Exponential Stability of Impulsive Stochastic Functional Differential Systems

Abstract

This paper is concerned with stabilization of impulsive stochastic delay differential systems. Based on the Razumikhin techniques and Lyapunov functions, several criteria on pth moment and almost sure exponential stability are established. Our results show that stochastic functional differential systems may be exponentially stabilized by impulses.

1. Introduction

In the past decades, many authors have obtained various results of deterministic functional differential systems (see [1–6] and the references therein). But it is well known that there are many stochastic factors in the realistic environment, and it is necessary to consider stochastic models. In fact, stochastic functional differential systems (SFDSs) have received more attention in recent years. The properties of SFDSs including stability have been studied in [7–10], which can be widely used in science and engineering (see [11] and the references therein). Furthermore, besides stochastic effects, impulsive effects likewise exist in many evolution processes in which system states change abruptly at certain moments of time, involving such fields as medicine and biology, economics, mechanics, electronics, and telecommunications, and so forth. The impulsive control theory comes to play an important role in science and industry [12]. So the stability investigation of impulsive stochastic differential systems (ISDSs) and impulsive stochastic functional differential systems (ISFDSs) is interesting to many authors [13–20].

Recently, the Razumikhin-type asymptotical stability theorems for ISFESs were established [21, 22]. However, little work has been done on generally exponential stability of ISFESs [23, 24]. In this paper, stability criteria for impulsive stochastic function differential systems are investigated by Razumikhin technique and Lyapunov functions. It is shown that an unstable stochastic delay system can be successfully stabilized by impulses and the results can be easily applied.

2. Preliminaries

Throughout this paper, unless otherwise specified, let (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ₀ contains all 𝑃-null sets). 𝑤(𝑡)=(𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑑(𝑡))^𝑇 means a 𝑑-dimensional Brownian motion defined on this probability space. 𝑅 denotes the set of real numbers, 𝑅₊ is the set of nonnegative real numbers, and 𝑅^𝑛 denotes the 𝑛-dimensional real space equipped with Euclidean norm |⋅|. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^𝑇 and its operator norm is denoted by ‖𝐴‖=sup{|𝐴𝑥|∶|𝑥|=1}. Moreover, let 𝜏>0 and denote by 𝐶([−𝜏,0];𝑅₊) the family of continuous functions from [−𝜏,0] to 𝑅₊. Let 𝑁 denote the set of positive integers, that is, 𝑁={1,2,…}.

For −∞<𝑎<𝑏<+∞, a function from [𝑎,𝑏] to 𝑅^𝑛 is called piecewise continuous, if the function has at most a finite number of jump discontinuities on (𝑎,𝑏], which is continuous from the right for all points in [𝑎,𝑏). Given 𝜏>0, 𝑃𝐶([−𝜏,0];𝑅^𝑛) denotes the family of piecewise continuous functions from [−𝜏,0] to 𝑅^𝑛. A norm on 𝑃𝐶([−𝜏,0];𝑅^𝑛) is defined as ‖𝜙‖=sup_{−𝜏≤𝑠≤0}|𝜙(𝑠)| for 𝜙∈𝑃𝐶([−𝜏,0];𝑅^𝑛).

For 𝑝>0 and 𝑡≥0, let 𝑃𝐶^𝑝_ℱ𝑡([−𝜏,0];𝑅^𝑛) denote the family of all ℱ_𝑡-measurable 𝑃𝐶([−𝜏,0];𝑅^𝑛)-value random variables 𝜙 such that sup_{−𝜏≤𝜃≤0}𝐸|𝜙(𝜃)|^𝑝<∞ and 𝑃𝐶^𝑏_ℱ𝑡([−𝜏,0];𝑅^𝑛) denote the family of 𝑃𝐶([−𝜏,0];𝑅^𝑛)-value random variables that are bounded and ℱ_𝑡-measurable.

