Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses

Abstract

A class of generalized impulsive stochastic functional differential systems with delayed impulses is considered. By employing piecewise continuous Lyapunov functions and the Razumikhin techniques, several criteria on the exponential stability and uniform stability in terms of two measures for the mentioned systems are obtained, which show that unstable stochastic functional differential systems may be stabilized by appropriate delayed impulses. Based on the stability results, delayed impulsive controllers which mean square exponentially stabilize linear stochastic delay systems are proposed. Finally, numerical examples are given to verify the effectiveness and advantages of our results.

1. Introduction

In recent years, the theory of impulsive functional differential systems (IFDSs) has attracted an increasing interest due to the wide existence of impulse effects and time delays in real-world systems. An area of particular interest has been the impulsive control of delay systems, with consequent emphasis on stability analysis of IFDSs; see [1–8] and reference therein. However, in the current literature on IFDSs, most authors have assumed that the impulses are only related to the present states. But in most cases, it is more realistic that the impulses depend not only on the present but, also the past states, and such impulses are called delayed impulses. Recently, several studies have attempted to investigate IFDSs with delayed impulses (IFDSs-DI); see [9–12]. By employing Lyapunov functions coupled with Razumikhin techniques, [9] investigated the asymptotic stability and practical stability for a class of generalized IFDSs-DI, while [10, 11] further established several criteria for the exponential stability of the systems. In [12], sufficient conditions for the stability of impulsive differential systems with linear delayed impulses were derived and then applied to impulsively synchronize two coupled chaotic systems by using delayed impulses.

In addition to impulse effects and time delays, as is well known, environment noise exists inevitably in real systems and may greatly affect the performance of systems. Recently, [13, 14] took environment noise into account and generalized delayed impulses to stochastic systems. In particular, applying the Lyapunov-Razumikhin techniques, [13] investigated both moment and almost sure exponential stability of impulsive stochastic functional differential systems with delayed impulses (ISFDSs-DI). In [14], the authors started the study of robust stability and state-feedback stabilization of uncertain impulsive stochastic delayed differential systems with linear delayed impulses.

On the other hand, impulsive control has become an active research area and found successful applications in a wide variety of areas, such as control and synchronization of chaotic systems [15–17], ecosystems management [18], secure communication [15], and orbital transfer of satellite [19, 20]. In the past few years, the impulsive control theory has been generalized from deterministic systems to stochastic systems; see [13, 21, 22]. But in many cases, some well-designed impulsive control schemes cannot work as expected. One reason is that the schemes designed are usually free of delays, but time delays do exist due to time spent in computation and transfer. As is well known, the presence of time delays of controllers may be the cause of serious deterioration of performance or even instability of the resulting controlled system if it is not considered in controller design [23]. One way to overcome this problem is to use delayed impulsive control laws.

Motivated by the above discussion, the present paper will employ the Razumikhin techniques to investigate the problems of stability analysis and impulsive stabilization of ISFDSs-DI. Obviously, it is more difficult to motivate ISFDSs-DI than the common ISFDSs with nondelayed impulses (ISFDSs-nDI), and the key difficulties and challenges lie in finding proper way to deal with the delayed impulses. Different from [13], we will exploit the notion of supremum to deal with the delayed impulses, which will relaxe the restriction imposed on impulses. The proof of the paper is enlightened by the idea from [6, 10]. Our results extend and generalize some results existing in the literature and show that appropriate delayed impulsive perturbations may make unstable stochastic functional differential systems stable. The rest of the paper is organized as follows. In Section 2, we introduce some notations and definitions. In Section 3, several criteria on the exponential stability and uniform stability in terms of two measures for ISFDSs-DI are established. Then, Section 4 is devoted to the discussion of two special types of ISFDSs-DI, which is followed by the stabilization of linear stochastic delay systems by using delay impulses in Section 5. Finally, illustrative examples and concluding remarks are given in Section 6 and Section 7, respectively.

2. Preliminaries

Throughout this paper, unless otherwise specified, we will employ the following notations. 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), and let 𝔼[⋅] be the expectation operator with respect to the probability measure. Let 𝐵(𝑡)=(𝐵₁(𝑡),…,𝐵_𝑚(𝑡))^T be an 𝑚-dimensional Brownian motion defined on the probability space. ℝ=(−∞,+∞), ℝ₊=[0,+∞), ℕ={1,2,3,…}, and ℝ^𝑛 denotes the 𝑛-dimensional real space equipped with the Euclidean norm |⋅|. Moreover, Γ={ℎ∶[𝑡₀−𝜏,∞)×ℝ^𝑛→ℝ₊∣ℎ(𝑡,𝑥) is continuous with inf_(𝑡,𝑥)ℎ(𝑡,𝑥)=0}.

Let 𝜏⩾0, and 𝑃𝐶([−𝜏,0];ℝ^𝑛)={𝜑∶[−𝜏,0]→ℝ^𝑛∣𝜑(𝑡) is continuous for all but at most a finite number of points 𝑡 at which 𝜑(𝑡⁺), 𝜑(𝑡⁻) exist, and 𝜑(𝑡⁺)=𝜑(𝑡)} with the norm ‖𝜑‖=sup{|𝜑(𝜃)|∶−𝜏⩽𝜃⩽0}, where 𝜑(𝑡⁺) and 𝜑(𝑡⁻) denote the right-hand and left-hand limits of 𝜑(𝑡) at 𝑡, respectively. Let 𝑃𝐶^𝑏_ℱ𝑡([−𝜏,0];ℝ^𝑛) be the family of all ℱ_𝑡-measurable and bounded 𝑃𝐶([−𝜏,0];ℝ^𝑛)-valued random variables 𝜉={𝜉(𝜃)∶−𝜏⩽𝜃⩽0}.

