# Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses

Authors: Fengqi Yao; Feiqi Deng; Pei Cheng
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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,{ℱt}t⩾0,ℙ) be a complete probability space with a filtration {ℱt}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),…,Bm(t))T be an m-dimensional Brownian motion defined on the probability space. ℝ=(-∞,+∞), ℝ+=[0,+∞), ℕ={1,2,3,…}, and ℝn denotes the n-dimensional real space equipped with the Euclidean norm |·|. Moreover, Γ={h:[t0-τ,∞)×ℝn→ℝ+∣h(t,x) is continuous with inf(t,x)h(t,x)=0}.Letτ⩾0, and PC([-τ,0];ℝn)={φ:[-τ,0]→ℝn∣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 PCℱtb([-τ,0];ℝn) be the family of all ℱt-measurable and bounded PC([-τ,0];ℝn)-valued random variables ξ={ξ(θ):-τ⩽θ⩽0}.IfA is a vector or matrix, its transpose is denoted by AT. 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. λmin(·) and λmax(·) 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:(1)dx(t)=f(t,xt)dt+σ(t,xt)dB(t),t⩾t0,t≠tk,x(tk)=Ik(tk,xtk-),k∈ℕ,
with initial value xt0=ξ={ξ(θ):-τ⩽θ⩽0}∈PCℱt0b([-τ,0];ℝn), where x∈ℝn; xt={x(t+θ):-τ⩽θ⩽0} is regarded as a PC([-τ,0];ℝn)-valued random variables. Moreover, f:ℝ+×PC([-τ,0];ℝn)→ℝn and σ:ℝ+×PC([-τ,0];ℝn)→ℝn×m are Borel measurable; Ik(tk,xtk-):ℝ+×PC([-τ,0];ℝn)→ℝn represents the impulsive perturbation of x at time tk. The fixed moments of time tk satisfy 0⩽t0<t1<⋯<tk<⋯,limk→∞tk=∞.As a standing hypothesis, we assume that for anyξ∈PCℱt0b([-τ,0];ℝn), there exists a unique solution to system (1) denoted by x(t;t0,ξ) [24], which is continuous except at t=tk, 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)=Ik(t,0)≡0 for all t⩾t0,k∈ℕ, then system (1) admits a trivial solution x(t)≡0.Definition 1 (see [25]).
A functionV(t,x):[t0-τ,∞)×ℝn→ℝ+ is said to belong to the class v0, if V is continuous on each of the sets [tk-1,tk)×ℝn, lim(t,y)→(tk-,x)V(t,y)=V(tk-,x) exists, and Vt, Vx, and Vxx are continuous for (t,x)∈(tk-1,tk)×ℝn,k∈ℕ, where
(2)Vt(t,x)=∂V(t,x)∂t,Vx(t,x)=[∂V(t,x)∂x1,…,∂V(t,x)∂xn],Vxx(t,x)=[∂2V(t,x)∂xi∂xj]n×n.For eachV∈v0 and (t,ϕ)∈[tk-1,tk)×PCℱtb([-τ,0];ℝn),(k∈ℕ), one defines the Kolmogorov operator ℒV associated with system (1) by
(3)ℒV(t,ϕ)=Vt(t,ϕ(0))+Vx(t,ϕ(0))f(t,ϕ)+12trace[σT(t,ϕ)Vxx(t,ϕ(0))σ(t,ϕ)].Definition 2 (see [4]).
Leth0∈Γ, φ∈PC([-τ,0];ℝn). One defines
(4)h0(t,φ)=sup-τ⩽s⩽0h0(t+s,φ(s)),t⩾t0.Definition 3.