In this paper, we consider the following ISFDS:𝑑𝑥(𝑡)=𝑓𝑥_𝑡,𝑡𝑑𝑡+𝑔𝑥_𝑡,𝑡𝑑𝑤(𝑡),𝑡≠𝑡_𝑘,𝑡≥𝑡₀,Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥_𝑡⁻_𝑘,𝑡_𝑘𝑥,𝑘∈𝑁,_𝑡0=𝜉, (2.1) where the initial value 𝜉∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0];𝑅^𝑛),𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇,𝑥_𝑡 is regarded as a 𝑃𝐶([−𝜏,0];𝑅^𝑛)-value process and 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃),𝜃∈[−𝜏,0]. Similarly, 𝑥_𝑡− is defined by 𝑥_𝑡−(𝜃)=𝑥(𝑡+𝜃),𝜃∈[−𝜏,0) and 𝑥_𝑡−(0)=lim_{𝑠→𝑡−}𝑥(𝑠). Both 𝑓∶𝑃𝐶^𝑏_ℱ𝑡([−𝜏,0];𝑅^𝑛)×𝑅₊→𝑅^𝑛and𝑔∶𝑃𝐶^𝑏_ℱ𝑡([−𝜏,0];𝑅^𝑛)×𝑅₊→𝑅^𝑛×𝑑 are Borel measurable, and 𝐼_𝑘∶𝑃𝐶^𝑏_ℱ𝑡([−𝜏,0];𝑅^𝑛)×𝑅₊→𝑅^𝑛 represents the impulsive perturbation of 𝑥 at time 𝑡_𝑘. The fixed moments of impulse times 𝑡_𝑘 satisfy 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯,𝑡_𝑘→∞(as𝑘→∞), Δ𝑥(𝑡_𝑘)=𝑥(𝑡_𝑘)−𝑥(𝑡⁻_𝑘). Moreover, 𝑓, 𝑔, and 𝐼_𝑘 are assumed to satisfy necessary assumptions so that, for any initial data 𝜉∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0];𝑅^𝑛), system (2.1) has a unique global solution, denoted by 𝑥(𝑡;𝑡₀,𝜉) (e.g., see [25] for existence and uniqueness results for general impulsive hybrid stochastic delay systems including (2.1)). For the purpose of stability in this note, we also assume the 𝑓(0,𝑡)≡0, 𝑔(𝑡,0)≡0 and 𝐼_𝑘(0,𝑡)≡0 for all 𝑡≥𝑡₀, 𝑘∈𝑁, then system (2.1) admits a trivial solution.

Definition 2.1. The trivial solution of system (2.1) is said to be 𝑝th (𝑝>0) moment exponentially stable if there is a pair of positive constants 𝜆,𝐶 such that 𝐸𝑥𝑡;𝑡₀,𝜉^𝑝≤𝐶𝜉^𝑝𝑒^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀, (2.2) for all 𝜉∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0];𝑅^𝑛). When 𝑝=2, it is usually said to be exponentially stable in mean square. It follows from (2.2) that limsup_𝑡→∞1𝑡log𝐸𝑥𝑡;𝑡₀,𝜉^𝑝≤−𝜆. (2.3) The left-hand side of (2.3) is called the 𝑝th moment Lyapunov exponent of the solution.

Definition 2.2. The trivial solution of system (2.1) is said to be almost exponentially stable if there is a pair of positive constants 𝜆, 𝐶 such that for 𝑡≥𝑡₀𝑥𝑡;𝑡₀,𝜉^𝑝≤𝐶𝜉𝑒^{−𝜆(𝑡−𝑡0)},a.s., (2.4) for all  𝜉∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0];𝑅^𝑛). It follows from  (2.4)  that limsup_𝑡→∞1𝑡log𝑥𝑡;𝑡₀,𝜉≤−𝜆. (2.5) The left-hand side of (2.5) is called the Lyapunov exponent of the solution.

Definition 2.3. Let 𝐶^2,1(𝑅^𝑛×[𝑡₀,∞);𝑅₊) denote the family of all nonnegative functions 𝑉(𝑥,𝑡) on 𝑅^𝑛×[𝑡₀−𝜏,∞) that are continuously twice differential in 𝑥 and once in 𝑡. If 𝑉∈𝐶^2,1(𝑅^𝑛×[𝑡₀,∞);𝑅₊), define the operator ℒ𝑉∶𝑃𝐶([−𝜏,0];𝑅^𝑛)×[𝑡₀,∞)→𝑅 for system (2.1) by ℒ𝑉𝑥_𝑡,𝑡=𝑉_𝑡(𝑥,𝑡)+𝑉_𝑥(𝑥,𝑡)𝑓𝑥_𝑡+1,𝑡2trace𝑔^𝑇𝑥_𝑡,𝑡𝑉_𝑥𝑥(𝑥,𝑡)𝑔𝑥_𝑡,𝑡, (2.6) where 𝑉_𝑡(𝑥,𝑡)=𝜕𝑉(𝑥,𝑡)/𝜕𝑡,𝑉_𝑥(𝑥,𝑡)=(𝜕𝑉(𝑥,𝑡)/𝜕𝑥₁,…,𝜕𝑉(𝑥,𝑡)/𝜕𝑥_𝑛),𝑉_𝑥𝑥(𝑥,𝑡)=(𝜕²𝑉(𝑥,𝑡)/𝜕𝑥_𝑖𝜕𝑥_𝑗)_𝑛×𝑛.

3. Main Results

In this section, we will establish some criteria on the 𝑝th moment exponential stability and almost exponential stability for system (2.1) by using the Razumikhin technique and Lyapunov functions. We begin with the following lemma, which concerns with the continuity of 𝐸𝑉(𝑥(𝑡),𝑡).

Lemma 3.1. Let 𝑉(𝑥,𝑡)∈𝐶^2,1(𝑅^𝑛×[𝑡₀,∞);𝑅₊), and let 𝑥(𝑡) be a solution of system (2.1). If there exists 𝑐>0 such that 𝑉(𝑥,𝑡)≤𝑐|𝑥|^𝑝, then 𝐸𝑉(𝑥(𝑡),𝑡) is continuous on [𝑡_𝑘−1,𝑡_𝑘),𝑘∈𝑁.