If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^T. If 𝑃 is a square matrix, then 𝑃>0(𝑃<0) means that 𝑃 is a symmetric positive (negative) definite matrix, while 𝑃⩾0(𝑃⩽0) is a symmetric positive (negative) semidefinite matrix. 𝜆_min(⋅) and 𝜆_max(⋅) represent the minimum and maximum eigenvalues of the corresponding matrix, respectively, and 𝐼 stands for the identity matrix. Unless explicitly stated, all matrices are assumed to have real entries and compatible dimensions.

Let us consider the following ISFDSs-DI as follows: d𝑥(𝑡)=𝑓𝑡,𝑥_𝑡d𝑡+𝜎𝑡,𝑥_𝑡d𝐵(𝑡),𝑡⩾𝑡₀,𝑡≠𝑡_𝑘,𝑥𝑡_𝑘=𝐼_𝑘𝑡_𝑘,𝑥_𝑡⁻_𝑘,𝑘∈ℕ,(1) with initial value 𝑥_𝑡0=𝜉={𝜉(𝜃)∶−𝜏⩽𝜃⩽0}∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0];ℝ^𝑛), where 𝑥∈ℝ^𝑛; 𝑥_𝑡={𝑥(𝑡+𝜃)∶−𝜏⩽𝜃⩽0} is regarded as a 𝑃𝐶([−𝜏,0];ℝ^𝑛)-valued random variables. Moreover, 𝑓∶ℝ₊×𝑃𝐶([−𝜏,0];ℝ^𝑛)→ℝ^𝑛 and 𝜎∶ℝ₊×𝑃𝐶([−𝜏,0];ℝ^𝑛)→ℝ^𝑛×𝑚 are Borel measurable; 𝐼_𝑘(𝑡_𝑘,𝑥_𝑡⁻_𝑘)∶ℝ₊×𝑃𝐶([−𝜏,0];ℝ^𝑛)→ℝ^𝑛 represents the impulsive perturbation of 𝑥 at time 𝑡_𝑘. The fixed moments of time 𝑡_𝑘 satisfy 0⩽𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯,lim_𝑘→∞𝑡_𝑘=∞.

As a standing hypothesis, we assume that for any 𝜉∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0];ℝ^𝑛), there exists a unique solution to system (1) denoted by 𝑥(𝑡;𝑡₀,𝜉) [24], which is continuous except at 𝑡=𝑡_𝑘, at which it is right continuous and left limitable. For the purpose of stability analysis, we further assume that 𝑓(𝑡,0)=𝜎(𝑡,0)=𝐼_𝑘(𝑡,0)≡0 for all 𝑡⩾𝑡₀,𝑘∈ℕ, then system (1) admits a trivial solution 𝑥(𝑡)≡0.

Definition 1 (see [25]). A function 𝑉(𝑡,𝑥)∶[𝑡₀−𝜏,∞)×ℝ^𝑛→ℝ₊ is said to belong to the class 𝑣₀, if 𝑉 is continuous on each of the sets [𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛, lim_{(𝑡,𝑦)→(𝑡}⁻_𝑘,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑡⁻_𝑘,𝑥) exists, and 𝑉_𝑡, 𝑉_𝑥, and 𝑉_𝑥𝑥 are continuous for (𝑡,𝑥)∈(𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛,𝑘∈ℕ, where 𝑉_𝑡()=()𝑡,𝑥𝜕𝑉𝑡,𝑥,𝑉𝜕𝑡_𝑥()=()𝑡,𝑥𝜕𝑉𝑡,𝑥𝜕𝑥₁(),…,𝜕𝑉𝑡,𝑥𝜕𝑥_𝑛,𝑉_𝑥𝑥()=𝜕𝑡,𝑥²𝑉(𝑡,𝑥)𝜕𝑥_𝑖𝜕𝑥_{𝑗𝑛×𝑛}.(2)

For each 𝑉∈𝑣₀ and (𝑡,𝜙)∈[𝑡_𝑘−1,𝑡_𝑘)×𝑃𝐶^𝑏_ℱ𝑡([−𝜏,0];ℝ^𝑛),(𝑘∈ℕ), one defines the Kolmogorov operator ℒ𝑉 associated with system (1) by ℒ𝑉𝑡,𝜙=𝑉_𝑡𝑡,𝜙(0)+𝑉_𝑥(0)𝑓+1𝑡,𝜙𝑡,𝜙2trace𝜎^T𝑡,𝜙𝑉_𝑥𝑥𝑡,𝜙(0)𝜎𝑡,𝜙.(3)

Definition 2 (see [4]). Let ℎ⁰∈Γ, 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛). One defines ℎ₀𝑡,𝜑=sup_{−𝜏⩽𝑠⩽0}ℎ⁰𝑡+𝑠,𝜑(𝑠),𝑡⩾𝑡₀.(4)

Definition 3. Let ℎ⁰,ℎ∈Γ. Then, system (1) is said to be (S1)(ℎ₀,ℎ)-uniformly stable, if for any given 𝜀>0 and 𝑡₀∈ℝ₊, there exists a 𝛿=𝛿(𝜀)>0 such that 𝔼ℎ₀𝑡₀,𝜉<𝛿implies𝔼ℎ𝑡,𝑥𝑡;𝑡₀,𝜉<𝜀,𝑡⩾𝑡₀,(5)(S2)(ℎ₀,ℎ)-globally exponentially stable, if there exist constants 𝛼>0 and 𝐶⩾1 such that, for all 𝜉∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0];ℝ^𝑛) and 𝑡₀∈ℝ₊, 𝔼ℎ𝑡,𝑥𝑡;𝑡₀,𝜉⩽𝐶𝔼ℎ₀𝑡₀,𝜉e^{−𝛼(𝑡−𝑡0)},𝑡⩾𝑡₀.(6)

