# Stability and Stabilization of Impulsive Stochastic Delay Differential Equations

Authors: Kaining Wu; Xiaohua Ding
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

We consider the stability and stabilization of impulsive stochastic delay differential equations (ISDDEs). Using the Lyapunov-Razumikhin method, we obtain the sufficient conditions to guarantee thepth moment exponential stability of ISDDEs. Then the almost sure exponential stability is considered and the sufficient conditions of the almost sure exponential stability are obtained. Moreover, the stabilization problem of ISDDEs is studied and the criterion on impulsive stabilization of ISDDEs is established. At last, examples are presented to illustrate the correctness of our results.


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## Body

## 1. Introduction

In recent years, the systems with stochastic or impulsive effects were studied by many authors due to their importance in many branches of science and industry, see [1–10] and references therein. In practice, a given system may be with stochastic, delay and impulsive effects simultaneously, so it is necessary to investigate the properties of impulsive stochastic delay differential equations.There are a lot of papers discussing ISDDEs, for example, see [11–17] and the references therein. In [11, 13], the authors studied the stability of a class of impulsive delay differential equations where the impulsive effects are nonlinear. In [12], the stability of a nonlinear ISDDE was studied and the equivalent relation between the stability of the nonlinear stochastic differential delay system under impulsive control and that of a corresponding nonlinear stochastic differential delay system without impulses was established. In [14], the authors studied the stability of nonlinear impulsive stochastic differential equations in terms of two measures and the concept of perturbing Lyapunov functions is introduced to discuss stability properties. In [15], the pth exponential stability and almost sure exponential stability were studied by the Lyapunov-Krasovskii functional method. In [16], the authors considered the pth moment exponential stability by using an inequality and the propertied of M-cone. In [17], the authors studied the mean square exponential stability of ISDDEs by the formula for the variation of parameters and Cauchy matrix.From the existing lectures, we can see the stability of ISDDEs is a main research direction. For ISDDEs, there are at least two questions on stability that need be answered, one is that if the stochastic delay differential equation (SDDE) without impulse is stable, what kinds of the impulses can the system tolerate so that it remains stable? The other is if a SDDE without impulse is unstable, what kind of impulsive effects should we take to make the ISDDEs be stable? The first one we call the question of stability, the second one we call the question of stabilization.As we all know, the Lyapunov-Razumikhin method is a powerful tool to investigate the stability; however, to our best knowledge, there is few work on ISDDEs by using Lyapunov-Razumikhin method.In this paper, we use the Lyapunov-Razumikhin method to answer the questions of stability and stabilization of ISDDEs, give the sufficient conditions ensuring thepth moment exponential stability of ISDDEs, and present the criteria of almost sure exponential stability of ISDDEs. The Lyapunov-Razumikhin method does not require that the formal derivative of the Lyapunov function falls into a restriction in all time; it just need to satisfy the restriction under some situation; therefore, our results relax the restrictions in some existing lectures. At last, examples are given to illustrate the efficiency of our results.