Proof. By the Itô formula, 𝑉(𝑥(𝑡),𝑡)=𝑉𝑥𝑡_𝑘−1,𝑡_𝑘−1+^𝑡_𝑡𝑘−1ℒ𝑉𝑥_𝑠,𝑠𝑑𝑠+^𝑡_𝑡𝑘−1𝑉_𝑥(𝑥(𝑠),𝑠)𝑔𝑥_𝑠,𝑠𝑑𝑤(𝑠) (3.1) for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), where 𝑘∈𝑁. Since 𝑥_𝑡𝑘−1∈𝑃𝐶^𝑏_{ℱ𝑡𝑘−1}([−𝜏,0];𝑅^𝑛), we can find an integer 𝑙₀ such that ‖𝑥_𝑡𝑘−1‖<𝑙₀a.s. For any integer 𝑙>𝑙₀, define the stopping time 𝜌_𝑙=inf𝑡∈𝑡_𝑘−1,𝑡_𝑘∶|𝑥(𝑡)|≥𝑙, (3.2) where inf∅=∞ as usual. Since 𝑥(𝑡) is continuous on [𝑡_𝑘−1,𝑡_𝑘),|𝑥(𝑡)| is also continuous on [𝑡_𝑘−1,𝑡_𝑘). Clearly, 𝜌_𝑙→∞ a.s. as 𝑙→∞. Moreover, it has 𝐸𝑉(𝑥(𝑡_𝑘−1),𝑡_𝑘)≤𝑐𝑙₀, following from 𝑥_𝑡𝑘−1∈𝑃𝐶^𝑏_{ℱ𝑡𝑘−1}([−𝜏,0];𝑅^𝑛). It then follows from the definition of 𝜌_𝑙 above that 𝐸𝑉𝑥𝑡^′_𝑙,𝑡^′_𝑙=𝐸𝑉𝑥𝑡_𝑘−1,𝑡_𝑘−1+𝐸^𝑡′𝑙_𝑡𝑘−1ℒ𝑉𝑥_𝑠,𝑠𝑑𝑠, (3.3) where 𝑡^′_𝑙=𝑡∧𝜌_𝑙. So, letting 𝑙→∞, by the dominated convergence theorem and Fubini’s theorem, we have 𝐸𝑉(𝑥(𝑡),𝑡)=𝐸𝑉𝑥𝑡_𝑘−1,𝑡_𝑘−1+𝐸^𝑡_𝑡𝑘−1ℒ𝑉𝑥_𝑠𝑥𝑡,𝑠𝑑𝑠=𝐸𝑉_𝑘−1,𝑡_𝑘−1+^𝑡_𝑡𝑘−1𝐸ℒ𝑉𝑥_𝑠,𝑠𝑑𝑠, (3.4) for 𝑡∈[𝑡_𝑘−1,𝑡_𝑘). This implies that 𝐸𝑉(𝑥(𝑡),𝑡) is continuous on [𝑡_𝑘−1,𝑡_𝑘),𝑘∈𝑁.

Theorem 3.2. Let 𝑉∈𝐶^2,1(𝑅^𝑛×[𝑡₀−𝜏,∞);𝑅₊) and 𝑢∶[𝑡₀,∞)→𝑅₊ be a piecewise continuous function. Suppose there exist some positive constants 𝑝,𝑐₁,𝑐₂, and 𝜆 such that(i) for all (𝑥,𝑡)∈𝑅^𝑛×[𝑡₀−𝜏,∞), 𝑐₁|𝑥|^𝑝≤𝑉(𝑥,𝑡)≤𝑐₂|𝑥|^𝑝, (3.5)(ii)for all 𝑘∈𝑁, and 𝜙∈𝑃𝐶^𝑝_ℱ𝑡([−𝜏,0];𝑅^𝑛), 𝐸𝑉𝜙(0⁻)+𝐼𝑡_𝑘,𝜙,𝑡_𝑘≤𝑑_𝑘𝐸𝑉𝜙(0⁻),𝑡⁻_𝑘, (3.6) where 0<𝑑_𝑘<exp{−𝜆(𝑡_𝑘+1−𝑡_𝑘)−∫^𝑡𝑘+1_𝑡𝑘𝑢(𝑠)𝑑𝑠},(iii)for all 𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑘∈𝑁 and 𝜙∈𝑃𝐶^𝑝_ℱ𝑡([−𝜏,0];𝑅^𝑛), 𝐸ℒ𝑉𝜙,𝑡≤𝑢(𝑡)𝐸𝑉𝜙(0),𝑡 (3.7) whenever 𝐸𝑉𝜙,𝑡+𝜃<𝑞𝐸𝑉𝜙(0),𝑡,𝜃∈[−𝜏,0], (3.8) where 𝑞>max_𝑘∈𝑁{𝑑_𝑘⁻¹𝑒^𝜆𝜏}∨exp{∫^𝑡1_𝑡0𝑢(𝑠)𝑑𝑠}.

