# Stability and Stabilization of Impulsive Stochastic Delay Difference Equations

Authors: Kaining Wu; Xiaohua Ding; Liming Wang
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

When an impulsive control is adopted for a stochastic delay difference
system (SDDS), there are at least two situations that should be contemplated. If the
SDDS is stable, then what kind of impulse can the original system tolerate to keep
stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken
to make the system stable? Using the Lyapunov-Razumikhin technique, we
establish criteria for the stability of impulsive stochastic delay difference equations
and these criteria answer those questions. As for applications, we consider a kind of
impulsive stochastic delay difference equation and present some corollaries to our
main results.


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## Body

## 1. Introduction

In recent years, stochastic delay difference equations (SDDEs) have been studied by many researchers; a number of results have been reported [1–7]. In these literatures, stability analysis stays on the focus of attention; see [1, 2, 4–6] and the references therein. As we all know, when we adopt an impulsive strategy to an SDDE, the stability of the SDDE may be destroyed or strengthen. Impulsive phenomena exist widely in the real world; therefore, it is important to study the stability problem for SDDEs with impulsive effects [8–10], that is to say, the stability problem for impulsive stochastic delay difference equations (ISDDEs).For SDDEs, when we take impulsive effects into account, we have at least two problems to deal with.Problem 1. When a SDDE is stable, what kind of impulsive effect can the system tolerate so that it remain stable? Problem 2. If the SDDE is unstable, then what kind of impulsive effect should be taken to make the system stable? Problems 1 and 2 are called the problem of impulsive stability and the problem of impulsive stabilization, respectively.As well known, Lyapunov-Razumikhin technique is one of main methods to investigate the stability of delay systems [11, 12]. There are little papers on the stability of ISDDEs [13, 14], and up to our knowledge, there is no paper on the stability of ISDDEs using Lyapunov-Razumikhin technique. In this paper, we study the stability of ISDDEs by Lyapunov-Razumikhin technique. We establish criteria for the r-moment exponential stability; these criteria present the answers to Problems 1 and 2. As for applications, we consider a kind of ISDDE and present some corollaries to our main theorems.


## 2. Preliminaries

In the sequel,ℝ denotes the field of real numbers, and ℕ represents the natural numbers. For some positive integer m and n0, let N-m={-m,-m+1,…,-1,0} and Nn0-m={n0-m,n0-m+1,…,n0-1,n0}. Given a matrix A, ∥A∥ denotes the norm of A induced by the Euclidean vector norm. Let C([-r,0],ℝn)={ψ:[-r,0]→ℝn,ψiscontinuous}. Given a positive integer m, we define ∥φ∥m=maxθ∈N-m{∥φ(s)∥} for any φ∈C([-m,0],ℝn). Let (Ω,ℱ,P) be a complete probability space and let {ℱn,n∈ℤ} be a nondecreasing family of sub-σ-algebra of ℱ, that is, ℱn1⊂ℱn2 for n1<n2.Consider the impulsive stochastic delay difference equations of the form(2.1)x(n+1)=f(n,xn)+g(n,xn)ξn,n≠ηk-1,n⩾n0,n,k∈ℕ,x(ηk)=Hk(x(ηk-1)),k∈ℕ,xn0=φ,
where n0∈ℕ, f,g∈C(ℕ×C((-m,0),ℝn),ℝn), and m∈ℕ represents the delay in system (2.1), m⩾2. xn∈C([-m,0],ℝn) is defined by xn(s)=x(n+s) for any s∈[-m,0]. {ξn} are ℱn+1-adapted mutually independent random variables and satisfy Eξn=0, Eξn2=1, where E denotes the mathematical expectation. Hk∈C(ℝn,ℝn). Impulsive moment ηk∈ℕ satisfies: n0<η1<η2<⋯<ηn<⋯, and ηk→∞ as k→∞. Let η0=n0.Assume thatf(n,0)≡0, g(n,0)≡0, and Hk(0)=0, then system (2.1) admits the trivial solution. We also assume there exists a unique solution of system (2.1), denoted by x(n)=x(n,n0,φ), for any given initial data xn0=φ.Definition 2.1.
One calls the trivial solution of system (2.1) r-moment exponentially stable if for any initial data xn0=φ there exist two positive constants α and M, such that for all n⩾n0, n∈ℕ, the following inequality holds:
(2.2)E∥x(n)∥r⩽M∥φ∥mre-αn.If the trivial solution of system (2.1) is r-moment exponentially stable, then we also call the system (2.1)  r-moment exponentially stable.


