# Exponential Stability of Impulsive Stochastic Functional Differential Systems

Authors: Zheng Wu; Hao Huang; Lianglong Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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


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## Body

## 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,{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). w(t)=(w1(t),w2(t),…,wd(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 Rn denotes the n-dimensional real space equipped with Euclidean norm |·|. If A is a vector or matrix, its transpose is denoted by AT 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 Rn 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];Rn) denotes the family of piecewise continuous functions from [-τ,0] to Rn. A norm on PC([-τ,0];Rn) is defined as ∥ϕ∥=sup-τ≤s≤0|ϕ(s)| for ϕ∈PC([-τ,0];Rn).Forp>0 and t≥0, let PCℱtp([-τ,0];Rn) denote the family of all ℱt-measurable PC([-τ,0];Rn)-value random variables ϕ such that sup-τ≤θ≤0E|ϕ(θ)|p<∞ and PCℱtb([-τ,0];Rn) denote the family of PC([-τ,0];Rn)-value random variables that are bounded and ℱt-measurable.In this paper, we consider the following ISFDS:(2.1)dx(t)=f(xt,t)dt+g(xt,t)dw(t),t≠tk,t≥t0,Δx(tk)=Ik(xtk-,tk),k∈N,xt0=ξ,
where the initial value ξ∈PCℱt0b([-τ,0];Rn),x(t)=(x1(t),…,xn(t))T,xt is regarded as a PC([-τ,0];Rn)-value process and xt(θ)=x(t+θ),θ∈[-τ,0]. Similarly, xt- is defined by xt-(θ)=x(t+θ),θ∈[-τ,0) and xt-(0)=lims→t-x(s). Both f:PCℱtb([-τ,0];Rn)×R+→Rnandg:PCℱtb([-τ,0];Rn)×R+→Rn×d are Borel measurable, and Ik:PCℱtb([-τ,0];Rn)×R+→Rn represents the impulsive perturbation of x at time tk. The fixed moments of impulse times tk satisfy 0≤t0<t1<⋯<tk<⋯,tk→∞(ask→∞), Δx(tk)=x(tk)-x(tk-). Moreover, f, g, and Ik are assumed to satisfy necessary assumptions so that, for any initial data ξ∈PCℱt0b([-τ,0];Rn), 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 Ik(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
(2.2)E|x(t;t0,ξ)|p≤C‖ξ‖pe-λ(t-t0),t≥t0,
for all ξ∈PCℱt0b([-τ,0];Rn). When p=2, it is usually said to be exponentially stable in mean square. It follows from (2.2) that
(2.3)limsup⁡t→∞1tlog⁡E|x(t;t0,ξ)|p≤-λ.
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≥t0(2.4)|x(t;t0,ξ)|p≤C‖ξ‖e-λ(t-t0),a.s.,
for all  ξ∈PCℱt0b([-τ,0];Rn). It follows from  (2.4)  that
(2.5)limsup⁡t→∞1tlog⁡|x(t;t0,ξ)|≤-λ.
The left-hand side of (2.5) is called the Lyapunov exponent of the solution.Definition 2.3.
LetC2,1(Rn×[t0,∞);R+) denote the family of all nonnegative functions V(x,t) on Rn×[t0-τ,∞) that are continuously twice differential in x and once in t. If V∈C2,1(Rn×[t0,∞);R+), define the operator ℒV:PC([-τ,0];Rn)×[t0,∞)→R for system (2.1) by
(2.6)LV(xt,t)=Vt(x,t)+Vx(x,t)f(xt,t)+12trace[gT(xt,t)Vxx(x,t)g(xt,t)],
where Vt(x,t)=∂V(x,t)/∂t,Vx(x,t)=(∂V(x,t)/∂x1,…,∂V(x,t)/∂xn),Vxx(x,t)=(∂2V(x,t)/∂xi∂xj)n×n.


## 3. Main Results

In this section, we will establish some criteria on thepth 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.
LetV(x,t)∈C2,1(Rn×[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 [tk-1,tk),k∈N.Proof.