## 2. Preliminaries

Let(Ω,ℱ,P) be a complete probability space with a natural filtration {ℱt}t⩾0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all P-null sets). ℝn denotes the n-dimensional Euclidean space with the Euclidean norm |·|. Let PC([-τ,0],ℝn) to denote the set of piecewise right continuous functions ψ:[-τ,0]→ℝn with the sup-norm ∥ψ∥=sup-τ⩽θ⩽0|ψ(θ)|. Take PCℱ0b([-τ,0],ℝn) to denote the family of all bounded ℱ0-measurable PC([-τ,0],ℝn)-valued random variables ψ satisfying ∥ψ∥=sup-τ⩽θ⩽0𝔼|ψ(θ)|<∞, and PCℱtp([-τ,0],ℝn) denote the family of all ℱt-measurable PC([-τ,0],ℝn)-valued random variables ψ satisfying ∫-τ0𝔼|ψ(θ)|pdθ<∞, where 𝔼 denotes the expectation with respect to the given probability measure P.In this paper, we consider the following impulsive stochastic delay differential system:(2.1)dx(t)=f(x(t),x(t-τ),t)dt+g(x(t),x(t-τ),t)dB(t),t≠tk,x(tk)=Hk(x(tk-)),
where x(tk-)=limh→0+x(tk-h), Hk(x(tk-))=(H1k(x(tk-)),…,Hnk(x(tk-)))T represents the impulsive perturbation of x at time tk and satisfies |Hk(x(tk-))|⩽Γ|x(tk-)|,Γ⩾0, k=1,2,…. The impulsive moments tk satisfy 0=t0<t1<···<tk<··· and limk→∞tk=∞. The functions f:ℝn×ℝn×ℝ+→ℝn and g:ℝn×ℝn×ℝ+→ℝn×m are continuous functions; B(t)=(B1(t),B2(t),…,Bm(t))T is an m-dimensional standard Brownian motion defined on (Ω,ℱ,P).The following initial value is imposed on system (2.1)(2.2)x(s)=ξ(s),s∈[-τ,0],whereξ(t)∈PCℱ0b([-τ,0],ℝn).As a standing hypothesis, we suppose that system (2.1) has a unique solution x(t,ξ) for any given initial value ξ(t)∈PCℱ0b([-τ,0],ℝn), and there exists an M(p) for any p>0 such that 𝔼|x(t,ξ)|p<M(p). Suppose x(t,ξ) is left continuous and right limitable. Moreover, we assume that f(0,0,t)≡0, g(0,0,t)≡0 and Hk(0)=0 for any positive integer k, then system (2.1) admits a trivial solution.Letv0 denote the set of nonnegative functions V(x,t) on ℝn×([-τ,0]∪ℝ+), which are twice continuously differential in x and once in t on ℝn×[tk,tk+1).IfV(x,t)∈v0, define an operator LV:ℝn×ℝn×[tk,tk+1)→ℝ associated with system (2.1) as follows:(2.3)LV(x,y,t)=Vt(x,t)+Vx(x,t)f(x,y,t)+12trace[gT(x,y,t)Vxx(x,t)g(x,y,t)],where(2.4)Vt(x,t)=∂V(x,t)∂t,Vx(x,t)=(∂V(x,t)∂x1,…,∂V(x,t)∂xn),Vxx=(∂2V(x,t)∂xixj)n×n.Definition 2.1 (see [15, 16]).
The trivial solution of system (2.1) is said to be pth moment exponentially stable (p>0) if there exist positive constants μ and M such that for any initial value ξ(t)∈PCℱ0b([-τ,0],ℝn),(2.5)E|x(t,ξ)|p⩽ME‖ξ‖pe-μt,t⩾0.Definition 2.2 (see [15]).
The trivial solution of system (2.1) is said to be almost surely exponentially stable if there exists a positive constant γ such that for any initial value ξ(t)∈PCℱ0b([-τ,0],ℝn) and t⩾0,
(2.6)limsup⁡t→∞1tln⁡|x(t,ξ)|⩽-γa.s.
If the trivial solution of system (2.1) is pth moment exponentially stable or almost surely exponentially stable, we also say the system (2.1) is pth moment exponentially stable or almost surely exponentially stable.


## 3. Stability and Stabilization of ISDDEs

In this section, we establish the criteria ofpth moment exponential stability for system (2.1) by using the Lyapunov-Razumikhin technique, and the almost sure exponential stability is also considered. Moreover, the stabilization theorem is presented for system (2.1). The results show that impulses may change the stability of a given system. Some techniques used in the proof are motivated by the paper [5].Theorem 3.1.