Leth0,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
(5)𝔼h0(t0,ξ)<δimplies𝔼h(t,x(t;t0,ξ))<ε,t⩾t0,(S2)
(h0,h)-globally exponentially stable, if there exist constants α>0 and C⩾1 such that, for all ξ∈PCℱt0b([-τ,0];ℝn) and t0∈ℝ+,
(6)𝔼h(t,x(t;t0,ξ))⩽C𝔼h0(t0,ξ)e-α(t-t0),t⩾t0.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)
thepth-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)
thepth-moment exponentially partial stability of the trivial solution, if h(t,x)=|(x1,…,xs)|p,1⩽s⩽n and h0(t,x)=|x|p;(3)
the global exponential stability of the prescribed motiony(t), if h(t,x)=h0(t,x)=|x-y|;(4)
the global exponential stability of the invariant setA∈ℝn, 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, ifh(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 functionsV∈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,∞)×ℝn;(ii)
𝔼ℒV(t,ϕ)⩽b(t)c(𝔼V(t,ϕ(0))) for all t∈[tk-1,tk),k∈ℕ and those ϕ∈PCℱt([-τ,0];ℝn) satisfying
(7)𝔼V(t+θ,ϕ(θ))⩽γ𝔼V(t,ϕ(0))on-τ⩽θ⩽0,
where b,c:ℝ+→ℝ+ are continuous;(iii)
𝔼V(tk,Ik(tk,ϕ))⩽(1/γ)(1+βk)sups∈[-τ,0]𝔼V(tk-+s,ϕ(s)), where βk⩾0 and ∑k=1∞‍βk<∞;(iv)
lnq>M1M2, where M1=supt⩾0∫tt+μ‍b(s)ds,M2=sups>0{c(s)/s}, and μ=supk∈ℕ{tk-tk-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 dataξ∈PCℱt0b([-τ,0);ℝn), 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
(8)𝔼V(t,x(t))⩽qc2𝔼h0(t0,ξ)H(t)e-α(t-t0),t⩾t0,
where
(9)H(t)={1,t0-τ⩽t<t1,∏t0<tk⩽t‍(1+βk),t⩾t1.
Obviously, H(t) is nondecreasing and
(10)H(t)=H(tm-1)fort∈[tm-1,tm),m∈ℕ,H(t)=1fort∈[t0,t1).
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
(11)v(t)⩽w(t),t⩾t0.
We first prove that(12)v(t)⩽w(t),t∈[t0,t1).
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
(13)v(t0+θ)⩽c2𝔼h0(t0,ξ)⩽1qw(t0+θ)<w(t0+θ),θ∈[-τ,0],
we get
(14)t*>t0,v(t*)=w(t*),v(t)⩽w(t)fort∈[t0,t*).
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
(15)t*>t**,v(t**)=1qw(t**),v(t)>1qw(t)fort∈(t**,t*].
Hence,
(16)1qw(t)⩽v(t)⩽w(t),t∈[t**,t*].
Consequently, since t+θ∈[t0-τ,t*] for any t∈[t**,t*] and θ∈[-τ,0], it follows from  (13),  (14), and (16)  that
(17)v(t+θ)⩽w(t+θ)⩽eατw(t)⩽qeατv(t)⩽γv(t),t∈[t**,t*],θ∈[-τ,0],
which, by condition (ii), implies that
(18)𝔼ℒV(t,xt)⩽b(t)c(v(t))⩽M2b(t)v(t),t∈[t**,t*].
By the Itô formula and applying the Gronwall inequality, we can derive that
(19)v(t*)⩽v(t**)e∫t**t*‍M2b(s)ds⩽v(t**)eM1M2.
That is,
(20)qc2𝔼h0(t0,ξ)e-α(t*-t0)⩽c2𝔼h0(t0,ξ)e-α(t**-t0)+M1M2,
which leads to
(21)lnq⩽αμ+M1M2.
This contradicts the definition of α. Hence, (12) holds.
Now, we assume that(22)v(t)⩽w(t),t∈[tk-1,tk),k=1,2,…,m(m∈ℕ,m⩾1).
Then, the following inequality holds by condition, (iii), (13), and (22) as follows:
(23)v(tm)⩽1γ(1+βm)sups∈[-τ,0]v(tm-+s)⩽1γ(1+βm)sups∈[-τ,0]w(tm-+s)⩽1γ(1+βm)qc2𝔼h0(t0,ξ)eατH(tm-)e-α(tm-t0)⩽c2𝔼h0(t0,ξ)H(tm)e-α(tm-t0)=1qw(tm)<w(tm).
Next, we proceed to prove that
(24)v(t)⩽w(t),t∈[tm,tm+1).
