Stability and Stabilization of Impulsive Stochastic Delay Difference Equations

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.

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 𝑁_𝑛0−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],ℝ^𝑛)={𝜓∶[−𝑟,0]→ℝ^𝑛,𝜓iscontinuous}. Given a positive integer 𝑚, we define ‖𝜑‖_𝑚=max_{𝜃∈𝑁−𝑚}{‖𝜑(𝑠)‖} for any 𝜑∈𝐶([−𝑚,0],ℝ^𝑛). Let (Ω,ℱ,𝑃) be a complete probability space and let {ℱ_𝑛,𝑛∈ℤ} be a nondecreasing family of sub-𝜎-algebra of ℱ, that is, ℱ_𝑛1⊂ℱ_𝑛2 for 𝑛₁<𝑛₂.

Consider the impulsive stochastic delay difference equations of the form

𝑥(𝑛+1)=𝑓(𝑛,𝑥_𝑛)+𝑔(𝑛,𝑥_𝑛)𝜉_𝑛,𝑛≠𝜂_𝑘−1,𝑛⩾𝑛₀,𝑛,𝑘∈ℕ,𝑥(𝜂_𝑘)=𝐻_𝑘(𝑥(𝜂_𝑘𝑥−1)),𝑘∈ℕ,_𝑛0=𝜑, (2.1) where 𝑛₀∈ℕ, 𝑓,𝑔∈𝐶(ℕ×𝐶((−𝑚,0),ℝ^𝑛),ℝ^𝑛), and 𝑚∈ℕ represents the delay in system (2.1), 𝑚⩾2. 𝑥_𝑛∈𝐶([−𝑚,0],ℝ^𝑛) is defined by 𝑥_𝑛(𝑠)=𝑥(𝑛+𝑠) for any 𝑠∈[−𝑚,0]. {𝜉_𝑛} are ℱ_𝑛+1-adapted mutually independent random variables and satisfy 𝐸𝜉_𝑛=0, 𝐸𝜉²_𝑛=1, where 𝐸 denotes the mathematical expectation. 𝐻_𝑘∈𝐶(ℝ^𝑛,ℝ^𝑛). Impulsive moment 𝜂_𝑘∈ℕ satisfies: 𝑛₀<𝜂₁<𝜂₂<⋯<𝜂_𝑛<⋯, and 𝜂_𝑘→∞ as 𝑘→∞. Let 𝜂₀=𝑛₀.

Assume that 𝑓(𝑛,0)≡0, 𝑔(𝑛,0)≡0, and 𝐻_𝑘(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 𝑥_𝑛0=𝜑.

Definition 2.1. One calls the trivial solution of system (2.1) 𝑟-moment exponentially stable if for any initial data 𝑥_𝑛0=𝜑 there exist two positive constants 𝛼 and 𝑀, such that for all 𝑛⩾𝑛₀, 𝑛∈ℕ, the following inequality holds: 𝐸‖𝑥(𝑛)‖^𝑟⩽𝑀‖𝜑‖^𝑟_𝑚𝑒^−𝛼𝑛. (2.2)

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 on 𝑟-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 function 𝑉(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∈𝑁_𝑛0−𝑚∪ℕ and 𝑥∈ℝ^𝑛.(𝐶₂) For 𝑛≠𝜂_𝑘−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) For 𝑛≠𝜂_𝑘−1, some 𝑠∈𝑁_−𝑚−{0}, 𝐸𝑉(𝑛+1,𝑥(𝑛+1)⩽(1/𝑝)max_{𝜃∈𝑁−𝑚}{𝐸𝑉(𝑛+𝜃,𝑥(𝑛+𝜃))} whenever 𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))>𝑒^𝛼𝐸𝑉(𝑛,𝑥(𝑛)), where 𝛼=min{−ln𝜆,ln𝑝/(𝑚+1)}.(𝐶₄)𝐸𝑉(𝜂_𝑘,𝑥(𝜂_𝑘))⩽𝑑_𝑘𝐸𝑉(𝜂_𝑘−1,𝑥(𝜂_𝑘−1)), where 𝑑_𝑘>1 and 𝑑=max_𝑘∈𝐍{𝑑_𝑘}<∞.(𝐶₅)𝜂_𝑘+1−𝜂_𝑘>𝑚, 𝛼(1−1/𝑚)−ln𝑑/𝑚=𝛽>0.Then for any initial data 𝑥_𝑛0=𝜑, 𝐸‖𝑥(𝑛)‖^𝑟⩽𝑐₂𝑐₁𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝛽𝑛. (3.1) That is the trivial solution of system (2.1) that is 𝑟-moment exponentially stable.