## 3. Main Results

In this section, we will establish two theorems onr-moment exponential stability of system (2.1); these theorems give the answers to Problems 1 and 2.First, we present the theorem on impulsive stability. The technique adopted in the proof is motivated by [15].Theorem 3.1.
Assume that there exist a positive functionV(n,x) for system (2.1) and positive constants r,p,c1,c2,andλ, where p>1, 0<λ<1, such that.(C1)
c1∥x∥r⩽V(n,x)⩽c2∥x∥r for any n∈Nn0-m∪ℕ and x∈ℝn.(C2)
Forn≠ηk-1, any s∈N-m, EV(n+1,x(n+1))⩽λEV(n,x(n)) whenever EV(n+s,x(n+s))⩽pEV(n,x(n)).(C3)
Forn≠ηk-1, some s∈N-m-{0}, EV(n+1,x(n+1)⩽(1/p)maxθ∈N-m{EV(n+θ,x(n+θ))} whenever EV(n+s,x(n+s))>eαEV(n,x(n)), where α=min{-lnλ,lnp/(m+1)}.(C4)
EV(ηk,x(ηk))⩽dkEV(ηk-1,x(ηk-1)), where dk>1 and d=maxk∈N{dk}<∞.(C5)
ηk+1-ηk>m, α(1-1/m)-lnd/m=β>0.Then for any initial data xn0=φ,
(3.1)E∥x(n)∥r⩽c2c1E∥φ∥mre-βn.
That is the trivial solution of system (2.1) that is r-moment exponentially stable.Proof.
LetU(n)=maxθ∈N-m{eα(n+θ)EV(n+θ,x(n+θ))}. For any n⩾n0, n∈[ηk,ηk+1-1), k∈ℕ, define
(3.2)θ̅n=max{θ∈N-m:eα(n+θ)EV(n+θ,x(n+θ))=U(n)},
then U(n)=eα(n+θ̅n)EV(n+θ̅n,x(n+θ̅n)).
Next, we will show that, for anyn∈[ηk,ηk+1-1),
(3.3)U(n+1)⩽U(n).
For a givenn, we have two situations to contemplate: θ̅n⩽-1 and θ̅n=0.Case 1 (θ̅n⩽-1).
Under this situation, we haveeαnEV(n,x(n))<eα(n+θ̅n)EV(n+θ̅n,x(n+θ̅n)), then
(3.4)EV(n+θ̅n,x(n+θ̅n))>eα(-θ̅n)EV(n,x(n))⩾eαEV(n,x(n)).
Using condition (C3) and noticing p⩾eα(m+1), we obtain
(3.5)maxs∈N-m{EV(n+s,x(n+s))}⩾eα(m+1)EV(n+1,x(n+1)).
Multiplying both sides by eαn and rearranging yield
(3.6)eα(n-m)maxs∈N-m{EV(n+s,x(n+s))}⩾eα(n+1)EV(n+1,x(n+1)).
Then we get
(3.7)eα(n+1)EV(n+1,x(n+1))⩽maxs∈N-m{eα(n+s)EV(n+s,x(n+s))}=U(n),
which implies that
(3.8)U(n+1)⩽U(n).Case 2 (θ̅n=0).
Making use of the definition ofU(n) and θ̅n, noticing that p>e-αθ for any θ∈N-m, we have
(3.9)EV(n+θ,x(n+θ))⩽eα(-θ)EV(n,x(n))<pEV(n,x(n)).
Under condition (C2), the above inequality implies that
(3.10)EV(n+1,x(n+1))⩽λEV(n,x(n)).
Multiplying both sides by eα(n+1), we have
(3.11)eα(n+1)EV(n+1,x(n+1))⩽eα(n+1)λEV(n,x(n))=eαnEV(n,x(n))eαλ⩽eαnEV(n,x(n))=U(n).
Thus
(3.12)U(n+1)⩽U(n),
which is the desired assertion.
Whenn=ηk+1, under condition (C4) and using the definition of U(n), we get