Then the trivial solution of system (2.1) is 𝑝th moment exponentially stable and its 𝑝th moment Lyapunov exponent is not greater than −𝜆.

Proof. Given any initial data 𝜉∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0];𝑅^𝑛), the global solution 𝑥(𝑡;𝑡₀,𝜉)=𝑥(𝑡) of (2.1) is written as 𝑥(𝑡) in this proof. Without loss of generality, assume that the initial date 𝜉 is nontrivial so that 𝑥(𝑡) is not a trivial solution. Choose 𝑀 such that 𝑐₂𝑒^{𝜆(𝑡1−𝑡0)+∫𝑡1𝑡0𝑢(𝑠)𝑑𝑠}<𝑀<𝑐₂𝑞𝑒^{𝜆(𝑡1−𝑡0)}. (3.9) Then it follows from condition (i) and (3.9) that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑐₂𝜉^𝑝<𝑀𝜉^𝑝𝑒^{−𝜆(𝑡1−𝑡0)},𝑡∈𝑡₀−𝜏,𝑡₀. (3.10) In the following, we will show that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑘−𝑡0)},𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈𝑁. (3.11) In order to do so, we first prove that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑀𝜉^𝑝𝑒^{−𝜆(𝑡1−𝑡0)},𝑡∈𝑡₀,𝑡₁. (3.12) If (3.12) is not true, then there exist some 𝑡∈[𝑡₀,𝑡₁) such that 𝐸𝑉(𝑥(𝑡),𝑡)>𝑀‖𝜉‖^𝑝𝑒^{−𝜆(𝑡1−𝑡0)}. Set 𝑡^∗=inf{𝑡∈[𝑡₀,𝑡₁)∶𝐸𝑉(𝑥(𝑡),𝑡)>𝑀‖𝜉‖^𝑝𝑒^{−𝜆(𝑡1−𝑡0)}}. Then 𝑡^∗∈(𝑡₀,𝑡₁) and also, by the continuity of 𝐸𝑉(𝑥(𝑡),𝑡)) (see Lemma 3.1), 𝐸𝑉(𝑥(𝑡),𝑡)<𝐸𝑉𝑥𝑡^∗,𝑡^∗=𝑀𝜉^𝑝𝑒^{−𝜆(𝑡1−𝑡0)},𝑡∈𝑡₀−𝜏,𝑡_∗. (3.13) In view of (3.10), define 𝑡_∗=sup{𝑡∈[𝑡₀−𝜏,𝑡^∗)∶𝐸𝑉(𝑥(𝑡),𝑡)≤𝑐₂‖𝜉‖^𝑝}. Then 𝑡_∗∈[𝑡₀,𝑡^∗) and, by the continuity of 𝐸𝑉(𝑥(𝑡),𝑡), 𝐸𝑉(𝑥(𝑡),𝑡)>𝐸𝑉𝑥𝑡_∗,𝑡_∗=𝑐₂𝜉^𝑝,𝑡∈𝑡_∗,𝑡^∗. (3.14) Now in view of (3.9), (3.13), and (3.14), one has, for 𝑡∈[𝑡_∗,𝑡^∗] and 𝜃∈[−𝜏,0], 𝐸𝑉(𝑥(𝑡+𝜃),𝑡+𝜃)≤𝑀𝜉^𝑝𝑒^{−𝜆(𝑡1−𝑡0)}<𝑞𝐸𝑉𝑥𝑡_∗,𝑡_∗≤𝑞𝐸𝑉(𝑥(𝑡),𝑡). (3.15) By the Razumikhin-type condition (iii), 𝐸ℒ𝑉𝑥_𝑡,𝑡≤𝑢(𝑡)𝐸𝑉(𝑥(𝑡),𝑡),∀𝑡∈𝑡_∗,𝑡^∗. (3.16) Applying Itô formula and by (3.16), one obtains that 𝐸𝑉𝑥𝑡^∗,𝑡^∗≤𝐸𝑉𝑥𝑡_∗,𝑡_∗+^𝑡∗_𝑡∗𝑢(𝑠)𝐸𝑉(𝑥(𝑠),𝑠)𝑑𝑠. (3.17) Finally, by (3.9), (3.13), (3.14), and the Gronwall inequality, 𝐸𝑉𝑥𝑡^∗,𝑡^∗≤𝐸𝑉𝑥𝑡_∗,𝑡_∗𝑒^{∫𝑡∗𝑡∗𝑢(𝑠)𝑑𝑠}≤𝑐₂𝜉^𝑝𝑒^{∫𝑡1𝑡0𝑢(𝑠)𝑑𝑠}<𝑀𝜉^𝑝𝑒^{−𝜆(𝑡1−𝑡0)}=𝐸𝑉𝑥𝑡^∗,𝑡^∗, (3.18) which is a contradiction. So inequality (3.12) holds and (3.11) is true for 𝑘=1.