Remark 4. The (ℎ₀,ℎ)-stability notions are considered here in the spirit of the work by Lakshmikantham and Liu [26] to unify different stability concepts found in the literature such as the stability of the trivial solution, the partial stability, the conditional stability, and the stability of invariant sets, which would otherwise be treated separately. For example, it is easy to see that (S2) in Definition 3 gives (1)the 𝑝th-moment exponential stability of the trivial solution 𝑥(𝑡)≡0, if ℎ(𝑡,𝑥)=ℎ⁰(𝑡,𝑥)=|𝑥|^𝑝. When 𝑝=2, it is usually called mean square exponential stability; (2)the 𝑝th-moment exponentially partial stability of the trivial solution, if ℎ(𝑡,𝑥)=|(𝑥₁,…,𝑥_𝑠)|^𝑝,1⩽𝑠⩽𝑛 and ℎ⁰(𝑡,𝑥)=|𝑥|^𝑝;(3)the global exponential stability of the prescribed motion 𝑦(𝑡), if ℎ(𝑡,𝑥)=ℎ⁰(𝑡,𝑥)=|𝑥−𝑦|; (4)the global exponential stability of the invariant set 𝐴∈ℝ^𝑛, if ℎ(𝑡,𝑥)=ℎ⁰(𝑡,𝑥)=𝑑(𝑥,𝐴), where 𝑑(𝑥,𝐴) is the distance of 𝑥 from the set 𝐴; (5)the global exponential orbital stability of a periodic solution, if ℎ(𝑡,𝑥)=ℎ⁰(𝑡,𝑥)=𝑑(𝑥,𝐶), where 𝐶 is the closed orbit in the phase space.

3. Stability Results

In this section, we will develop Lyapunov-Razumikhin methods and establish some criteria which provide sufficient conditions for the (ℎ₀,ℎ)-exponential stability and (ℎ₀,ℎ)-uniform stability of ISFDSs-DI. Our results illustrate that the impulses play a positive role in making the continuous flow stable.

Theorem 5. Assume that there exist functions 𝑉∈𝑣₀, ℎ⁰,ℎ∈Γ and constants 𝑐₁>0, 𝑐₂>0, 𝛾>𝑞>1 such that (i)𝑐₁ℎ(𝑡,𝑥)⩽𝑉(𝑡,𝑥)⩽𝑐₂ℎ⁰(𝑡,𝑥) for any (𝑡,𝑥)∈[𝑡₀−𝜏,∞)×ℝ^𝑛; (ii)𝔼ℒ𝑉(𝑡,𝜙)⩽𝑏(𝑡)𝑐(𝔼𝑉(𝑡,𝜙(0))) for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘),𝑘∈ℕ and those 𝜙∈𝑃𝐶_ℱ𝑡([−𝜏,0];ℝ^𝑛) satisfying 𝔼𝑉𝑡+𝜃,𝜙(𝜃)⩽𝛾𝔼𝑉𝑡,𝜙(0)𝑜𝑛−𝜏⩽𝜃⩽0,(7) where 𝑏,𝑐∶ℝ₊→ℝ₊ are continuous; (iii)𝔼𝑉(𝑡_𝑘,𝐼_𝑘(𝑡_𝑘,𝜙))⩽(1/𝛾)(1+𝛽_𝑘)sup_{𝑠∈[−𝜏,0]}𝔼𝑉(𝑡⁻_𝑘+𝑠,𝜙(𝑠)), where 𝛽_𝑘⩾0 and ∑^∞_𝑘=1𝛽_𝑘<∞; (iv)ln𝑞>𝑀₁𝑀₂, where 𝑀₁=sup_𝑡⩾0∫_𝑡^𝑡+𝜇𝑏(𝑠)d𝑠,𝑀₂=sup_𝑠>0{𝑐(𝑠)/𝑠}, and 𝜇=sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}. Then, system (1) is (ℎ₀,ℎ)-exponentially stable for any time delay 𝜏∈(0,∞), and the convergence rate is not greater than min{ln(𝛾/𝑞)/𝜏,(ln𝑞−𝑀₁𝑀₂)/𝜇}.