On the contrary, we suppose that (24) is not true. Then, there exist some t∈[tm,tm+1) such that v(t)>w(t). Setting t-=inf{t∈[tm,tm+1):v(t)>w(t)}, it follows from (23) and the continuity of v(t) on [tm,tm+1) that
(25)t->tm,v(t-)=w(t-),v(t)⩽w(t)fort∈[tm,t-).
In view of v(t-)=w(t-)>(1/q)w(t) and v(tm)⩽(1/q)w(tm), we further define t_=sup{t∈[tm,t-):v(t)⩽(1/q)w(t)}, then
(26)t_<t-,v(t_)=1qw(t_),v(t)>1qw(t)fort∈(t_,t-].
Thus,
(27)1qw(t)⩽v(t)⩽w(t),t∈[t_,t-].
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
(28)v(t+θ)⩽w(t+θ)⩽eατw(t)⩽qeατv(t)⩽γv(t),t∈[t_,t-],θ∈[-τ,0].
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
(29)𝔼h(t,x(t))⩽C𝔼h0(t0,ξ)e-α(t-t0),t⩾t0,
where C=max{qc2/c1∏k=1∞‍(1+βk),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 Theorem5 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
(30)𝔼V(tk,Ik(tk,ϕ))⩽ρdk𝔼V(tk-,ϕ(0)),
where 0<ρ<1,dk>0,∏k=1∞‍dk<∞, our result has a wider adaptive range.
Especially, the convergence rateα=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 Theorem5 hold, while condition (iv) is replaced by
(iv*)lnγ>M1M2, where M1=supt⩾0∫tt+μ‍b(s)ds, M2=sups>0{c(s)/s}, and μ=supk∈ℕ{tk-tk-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
(31)dx(t)=f(t,xt)dt+σ(t,xt)dB(t),t⩾t0,t≠tk,x(tk)=Ik(tk,x(tk-)),k∈ℕ.Theorem 8.
Assume that all conditions of Theorem5 hold with the following change:  (iii*)
𝔼V(tk,Ik(tk,x))⩽1/q(1+βk)𝔼V(tk-,x), where βk⩽0 and ∏k=1∞‍βk<∞. 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 Theorem5 only replacing (23) by
(32)v(tm)⩽1q(1+βm)v(tm-)⩽1q(1+βm)w(tm-)=c2𝔼h0(t0,ξ)H(tm)e-α(tm-t0)=1qw(tm)<w(tm).Remark 9.
In a special case whenh(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:
(33)x˙(t)=f(t,xt),t⩾t0,t≠tk,x(tk)=Ik(tk,xtk-),k∈ℕ.
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 functionsV∈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,∞)×ℝn;(ii)
D+V(t,ϕ(0))⩽b(t)c(V(t,ϕ(0))) for all t∈[tk-1,tk),k∈ℕ and those ϕ∈PC([-τ,0];ℝn) satisfying
(34)V(t+θ,ϕ(θ))⩽γV(t,ϕ(0))on-τ⩽θ⩽0;(iii)
V(tk,Ik(tk,ϕ))⩽(1/γ)(1+βk)sups∈[-τ,0]V(tk-+s,ϕ(s)), where βk⩾0 and ∑k=1∞‍βk<∞;(iv)
lnq>M1M2, where M1=supt⩾0∫tt+μ‍b(s)ds,M2=sups>0{c(s)/s}, and μ=supk∈ℕ{tk-tk-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, Theorem10 includes Theorem 3.1 in [10] as a special case. While in [11], the exponential stability theorem was obtained under the following condition:
(35)V(tk,Ik(tk,ϕ))⩽dkV(tk-,ϕ(0)),where0<dk-1⩽1areconstants,
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 ann-dimensional linear stochastic system with time-varying delay
(36)dx(t)=[A0x(t)+A1x(t-τ(t))]dt+[D0x(t)+D1x(t-τ(t))]dB(t),t⩾t0,
where x(t)∈ℝn 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
(37)x(tk)=C0x(tk-)+C1x(tk--τ(tk-)),k∈ℕ.In other words, we need to find appropriate impulsive gain matricesC0, C1 and impulse time sequence {tk}k∈ℕ such that the corresponding linear impulsive stochastic delay system
(38)dx(t)=[A0x(t)+A1x(t-τ(t))]dt+[D0x(t)+D1x(t-τ(t))]dB(t),t⩾t0,t≠tk,x(tk)=C0x(tk-)+C1x(tk--τ(tk-)),k∈ℕ.