Now assume that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑘−𝑡0)},∀𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈𝑁, (3.19) for all 𝑘≤𝑚, where 𝑘,𝑚∈𝑁. We proceed to show that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)},∀𝑡∈𝑡_𝑚,𝑡_𝑚+1. (3.20) Suppose (3.20) is not true, set 𝑡=inf{𝑡∈[𝑡_𝑚,𝑡_𝑚+1)∶𝐸𝑉(𝑥(𝑡),𝑡)>𝑀‖𝜉‖^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)}}. By condition (ii) and (3.20), we know 𝐸𝑉𝑥𝑡_𝑚,𝑡_𝑚≤𝑑_𝑚𝐸𝑉𝑥𝑡⁻_𝑚,𝑡⁻_𝑚≤𝑑_𝑚𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑚−𝑡0)}<𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)}. (3.21) From this, together with 𝐸𝑉(𝑥(𝑡),𝑡) being continuous on 𝑡∈[𝑡_𝑚,𝑡_𝑚+1), we know that 𝑡∈(𝑡_𝑚,𝑡_𝑚+1) and 𝐸𝑉(𝑥(𝑡),𝑡)<𝐸𝑉𝑥𝑡,𝑡=𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)},∀𝑡∈𝑡_𝑚,𝑡. (3.22) Define 𝑡=sup{𝑡∈[𝑡₀,𝑡]∶𝐸𝑉(𝑥(𝑡),𝑡)≤𝑑_𝑚𝑀‖𝜉‖^𝑝𝑒^{−𝜆(𝑡𝑚−𝑡0)}}, then 𝑡∈[𝑡_𝑚,𝑡) and 𝐸𝑉(𝑥(𝑡),𝑡)>𝐸𝑉𝑥𝑡,𝑡=𝑑_𝑚𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑚−𝑡0)},∀𝑡∈𝑡,𝑡. (3.23) For 𝑡∈[𝑡,𝑡] and 𝜃∈[−𝜏,0], when 𝑡+𝜃≥𝑡_𝑚, then (3.22) and (3.23) imply that 𝐸𝑉(𝑥(𝑡+𝜃),𝑡+𝜃)≤𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)}<𝑀𝜉^𝑝𝑒^{−𝜆(𝑡+𝜃−𝑡0)}≤𝑀𝑒^𝜆𝜏𝜉^𝑝𝑒^{−𝜆(𝑡−𝑡0)}≤𝑀𝑒^𝜆𝜏𝜉^𝑝𝑒^{−𝜆(𝑡𝑚−𝑡0)}≤𝑞𝐸𝑉𝑥𝑡,𝑡. (3.24) If 𝑡+𝜃<𝑡_𝑚 for some 𝜃∈[−𝜏,0), we assume that, without loss of generality, 𝑡+𝜃∈[𝑡_𝑙,𝑡_𝑙+1) for some 𝑙∈𝑁,𝑙≤𝑚−1, then from (3.19) and (3.23), 𝐸𝑉(𝑥(𝑡+𝜃),𝑡+𝜃)≤𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑙+1−𝑡0)}<𝑀𝜉^𝑝𝑒^{−𝜆(𝑡+𝜃−𝑡0)}≤𝑀𝑒^𝜆𝜏𝜉^𝑝𝑒^{−𝜆(𝑡−𝑡0)}≤𝑀𝑒^𝜆𝜏𝜉^𝑝𝑒^{−𝜆(𝑡𝑚−𝑡0)}≤𝑞𝐸𝑉𝑥𝑡,𝑡. (3.25) Therefore, 𝐸𝑉(𝑥(𝑡+𝜃),𝑡+𝜃)<𝑞𝐸𝑉(𝑥(𝑡),𝑡),𝑡∈𝑡,𝑡,𝜃∈[−𝜏,0]. (3.26) Then, it follows from condition (iii) that 𝐸ℒ𝑉𝑥_𝑡,𝑡≤𝑢(𝑡)𝐸𝑉(𝑥(𝑡),𝑡),∀𝑡∈𝑡,𝑡. (3.27) Combining Itô formula with (3.27), we can check that 𝐸𝑉𝑥𝑡,𝑡≤𝐸𝑉𝑥𝑡,𝑡+^𝑡_𝑡𝑢(𝑠)𝐸𝑉(𝑥(𝑠),𝑠)𝑑𝑠. (3.28) Finally, by (3.22), (3.23), and the Gronwall inequality, 𝐸𝑉𝑥𝑡,𝑡≤𝐸𝑉𝑥𝑡,𝑡𝑒^{∫𝑡𝑡𝑢(𝑠)𝑑𝑠}≤𝐸𝑉𝑥𝑡,𝑡𝑒^{∫𝑡𝑡𝑚𝑚+1𝑢(𝑠)𝑑𝑠}=𝑑_𝑚𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑚−𝑡0)}𝑒^{∫𝑡𝑡𝑚𝑚+1𝑢(𝑠)𝑑𝑠}<𝐸𝑉𝑥𝑡,𝑡, (3.29) which is a contradiction. So inequality (3.20) holds. By mathematical induction, we obtain that (3.11) holds for all 𝑘∈𝑁. Furthermore, from condition (i), we have 𝐸|𝑥(𝑡)|^𝑝≤𝑐₁𝑐₂𝑀𝜉^𝑝𝑒^{−𝜆(𝑡𝑘−𝑡0)}≤𝑐₁𝑐₂𝑀𝜉^𝑝𝑒^{−𝜆(𝑡−𝑡0)},𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈𝑁, (3.30) which implies 𝐸‖𝑥‖^𝑝≤𝑐₁𝑐₂𝑀𝜉^𝑝𝑒^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀, (3.31) that is, system (2.1) is 𝑝th moment exponentially stable. The proof is complete.