Assume that there exist a functionV(x,t)∈v0 and positive constants p,c1,c2,λ,μ,γ, q>1,dk⩾0, and k is any positive integer, such that the following conditions hold:(1)
c1|x|p⩽V(x,t)⩽c2|x|p for any x∈ℝn and t∈[-τ,0]∪ℝ+;(2)
LV(φ(t),φ(t-τ),t)⩽-λV(φ(t),t) for all t≠tk in ℝ+ whenever V(φ(t+s),t+s)⩽qV(φ(t),t) for any s∈[-τ,0], q⩾eλτ;(3)
V(Hk(x(tk-)),tk)⩽(1+dk)V(x(tk-),tk-);(4)
μ⩽tk-tk-1, dk⩽Γ, and ln(1+Γ)/μ<λ.Then, for any ξ(t)∈PCℱ0b([-τ,0];ℝn) and t⩾0, the solution x(t,ξ) of system (2.1) satisfies
(3.1)E|x(t,ξ)|p⩽c2c1E‖ξ‖pe-γt,where γ=min{λ-ln(1+Γ)/μ,lnq/τ}, that is, system (2.1) is pth moment exponentially stable.Proof.
For a givenξ, let x(t)=x(t,ξ) and write V(x(t),t)=V(t) for the simplicity.
We claim(3.2)EV(t)⩽c2∏i=0n-1(1+di)E‖ξ‖pe-λt,
for any t∈[tn-1,tn), where d0=0.
Let(3.3)Q(t)={EV(t)-c2(∏i=0n-1(1+di))E‖ξ‖pe-λt,t∈[tn-1,tn),EV(t)-c2E‖ξ‖pe-λt,t∈[-τ,0].
It is easy to check Q(t) is continuous and differentiable in [tn-1,tn), and
(3.4)Q′(t)=ELV(t)+λc2(∏i=0n-1(1+di))E‖ξ‖pe-λt,for t∈[tn-1,tn).
To verify (3.2), we just need to show that Q(t)⩽0 for all t⩾0.
We first show thatQ(t)⩽0 for t∈(0,t1).
Fort∈[-τ,0], we have e-λt⩾1; using condition 1, we can get Q(t)⩽0. Let α be an arbitrary positive constant; we claim(3.5)Q(t)⩽α,
for t∈(0,t1).
If (3.5) is not true, then there must exist a t¯∈(0,t1), such that Q(t)>α, which implies that there exists a t*∈(0,t¯) such that Q(t*)=α and Q(t)⩽α for t∈[-τ,t*] being the continuity of Q(t) in [-τ,t*]. Noting(3.6)EV(t*)=Q(t*)+c2E‖ξ‖pe-λt*=α+c2E‖ξ‖pe-λt*,using the fact q⩾eλτ, Q(t*+s)=𝔼V(t*+s)-c2𝔼∥ξ∥pe-λ(t*+s), Q(t*+s)⩽α for s∈[-τ,0], we have, for any s∈[-τ,0],
(3.7)EV(t*+s)=Q(t*+s)+c2E‖ξ‖pe-λ(t*+s)⩽α+c2E‖ξ‖pe-λ(t*-τ)⩽(α+c2E‖ξ‖pe-λt*)eλτ=EV(t*)eλτ⩽qEV(t*).
By virtue of condition 2, we can obtain𝔼LV(t*)⩽-λ𝔼V(t*); then(3.8)Q′(t*)=ELV(t*)+λc2E‖ξ‖pe-λt*⩽-λ(EV(t*)-c2E‖ξ‖pe-λt*)=-λQ(t*)=-λα<0,
which contradicts the definition of t*. So we get Q(t)⩽α for all t∈[0,t1). Let α→0+; we obtain Q(t)⩽0 for t∈[0,t1).
Now, assumeQ(t)⩽0 for t∈[0,tm). In view of condition 3, we have(3.9)Q(tm)=EV(tm)+c2(∏i=0m(1+di))E‖ξ‖pe-λtm⩽(1+dm)(EV(tm-)+c2(∏i=0m-1(1+di))E‖ξ‖pe-λtm)=(1+dm)Q(tm-)⩽0.