Proof. Fix any initial data 𝜉∈𝑃𝐶^𝑏_ℱ𝑡0([−𝜏,0);ℝ^𝑛), and write 𝑥(𝑡;𝑡₀,𝜉)=𝑥(𝑡) simply. Let 𝛽>1 be an arbitrary constant, and define 𝛼=min{ln(𝛾/𝑞)/𝜏,(ln𝑞−𝑀₁𝑀₂)/𝛽𝜇}. We claim that 𝔼𝑉(𝑡,𝑥(𝑡))⩽𝑞𝑐₂𝔼ℎ₀𝑡₀,𝜉𝐻(𝑡)e^{−𝛼(𝑡−𝑡0)},𝑡⩾𝑡₀,(8) where 𝐻(𝑡)=1,𝑡₀−𝜏⩽𝑡<𝑡₁,_𝑡0<∏_{𝑡𝑘⩽𝑡}1+𝛽_𝑘,𝑡⩾𝑡₁.(9) Obviously, 𝐻(𝑡) is nondecreasing and 𝐻(𝑡)=𝐻𝑡_𝑚−1for𝑡∈𝑡_𝑚−1,𝑡_𝑚𝐻(𝑡)𝑡,𝑚∈ℕ,=1for𝑡∈₀,𝑡₁.(10) For convenience, we write 𝔼𝑉(𝑡,𝑥(𝑡))=𝑣(𝑡) and 𝑞𝑐₂𝔼ℎ₀(𝑡₀,𝜉)𝐻(𝑡)e^{−𝛼(𝑡−𝑡0)}=𝑤(𝑡). Then, (8) can be written as 𝑣(𝑡)⩽𝑤(𝑡),𝑡⩾𝑡₀.(11)
We first prove that 𝑣(𝑡)⩽𝑤(𝑡),𝑡∈𝑡₀,𝑡₁.(12) If (12) is not true, there would be some 𝑡∈[𝑡₀,𝑡₁) such that 𝑣(𝑡)>𝑤(𝑡). Let 𝑡^∗=inf{𝑡∈[𝑡₀,𝑡₁)∶𝑣(𝑡)>𝑤(𝑡)}. Due to the continuity of 𝑣(𝑡) on [𝑡₀,𝑡₁) and 𝑣𝑡₀+𝜃⩽𝑐₂𝔼ℎ₀𝑡₀⩽1,𝜉𝑞𝑤𝑡₀+𝜃<𝑤𝑡₀,[],+𝜃𝜃∈−𝜏,0(13) we get 𝑡^∗>𝑡₀,𝑣𝑡^∗=𝑤𝑡^∗(𝑡)(𝑡)𝑡,𝑣⩽𝑤for𝑡∈₀,𝑡^∗.(14) Noticing 𝑣(𝑡^∗)=𝑤(𝑡^∗)>(1/𝑞)𝑤(𝑡^∗), 𝑣(𝑡₀)⩽(1/𝑞)𝑤(𝑡₀), we further define 𝑡^∗∗=sup{𝑡∈[𝑡₀,𝑡^∗)∶𝑣(𝑡)⩽(1/𝑞)𝑤(𝑡)}, then 𝑡^∗>𝑡^∗∗,𝑣𝑡^∗∗=1𝑞𝑤𝑡^∗∗(𝑡)>1,𝑣𝑞𝑤(𝑡)for𝑡∈𝑡^∗∗,𝑡^∗.(15) Hence, 1𝑞𝑤(𝑡)⩽𝑣(𝑡)⩽𝑤(𝑡),𝑡∈𝑡^∗∗,𝑡^∗.(16) Consequently, since 𝑡+𝜃∈[𝑡₀−𝜏,𝑡^∗] for any 𝑡∈[𝑡^∗∗,𝑡^∗] and 𝜃∈[−𝜏,0], it follows from (13), (14), and (16) that 𝑣(𝑡+𝜃)⩽𝑤(𝑡+𝜃)⩽e^𝛼𝜏𝑤(𝑡)⩽𝑞e^𝛼𝜏𝑣(𝑡)(𝑡),𝑡⩽𝛾𝑣𝑡∈^∗∗,𝑡^∗,𝜃∈[−𝜏,0],(17) which, by condition (ii), implies that 𝔼ℒ𝑉𝑡,𝑥_𝑡⩽𝑏(𝑡)𝑐(𝑣(𝑡))⩽𝑀₂𝑏(𝑡)𝑣(𝑡),𝑡∈𝑡^∗∗,𝑡^∗.(18) By the Itô formula and applying the Gronwall inequality, we can derive that 𝑣𝑡^∗⩽𝑣𝑡^∗∗e^{∫𝑡∗𝑡∗∗𝑀2𝑏(𝑠)d𝑠}⩽𝑣𝑡^∗∗e^𝑀1𝑀2.(19) That is, 𝑞𝑐₂𝔼ℎ₀𝑡₀,𝜉e^{−𝛼(𝑡∗−𝑡0)}⩽𝑐₂𝔼ℎ₀𝑡₀,𝜉e^{−𝛼(𝑡∗∗−𝑡0)+𝑀1𝑀2},(20) which leads to ln𝑞⩽𝛼𝜇+𝑀₁𝑀₂.(21) This contradicts the definition of 𝛼. Hence, (12) holds.
Now, we assume that 𝑣(𝑡)⩽𝑤(𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘=1,2,…,𝑚(𝑚∈ℕ,𝑚⩾1).(22) Then, the following inequality holds by condition, (iii), (13), and (22) as follows: 𝑣𝑡_𝑚⩽1𝛾1+𝛽_𝑚sup_{[]𝑠∈−𝜏,0}𝑣𝑡⁻_𝑚⩽1+𝑠𝛾1+𝛽_𝑚sup_{[]𝑠∈−𝜏,0}𝑤𝑡⁻_𝑚⩽1+𝑠𝛾1+𝛽_𝑚𝑞𝑐₂𝔼ℎ₀𝑡₀,𝜉e^𝛼𝜏𝐻𝑡⁻_𝑚e^{−𝛼(𝑡𝑚−𝑡0)}⩽𝑐₂𝔼ℎ₀𝑡₀,𝜉𝐻𝑡_𝑚e^{−𝛼(𝑡𝑚−𝑡0)}=1𝑞𝑤𝑡_𝑚<𝑤𝑡_𝑚.(23) Next, we proceed to prove that 𝑣(𝑡)⩽𝑤(𝑡),𝑡∈𝑡_𝑚,𝑡_𝑚+1.(24) On the contrary, we suppose that (24) is not true. Then, there exist some 𝑡∈[𝑡_𝑚,𝑡_𝑚+1) such that 𝑣(𝑡)>𝑤(𝑡). Setting 𝑡=inf{𝑡∈[𝑡_𝑚,𝑡_𝑚+1)∶𝑣(𝑡)>𝑤(𝑡)}, it follows from (23) and the continuity of 𝑣(𝑡) on [𝑡_𝑚,𝑡_𝑚+1) that 𝑡>𝑡_𝑚,𝑣𝑡=𝑤𝑡,𝑣(𝑡)⩽𝑤(𝑡)for𝑡∈𝑡_𝑚,𝑡.(25) In view of 𝑣(𝑡)=𝑤(𝑡)>(1/𝑞)𝑤(𝑡) and 𝑣(𝑡_𝑚)⩽(1/𝑞)𝑤(𝑡_𝑚), we further define 𝑡=sup{𝑡∈[𝑡_𝑚,𝑡)∶𝑣(𝑡)⩽(1/𝑞)𝑤(𝑡)}, then 𝑡<𝑡,𝑣𝑡=1𝑞𝑤𝑡(𝑡)>1,𝑣𝑞𝑤(𝑡)for𝑡∈𝑡,𝑡.(26) Thus, 1𝑞𝑤(𝑡)⩽𝑣(𝑡)⩽𝑤(𝑡),𝑡∈𝑡,𝑡.(27) Consequently, since 𝑡+𝜃∈[𝑡₀−𝜏,𝑡] for all 𝑡∈[𝑡,𝑡] and 𝜃∈[−𝜏,0], we get from (13), (22), (25), and (27) that 𝑣(𝑡+𝜃)⩽𝑤(𝑡+𝜃)⩽e^𝛼𝜏𝑤(𝑡)⩽𝑞e^𝛼𝜏𝑣(𝑡)(𝑡),𝑡⩽𝛾𝑣𝑡∈,𝑡,𝜃∈[−𝜏,0].(28) By similar argument, we will be led to contradiction (21) once again, which verifies the validity of (24).
By mathematical induction, we conclude that (11) or equivalently, (8), is true. Then, it follows from condition (i) that 𝔼ℎ(𝑡,𝑥(𝑡))⩽𝐶𝔼ℎ₀𝑡₀,𝜉e^{−𝛼(𝑡−𝑡0)},𝑡⩾𝑡₀,(29) where 𝐶=max{𝑞𝑐₂/𝑐₁∏^∞_𝑘=1(1+𝛽_𝑘),1}. Thus, system (1) is (ℎ₀,ℎ)-exponentially stable with convergence rate 𝛼. Furthermore, in view of the fact that 𝛽>1 is arbitrary, we must have 𝛼=min{ln(𝛾/𝑞)/𝜏,(ln𝑞−𝑀₁𝑀₂)/𝜇}>0. The proof is therefore complete.