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 Ci,i=0,1, P>0 and positive scalars ε,εj,j=1,2,3,4 such that the following matrix inequalities hold
(39)[PA0+A0TP+D0TPD0-ε1PPA1+D0TPD1D1TPD1-ε2P]⩽0,(40)[-ε3P0C0T*-ε4PC1T**-P-1]⩽0,(41)ε3+ε4<ε<1,(42)μ=supk∈ℕ{tk-tk-1}<(ε3+ε4)ln(ε/(ε3+ε4))ε1(ε3+ε4)+ε2,
where entries denoted by * can be readily inferred from the symmetry of a matrix.Proof.
The conclusion follows from Theorem5 by considering V(t,x)=xTPx and h0(t,x)=h(t,x)=|x|2.
Although (39)-(40) are linear with respect to (Ci,εj),i=0,1,j=1,2,3,4 for fixed P, it is not linear with the combined variables (Ci,εj,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 εi,i=1,2,3,4 and C0,C1 by solving LMI (39), (40), and
(43)εi>0,i=1,2,3,4,ε3+ε4<1,Cj>(<)0,j=0,1.Step  3. Choose {tk}k∈ℕ such that tk-tk-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  IIStep  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
(44)εi>0,i=3,4,ε3+ε4<1,Cj>(<)0,j=0,1.Step  3. Choose {tk}k∈ℕ such that tk-tk-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 {tk}k∈ℕ with the first impulse interval being finite will be a simple one. In the above two methods, we impose conditions Ci>0 or Ci<0 to get impulsive controllers with Ci≠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(45)dx1(t)=x1(t)dt+x1(t)dB(t),t⩾t0,t≠tk,dx2(t)=[-x12(t-τ)x2(t)-12ax2(t)]dt-ax2(t-τ)dB(t),t⩾t0,t≠tk,x1(tk)=x1(tk-),k∈ℕ,x2(tk)=12γ(1+k-2)[x2(tk-)+x2(tk--τ)],k∈ℕ,
where constants a>0, τ>0, and γ>1. If
(46)μ=supk∈ℕ{tk-tk-1}<lnqa(γ-1),
for any q satisfying 1<q<γ, then system (45) is mean square exponentially stable with respect to x2. If
(47)μ<lnγa(γ-1),
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)=x12+x22,h(t,x)=x22. Choose V(x)=x22, then
(48)𝔼V(x(tk))=14γ(1+k-2)2𝔼[x2(tk-)+x2(tk--τ)]2⩽1γ(1+k-4+2k-2)sups∈[-τ,0]𝔼V(x(tk-+s)),𝔼ℒV(xt)=𝔼[-2x12(t-τ)x22(t)-ax22(t)+ax22(t-τ)]⩽a(γ-1)𝔼x22(t)=a(γ-1)𝔼V(x(t)),
whenever 𝔼V(x(t+s))⩽γ(𝔼V(x(t))), -τ⩽s⩽0.
Takingc1=c2=1, b(t)≡1, c(s)=a(γ-1)s, and βk=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(49)x1(t)=x10exp[12(t-t0)+B(t)-B(t0)],
where x10 is the initial data of x1(t). Thus,
(50)𝔼|x1(t)|2=|x10|2et-t0,
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(51)dx(t)=[A0x(t)+A1x(t-τ)]dt+[D0x(t)+D1x(t-τ)]dB(t),t⩾t0,
where A0=[0.50.300.4], A1=[1.30.30.231.5], D0=[0.20.10.30.1], 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
(52)ε1=3.6290,ε2=3.0558,ε3=0.7334,ε4=0.1929,C0=0.8089I,C1=0.1110I
is a group of feasible solution to linear matrix inequalities (39)–(41), and the maximum impulse interval can be chosen as
(53)μ=0.01<(ε3+ε4)ln(ε/(ε3+ε4))ε1(ε3+ε4)+ε2=0.011.
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 tk-tk-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.



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