Next, we will show, for arbitraryα>0,(3.10)Q(t)⩽αfort∈[tm,tm+1).
For the sake of contradiction, suppose (3.10) is not true. Define(3.11)t̃=inf⁡{t∈[tm,tm+1)∣Q(t)>α}.From (3.9), we have t̃≠tm. The continuity of Q(t) in [tm,tm+1) yields Q(t̃)=α and Q(t)⩽α for t∈[tm,t̃].
Since𝔼V(t̃)=Q(t̃)+c2∏i=0m(1+di)𝔼∥ξ∥pe-λt̃, then, for s∈[-τ,0],(3.12)EV(t̃+s)=Q(t̃+s)+c2(∏i=0m(1+di))E‖ξ‖pe-λ(t̃+s)⩽α+c2(∏i=0m(1+di))E‖ξ‖pe-λ(t̃-τ)⩽[α+c2(∏i=0m(1+di))E‖ξ‖pe-λt̃]eλτ=EV(t̃)eλτ⩽qEV(t̃).
In view of condition 2, we obtain 𝔼LV(t̃)⩽-λ𝔼V(t̃), then
(3.13)Q′(t̃)=ELV(t̃)+λc2(∏i=0m(1+di))E‖ξ‖pe-λt̃⩽-λ[EV(t̃)-c2(∏i=0m(1+di))E‖ξ‖pe-λt̃]=-λQ(t̃)=-λα<0,
which contradicts the definition of t̃. Therefore, Q(t)⩽α for all t∈[tm,tm+1). Let α→0+; we have Q(t)⩽0 for t∈[tm,tm+1). Thus, by induction, we obtain Q(t)⩽0 holds for t⩾0; hence
(3.14)EV(t)⩽c2(∏i=0n-1(1+di))E‖ξ‖pe-λtfort∈[tn-1,tn).Then by condition 1, we have
(3.15)E|x|p⩽c2c1(∏i=0n-1(1+di))E‖ξ‖pe-λt⩽c2c1E‖ξ‖pexp⁡{nln⁡(1+Γ)-λt}⩽c2c1E‖ξ‖pexp⁡{ln⁡(1+Γ)μt-λt}⩽c2c1E‖ξ‖pe-γt.
This completes the proof.The following theorem states the almost sure exponential stability of system (2.1). In the proof, the classical method used in [4] is borrowed and this method was also adopted in paper [15].Theorem 3.2.
Suppose all of the conditions of Theorem3.1 are satisfied and in addition p⩾2. If there exist positive constants T and K such that 0<tk-tk-1⩽T and for all t⩾0(3.16)E|f(ϕ(t),ϕ(t-τ),t)|p+E(trace[gT(ϕ(t),ϕ(t-τ),t)g(ϕ(t),ϕ(t-τ),t)])p/2⩽Ksup⁡-τ⩽θ⩽0E|ϕ(t+θ)|p,
then for all ξ∈PCℱ0b([-τ,0];ℝn),
(3.17)limsup⁡t→∞1tln⁡|x(t,ξ)|⩽-γpa.s.Proof.
From system (2.1), we can get, for t∈(tn-1,tn),
(3.18)E(sup⁡tn-1<t<tn|x(t)|p)⩽3p-1(E|x(tn-1)|p+E[∫tn-1tn|f(x(s),x(s-τ),s)|ds]p+E[sup⁡tn-1<t<tn|∫tn-1tg(x(s),x(s-τ),s)dB(s)|p]).