Remark 3.3. If 𝑢(𝑡)≡𝑐>0, then Theorem 3.1 of [23] follows from Theorem 3.2 immediately.

Theorem 3.4. Let 𝑉∈𝐶^2,1(𝑅^𝑛×[𝑡₀−𝜏,∞);𝑅₊), and let 𝑢∶[𝑡₀,∞)→𝑅₊ be a piecewise continuous function. Suppose there exist some positive constants 𝑝,𝑐₁,𝑐₂, and 𝜆 such that(i) for all (𝑥,𝑡)∈𝑅^𝑛×[𝑡₀−𝜏,∞), 𝑐₁|𝑥|^𝑝≤𝑉(𝑥,𝑡)≤𝑐₂|𝑥|^𝑝, (3.32)(ii) for all 𝑘∈𝑁 and 𝜙∈𝑃𝐶^𝑝_ℱ𝑡([−𝜏,0];𝑅^𝑛), 𝐸𝑉𝜙(0⁻)+𝐼𝑡_𝑘,𝜙≤𝜌𝑑_𝑘𝐸𝑉𝜙(0⁻),𝑡⁻_𝑘, (3.33) where 0<𝜌<max{𝑒^{−𝜆(𝑡𝑚+1−𝑡𝑚)}}and𝑑_𝑘>0 with 𝑑=sup_𝑛∈𝑁Π^𝑛_𝑘=1𝑑_𝑘<∞,(iii)for all 𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑘∈𝑁, and 𝜙∈𝑃𝐶^𝑝_ℱ𝑡([−𝜏,0];𝑅^𝑛), 𝐸ℒ𝑉𝜙,𝑡≤𝑢(𝑡)𝐸𝑉𝜙(0),𝑡 (3.34) whenever 𝐸𝑉𝜙,𝑡+𝜃<𝑞𝐸𝑉𝜙(0),𝑡,𝜃∈[−𝜏,0], (3.35) where 𝑞>(𝜌⁻¹𝑒^𝜆𝜏)∨(𝜌⁻¹𝑒^𝜆𝜏𝑑⁻¹).

Then the trivial solution of system (2.1) is 𝑝th moment exponentially stable and its 𝑝th moment Lyapunov exponent is not greater than −𝜆.