Remark 6. It is noted that Theorem 5 allows for significant increases in 𝑉 between impulses as long as the decrease of 𝑉 at impulse times balance it properly. We can see that the impulses do contribute to the stability behavior of the system. Moreover, compared with the 𝑝th-moment exponential stability theorem in [13] which assumed that 𝔼𝑉𝑡_𝑘,𝐼_𝑘𝑡_𝑘,𝜙⩽𝜌𝑑_𝑘𝔼𝑉𝑡⁻_𝑘,𝜙(0),(30) where 0<𝜌<1,𝑑_𝑘>0,∏^∞_𝑘=1𝑑_𝑘<∞, our result has a wider adaptive range.
Especially, the convergence rate 𝛼=0 if 𝑞=𝛾 in Theorem 5, which implies that system (1) is (ℎ₀,ℎ)-uniformly stable. Therefore, letting 𝑞 tend to 𝛾 in Theorem 5 will immediately yield the following corollary.

Corollary 7. Assume that conditions (i)–(iii) of Theorem 5 hold, while condition (iv) is replaced by
(iv*) ln𝛾>𝑀₁𝑀₂, where 𝑀₁=sup_𝑡⩾0∫_𝑡^𝑡+𝜇𝑏(𝑠)d𝑠, 𝑀₂=sup_𝑠>0{𝑐(𝑠)/𝑠}, and 𝜇=sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}.
Then, system (1) is (ℎ₀,ℎ)-uniformly stable for any time delay 𝜏∈(0,∞).

4. Special Cases

In this section, we will apply the general Razumikhin-type theorems established in previous section to deal with the stability of two special types of system (1).

Case 1 (ISFDSs-nDI). An important special case of system (1) is the following ISFDSs-nDI, in which the state variables on impulses are not related to the time delay d𝑥(𝑡)=𝑓𝑡,𝑥_𝑡d𝑡+𝜎𝑡,𝑥_𝑡d𝐵(𝑡),𝑡⩾𝑡₀,𝑡≠𝑡_𝑘,𝑥𝑡_𝑘=𝐼_𝑘𝑡_𝑘,𝑥𝑡⁻_𝑘,𝑘∈ℕ.(31)

Theorem 8. Assume that all conditions of Theorem 5 hold with the following change: (iii*)𝔼𝑉(𝑡_𝑘,𝐼_𝑘(𝑡_𝑘,𝑥))⩽1/𝑞(1+𝛽_𝑘)𝔼𝑉(𝑡⁻_𝑘,𝑥), where 𝛽_𝑘⩽0 and ∏^∞_𝑘=1𝛽_𝑘<∞. Then, system (31) is (ℎ₀,ℎ)-exponentially stable for any time delay 𝜏∈(0,∞), and the convergence rate is not greater than min{ln(𝑟/𝑞)/𝜏,(ln𝑞−𝑀₁𝑀₂)/𝜇}.

