we have(20)EVt−=MEX0τpe−λtm+1−t0,and when t∈tm,t−,(21)EVt≤MEX0τpe−λtm+1−t0,(22)EVtm<MEX0τpe−λtm+1−t0−λα<EVt−hold from (17). Then, there exists t∗∈tm,t−, such that(23)EVt∗=MEX0τpe−λtm+1−t0−λα.
Furthermore,(24)EVt∗≤EVt≤EVt−holds for t∈t∗,t−. As τ≤tk−tk−1≤α, t+s∈tm−1,t− holds for t∈t∗,t− and s∈−τ,0. We have the following inequality for t∈t∗,t− from (16) and (21):(25)EVt+s≤MEX0τpe−λtm−t0=MEX0τpe−λtm+1−t0eλtm+1−tm≤MEX0τpeλα−λtm+1−t0=e2λαEVt∗≤qEVt,s∈−τ,0.
But from condition (II), (24), and λ>c, we know that(26)EVt−≤EVt∗eαc=e−λtm+1−t0−λαMEX0τpeαc<EVt−,which leads to a conflict, so (16) holds for k=m+1.
Therefore, (16) holds for all k∈N. From condition (I),(27)EX≤M∗EX0τe−λ/pt−t0holds for t∈tk−1,tk,k∈N, where M∗=max1,M/c11/p. In other words, the trivial solution of system (1) is exponentially stable, and the convergence rate is λ/p.


## 4. Numerical Example

In order to show the validity and effectiveness of Theorem1, we investigate chaos controllability of the Lorenz system with white noise and delay using the impulsive method in the sense of exponential stability.The Lorenz system with delay and white-noise perturbation can be written as(28)x•=σy−x+λ1xt−t1+βxdwtdt,y•=rx−y−xz+λ2xt−t2+βydwtdt,z•=−bz+xy+λ3zt−t3+βzdwtdt,where σ, r, b, β, λ1,λ2,andλ3 are all nonnegative parameters, t1,t2,andt3 are nonnegative time delay parameters, and wt is the normal Wiener process. When σ=10, r=28,b=8/3,β=0.5,λ1=λ2=λ3=0.1, and t1=t2=t3=0.05, (28) is chaotic, and time history is given in Figure 1.Figure 1
Time history of (28) under chaotic state.In the impulsive control scheme, signals are added to system (28) at discrete time τi,i=1,2,…, and system (28) suffers sudden changes at these instants, and this impulsive control scheme can be written as(29)x•=−σx+σy+λ1xt−t1+βxdwtdt,y•=rx−y−xz+λ2xt−t2+βydwtdt,z•=−bz+xy+λ3xt−t3+βzdwtdt,xτi+=1+k1xτi,yτi+=1+k2yτi,zτi+=1+k3zτi,where k1,k2,andk3 are control parameters and t≠τi, i=1,2,…. Suppose that impulses are equidistant, and τi+1−τi=δ, i=0,1,2,…,δ>0. In the following, we will find out whether system (29) satisfies the conditions of Theorem 1, from which we can judge whether system (28) can be controlled using the impulsive method. Let Vt,X=x2+y2+z2, and it is easy to find that condition (I) of Theorem 1 holds.(30)ELVt,X=β2−2σx2+β2−2y2+β2−2bz2+2σ+rxy+2λ1xt−t1x+2λ2yt−t2y+2λ3zt−t3z≤β2−σ+r+λ1x2+β2−2+σ+r+λ2y2+β2−2b+λ3z2+λ1x2t−t1+λ2y2t−t2+λ3z2t−t3.LetK = maxβ2−σ+r+λ1,β2+σ+r−2+λ2,β2−2b+λ3,(31)λ∗=maxλ1,λ2,λ3,and we have(32)ELVt,e≤λ∗+KEVt,e,so condition (II) of Theorem 1 holds. When t=tk,k=1,2,…,EVtk,etk+Iktk+etk=1+k12Eex2tk+1+k22Eey2tk+1+k32Eez2tk, and let dk=max1+k12,1+k22,1+k32; then,(33)EVtk,etk+Iktk+etk≤dkEVtk−,etk,so condition (III) of Theorem 1 holds. In this example, let k1=−1.6,k2=k3=−1.1, and we have dk=0.36. Let λ=4, and thus, lndk≤−2λδ, and then condition (IV) of Theorem 1 holds. According to Theorem 1, system (29) is exponentially stable, and the convergence rate is 2. Figure 2 shows the evolution of xt,yt,andzt, from which we can find that system (29) approaches the origin as time goes by. In other words, system (28) can be controlled using the impulsive method.Figure 2
Time history of system (29) under stable state.
(a)(b)(c)


## 5. Conclusion

In this paper, exponential stability of impulsive stochastic delay differential equations is studied using the Lyapunov–Razumikhin method, and a judgement theorem of Razumikhin form is given, which gives the theoretical foundation to chaos control and synchronization for the stochastic delay system using the impulsive method. The convergence rate of the control system can be deduced too. Impulsive control for the Lorenz system with white noise and time delay is taken as an example to show the validity and effectiveness of our theorem, and numerical simulation is employed to show the feasibility of the theoretical result.



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