By the Itô formula,(3.1)V(x(t),t)=V(x(tk-1),tk-1)+∫tk-1tLV(xs,s)ds+∫tk-1tVx(x(s),s)g(xs,s)dw(s)
for all t∈[tk-1,tk), where k∈N. Since xtk-1∈PCℱtk-1b([-τ,0];Rn), we can find an integer l0 such that ∥xtk-1∥<l0a.s. For any integer l>l0, define the stopping time
(3.2)ρl=inf⁡{t∈[tk-1,tk):|x(t)|≥l},
where inf∅=∞ as usual. Since x(t) is continuous on [tk-1,tk),|x(t)| is also continuous on [tk-1,tk). Clearly, ρl→∞ a.s. as l→∞. Moreover, it has EV(x(tk-1),tk)≤cl0, following from xtk-1∈PCℱtk-1b([-τ,0];Rn). It then follows from the definition of ρl above that
(3.3)EV(x(tl′),tl′)=EV(x(tk-1),tk-1)+E[∫tk-1tl′LV(xs,s)ds],
where tl′=t∧ρl. So, letting l→∞, by the dominated convergence theorem and Fubini’s theorem, we have
(3.4)EV(x(t),t)=EV(x(tk-1),tk-1)+E[∫tk-1tLV(xs,s)ds]=EV(x(tk-1),tk-1)+∫tk-1tE[LV(xs,s)]ds,
for t∈[tk-1,tk). This implies that EV(x(t),t) is continuous on [tk-1,tk),k∈N.Theorem 3.2.
LetV∈C2,1(Rn×[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)∈Rn×[t0-τ,∞),
(3.5)c1|x|p≤V(x,t)≤c2|x|p,(ii)
for allk∈N, and ϕ∈PCℱtp([-τ,0];Rn),
(3.6)EV(ϕ(0-)+I(tk,ϕ),tk)≤dkEV(ϕ(0-),tk-),
where 0<dk<exp{-λ(tk+1-tk)-∫tktk+1u(s)ds},(iii)
for allt≥t0,t≠tk,k∈N and ϕ∈PCℱtp([-τ,0];Rn),
(3.7)E[LV(ϕ,t)]≤u(t)EV(ϕ(0),t)
whenever
(3.8)EV(ϕ,t+θ)<qEV(ϕ(0),t),θ∈[-τ,0],
where q>maxk∈N{dk-1eλτ}∨exp{∫t0t1u(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 dataξ∈PCℱt0b([-τ,0];Rn), 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
(3.9)c2eλ(t1-t0)+∫t0t1u(s)ds<M<c2qeλ(t1-t0).
Then it follows from condition (i) and (3.9) that
(3.10)EV(x(t),t)≤c2‖ξ‖p<M‖ξ‖pe-λ(t1-t0),t∈[t0-τ,t0].
In the following, we will show that
(3.11)EV(x(t),t)≤M‖ξ‖pe-λ(tk-t0),t∈[tk-1,tk),k∈N.
In order to do so, we first prove that
(3.12)EV(x(t),t)≤M‖ξ‖pe-λ(t1-t0),t∈[t0,t1).
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),
(3.13)EV(x(t),t)<EV(x(t*),t*)=M‖ξ‖pe-λ(t1-t0),t∈[t0-τ,t*).
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),
(3.14)EV(x(t),t)>EV(x(t*),t*)=c2‖ξ‖p,t∈(t*,t*].
Now in view of (3.9), (3.13), and (3.14), one has, for t∈[t*,t*] and θ∈[-τ,0],
(3.15)EV(x(t+θ),t+θ)≤M‖ξ‖pe-λ(t1-t0)<qEV(x(t*),t*)≤qEV(x(t),t).
By the Razumikhin-type condition (iii),
(3.16)E[LV(xt,t)]≤u(t)EV(x(t),t),∀t∈[t*,t*].
Applying Itô formula and by (3.16), one obtains that
(3.17)EV(x(t*),t*)≤EV(x(t*),t*)+∫t*t*u(s)EV(x(s),s)ds.
Finally, by (3.9), (3.13), (3.14), and the Gronwall inequality,
(3.18)EV(x(t*),t*)≤EV(x(t*),t*)e∫t*t*u(s)ds≤c2‖ξ‖pe∫t0t1u(s)ds<M‖ξ‖pe-λ(t1-t0)=EV(x(t*),t*),
which is a contradiction. So inequality (3.12) holds and (3.11) is true for k=1.
Now assume that(3.19)EV(x(t),t)≤M‖ξ‖pe-λ(tk-t0),∀t∈[tk-1,tk),k∈N,
for all k≤m, where k,m∈N. We proceed to show that
(3.20)EV(x(t),t)≤M‖ξ‖pe-λ(tm+1-t0),∀t∈[tm,tm+1).
Suppose (3.20) is not true, set t̅=inf{t∈[tm,tm+1):EV(x(t),t)>M∥ξ∥pe-λ(tm+1-t0)}. By condition (ii) and (3.20), we know
(3.21)EV(x(tm),tm)≤dmEV(x(tm-),tm-)≤dmM‖ξ‖pe-λ(tm-t0)<M‖ξ‖pe-λ(tm+1-t0).
From this, together with EV(x(t),t) being continuous on t∈[tm,tm+1), we know that t̅∈(tm,tm+1) and
(3.22)EV(x(t),t)<EV(x(t̅),t̅)=M‖ξ‖pe-λ(tm+1-t0),∀t∈[tm,t̅).