Proof. Given any initial data 𝜉∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0];𝑅^𝑛), the global solution 𝑥(𝑡;𝑡₀;𝜉)=𝑥(𝑡) of (2.1) is written as 𝑥(𝑡) in this proof. Without loss of generality, assume that the initial date 𝜉 is nontrivial so that 𝑥(𝑡) is not a trivial solution. Choose 𝑀 such that 𝑐₂𝑒^{𝜆(𝑡1−𝑡0)+∫𝑡1𝑡0𝑢(𝑠)𝑑𝑠}<𝑀<𝑐₂𝑞𝑒^{𝜆(𝑡1−𝑡0)}. (3.36) Then it follows from condition (i) and (3.36) that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑐₂𝜉^𝑝<𝑀𝜉^𝑝𝑒^{−𝜆(𝑡1−𝑡0)},𝑡∈𝑡₀−𝜏,𝑡₀. (3.37) In the following, we will show that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑀_𝑘𝜉^𝑝𝑒^{−𝜆(𝑡𝑘−𝑡0)},𝑡∈𝑡_𝑘−1,𝑡_𝑘, (3.38) where 𝑘∈𝑁 and 𝑀_𝑘 is defined as 𝑀₁=𝑀 and 𝑀_𝑘=𝑀Π_{1≤𝑙≤𝑘−1}𝑑_𝑙. Similarly, as the proof in Theorem 3.2, one can prove that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑀𝜉^𝑝𝑒^{−𝜆(𝑡1−𝑡0)},𝑡∈𝑡₀,𝑡₁. (3.39) Now assume that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑀_𝑘𝜉^𝑝𝑒^{−𝜆(𝑡𝑘−𝑡0)},∀𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈𝑁, (3.40) for all 𝑘≤𝑚, where 𝑘,𝑚∈𝑁. We proceed to show that 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑀_𝑚+1𝜉^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)},∀𝑡∈𝑡_𝑚,𝑡_𝑚+1. (3.41) Suppose (3.41) is not true, set 𝑡=inf{𝑡∈[𝑡_𝑚,𝑡_𝑚+1)∶𝐸𝑉(𝑥(𝑡),𝑡)>𝑀_𝑘‖𝜉‖^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)}}. By condition (ii), 𝐸𝑉𝑥𝑡_𝑚,𝑡_𝑚≤𝜌𝑑_𝑚𝐸𝑉𝑥𝑡⁻_𝑚,𝑡⁻_𝑚≤𝜌𝑀_𝑚+1𝜉^𝑝𝑒^{−𝜆(𝑡𝑚−𝑡0)}<𝑀_𝑚+1𝜉^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)}. (3.42) From this, together with 𝐸𝑉(𝑥(𝑡),𝑡) being continuous on 𝑡∈[𝑡_𝑚,𝑡_𝑚+1), we know that 𝑡∈(𝑡_𝑚,𝑡_𝑚+1) and 𝐸𝑉(𝑥(𝑡),𝑡)<𝐸𝑉𝑥𝑡,𝑡=𝑀_𝑚+1𝜉^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)},∀𝑡∈𝑡_𝑚,𝑡. (3.43) Define 𝑡=sup{𝑡∈[𝑡₀,𝑡]∶𝐸𝑉(𝑥(𝑡),𝑡)≤𝜌𝑀_𝑚+1‖𝜉‖^𝑝𝑒^{−𝜆(𝑡𝑚−𝑡0)}}, then 𝑡∈[𝑡_𝑚,𝑡) and 𝐸𝑉(𝑥(𝑡),𝑡)>𝐸𝑉𝑥𝑡,𝑡=𝜌𝑀_𝑚+1𝜉^𝑝𝑒^{−𝜆(𝑡𝑚−𝑡0)},∀𝑡∈𝑡,𝑡. (3.44) For 𝑡∈[𝑡,𝑡] and 𝜃∈[−𝜏,0], when 𝑡+𝜃≥𝑡_𝑚, then (3.44) implies that 𝐸𝑉(𝑥(𝑡+𝜃),𝑡+𝜃)≤𝑀_𝑚+1𝜉^𝑝𝑒^{−𝜆(𝑡𝑚+1−𝑡0)}=𝜌⁻¹𝑒^{−𝜆(𝑡𝑚+1−𝑡𝑚)}𝐸𝑉𝑥𝑡,𝑡<𝑞𝐸𝑉𝑥𝑡,𝑡. (3.45) If 𝑡+𝜃<𝑡_𝑚 for some 𝜃∈[−𝜏,0), we assume that, without loss of generality, 𝑡+𝜃∈[𝑡_𝑙,𝑡_𝑙+1) for some 𝑙∈𝑁,𝑙≤𝑚−1, then from (3.41) and (3.44), we obtain 𝐸𝑉(𝑥(𝑡+𝜃),𝑡+𝜃)≤𝑀_𝑙+1𝜉^𝑝𝑒^{−𝜆(𝑡𝑙+1−𝑡0)}<𝑀_𝑙+1𝜉^𝑝𝑒^{−𝜆(𝑡+𝜃−𝑡0)}≤𝑀_𝑙+1𝑒^𝜆𝜏𝜉^𝑝𝑒^{−𝜆(𝑡−𝑡0)}≤𝜌⁻¹𝑒^𝜆𝜏𝑀_𝑙+1𝑀_𝑚+1𝐸𝑉𝑥𝑡,𝑡=𝜌⁻¹𝑒^𝜆𝜏𝑑⁻¹𝐸𝑉𝑥𝑡,𝑡≤𝑞𝐸𝑉𝑥𝑡,𝑡. (3.46) Therefore, 𝐸𝑉(𝑥(𝑡+𝜃),𝑡+𝜃)<𝑞𝐸𝑉(𝑥(𝑡),𝑡),𝑡∈𝑡,𝑡,𝜃∈[−𝜏,0]. (3.47) The rest of the proof is similar to that of Theorem 3.2 and omitted here.

Remark 3.5. Let 𝑢 and 𝛿 be positive constants. Assume that the conditions of Theorem 3.4 hold, function 𝑢∶[𝑡₀,∞)→𝑅₊ satisfies ∫_𝑡^𝑡+𝛿𝑢(𝑠)𝑑𝑠≤𝑢𝛿 and sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}=𝛿<−(ln𝜌/(𝜆+𝑢)). Then Theorem 3.1 of [24] follows immediately.

Remark 3.6. It is not strictly required by condition (ii) of Theorem 3.4 that each impulse contributes to stabilize the system, as long as the overall contribution of the impulses are stabilizing. Without these 𝑑_𝑘 (i.e., 𝑑_𝑘≡1), it is required that each impulse is a stabilizing factor (𝜌<1), which is more restrictive.

Remark 3.7. It is clear that Theorems 3.2 and 3.4 allow the continuous dynamics of system (2.1) to be unstable, since the function 𝑢(𝑡), which characterizes the changing rate of 𝑉(𝑥(𝑡),𝑡) at 𝑡, is assumed to be nonnegative. Theorems 3.2 and 3.4 show that an unstable stochastic delay system can be successfully stabilized by impulses.