Proof. The proof is similar to that of Theorem 5 only replacing (23) by 𝑣𝑡_𝑚⩽1𝑞1+𝛽_𝑚𝑣𝑡⁻_𝑚⩽1𝑞1+𝛽_𝑚𝑤𝑡⁻_𝑚=𝑐₂𝔼ℎ₀𝑡₀,𝜉𝐻𝑡_𝑚e^{−𝛼(𝑡𝑚−𝑡0)}=1𝑞𝑤𝑡_𝑚<𝑤𝑡_𝑚.(32)

Remark 9. In a special case when ℎ(𝑡,𝑥)=ℎ⁰(𝑡,𝑥)=|𝑥|^𝑝,𝑏(𝑡)≡1,𝑐(𝑠)=𝛾₁(𝑠), Theorem 8 can be expressed as Theorem 3.1 in [25], which demonstrates the generality of our result.

Case 2 (IFDSs-DI). Deterministic systems may be regarded as a special class of stochastic systems. The following deterministic IFDSs-DI, which have been investigated in [10, 11], are exactly system (1) with 𝜎(𝑡,𝜙)≡0: 𝑥̇(𝑡)=𝑓𝑡,𝑥_𝑡,𝑡⩾𝑡₀,𝑡≠𝑡_𝑘,𝑥𝑡_𝑘=𝐼_𝑘𝑡_𝑘,𝑥_𝑡⁻_𝑘,𝑘∈ℕ.(33) Based on Theorem 5, one can easily get the following exponential stability result for system (33). For the notation of 𝑣₀ function family and the upper right-hand derivative appeared in the following theorem, we refer to [6].

Theorem 10. Assume that there exist functions 𝑉∈𝑣₀, ℎ⁰,ℎ∈Γ and constants 𝑐₁>0, 𝑐₂>0, 𝛾>𝑞>1 such that (i)𝑐₁ℎ(𝑡,𝑥)⩽𝑉(𝑡,𝑥)⩽𝑐₂ℎ⁰(𝑡,𝑥) for any (𝑡,𝑥)∈[𝑡₀−𝜏,∞)×ℝ^𝑛; (ii)𝐷⁺𝑉(𝑡,𝜙(0))⩽𝑏(𝑡)𝑐(𝑉(𝑡,𝜙(0))) for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘),𝑘∈ℕ and those 𝜙∈𝑃𝐶([−𝜏,0];ℝ^𝑛) satisfying 𝑉𝑡+𝜃,𝜙(𝜃)⩽𝛾𝑉𝑡,𝜙(0)𝑜𝑛−𝜏⩽𝜃⩽0;(34)(iii)𝑉(𝑡_𝑘,𝐼_𝑘(𝑡_𝑘,𝜙))⩽(1/𝛾)(1+𝛽_𝑘)sup_{𝑠∈[−𝜏,0]}𝑉(𝑡⁻_𝑘+𝑠,𝜙(𝑠)), where 𝛽_𝑘⩾0 and ∑^∞_𝑘=1𝛽_𝑘<∞; (iv)ln𝑞>𝑀₁𝑀₂, where 𝑀₁=sup_𝑡⩾0∫_𝑡^𝑡+𝜇𝑏(𝑠)d𝑠,𝑀₂=sup_𝑠>0{𝑐(𝑠)/𝑠}, and 𝜇=sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}. Then, system (33) is (ℎ₀,ℎ)-exponentially stable for any time delay 𝜏∈(0,∞), and the convergence rate is not greater than min{ln(𝛾/𝑞)/𝜏,(ln𝑞−𝑀₁𝑀₂)/𝜇}.

Remark 11. Obviously, Theorem 10 includes Theorem 3.1 in [10] as a special case. While in [11], the exponential stability theorem was obtained under the following condition: 𝑉𝑡_𝑘,𝐼_𝑘𝑡_𝑘,𝜙⩽𝑑_𝑘𝑉𝑡⁻_𝑘,𝜙(0),where0<𝑑_𝑘−1⩽1areconstants,(35) which is simpler but stronger. Therefore, our results are more general and considerably less conservative.

5. Impulsive Stabilization

This section is devoted to designing a delayed impulsive controller which mean square exponentially stabilize a linear stochastic delay system.

Suppose that we are given an 𝑛-dimensional linear stochastic system with time-varying delay d𝑥(𝑡)=𝐴₀𝑥(𝑡)+𝐴₁𝑥((𝑡+𝐷𝑡−𝜏))d𝑡₀𝑥(𝑡)+𝐷₁𝑥(𝑡−𝜏(𝑡))d𝐵(𝑡),𝑡⩾𝑡₀,(36) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector; time delay of the system 𝜏(𝑡) is a Borel-measurable function on 𝑡⩾0 with 0⩽𝜏(𝑡)⩽𝜏, where 𝜏 is a positive constant; and 𝐵(𝑡) is a standard one-dimensional Brownian motion. We are required to stabilize system (36) by delayed impulses of the form 𝑥𝑡_𝑘=𝐶₀𝑥𝑡⁻_𝑘+𝐶₁𝑥𝑡⁻_𝑘−𝜏𝑡⁻_𝑘,𝑘∈ℕ.(37)

In other words, we need to find appropriate impulsive gain matrices 𝐶₀, 𝐶₁ and impulse time sequence {𝑡_𝑘}_𝑘∈ℕ such that the corresponding linear impulsive stochastic delay system d𝑥(𝑡)=𝐴₀𝑥(𝑡)+𝐴₁𝑥((𝑡+𝐷𝑡−𝜏))d𝑡₀𝑥(𝑡)+𝐷₁𝑥(𝑡−𝜏(𝑡))d𝐵(𝑡),𝑡⩾𝑡₀,𝑡≠𝑡_𝑘,𝑥𝑡_𝑘=𝐶₀𝑥𝑡⁻_𝑘+𝐶₁𝑥𝑡⁻_𝑘−𝜏𝑡⁻_𝑘,𝑘∈ℕ.(38) Will be mean square exponentially stable.