Define t̲=sup{t∈[t0,t̅]:EV(x(t),t)≤dmM∥ξ∥pe-λ(tm-t0)}, then t̲∈[tm,t̅) and
(3.23)EV(x(t),t)>EV(x(t̲),t̲)=dmM‖ξ‖pe-λ(tm-t0),∀t∈(t̲,t¯].
For t∈[t̲,t̅] and θ∈[-τ,0], when t+θ≥tm, then (3.22) and (3.23) imply that
(3.24)EV(x(t+θ),t+θ)≤M‖ξ‖pe-λ(tm+1-t0)<M‖ξ‖pe-λ(t+θ-t0)≤Meλτ‖ξ‖pe-λ(t-t0)≤Meλτ‖ξ‖pe-λ(tm-t0)≤qEV(x(t̲),t̲).
If t+θ<tm for some θ∈[-τ,0), we assume that, without loss of generality, t+θ∈[tl,tl+1) for some l∈N,l≤m-1, then from (3.19) and (3.23),
(3.25)EV(x(t+θ),t+θ)≤M‖ξ‖pe-λ(tl+1-t0)<M‖ξ‖pe-λ(t+θ-t0)≤Meλτ‖ξ‖pe-λ(t-t0)≤Meλτ‖ξ‖pe-λ(tm-t0)≤qEV(x(t̲),t̲).
Therefore,
(3.26)EV(x(t+θ),t+θ)<qEV(x(t),t),t∈[t̲,t̅],θ∈[-τ,0].
Then, it follows from condition (iii) that
(3.27)E[LV(xt,t)]≤u(t)EV(x(t),t),∀t∈[t̲,t̅].
Combining Itô formula with (3.27), we can check that
(3.28)EV(x(t̅,t̅))≤EV(x(t̲),t̲)+∫t̅t̲u(s)EV(x(s),s)ds.
Finally, by (3.22), (3.23), and the Gronwall inequality,
(3.29)EV(x(t̅),t̅)≤EV(x(t̲),t̲)e∫t̲t̅u(s)ds≤EV(x(t̲),t̲)e∫tmtm+1u(s)ds=dmM‖ξ‖pe-λ(tm-t0)e∫tmtm+1u(s)ds<EV(x(t̅),t̅),
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
(3.30)E|x(t)|p≤c1c2M‖ξ‖pe-λ(tk-t0)≤c1c2M‖ξ‖pe-λ(t-t0),t∈[tk-1,tk),k∈N,
which implies
(3.31)E‖x‖p≤c1c2M‖ξ‖pe-λ(t-t0),t≥t0,
that is, system (2.1) is pth moment exponentially stable. The proof is complete.Remark 3.3.
Ifu(t)≡c>0, then Theorem 3.1 of [23] follows from Theorem 3.2 immediately.Theorem 3.4.
LetV∈C2,1(Rn×[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)∈Rn×[t0-τ,∞),
(3.32)c1|x|p≤V(x,t)≤c2|x|p,(ii)
for allk∈N and ϕ∈PCℱtp([-τ,0];Rn),
(3.33)EV(ϕ(0-)+I(tk,ϕ))≤ρdkEV(ϕ(0-),tk-),
where 0<ρ<max{e-λ(tm+1-tm)}anddk>0 with d̂=supn∈NΠk=1ndk<∞,(iii)
for allt≥t0,t≠tk,k∈N, and ϕ∈PCℱtp([-τ,0];Rn),
(3.34)E[LV(ϕ,t)]≤u(t)EV(ϕ(0),t)
whenever
(3.35)EV(ϕ,t+θ)<qEV(ϕ(0),t),θ∈[-τ,0],
where q>(ρ-1eλτ)∨(ρ-1eλτd̂-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 dataξ∈PCℱt0b([-τ,0];Rn), 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
(3.36)c2eλ(t1-t0)+∫t0t1u(s)ds<M<c2qeλ(t1-t0).
Then it follows from condition (i) and (3.36) that
(3.37)EV(x(t),t)≤c2‖ξ‖p<M‖ξ‖pe-λ(t1-t0),t∈[t0-τ,t0].
In the following, we will show that
(3.38)EV(x(t),t)≤Mk‖ξ‖pe-λ(tk-t0),t∈[tk-1,tk),
where k∈N and Mk is defined as M1=M and Mk=MΠ1≤l≤k-1dl. Similarly, as the proof in Theorem 3.2, one can prove that
(3.39)EV(x(t),t)≤M‖ξ‖pe-λ(t1-t0),t∈[t0,t1).