(3.13)U(ηk+1)=maxθ∈N-m{eα(ηk+1+θ)EV(ηk+1+θ,x(ηk+1+θ))}=max{eαηk+1EVmaxθ∈N-m-{0}{eα(ηk+1+θ)EV(ηk+1+θ,x(ηk+1+θ))}(ηk+1,x(ηk+1)),maxθ∈N-m-{0}{eα(ηk+1+θ)EV(ηk+1+θ,x(ηk+1+θ))}}⩽max{dk+1eαeα(ηk+1-1)EVmaxθ∈N-m-{0}{eα(ηk+1+θ)EV(ηk+1+θ,x(ηk+1+θ))}(ηk+1-1,x(ηk+1-1)),maxθ∈N-m-{0}{eα(ηk+1+θ)EV(ηk+1+θ,x(ηk+1+θ))}}⩽dk+1eαmaxθ∈N-m-{0}{eα(ηk+1+θ)EV(ηk+1+θ,x(ηk+1+θ))}⩽dk+1eαU(ηk+1-1)⩽dk+1eαU(ηk).
By induction and taking (3.3) into account, when n∈[ηl,ηl+1), for all l∈ℕ, we have
(3.14)U(n)⩽U(ηl)⩽dleαU(ηl-1)⩽∏i=1l(dieα)U(n0),
which yields
(3.15)eαnEV(n,x(n))⩽∏i=1l(dieα)c2E∥φ∥mr.
By virtue of condition (C5),
(3.16)EV(n,x(n))⩽e-n(α(1-l/n)(-1/n)∑i=1llndi)c2E∥φ∥mr⩽c2E∥φ∥mre-βn.
The desired result follows when we take condition (C1) into account.Now, we are in position to state the theorem on impulsive stabilization. The method used in the proof is motivated by [16].Theorem 3.2.
Assume that there exist a functionV(n,x) for system (2.1) and constants r>0, c1>0, c2>0, λ>0, and natural number α>1, such that the following conditions hold. (C1)
c1∥x∥r⩽V(n,x)⩽c2∥x∥r for any n∈Nn0-m∪ℕ and x∈ℝn.(C2)
Forn≠ηk-1, any s∈N-m, EV(n+1,x(n+1))⩽(1+λ)EV(n,x(n)) whenever qEV(n+1,x(n+1))⩾EV(n+s,x(n+s)), where q>e2λα.(C3)
EV(ηk,x(ηk))⩽dkEV(ηk-1,x(ηk-1)), where dk>0.(C4)
m⩽ηk+1-ηk⩽α, lndk+αλ<-λ(ηk+1-ηk).Then for any initial data xn0=φ there exists positive constant C; for any n∈ℕ, the following inequality holds:
(3.17)E∥x(n)∥r⩽CE∥φ∥mre-λn,
that is, the trivial solution of system (2.1) is r-moment exponentially stable.Proof.
ChooseM>1 such that
(3.18)(1+λ)c2E∥φ∥mr⩽ME∥φ∥mre-λη1e-αλ<ME∥φ∥mre-λη1⩽qc2E∥φ∥mr.
We will show that, for any n∈[ηk,ηk+1), k=1,2,…,
(3.19)EV(n,x(n))⩽ME∥φ∥mre-ληk+1.
WriteEV(n,x(n))=EV(n) for the sake of brevity.
First we will show that, for anyn∈[0,η1),
(3.20)EV(n)⩽ME∥φ∥mre-λη1.
Obviously, when n∈[-m,0], EV(n)⩽ME∥ξ∥mre-λη1. If (3.20) is not true, then there exists n̅∈[0,η1-1) such that
(3.21)EV(n̅+1)>ME∥φ∥mre-λη1.
And when n⩽n̅, EV(n)⩽ME∥φ∥mre-λη1. At the same time there exists n*⩾0 such that EV(n*)⩽c2E∥φ∥mr, and when n*<n⩽n̅,
(3.22)c2E∥φ∥mr<EV(n)⩽ME∥φ∥mre-λη1.
Note that there may not exist the natural numbern that satisfies n*<n⩽n̅ such that (3.22) holds. However, we claim that there must be a natural number n satisfing n*<n⩽n̅ such that (3.22) holds. If not, we have n*=n̅; then we get
(3.23)EV(n)⩽c2E∥φ∥mr,n⩽n̅.
Obviously,
(3.24)qEV(n̅+1)⩾EV(n̅+s),∀s∈N-m.
Under condition (C2) we get
(3.25)EV(n̅+1)⩽(1+λ)EV(n̅).
That is
(3.26)EV(n̅)⩾11+λEV(n̅+1)>11+λME∥φ∥mre-λη1=eαλ1+λME∥φ∥mre-λη1e-αλ>ME∥φ∥mre-λη1e-αλ⩾c2E∥φ∥mr,