Theorem 12. System (36) can be mean square exponentially stabilized by impulses in the form of (37) if there exist matrices 𝐶_𝑖,𝑖=0,1, 𝑃>0 and positive scalars 𝜀,𝜀_𝑗,𝑗=1,2,3,4 such that the following matrix inequalities hold 𝑃𝐴₀+𝐴^T₀𝑃+𝐷^T₀𝑃𝐷₀−𝜀₁𝑃𝑃𝐴₁+𝐷^T₀𝑃𝐷₁𝐷^T₁𝑃𝐷₁−𝜀₂𝑃⩽0,(39)−𝜀₃𝑃0𝐶^T₀∗−𝜀₄𝑃𝐶^T₁∗∗−𝑃⁻¹𝜀⩽0,(40)₃+𝜀₄<𝜀<1,(41)𝜇=sup_𝑘∈ℕ𝑡_𝑘−𝑡_𝑘−1<𝜀₃+𝜀₄ln𝜀/𝜀₃+𝜀₄𝜀₁𝜀₃+𝜀₄+𝜀₂,(42) where entries denoted by ∗ can be readily inferred from the symmetry of a matrix.

Proof. The conclusion follows from Theorem 5 by considering 𝑉(𝑡,𝑥)=𝑥^T𝑃𝑥 and ℎ⁰(𝑡,𝑥)=ℎ(𝑡,𝑥)=|𝑥|².
Although (39)-(40) are linear with respect to (𝐶_𝑖,𝜀_𝑗),𝑖=0,1,𝑗=1,2,3,4 for fixed 𝑃, it is not linear with the combined variables (𝐶_𝑖,𝜀_𝑗,𝑃), and, therefore, it is not LMIs. This makes the computation difficult but flexible. Two different impulsive controller designing steps are given as follows.
Method  I
Step  1. Choose matrix 𝑃>0.
Step  2. Get 𝜀_𝑖,𝑖=1,2,3,4 and 𝐶₀,𝐶₁ by solving LMI (39), (40), and 𝜀_𝑖>0,𝑖=1,2,3,4,𝜀₃+𝜀₄𝐶<1,_𝑗>(<)0,𝑗=0,1.(43)Step  3. Choose {𝑡_𝑘}_𝑘∈ℕ such that 𝑡_𝑘−𝑡_𝑘−1<((𝜀₃+𝜀₄)ln(𝜀/(𝜀₃+𝜀₄)))/(𝜀₁(𝜀₃+𝜀₄)+𝜀₂),𝑘∈ℕ, where 𝜀 is an arbitrary number satisfying 𝜀₃+𝜀₄<𝜀<1.

Method II

Step  1. Give appropriate positive scalars 𝜀₁ and 𝜀₂ such that LMI (39) is feasible and get matrix 𝑃>0.

Step  2. Get 𝜀₃,𝜀₄ and 𝐶₀,𝐶₁ by solving LMI (40) and 𝜀_𝑖>0,𝑖=3,4,𝜀₃+𝜀₄𝐶<1,_𝑗>(<)0,𝑗=0,1.(44)

Step  3. Choose {𝑡_𝑘}_𝑘∈ℕ such that 𝑡_𝑘−𝑡_𝑘−1<((𝜀₃+𝜀₄)ln(𝜀/(𝜀₃+𝜀₄)))/(𝜀₁(𝜀₃+𝜀₄)+𝜀₂),𝑘∈ℕ, where 𝜀 is an arbitrary number satisfying 𝜀₃+𝜀₄<𝜀<1.

Remark 13. Obviously, the stabilizing impulses (37) are not unique. For example, 𝐶₀=𝐶₁=0 and {𝑡_𝑘}_𝑘∈ℕ with the first impulse interval being finite will be a simple one. In the above two methods, we impose conditions 𝐶_𝑖>0 or 𝐶_𝑖<0 to get impulsive controllers with 𝐶_𝑖≠0,𝑖=0,1.

6. Illustrative Examples

In this section, examples are given to verify the effectiveness and advantages of our results. For simplicity, we consider the case of constant delay in the following examples. It should be pointed out that our results can be applied to systems with time-varying delay.