Using Hölder’s inequality, condition (3.16), and Theorem 3.1, we derive that
(3.19)E[∫tn-1tn|f(x(s),x(s-τ),s)|ds]p⩽(tn-tn-1)p-1∫tn-1tnE|f(x(s),x(s-τ),s)|pds⩽Tp-1K∫tn-1tnsup⁡-τ⩽θ⩽0E|x(t+θ)|pds⩽Tp-1K∫tn-1tn(c2c1)E‖ξ‖pe-γsds⩽c2c1TpKE‖ξ‖pe-γtn-1⩽c2c1TpeTKE‖ξ‖pe-γtn.
In virtue of Burkholder-Davis-Gundy inequality, we have
(3.20)E[sup⁡tn-1<t<tn∫tn-1t|g(x(s),x(s-τ),s)dB(s)|p]⩽CpE(∫tn-1tntrace[gTg]ds)p/2,where Cp is a positive constant dependent on p only. One can then show in the same way as in (3.19) that
(3.21)E[sup⁡tn-1<t<tn∫tn-1t|g(x(s),x(s-τ),s)dB(s)|p]⩽c2c1CpTp/2eTKE‖ξ‖pe-γtn.
Substituting (3.15), (3.19), and (3.21) into (3.18) yields
(3.22)E(sup⁡tn-1<t<tn|x(t)|p)⩽K1e-γtn,
where
(3.23)K1=3p-1c2c1E‖ξ‖p[1+KeT(Tp+CpTp/2)].
Whent=tn-1, keeping |Hn-1(x(tn-1-))|⩽Γ|x(tn-1-)| and (3.15) in mind, we can get(3.24)E|x(tn-1)|=E|Hn-1(x(tn-1-))|⩽ΓE|x(tn-1-)|⩽c2c1ΓeTE‖ξ‖pe-γtn.
Taking K2=max{K1,c2/c1ΓeT+t0𝔼∥ξ∥p}, we have
(3.25)E(sup⁡tn-1⩽t<tn|x(t)|p)⩽K2e-γtn.
We now show that (3.25) implies the required (3.17).
Letϵ be an arbitrary constant satisfying 0<ϵ<γ. By virtue of (3.25) and Markovian inequality, we have(3.26)P(ω:sup⁡tn-1⩽t<tn|x(t)|p>e-(γ-ϵ)tn)⩽e(γ-ϵ)tnE(sup⁡tn-1⩽t<tn|x(t)|p)⩽K2e-ϵtn⩽K2et0e-nμϵ.In view of Borel-Cantelli lemma, we can obtain that for almost all ω∈Ω,
(3.27)sup⁡tn-1⩽t<tn|x(t)|p⩽e-(γ-ϵ)tn
holds for all but finitely many n. Hence there exists an N(ω), for all ω∈Ω but a P-null set, such that (3.27) holds when n>N(ω). Then we have, for almost all ω∈Ω, if tn-1⩽t<tn, n>N(ω),
(3.28)|x(t)|p⩽e-(γ-ϵ)tn⩽e-(γ-ϵ)t,that is,
(3.29)1tln⁡|x(t)|⩽-γ-ϵpand (3.17) follows by letting ϵ→0.In the following, we give two corollaries.Corollary 3.3.
Assume there exist positive constantsλ,α,μ,p,c1,c2 and a function V(x,t)∈ν0 such that(1)
c1|x|p⩽V(x,t)⩽c2|x|p for all (x,t)∈ℝn×[-τ,∞);(2)
Vt(x,t)+Vx(x,t)f(x,x(t-τ),t)+(1/2)trace[gT(x,x(t-τ),t)Vxx(x(t),t)g(x,x(t-τ),t)]⩽-λV(x,t)+αV(x(t-τ),t-τ) for all t∈(tk-1,tk), where λ>eλτα;(3)
V(Hk(x(tk-)),tk)⩽(1+dk)V(x(tk-),tk-);(4)
μ⩽tk-tk-1, dk⩽Γ, and ln(1+Γ)/μ<λ-eλτα.Then, the trivial solution of system (2.1) is pth moment exponentially stable.