Proof. Let 𝑈(𝑛)=max_{𝜃∈𝑁−𝑚}{𝑒^{𝛼(𝑛+𝜃)}𝐸𝑉(𝑛+𝜃,𝑥(𝑛+𝜃))}. For any 𝑛⩾𝑛₀, 𝑛∈[𝜂_𝑘,𝜂_𝑘+1−1), 𝑘∈ℕ, define 𝜃_𝑛=max{𝜃∈𝑁_−𝑚∶𝑒^{𝛼(𝑛+𝜃)}𝐸𝑉(𝑛+𝜃,𝑥(𝑛+𝜃))=𝑈(𝑛)}, (3.2) then 𝑈(𝑛)=𝑒^{𝛼(𝑛+𝜃𝑛)}𝐸𝑉(𝑛+𝜃_𝑛,𝑥(𝑛+𝜃_𝑛)).
Next, we will show that, for any 𝑛∈[𝜂_𝑘,𝜂_𝑘+1−1), 𝑈(𝑛+1)⩽𝑈(𝑛). (3.3)
For a given 𝑛, we have two situations to contemplate: 𝜃_𝑛⩽−1 and 𝜃_𝑛=0.Case 1 (𝜃_𝑛⩽−1). Under this situation, we have 𝑒^𝛼𝑛𝐸𝑉(𝑛,𝑥(𝑛))<𝑒^{𝛼(𝑛+𝜃𝑛)}𝐸𝑉(𝑛+𝜃_𝑛,𝑥(𝑛+𝜃_𝑛)), then 𝐸𝑉(𝑛+𝜃_𝑛,𝑥(𝑛+𝜃_𝑛))>𝑒^{𝛼(−𝜃𝑛)}𝐸𝑉(𝑛,𝑥(𝑛))⩾𝑒^𝛼𝐸𝑉(𝑛,𝑥(𝑛)). (3.4) Using condition (𝐶₃) and noticing 𝑝⩾𝑒^𝛼(𝑚+1), we obtain max_{𝑠∈𝑁−𝑚}{𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))}⩾𝑒^𝛼(𝑚+1)𝐸𝑉(𝑛+1,𝑥(𝑛+1)). (3.5) Multiplying both sides by 𝑒^𝛼𝑛 and rearranging yield 𝑒^{𝛼(𝑛−𝑚)}max_{𝑠∈𝑁−𝑚}{𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))}⩾𝑒^𝛼(𝑛+1)𝐸𝑉(𝑛+1,𝑥(𝑛+1)). (3.6) Then we get 𝑒^𝛼(𝑛+1)𝐸𝑉(𝑛+1,𝑥(𝑛+1))⩽max_{𝑠∈𝑁−𝑚}{𝑒^{𝛼(𝑛+𝑠)}𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))}=𝑈(𝑛), (3.7) which implies that 𝑈(𝑛+1)⩽𝑈(𝑛). (3.8)Case 2 (𝜃_𝑛=0). Making use of the definition of 𝑈(𝑛) and 𝜃_𝑛, noticing that 𝑝>𝑒^−𝛼𝜃 for any 𝜃∈𝑁_−𝑚, we have 𝐸𝑉(𝑛+𝜃,𝑥(𝑛+𝜃))⩽𝑒^{𝛼(−𝜃)}𝐸𝑉(𝑛,𝑥(𝑛))<𝑝𝐸𝑉(𝑛,𝑥(𝑛)). (3.9) Under condition (𝐶₂), the above inequality implies that 𝐸𝑉(𝑛+1,𝑥(𝑛+1))⩽𝜆𝐸𝑉(𝑛,𝑥(𝑛)). (3.10) Multiplying both sides by 𝑒^𝛼(𝑛+1), we have 𝑒^𝛼(𝑛+1)𝐸𝑉(𝑛+1,𝑥(𝑛+1))⩽𝑒^𝛼(𝑛+1)𝜆𝐸𝑉(𝑛,𝑥(𝑛))=𝑒^𝛼𝑛𝐸𝑉(𝑛,𝑥(𝑛))𝑒^𝛼𝜆⩽𝑒^𝛼𝑛𝐸𝑉(𝑛,𝑥(𝑛))=𝑈(𝑛). (3.11) Thus 𝑈(𝑛+1)⩽𝑈(𝑛), (3.12) which is the desired assertion.