which contradicts with (3.23). Then there must be an n satisfing n*<n⩽n̅ such that (3.22) holds.
For anyn∈[n*+1,n̅],
(3.27)EV(n+s)⩽ME∥φ∥mre-λη1<qc2E∥φ∥mr<qEV(n).
By virtue of (C2), for any n∈[n*+1,n̅],
(3.28)EV(n)⩽(1+λ)EV(n-1),
and for s∈N-m, we have
(3.29)qEV(n̅+1)⩾EV(n̅+s),qEV(n*+1)⩾EV(n*+s).
Making use of (3.28), we get
(3.30)EV(n̅+1)⩽(1+λ)EV(n̅)⩽(1+λ)n̅-n*EV(n*+1)⩽(1+λ)αEV(n*)<eαλc2E∥φ∥mr.
Taking (3.3) into account, the above inequality yields
(3.31)EV(n̅+1)>ME∥φ∥mre-λη1,
which implies that
(3.32)ME∥φ∥mre-λη1<eαλc2E∥φ∥mr.
It contradicts with (3.18); then (3.20) holds, that is, (3.19) holds for k=1.
Assume that (3.19) holds for k=1,2,…,h, that is, when n∈[ηk-1,ηk), k=1,2,…,h,
(3.33)EV(n)⩽ME∥φ∥mre-ληk.
Under conditions (C3) and (C4), we have
(3.34)EV(ηh)⩽dhEV(ηh-1)⩽dhME∥φ∥mre-ληh⩽ME∥φ∥mre-ληh+1e-αλ⩽ME∥φ∥mre-ληh+1.
Now we will show that, whenn∈[ηh,ηh+1),
(3.35)EV(n)⩽ME∥φ∥mre-ληh+1.
If (3.35) is not true, then there must be an n̅∈(ηh,ηh+1-1), such that
(3.36)EV(n̅+1)>ME∥φ∥mre-ληh+1,
and for n∈[ηh,n̅](3.37)EV(n)⩽ME∥φ∥mre-ληh+1.
At the same time, there exists an n*∈[ηh,n̅] such that
(3.38)EV(n*)⩽ME∥φ∥mre-ληh+1e-αλ,
And, when n*<n⩽n̅,
(3.39)EV(n)>ME∥φ∥mre-ληh+1e-αλ.
If there does not exist an n satisfing n*<n⩽n̅ such that (3.39) holds, then n*=n̅. Obviously, for any s∈N-m, qEV(n̅+1)⩾EV(n̅+s). Using condition (C2) yields EV(n̅+1)⩽(1+λ)EV(n̅), that is,
(3.40)EV(n̅)⩾11+λEV(n̅+1)⩾eλα1+λME∥φ∥mre-ληh+1e-αλ>ME∥φ∥mre-ληh+1e-αλ,
which contradicts with the definition of n̅; then there exists at least one number n satisfing n*<n⩽n̅ such that (3.39) holds.
Forn∈[n*+1,n̅] and s∈N-m, we have
(3.41)EV(n+s)⩽ME∥φ∥mre-ληh=eλ(ηh+1-ηh)ME∥φ∥mre-ληh+1⩽e2λαME∥φ∥mre-ληh+1e-αλ<qEV(n),
which implies that, under condition (C2),
(3.42)EV(n)⩽(1+λ)EV(n-1).
Obviously, qEV(n̅+1)⩾EV(n̅). Using condition (C2) again, we get
(3.43)EV(n̅+1)⩽(1+λ)EV(n̅).
since qEV(n*+1)>EV(n*+s), s∈N-m, we have, under condition (C2)
(3.44)EV(n*+1)⩽(1+λ)EV(n*).
Then
(3.45)EV(n̅+1)⩽(1+λ)EV(n̅)⩽(1+λ)n̅-n*EV(n*+1)⩽(1+λ)n̅-n*+1EV(n*)⩽(1+λ)αEV(n*)<eαλME∥φ∥mre-ληh+1e-αλ=ME∥φ∥mre-ληh+1<EV(n̅),
which conflicts with the definition of n̅. Then (3.19) holds for k=h+1.
By induction, we know that (3.19) holds for n∈[ηk,ηk+1), k∈ℕ.
Using condition (C1), for any n∈[ηk,ηk+1), k∈ℕ, we have
(3.46)c1E∥x(n)∥r⩽EV(n)⩽ME∥φ∥mre-ληk+1⩽ME∥φ∥mre-λn.
That is the desired result.