Now assume that
(3.40)EV(x(t),t)≤Mk‖ξ‖pe-λ(tk-t0),∀t∈[tk-1,tk),k∈N,
for all k≤m, where k,m∈N. We proceed to show that
(3.41)EV(x(t),t)≤Mm+1‖ξ‖pe-λ(tm+1-t0),∀t∈[tm,tm+1).
Suppose (3.41) is not true, set t̅=inf{t∈[tm,tm+1):EV(x(t),t)>Mk∥ξ∥pe-λ(tm+1-t0)}. By condition (ii),
(3.42)EV(x(tm),tm)≤ρdmEV(x(tm-),tm-)≤ρMm+1‖ξ‖pe-λ(tm-t0)<Mm+1‖ξ‖pe-λ(tm+1-t0).
From this, together with EV(x(t),t) being continuous on t∈[tm,tm+1), we know that t̅∈(tm,tm+1) and
(3.43)EV(x(t),t)<EV(x(t̅),t̅)=Mm+1‖ξ‖pe-λ(tm+1-t0),∀t∈[tm,t̅).
Define t̲=sup{t∈[t0,t̅]:EV(x(t),t)≤ρMm+1∥ξ∥pe-λ(tm-t0)}, then t̲∈[tm,t̅) and
(3.44)EV(x(t),t)>EV(x(t̲),t̲)=ρMm+1‖ξ‖pe-λ(tm-t0),∀t∈(t̲,t¯].
For t∈[t̲,t̅] and θ∈[-τ,0], when t+θ≥tm, then (3.44) implies that
(3.45)EV(x(t+θ),t+θ)≤Mm+1‖ξ‖pe-λ(tm+1-t0)=ρ-1e-λ(tm+1-tm)EV(x(t̲),t̲)<qEV(x(t̲),t̲).
If t+θ<tm for some θ∈[-τ,0), we assume that, without loss of generality, t+θ∈[tl,tl+1) for some l∈N,l≤m-1, then from (3.41) and (3.44), we obtain
(3.46)EV(x(t+θ),t+θ)≤Ml+1‖ξ‖pe-λ(tl+1-t0)<Ml+1‖ξ‖pe-λ(t+θ-t0)≤Ml+1eλτ‖ξ‖pe-λ(t-t0)≤ρ-1eλτMl+1Mm+1EV(x(t̲),t̲)=ρ-1eλτd̂-1EV(x(t̲),t̲)≤qEV(x(t̲),t̲).
Therefore,
(3.47)EV(x(t+θ),t+θ)<qEV(x(t),t),t∈[t̲,t̅],θ∈[-τ,0].
The rest of the proof is similar to that of Theorem 3.2 and omitted here.Remark 3.5.
Letu̅ and δ be positive constants. Assume that the conditions of Theorem 3.4 hold, function u:[t0,∞)→R+ satisfies ∫tt+δu(s)ds≤u̅δ and supk∈ℕ{tk-tk-1}=δ<-(lnρ/(λ+u̅)). Then Theorem 3.1 of [24] follows immediately.Remark 3.6.
It is not strictly required by condition (ii) of Theorem3.4 that each impulse contributes to stabilize the system, as long as the overall contribution of the impulses are stabilizing. Without these dk (i.e., dk≡1), it is required that each impulse is a stabilizing factor (ρ<1), which is more restrictive.Remark 3.7.
It is clear that Theorems3.2 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 instancestk satisfy
(3.48)sup⁡k∈N{tk-tk-1}<∞,inf⁡k∈N{tk-tk-1}>0.Assumption 3.9.
Assume that there is a constantL>0 such that, for all (ϕ,t)∈PCℱtp([-τ,0];Rn)  × [t0,∞),
(3.49)E[|f(ϕ,t)|p+|g(ϕ,t)|p]≤Lsup⁡-τ≤θ≤0E|ϕ(θ)|.Lemma 3.10 (see [23]).
Letp≥1, and let Assumptions 3.8 and 3.9 hold. Then (3.31) implies that, for all t≥t0,
(3.50)|x(t;ξ,t0)|≤Ce-(λ/p)(t-t0)‖ξ‖pa.s.,
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 Theorems3.2 and 3.4 and Lemma 3.10, it is easy to show the following conclusions.Theorem 3.11.
Suppose thatp≥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 thatp≥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(4.1)dx(t)=x(t)dt+14x2(t)+x2(t-2)dw(t),t≠tk,t≥t0,Δx(tk)=-0.4x(tk-),k∈N.
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)e2t=E|x(0)|2e2t for all t≥0. But, in the following, we will show that system (4.1) is mean square exponentially stable and almost exponentially stable.IfV(x(t),t)=x2, then condition (i) of Theorem 3.2 holds with c1=c2=1,p=2, and condition (ii) holds with dk=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 tk-tk-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.



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*Source: 678536-2012-05-09.xml*