When 𝑛=𝜂_𝑘+1, under condition (𝐶₄) and using the definition of 𝑈(𝑛), we get 𝑈(𝜂_𝑘+1)=max_{𝜃∈𝑁−𝑚}{𝑒^{𝛼(𝜂𝑘+1+𝜃)}𝐸𝑉(𝜂_𝑘+1+𝜃,𝑥(𝜂_𝑘+1+𝜃))}=max{𝑒^{𝛼𝜂𝑘+1}𝐸𝑉(𝜂_𝑘+1,𝑥(𝜂_𝑘+1)),max_{𝜃∈𝑁−𝑚−{0}}{𝑒^{𝛼(𝜂𝑘+1+𝜃)}𝐸𝑉(𝜂_𝑘+1+𝜃,𝑥(𝜂_𝑘+1+𝜃))}}⩽max{𝑑_𝑘+1𝑒^𝛼𝑒^{𝛼(𝜂𝑘+1−1)}𝐸𝑉(𝜂_𝑘+1−1,𝑥(𝜂_𝑘+1−1)),max_{𝜃∈𝑁−𝑚−{0}}{𝑒^{𝛼(𝜂𝑘+1+𝜃)}𝐸𝑉(𝜂_𝑘+1+𝜃,𝑥(𝜂_𝑘+1+𝜃))}}⩽𝑑_𝑘+1𝑒^𝛼max_{𝜃∈𝑁−𝑚−{0}}{𝑒^{𝛼(𝜂𝑘+1+𝜃)}𝐸𝑉(𝜂_𝑘+1+𝜃,𝑥(𝜂_𝑘+1+𝜃))}⩽𝑑_𝑘+1𝑒^𝛼𝑈(𝜂_𝑘+1−1)⩽𝑑_𝑘+1𝑒^𝛼𝑈(𝜂_𝑘). (3.13) By induction and taking (3.3) into account, when 𝑛∈[𝜂_𝑙,𝜂_𝑙+1), for all 𝑙∈ℕ, we have 𝑈(𝑛)⩽𝑈(𝜂_𝑙)⩽𝑑_𝑙𝑒^𝛼𝑈(𝜂_𝑙−1)⩽^𝑙_𝑖=1(𝑑_𝑖𝑒^𝛼)𝑈(𝑛₀), (3.14) which yields 𝑒^𝛼𝑛𝐸𝑉(𝑛,𝑥(𝑛))⩽^𝑙_𝑖=1(𝑑_𝑖𝑒^𝛼)𝑐₂𝐸‖𝜑‖^𝑟_𝑚. (3.15) By virtue of condition (𝐶₅), 𝐸𝑉(𝑛,𝑥(𝑛))⩽𝑒^{−𝑛(𝛼(1−𝑙/𝑛)(−1/𝑛)∑𝑙𝑖=1ln𝑑𝑖)}𝑐₂𝐸‖𝜑‖^𝑟_𝑚⩽𝑐₂𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝛽𝑛. (3.16) 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 function 𝑉(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∈𝑁_𝑛0−𝑚∪ℕ and 𝑥∈ℝ^𝑛.(𝐶₂) For 𝑛≠𝜂_𝑘−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>𝑒^2𝜆𝛼.(𝐶₃)𝐸𝑉(𝜂_𝑘,𝑥(𝜂_𝑘))⩽𝑑_𝑘𝐸𝑉(𝜂_𝑘−1,𝑥(𝜂_𝑘−1)), where 𝑑_𝑘>0.(𝐶₄)𝑚⩽𝜂_𝑘+1−𝜂_𝑘⩽𝛼, ln𝑑_𝑘+𝛼𝜆<−𝜆(𝜂_𝑘+1−𝜂_𝑘).Then for any initial data 𝑥_𝑛0=𝜑 there exists positive constant 𝐶; for any 𝑛∈ℕ, the following inequality holds: 𝐸‖𝑥(𝑛)‖^𝑟⩽𝐶𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝑛, (3.17) that is, the trivial solution of system (2.1) is 𝑟-moment exponentially stable.