Furthermore, if there exists a positive constantT such that tk-tk-1⩽T and the following inequality holds for any x,y∈ℝn:(3.30)|f(x,y,t)|∨|g(x,y,t)|⩽K(|x|+|y|),then the trivial solution of system (2.1) is almost surely exponentially stable.Proof.
Takeq=eλτ. Obviously, we just need to verify the condition 2 of Theorem 3.1. 							(3.31)LV(x(t),x(t-τ),t)=Vt(x(t),t)+Vx(x(t),t)f(x(t),x(t-τ),t)+12trace(gT(x(t),x(t-τ),t)Vxx(x(t),t)g(x(t),x(t-τ),t))⩽-λV(x(t),t)+αV(x(t-τ),t-τ).
If t⩾0 and t∈(tk,tk+1), s∈[-τ,0], the following inequality holds
(3.32)V(x(t+s),t+s)⩽qV(x(t),t),then
(3.33)LV(x(t),x(t-τ),t)⩽-λV(x(t),t)+αqV(x(t),t)=-(λ-αq)V(x(t),t)=-(λ-αeλτ)V(x(t),t).
Condition 2 of Theorem 3.1 is verified, then the pth moment exponential stability for the trivial solution of system (2.1) is obtained. The almost sure exponential stability is followed directly by virtue of Theorem 3.2.The 2th moment exponential stability; is also called mean square exponential stability, the following corollary presents the criteria of mean square exponential stability of system (2.1).Corollary 3.4.
For system (2.1), assume there exist positive constants a,b,c,d,μ,βk such that(1)
xTf(x,y,t)⩽-a|x|2+b|y|2, and |g(x,y,t)|2⩽c|x|2+d|y|2, 2a-c>0 and 2a-c-e(2a-c)τ(2b+d)>0;(2)
|Hk(x)|⩽βk|x|;(3)
μ⩽tk+1-tk, βk2⩽Γ+1, ln(Γ+2)/μ<2a-c-e(2a-c)τ(2b+d).Then the trivial solution of system (2.1) is mean square exponentially stable.Proof.
LetV(x,t)=|x|2; then
(3.34)LV(x(t),x(t-τ),t)=2xTf(x(t),x(t-τ),t)+|g(x(t),x(t-τ),t)|2⩽-2a|x|2+2b|x(t-τ)|2+c|x|2+d|x(t-τ)|2=(-2a+c)|x|2+(2b+d)|x(t-τ)|2.The conditions of Corollary 3.3 are easily to be verified, and the required result can be obtained.Now we are on the position to state the stabilization theorem.Theorem 3.5.
Assume there exist a functionV(x,t)∈v0 and positive constants p, c1, c2, c, λ, α such that(1)
c1|x|p⩽V(x,t)⩽c2|x|p, for any x∈ℝn and t∈ℝ+;(2)
LV(x(t),x(t-τ),t)⩽cV(x(t),t), for all t∈[tk,tk+1), whenever qV(x(t),t)⩾V(t+s,x(t+s)), for s∈[-τ,0], where q⩾max{e2λα,ecα};(3)
V(Hk(x(tk-)),tk)⩽dkV(tk-,x(tk-)), where dk>0;(4)
τ⩽tk+1-tk⩽α and lndk+λα<-λ(tk+1-tk).Then the trivial solution of system (2.1) is pth moment exponentially stable.Remark 3.6.
From condition 2, we can see that the formal derivative ofV(x,t) can be positive since c is a positive constant; this means that the original system without impulses may be unstable. Therefore, this theorem is called the stabilization theorem.Proof.
Letx(t)=x(t,ξ) be a solution of (2.1) with x(t)=ξ(t), t∈[-τ,0] and write V(x(t),t)=V(t) for simplicity. Choose M⩾1 such that
(3.35)c2E‖ξ‖p<ME‖ξ‖pe-λt1e-αc<ME‖ξ‖pe-λt1⩽qc2E‖ξ‖p.