The following theorems show that the trivial solutions of system (2.1) are also almost surely exponentially stable, under some additional conditions.

Assumption 3.8. Suppose the impulsive instances 𝑡_𝑘 satisfy sup_𝑘∈𝑁𝑡_𝑘−𝑡_𝑘−1<∞,inf_𝑘∈𝑁𝑡_𝑘−𝑡_𝑘−1>0. (3.48)

Assumption 3.9. Assume that there is a constant 𝐿>0 such that, for all (𝜙,𝑡)∈𝑃𝐶^𝑝_ℱ𝑡([−𝜏,0];𝑅^𝑛)  × [𝑡₀,∞), 𝐸𝑓𝜙,𝑡^𝑝+𝑔𝜙,𝑡^𝑝≤𝐿sup_{−𝜏≤𝜃≤0}𝐸𝜙(𝜃). (3.49)

Lemma 3.10 (see [23]). Let 𝑝≥1, and let Assumptions 3.8 and 3.9 hold. Then (3.31) implies that, for all 𝑡≥𝑡₀, 𝑥𝑡;𝜉,𝑡₀≤𝐶𝑒^{−(𝜆/𝑝)(𝑡−𝑡0)}𝜉^𝑝a.s., (3.50) where 𝐶 is a positive constant. In other words, under Assumptions 3.8 and 3.9, the 𝑝th moment exponential stability implies the almost exponential stability for system (2.1).

By using Theorems 3.2 and 3.4 and Lemma 3.10, it is easy to show the following conclusions.

Theorem 3.11. Suppose that 𝑝≥1, Assumptions 3.8 and 3.9 and the same conditions as in Theorem 3.2 hold. Then the trivial solution of system (2.1) is also almost surely exponentially stable, with its Lyapunov exponent not greater than −𝜆/𝑝.

Theorem 3.12. Suppose that 𝑝≥1, Assumptions 3.8 and 3.9 and the same conditions as in Theorem 3.4 hold. Then the trivial solution of system (2.1) is also almost surely exponentially stable, with its Lyapunov exponent not greater than −𝜆/𝑝.

4. An Example

Example 4.1. Consider a scalar ISDDs of the form (𝑡)(𝑡)1𝑑𝑥=𝑥𝑑𝑡+4𝑥²(𝑡)+𝑥²(𝑡−2)𝑑𝑤(𝑡),𝑡≠𝑡_𝑘,𝑡≥𝑡₀,Δ𝑥𝑡_𝑘=−0.4𝑥𝑡⁻_𝑘,𝑘∈𝑁. (4.1) It is easy to check that the corresponding system without impulses is not mean square exponentially stable. In fact, if 𝑉(𝑥,𝑡)=𝑥², then it follows from the Itô formula that 𝐸[ℒ𝑉(𝑥(𝑡),𝑥(𝑡−2),𝑡)]≥2𝐸|𝑥(𝑡)|²=2𝐸𝑉(𝑥(𝑡),𝑡). This leads to 𝐸|𝑥(𝑡)|²=𝐸𝑉(𝑥(𝑡),𝑡)≥𝐸𝑉(𝑥(0),0)𝑒^2𝑡=𝐸|𝑥(0)|²𝑒^2𝑡 for all 𝑡≥0. But, in the following, we will show that system (4.1) is mean square exponentially stable and almost exponentially stable.

If 𝑉(𝑥(𝑡),𝑡)=𝑥², then condition (i) of Theorem 3.2 holds with 𝑐₁=𝑐₂=1,𝑝=2, and condition (ii) holds with 𝑑_𝑘=0.36. By calculating, we have 𝐸[ℒ𝑉(𝑥(𝑡),𝑥(𝑡−2),𝑡)]≤(33/16)𝐸𝑉(𝑥(𝑡),𝑡)+(1/16)𝐸𝑉(𝑥(𝑡−2),𝑡). By taking 𝑞=5,𝜆=0.5, and 𝑡_𝑘−𝑡_𝑘−1=0.3, it is easy to verify that condition (iii) of Theorem 3.2 is satisfied, which means system (4.1) is mean square exponentially stable. Applying Theorem 3.11, we can derive that system (4.1) is almost exponentially stable.

Acknowledgments

The authors are grateful to Editor Professor Josef Diblík and anonymous referees for their helpful comments and suggestions which have improved the quality of this paper. This work is supported by Natural Science Foundation of China (no. 10771001), Research Fund for Doctor Station of Ministry of Education of China (no. 20113401110001, no. 20103401120002), TIAN YUAN Series of Natural Science Foundation of China (no. 11126177), Key Natural Science Foundation (no. KJ2009A49), Talent Foundation (no. 05025104) of Anhui Province Education Department, 211 Project of Anhui University (no. KJJQ1101), Anhui Provincial Nature Science Foundation (no. 090416237, no. 1208085QA15), and Foundation for Young Talents in College of Anhui Province (no. 2012SQRL021).