Proof. Choose 𝑀>1 such that (1+𝜆)𝑐₂𝐸‖𝜑‖^𝑟_𝑚⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1𝑒^−𝛼𝜆<𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1⩽𝑞𝑐₂𝐸‖𝜑‖^𝑟_𝑚. (3.18) We will show that, for any 𝑛∈[𝜂_𝑘,𝜂_𝑘+1), 𝑘=1,2,…, 𝐸𝑉(𝑛,𝑥(𝑛))⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂𝑘+1}. (3.19)
Write 𝐸𝑉(𝑛,𝑥(𝑛))=𝐸𝑉(𝑛) for the sake of brevity.
First we will show that, for any 𝑛∈[0,𝜂₁), 𝐸𝑉(𝑛)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1. (3.20) Obviously, when 𝑛∈[−𝑚,0], 𝐸𝑉(𝑛)⩽𝑀𝐸‖𝜉‖^𝑟_𝑚𝑒^−𝜆𝜂1. If (3.20) is not true, then there exists 𝑛∈[0,𝜂₁−1) such that 𝐸𝑉(𝑛+1)>𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1. (3.21) And when 𝑛⩽𝑛, 𝐸𝑉(𝑛)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1. At the same time there exists 𝑛^∗⩾0 such that 𝐸𝑉(𝑛^∗)⩽𝑐₂𝐸‖𝜑‖^𝑟_𝑚, and when 𝑛^∗<𝑛⩽𝑛, 𝑐₂𝐸‖𝜑‖^𝑟_𝑚<𝐸𝑉(𝑛)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1. (3.22)
Note that there may not exist the natural number 𝑛 that satisfies 𝑛^∗<𝑛⩽𝑛 such that (3.22) holds. However, we claim that there must be a natural number 𝑛 satisfing 𝑛^∗<𝑛⩽𝑛 such that (3.22) holds. If not, we have 𝑛^∗=𝑛; then we get 𝐸𝑉(𝑛)⩽𝑐₂𝐸‖𝜑‖^𝑟_𝑚,𝑛⩽𝑛. (3.23) Obviously, 𝑞𝐸𝑉(𝑛+1)⩾𝐸𝑉(𝑛+𝑠),∀𝑠∈𝑁_−𝑚. (3.24) Under condition (𝐶₂) we get 𝐸𝑉(𝑛+1)⩽(1+𝜆)𝐸𝑉(𝑛). (3.25) That is 𝐸𝑉(1𝑛)⩾1+𝜆𝐸𝑉(1𝑛+1)>1+𝜆𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1=𝑒^𝛼𝜆1+𝜆𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1𝑒^−𝛼𝜆>𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1𝑒^−𝛼𝜆⩾𝑐₂𝐸‖𝜑‖^𝑟_𝑚, (3.26) which contradicts with (3.23). Then there must be an 𝑛 satisfing 𝑛^∗<𝑛⩽𝑛 such that (3.22) holds.
For any 𝑛∈[𝑛^∗+1,𝑛], 𝐸𝑉(𝑛+𝑠)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1<𝑞𝑐₂𝐸‖𝜑‖^𝑟_𝑚<𝑞𝐸𝑉(𝑛). (3.27) By virtue of (𝐶₂), for any 𝑛∈[𝑛^∗+1,𝑛], 𝐸𝑉(𝑛)⩽(1+𝜆)𝐸𝑉(𝑛−1), (3.28) and for 𝑠∈𝑁_−𝑚, we have 𝑞𝐸𝑉(𝑛+1)⩾𝐸𝑉(𝑛+𝑠),𝑞𝐸𝑉(𝑛^∗+1)⩾𝐸𝑉(𝑛^∗+𝑠). (3.29) Making use of (3.28), we get 𝐸𝑉(𝑛+1)⩽(1+𝜆)𝐸𝑉(𝑛)⩽(1+𝜆)^{𝑛−𝑛∗}𝐸𝑉(𝑛^∗+1)⩽(1+𝜆)^𝛼𝐸𝑉(𝑛^∗)<𝑒^𝛼𝜆𝑐₂𝐸‖𝜑‖^𝑟_𝑚. (3.30) Taking (3.3) into account, the above inequality yields 𝐸𝑉(𝑛+1)>𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1, (3.31) which implies that 𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝜂1<𝑒^𝛼𝜆𝑐₂𝐸‖𝜑‖^𝑟_𝑚. (3.32) 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∈[𝜂_𝑘−1,𝜂_𝑘), 𝑘=1,2,…,ℎ, 𝐸𝑉(𝑛)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂𝑘}. (3.33) Under conditions (𝐶₃) and (𝐶₄), we have 𝐸𝑉(𝜂_ℎ)⩽𝑑_ℎ𝐸𝑉(𝜂_ℎ−1)⩽𝑑_ℎ𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ}⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}𝑒^−𝛼𝜆⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}. (3.34)