We will show, for any positive integerk,(3.36)EV(t)⩽ME‖ξ‖pe-λtk,t∈[tk-1,tk).
We first show that(3.37)EV(t)⩽ME‖ξ‖pe-λt1,t∈[0,t1).
From condition 1 and (3.35), we have, for t∈[-τ,0],
(3.38)EV(t)⩽c2E|x(t)|p⩽c2E‖ξ‖p<ME‖ξ‖pe-λt1e-αc.
If (3.37) is not true, then there must be a t¯∈(0,t1) such that(3.39)EV(t¯)>ME‖ξ‖pe-λt1>ME‖ξ‖pe-λt1e-αc>c2E‖ξ‖p⩾EV(s),
where s∈[-τ,0].
Then there exists at*∈(0,t¯) such that(3.40)EV(t*)=ME‖ξ‖pe-λt1,EV(t)⩽ME‖ξ‖pe-λt1,t∈[-τ,t*],
and there is a t**∈[0,t*) such that
(3.41)EV(t**)=c2E‖ξ‖p,EV(t)⩾c2E‖ξ‖P,t**⩽t⩽t*.
Then we have, for any t∈[t**,t*],
(3.42)EV(t+s)⩽ME‖ξ‖pe-λt1⩽qc2E‖ξ‖P⩽qEV(t),s∈[-τ,0].
From condition 2, we have
(3.43)ELV(t)⩽cEV(t)for t∈[t**,t*]. Since t*-t**<α, we get
(3.44)EV(t*)e-αc-EV(t**)⩽EV(t*)e-c(t*-t**)-EV(t**)=∫t**t*e-c(s-t**)(-cEV(s)+ELV(s))ds⩽0,that is,
(3.45)EV(t*)e-αc⩽EV(t**),then
(3.46)ME‖ξ‖pe-λt1e-αc⩽c2E‖ξ‖P,which is in conflict with (3.35). We obtain that (3.37) holds, that is, (3.36) holds when k=1.
Now we assume that (3.36) holds when k=1,2,3,…,m, m is a positive integer and m⩾1, then(3.47)EV(t)⩽ME‖ξ‖pe-λtk,t∈[tk-1,tk),especially,
(3.48)EV(tm-)⩽ME‖ξ‖pe-λtm.
From conditions 3, 4 and (3.48), we have(3.49)EV(tm)⩽dmEV(tm-)<e-λα-λ(tm+1-tm)ME‖ξ‖pe-λtm=e-λαME‖ξ‖pe-λtm+1<ME‖ξ‖pe-λtm+1.
Now we will show that (3.36) holds when k=m+1, that is,
(3.50)EV(t)⩽ME‖ξ‖pe-λtm+1,t∈[tm,tm+1).
If (3.50) is not true, we define
(3.51)t¯=inf⁡{t∈[tm,tm+1)∣EV(t)>ME‖ξ‖pe-λtm+1}.From (3.49), we know t¯≠tm; by the continuity of 𝔼V(t) in [tm,tm+1), we get
(3.52)EV(t¯)=ME‖ξ‖pe-λtm+1,EV(t)⩽ME‖ξ‖pe-λtm+1,t∈[tm,t¯].From (3.49), we have
(3.53)EV(tm)<e-λαME‖ξ‖pe-λtm+1<EV(t¯);
there must be a t*∈(tm,t¯) such that
(3.54)EV(t*)=e-λαME‖ξ‖pe-λtm+1,EV(t*)⩽EV(t)⩽EV(t¯),t∈[t*,t¯].Since τ⩽tk+1-tk⩽α, and s∈[-τ,0], when t∈[t*,t¯], we get t+s∈[tm+1,t¯]. From (3.48) and (3.53), when t∈[t*,t¯],s∈[-τ,0], we have
(3.55)EV(t+s)⩽ME‖ξ‖pe-λtm=ME‖ξ‖pe-λtm+1eλ(tm+1-tm)⩽ME‖ξ‖pe-λtm+1eλα=e2λαEV(t*)⩽qEV(t).From condition 2, we get
(3.56)ELV(t)⩽cEV(t).But
(3.57)EV(t¯)e-αc-EV(t*)⩽EV(t¯)e-(t¯-t*)c-EV(t*)=∫t*t¯e-c(s-t*)(-cEV(s)-ELV(s))ds<0,we get
(3.58)EV(t¯)⩽EV(t*)eαc=ME‖ξ‖pe-λtm+1e-λαeαc=ME‖ξ‖pe-λtm+1e-(λ-c)α<ME‖ξ‖pe-λtm+1=EV(t¯).Then (3.36) holds when k=m+1. By induction, we have that (3.36) holds, and
(3.59)EV(t)⩽ME‖ξ‖pe-λt,t∈[tk-1,tk).From condition 1, we have
(3.60)E|x(t)|p⩽M*E‖ξ‖pe-λt,t∈[tk,tk+1),k∈N,where M*⩾max{1,(M/c1)1/p}. This completes the proof.