## 4. Applications

In this section, we consider a kind of impulsive stochastic delay difference equation as follows:(4.1)x(n+1)=f(n,x(n),x(n-m))+g(n,x(n),x(n-m))ξn,n≠ηk-1,x(ηk)=Hk(x(ηk-1)),x(n0+s)=φ(s),s∈N-m.Using the obtained results, we present three corollaries for system (4.1).Corollary 4.1.
Assume that conditions(C1), (C4), and (C5) of Theorem 3.1 hold, but conditions (C2) and (C3) are replaced with the following conditions:(C2*)
There exist constantsλ1 and λ2, 0<λ1, λ2<1, such that
(4.2)EV(n+1,x(n+1))⩽λ1EV(n,x(n))+λ2EV(n-m,x(n-m)).If λ1+λ2<1, then the trivial solution of system (4.1) is r-moment exponentially stable.Proof.
Letx(n) be a solution of system (4.1). Take (p=λ12+4λ2+λ2-λ1)/3λ2. It is easy to see that, under the conditions 0<λ1, λ2<1, and 0<λ1+λ2<1,
(4.3)1<p<λ12+4λ2-λ12λ2<1λ1+λ2<1-λ1λ2.
If EV(n+θ,x(n+θ))⩽pEV(n,x(n)) for any θ∈N-m, it follows from the condition (C2*) that
(4.4)EV(n+1,x(n+1))⩽(λ1+pλ2)EV(n,x(n)).
Then condition (C2) of Theorem 3.1 follows under (4.3).
Letλ=λ1+pλ2; using inequality (4.3), we get α in Theorem 3.1: α=min{-lnλ,lnp/(m+1)}=lnp/(m+1).
Now we assume thatV(n+θ,x(n+θ))>eαV(n,x(n)) for some θ∈N-m; by virtue of (C2*) and inequality (4.3),
(4.5)EV(n+1,x(n+1))⩽λ1EV(n,x(n))+λ2EV(n-m,x(n-m))<λ1e-αEV(n+θ,x(n+θ))+λ2EV(n-m,x(n-m))<(λ1+λ2)maxs∈N-m{EV(n+s,x(n+s))}<1pmaxs∈N-m{EV(n+s,x(n+s))}.
Then condition (C3) of Theorem 3.1 follows, which completes the proof.From the above proof, we know that constantβ in Theorem 3.1 equals to(4.6)m-1m(m+1)lnλ12+4λ2+λ2-λ13λ2-lndm.Corollary 4.2.
Assume that conditions (C1), (C4), and (C5) of Theorem 3.1 hold, but conditions (C2) and (C3) are replaced with the following condition.(C2**)
There exists a constant0<λ<1 such that
(4.7)EV(n+1,x(n+1))⩽λmaxs∈N-m{EV(n+s,x(n+s))}.Then the trivial solution of system (4.1) is r-moment exponentially stable.Proof.
Letx(n) be a solution of system (4.1). Take
(4.8)p=(1λ)(m+1)/(m+2).
Since 0<λ<1, we have 1<p<1/λ and
(4.9)lnp(m+1)=ln(1/λ)(m+2)<ln(1λ).
For any s∈N-m, assume that V(n+s,x(n+s))⩽pV(n,x(n)); by virtue of condition (C2**), we get
(4.10)EV(n+1,x(n+1))⩽pλEV(n),
that is, condition (C2) of Theorem 3.1.
Since1<p<1/λ we have 1/p>λ. Under condition (C2**), for any n∈ℕ, we get
(4.11)EV(n+1,x(n+1))⩽λmaxs∈N-m{EV(n+s,x(n+s))}<(1p)maxs∈N-m{EV(n+s,x(n+s))},
that is condition (C3) of Theorem 3.1.From the above proof, we know that constantα in Theorem 3.1 equals to(4.12)min{-ln(λp),lnp(m+1)}=-ln(λp)=lnp(m+1)=-lnλ(m+2).
Then constant β in Theorem 3.1 equals(4.13)-lnλ(m+2)(1-1m)-lndm.Now, we present a corollary of Theorem3.2 which establishes a criterion of mean square exponential stability for system (4.1).Corollary 4.3.
Assume that there exist positive constantsλ, α, and q where α is a natural number and α>1, q⩾e2λαsuch that system (4.1) satisfies the following.(1)
(4.14)E∥f(n,x(n),x(n-m))∥2+E∥g(n,x(n),x(n-m))∥2⩽12(aE∥x(n)∥2+bE∥x(n-m)∥2),
where a,b are positive constants, b<1/q, and
(4.15)0<a+bq-11-bq⩽λ.(2)
∥Hk(x)∥⩽βk∥x∥, for any x∈ℝn, βk>0, and 2lnβk+λ(ηk+1-ηk)⩽-λα. The impulsive moments satisfy m⩽ηk+1-ηk⩽α.Then, for any initial data xn0=φ, the solution x(n) of system (4.1) satisfies
(4.16)E∥x(n)∥2≤E∥φ∥m2e-(λ/2)n.
That is to say, the trivial solution of system (4.1) is mean square exponentially stable.Proof.
LetV(n,x)=∥x∥2, then,
(4.17)EV(n+1,x(n+1))=E∥x(n+1)∥2=E∥f(n,x(n),x(n-m))+g(n,x(n),x(n-m))ξn∥2⩽2(E∥f(n,x(n),x(n-m))∥2+E(∥g(n,x(n),x(n-m))∥2ξn2))⩽aE∥x(n)∥2+bE∥x(n-m)∥2=aEV(n)+bEV(n-m).
Assume that qEV(n+1,x(n+1))⩾EV(n+s,x(n+s)) holds for any s∈N-m, then
(4.18)EV(n+1)=E∥x(n+1)∥2⩽a1-bqEV(n)⩽(1+λ)EV(n).
The other conditions of Theorem 3.2 are easy to be verified and the conclusion of this corollary now follows.