Now we will show that, when 𝑛∈[𝜂_ℎ,𝜂_ℎ+1), 𝐸𝑉(𝑛)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}. (3.35)
If (3.35) is not true, then there must be an 𝑛∈(𝜂_ℎ,𝜂_ℎ+1−1), such that 𝐸𝑉(𝑛+1)>𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}, (3.36) and for 𝑛∈[𝜂_ℎ,𝑛]𝐸𝑉(𝑛)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}. (3.37) At the same time, there exists an 𝑛^∗∈[𝜂_ℎ,𝑛] such that 𝐸𝑉(𝑛^∗)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}𝑒^−𝛼𝜆, (3.38) And, when 𝑛^∗<𝑛⩽𝑛, 𝐸𝑉(𝑛)>𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}𝑒^−𝛼𝜆. (3.39) If there does not exist an 𝑛 satisfing 𝑛^∗<𝑛⩽𝑛 such that (3.39) holds, then 𝑛^∗=𝑛. Obviously, for any 𝑠∈𝑁_−𝑚, 𝑞𝐸𝑉(𝑛+1)⩾𝐸𝑉(𝑛+s). Using condition (𝐶₂) yields 𝐸𝑉(𝑛+1)⩽(1+𝜆)𝐸𝑉(𝑛), that is, 𝐸𝑉(1𝑛)⩾1+𝜆𝐸𝑉(𝑒𝑛+1)⩾^𝜆𝛼1+𝜆𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}𝑒^−𝛼𝜆>𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}𝑒^−𝛼𝜆, (3.40) which contradicts with the definition of 𝑛; then there exists at least one number 𝑛 satisfing 𝑛^∗<𝑛⩽𝑛 such that (3.39) holds.
For 𝑛∈[𝑛^∗+1,𝑛] and 𝑠∈𝑁_−𝑚, we have 𝐸𝑉(𝑛+𝑠)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ}=𝑒^{𝜆(𝜂ℎ+1−𝜂ℎ)}𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}⩽𝑒^2𝜆𝛼𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}𝑒^−𝛼𝜆<𝑞𝐸𝑉(𝑛), (3.41) which implies that, under condition (𝐶₂), 𝐸𝑉(𝑛)⩽(1+𝜆)𝐸𝑉(𝑛−1). (3.42) Obviously, 𝑞𝐸𝑉(𝑛+1)⩾𝐸𝑉(𝑛). Using condition (𝐶₂) again, we get 𝐸𝑉(𝑛+1)⩽(1+𝜆)𝐸𝑉(𝑛). (3.43) since 𝑞𝐸𝑉(𝑛^∗+1)>𝐸𝑉(𝑛^∗+𝑠), 𝑠∈𝑁_−𝑚, we have, under condition (𝐶₂) 𝐸𝑉(𝑛^∗+1)⩽(1+𝜆)𝐸𝑉(𝑛^∗). (3.44) Then 𝐸𝑉(𝑛+1)⩽(1+𝜆)𝐸𝑉(𝑛)⩽(1+𝜆)^{𝑛−𝑛∗}𝐸𝑉(𝑛^∗+1)⩽(1+𝜆)^{𝑛−𝑛∗+1}𝐸𝑉(𝑛^∗)⩽(1+𝜆)^𝛼𝐸𝑉(𝑛^∗)<𝑒^𝛼𝜆𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}𝑒^−𝛼𝜆=𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂ℎ+1}<𝐸𝑉(𝑛), (3.45) which conflicts with the definition of 𝑛. Then (3.19) holds for 𝑘=ℎ+1.
By induction, we know that (3.19) holds for 𝑛∈[𝜂_𝑘,𝜂_𝑘+1), 𝑘∈ℕ.
Using condition (𝐶₁), for any 𝑛∈[𝜂_𝑘,𝜂_𝑘+1), 𝑘∈ℕ, we have 𝑐₁𝐸‖𝑥(𝑛)‖^𝑟⩽𝐸𝑉(𝑛)⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^{−𝜆𝜂𝑘+1}⩽𝑀𝐸‖𝜑‖^𝑟_𝑚𝑒^−𝜆𝑛. (3.46) That is the desired result.