## 4. Applications and Examples

In this section, we consider a nonlinear impulsive stochastic delay differential system. We present the stability criterion and stabilization criterion for this system, then we illustrate the correctness of our results using the numerical experiments.The following nonlinear impulsive stochastic delay differential system is considered:(4.1)dx(t)=(ax(t)+bx(t-τ)exp⁡[-x2(t-τ)])dt+cx(t-τ)dW(t),t≠tk,x(tk)=βkx(tk-).By virtue of Corollary3.4, we can get the following corollary directly.Corollary 4.1.
Assume there exist positive constantsμ and T such that the impulsive moments tk satisfy μ⩽tk-tk-1⩽T, and the following inequalities hold:
(4.2)2a+|b|+e(-2a-|b|)τ(2b+c2)<0,βk2<Γ+1,ln⁡(Γ+2)μ<-2a-|b|-e(-2a-|b|)τ(2b+c2),then the trivial solution of system (4.1) is mean square exponentially stable and almost surely exponentially stable.
Using Theorem3.5 and taking V(x,t)=|x|2, we can easily get the following stabilization corollary for system (4.1).Corollary 4.2.
Assume there exist constantsα,λ>0 satisfing(1)
there exists a constantq>0 such that q>max{e2λα,eγα}, where γ=2a+|b|+|b|q+c2q;(2)
τ⩽tk+1-tk⩽α and lnβk+λα/2<-(λ/2)(tk+1-tk).Then the trivial solution of system (4.1) is pth moment exponentially stable.Now let us illustrate the correctness of Corollaries4.1 and 4.2.Leta=-1,b=-1,c=1,τ=1,andβk=2. Take x(s)=s+1 when s∈[-1,0] and tk+1-tk=2. Let μ=1 and Γ=7; it is easy to verify the conditions of Corollary 4.1 are satisfied. Then the trivial solution of system (4.1) is mean square exponentially stable. The mean square exponential stability is showed in Figure 1. This illustrates the correctness of Corollary 4.1.Figure 1
Mean square exponential stability of system (4.1).Now we consider the stabilization of system (4.1). Take a=0.1,b=1,c=0.1,andτ=0.1. Let x(s)=s+1 when s∈[-0.1,0]. It is easy to see that the trivial solution of system (4.1) without impulsive effects is not mean square stable, see Figure 2. Then we take tk+1-tk=0.2,βk=0.8 and α=0.2,λ=1,q=2; it can be verified that the conditions of Corollary 4.2 are satisfied; the the trivial solution of system (4.1) is mean square exponentially stable, see Figure 3.Figure 2
Nonstability of system (4.1) without impulse.Figure 3
Mean square exponential stability of system (4.1).



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*Source: 176375-2012-05-28.xml*