## 5. Examples

Now we study some examples to illustrate our results.We consider a linear impulsive stochastic delay difference equation as following:(5.1)x(n+1)=ax(n)+bx(n-m)+cx(n-m)ξn,n≠ηk-1,n⩾0,x(ηk)=βkx(ηk-1),k∈ℕ,x(s)=φ(s),s∈N-m.First we takea=0.5, b=0.25, c=0.25, m=9, ηk=10k, βk=1.1, k=1,2,…, and φ(s)=1/(100+s2). By virtue of Corollary 4.1, taking V(x,n)=x2(n), we can get the mean square exponential stability of (5.1). The stability is shown in Figure 1.Figure 1
Mean square exponential stability of (5.1): a=0.5, b=0.25, c=0.25, m=9, ηk=10k, andβk=1.1.Now we takea=1.09, b=0, c=e-19, m=3, and φ(s)=1/(10+s2) in (5.1) without impulsive effects. It is easy to see that equation is unstable. This property is shown in Figure 2. Then we take an impulsive strategy: ηk=4k, βk=e-19. In light of Corollary 4.3, we can see that equation is mean square exponentially stable. The stability is shown in Figure 3.Figure 2
Instability without impulsive effects of (5.1): a=1.09, b=0, c=e-19, andm=3.Figure 3
Mean square exponential stability of (5.1): a=1.09, b=0, c=e-19, m=3, ηk=4k, andβk=e-19.It should be pointed that the conditions of Corollary4.3 are sufficient but not necessary. If we take a=1.09, b=0, c=e-19, m=3, ηk=4k, and βk=0.7 and φ(s)=1/(10+s2), then it is not difficult to show that the conditions of Corollary 4.3 are not satisfied again, but under this situation, the equation is still stable. The stability is shown in Figure 4.Figure 4
Stability of (5.1): a=1.09, b=0, c=e-19, m=3, ηk=4k, andβk=0.7.


## 6. Conclusions

In this paper, we considered ther-moment exponential stability for impulsive stochastic delay difference equations. Using the Lyapunov-Razumikhin method, we established criteria of r-moment exponential stability and these criteria presented the answers for the problem of impulsive stability and the problem of impulsive stabilization. As for applications, we considered a kind of impulsive stochastic delay difference equation and obtained three corollaries for our main theorems. The results we got may work in the study of stability of numerical method for the impulsive delay differential equations.



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