4. Applications

In this section, we consider a kind of impulsive stochastic delay difference equation as follows:

𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛),𝑥(𝑛−𝑚))+𝑔(𝑛,𝑥(𝑛),𝑥(𝑛−𝑚))𝜉_𝑛,𝑛≠𝜂_𝑘−1,𝑥(𝜂_𝑘)=𝐻_𝑘(𝑥(𝜂_𝑘−1)),𝑥(𝑛₀+𝑠)=𝜑(𝑠),𝑠∈𝑁_−𝑚. (4.1)

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:(𝐶^∗₂) There exist constants 𝜆₁ and 𝜆₂, 0<𝜆₁, 𝜆₂<1, such that 𝐸𝑉(𝑛+1,𝑥(𝑛+1))⩽𝜆₁𝐸𝑉(𝑛,𝑥(𝑛))+𝜆₂𝐸𝑉(𝑛−𝑚,𝑥(𝑛−𝑚)). (4.2)If 𝜆₁+𝜆₂<1, then the trivial solution of system (4.1) is 𝑟-moment exponentially stable.

Proof. Let 𝑥(𝑛) be a solution of system (4.1). Take (𝑝=𝜆²₁+4𝜆₂+𝜆₂−𝜆₁)/3𝜆₂. It is easy to see that, under the conditions 0<𝜆₁, 𝜆₂<1, and 0<𝜆₁+𝜆₂<1, 1<𝑝<𝜆²₁+4𝜆₂−𝜆₁2𝜆₂<1𝜆₁+𝜆₂<1−𝜆₁𝜆₂. (4.3) If 𝐸𝑉(𝑛+𝜃,𝑥(𝑛+𝜃))⩽𝑝𝐸𝑉(𝑛,𝑥(𝑛)) for any 𝜃∈𝑁_−𝑚, it follows from the condition (𝐶^∗₂) that 𝐸𝑉(𝑛+1,𝑥(𝑛+1))⩽(𝜆₁+𝑝𝜆₂)𝐸𝑉(𝑛,𝑥(𝑛)). (4.4) Then condition (𝐶₂) of Theorem 3.1 follows under (4.3).
Let 𝜆=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 that 𝑉(𝑛+𝜃,𝑥(𝑛+𝜃))>𝑒^𝛼𝑉(𝑛,𝑥(𝑛)) for some 𝜃∈𝑁_−𝑚; by virtue of (𝐶^∗₂) and inequality (4.3), 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)max_{𝑠∈𝑁−𝑚}<1{𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))}𝑝max_{𝑠∈𝑁−𝑚}{𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))}. (4.5) Then condition (𝐶₃) of Theorem 3.1 follows, which completes the proof.

From the above proof, we know that constant 𝛽 in Theorem 3.1 equals to

𝑚−1𝑚(𝑚+1)ln𝜆²₁+4𝜆₂+𝜆₂−𝜆₁3𝜆₂−ln𝑑𝑚. (4.6)

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 constant 0<𝜆<1 such that 𝐸𝑉(𝑛+1,𝑥(𝑛+1))⩽𝜆max_{𝑠∈𝑁−𝑚}{𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))}. (4.7)Then the trivial solution of system (4.1) is 𝑟-moment exponentially stable.

Proof. Let 𝑥(𝑛) be a solution of system (4.1). Take 1𝑝=(𝜆)^{(𝑚+1)/(𝑚+2)}. (4.8) Since 0<𝜆<1, we have 1<𝑝<1/𝜆 and ln𝑝=(𝑚+1)ln(1/𝜆)1(𝑚+2)<ln(𝜆). (4.9) For any 𝑠∈𝑁_−𝑚, assume that 𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))⩽𝑝𝑉(𝑛,𝑥(𝑛)); by virtue of condition (𝐶₂^∗∗), we get 𝐸𝑉(𝑛+1,𝑥(𝑛+1))⩽𝑝𝜆𝐸𝑉(𝑛), (4.10) that is, condition (𝐶₂) of Theorem 3.1.
Since 1<𝑝<1/𝜆 we have 1/𝑝>𝜆. Under condition (𝐶₂^∗∗), for any 𝑛∈ℕ, we get 𝐸𝑉(𝑛+1,𝑥(𝑛+1))⩽𝜆max_{𝑠∈𝑁−𝑚}1{𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))}<(𝑝)max_{𝑠∈𝑁−𝑚}{𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠))}, (4.11) that is condition (𝐶₃) of Theorem 3.1.