Example 14. Let us consider a two-dimensional impulsive stochastic delay system d𝑥₁(𝑡)=𝑥₁(𝑡)d𝑡+𝑥₁(𝑡)d𝐵(𝑡),𝑡⩾𝑡₀,𝑡≠𝑡_𝑘,d𝑥₂(𝑡)=−𝑥²₁(𝑡−𝜏)𝑥₂(𝑡)−12𝑎𝑥₂(𝑡−√)d𝑡𝑎𝑥₂(𝑡−𝜏)d𝐵(𝑡),𝑡⩾𝑡₀,𝑡≠𝑡_𝑘,𝑥₁𝑡_𝑘=𝑥₁𝑡⁻_𝑘𝑥,𝑘∈ℕ,₂𝑡_𝑘=12√𝛾1+𝑘⁻²𝑥₂𝑡⁻_𝑘+𝑥₂𝑡⁻_𝑘−𝜏,𝑘∈ℕ,(45) where constants 𝑎>0, 𝜏>0, and 𝛾>1. If 𝜇=sup_𝑘∈ℕ𝑡_𝑘−𝑡_𝑘−1<ln𝑞𝑎,𝛾−1(46) for any 𝑞 satisfying 1<𝑞<𝛾, then system (45) is mean square exponentially stable with respect to 𝑥₂. If 𝜇<ln𝛾𝑎,𝛾−1(47) then system (45) is mean square uniformly stable with respect to 𝑥₂.
We first note that the (ℎ₀,ℎ)-stability properties reduce to the mean square stability properties with respect to 𝑥₂ if ℎ⁰(𝑡,𝑥)=𝑥²₁+𝑥²₂,ℎ(𝑡,𝑥)=𝑥²₂. Choose 𝑉(𝑥)=𝑥²₂, then 𝔼𝑉𝑥𝑡_𝑘=14𝛾1+𝑘⁻²²𝔼𝑥₂(𝑡⁻_𝑘)+𝑥₂(𝑡⁻_𝑘−𝜏)²⩽1𝛾1+𝑘⁻⁴+2𝑘⁻²sup_{[]𝑠∈−𝜏,0}𝔼𝑉𝑥𝑡⁻_𝑘,𝑥+𝑠𝔼ℒ𝑉_𝑡=𝔼−2𝑥²₁(𝑡−𝜏)𝑥²₂(𝑡)−𝑎𝑥²₂(𝑡)+𝑎𝑥²₂(𝑡−𝜏)⩽𝑎𝛾−1𝔼𝑥²₂(𝑡)=𝑎𝛾−1𝔼𝑉(𝑥(𝑡)),(48) whenever 𝔼𝑉(𝑥(𝑡+𝑠))⩽𝛾(𝔼𝑉(𝑥(𝑡))), −𝜏⩽𝑠⩽0.
Taking 𝑐₁=𝑐₂=1, 𝑏(𝑡)≡1, 𝑐(𝑠)=𝑎(𝛾−1)𝑠, and 𝛽_𝑘=𝑘⁻⁴+2𝑘⁻², it is easy to check that all the conditions of Theorem 5 are satisfied under condition (46), which means that system (45) is mean square exponentially stable with respect to 𝑥₂. As well, under condition (47), one can derive the mean square uniform stability with respect to 𝑥₂ according to Corollary 7.
It is noted that 𝑥₁(𝑡)=𝑥₁₀1exp2𝑡−𝑡₀+𝐵(𝑡)−𝐵𝑡₀,(49) where 𝑥₁₀ is the initial data of 𝑥₁(𝑡). Thus, 𝔼𝑥₁(𝑡)|²=𝑥₁₀|²e^𝑡−𝑡0,(50) which demonstrates that the state 𝑥₁ is mean square exponentially unstable. Hence, the existing stability theorems for the trivial solution fail to work. This shows that the notion of stability in terms of two measures is more general, and our results have a wider adaptive range.

Example 15. Consider the following stochastic delay system d𝑥(𝑡)=𝐴₀𝑥(𝑡)+𝐴₁𝑥(+𝐷𝑡−𝜏)d𝑡₀𝑥(𝑡)+𝐷₁𝑥(𝑡−𝜏)d𝐵(𝑡),𝑡⩾𝑡₀,(51) where 𝐴₀=^0.50.3_00.4, 𝐴₁=^1.30.3_0.231.5, 𝐷₀=^0.20.1_0.30.1, and 𝐷₁=0.1𝐼.
It is noted that system (51) is not stable, and that its simulations with 𝜏=0.002, initial data 𝜉(𝜃)=[1−1]^T, and 𝜃∈[−0.002,0] are shown in Figures 1 and 2. In the following, we will design a delayed impulsive control law according to Theorem 12 to exponentially stabilize system (51). Adopting Method I, we choose 𝑃=3.25𝐼, 𝜀=0.9999, then 𝜀₁=3.6290,𝜀₂=3.0558,𝜀₃𝜀=0.7334,₄=0.1929,𝐶₀=0.8089𝐼,𝐶₁=0.1110𝐼(52) is a group of feasible solution to linear matrix inequalities (39)–(41), and the maximum impulse interval can be chosen as 𝜀𝜇=0.01<³+𝜀₄ln𝜀/𝜀₃+𝜀₄𝜀₁𝜀₃+𝜀₄+𝜀₂=0.011.(53) The simulations of system (51) under impulsive control are displayed in Figures 3 and 4, where 𝜏=0.002, initial data 𝜉(𝜃)=[1−1]^T,𝜃∈[−0.002,0], and impulse interval 𝑡_𝑘−𝑡_𝑘−1=0.01,𝑘∈ℕ. It is clearly demonstrated that the impulses we designed successfully stabilize an unstable stochastic delay system.

Figure 1 

The solution to system (51) without impulses (single sample).

Figure 2 

The mean square of the solution to system (51) without impulses (2000 samples).

Figure 3 

The solution to system (51) with impulses (single sample).

Figure 4 

The mean square of the solution to system (51) with impulses (2000 samples).

7. Conclusion

This paper has investigated the exponential stability in terms of two measures for ISFDSs-DI based on Razumikhin-type arguments. The results obtained improve and complement some recent works. Moreover, the stability criteria obtained are applied to stabilize linear stochastic delay systems, and delayed impulsive controllers that exponentially stabilize the systems are proposed.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 660874114, 61273126, and 11226247 and the Anhui Provincial Nature Science Foundation 1308085QA15.