From the above proof, we know that constant 𝛼 in Theorem 3.1 equals to

min{−ln(𝜆𝑝),ln𝑝(𝑚+1)}=−ln(𝜆𝑝)=ln𝑝(𝑚+1)=−ln𝜆.(𝑚+2) (4.12) Then constant 𝛽 in Theorem 3.1 equals

−ln𝜆1(𝑚+2)(1−𝑚)−ln𝑑𝑚. (4.13)

Now, we present a corollary of Theorem 3.2 which establishes a criterion of mean square exponential stability for system (4.1).

Corollary 4.3. Assume that there exist positive constants 𝜆, 𝛼, and 𝑞 where 𝛼 is a natural number and 𝛼>1, 𝑞⩾𝑒^2𝜆𝛼such that system (4.1) satisfies the following.(1)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), (4.14) where 𝑎,𝑏 are positive constants, 𝑏<1/𝑞, and 0<𝑎+𝑏𝑞−11−b𝑞⩽𝜆. (4.15)(2)‖𝐻_𝑘(𝑥)‖⩽𝛽_𝑘‖𝑥‖, for any 𝑥∈ℝ^𝑛, 𝛽_𝑘>0, and 2ln𝛽_𝑘+𝜆(𝜂_𝑘+1−𝜂_𝑘)⩽−𝜆𝛼. The impulsive moments satisfy 𝑚⩽𝜂_𝑘+1−𝜂_𝑘⩽𝛼.Then, for any initial data 𝑥_𝑛0=𝜑, the solution 𝑥(𝑛) of system (4.1) satisfies 𝐸‖𝑥(𝑛)‖²≤𝐸‖𝜑‖²_𝑚𝑒^{−(𝜆/2)𝑛}. (4.16) That is to say, the trivial solution of system (4.1) is mean square exponentially stable.

Proof. Let 𝑉(𝑛,𝑥)=‖𝑥‖², then, 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))𝜉_𝑛‖²⩽2(𝐸‖𝑓(𝑛,𝑥(𝑛),𝑥(𝑛−𝑚))‖²+𝐸(‖𝑔(𝑛,𝑥(𝑛),𝑥(𝑛−𝑚))‖²𝜉²_𝑛))⩽𝑎𝐸‖𝑥(𝑛)‖²+𝑏𝐸‖𝑥(𝑛−𝑚)‖²=𝑎𝐸𝑉(𝑛)+𝑏𝐸𝑉(𝑛−𝑚). (4.17) Assume that 𝑞𝐸𝑉(𝑛+1,𝑥(𝑛+1))⩾𝐸𝑉(𝑛+𝑠,𝑥(𝑛+𝑠)) holds for any 𝑠∈𝑁_−𝑚, then 𝐸𝑉(𝑛+1)=𝐸‖𝑥(𝑛+1)‖²⩽𝑎1−𝑏𝑞𝐸𝑉(𝑛)⩽(1+𝜆)𝐸𝑉(𝑛). (4.18) 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:

𝑥(𝑛+1)=𝑎𝑥(𝑛)+𝑏𝑥(𝑛−𝑚)+𝑐𝑥(𝑛−𝑚)𝜉_𝑛,𝑛≠𝜂_𝑘−1,𝑛⩾0,𝑥(𝜂_𝑘)=𝛽_𝑘𝑥(𝜂_𝑘−1),𝑘∈ℕ,𝑥(𝑠)=𝜑(𝑠),𝑠∈𝑁_−𝑚. (5.1)

First we take 𝑎=0.5, 𝑏=0.25, 𝑐=0.25, 𝑚=9, 𝜂_𝑘=10𝑘, 𝛽_𝑘=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 take 𝑎=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: 𝜂_𝑘=4𝑘, 𝛽_𝑘=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 Corollary 4.3 are sufficient but not necessary. If we take 𝑎=1.09, 𝑏=0, 𝑐=𝑒⁻¹⁹, 𝑚=3, 𝜂_𝑘=4𝑘, and 𝛽_𝑘=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 the 𝑟-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.

Acknowledgments

This work was supported by the Scientific Foundations of Harbin Institute of Technology at Weihai under Grants no.HIT(WH)20080008 and no.HIT(WH)2B200905. The authors would like to give their thanks to Professor Leonid Shaikhet for his valuable comments and to the referees who suggest that the authors give examples to illustrate the results.