We investigate the stability of stochastic delay differential systems with delayed impulses by Razumikhin methods. Some criteria on the pth moment and almost sure exponential stability are obtained. It is shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. Moreover, it is also shown that if a continuous dynamic system is stable, then, under some conditions, the delayed impulses do not destroy the stability of the systems. The effectiveness of the proposed results is illustrated by two examples.
1. Introduction
Impulsive dynamical systems have attracted considerable interest in science and engineering in recent years because they provide a natural framework for mathematical modeling of many real world problems where the reactions undergo abrupt changes [1–3]. These systems have found important applications in various fields, such as control systems with communication constraints [4], sampled-data systems [5, 6], and mechanical systems [7]. On the other hand, impulsive control based on impulsive systems can provide an efficient way to deal with plants that cannot endure continuous control inputs [3]. In recent years, the impulsive control theory has been generalized from deterministic systems to stochastic systems and has been shown to have wide applications [8].
Stability is one of the most important issues in the study of impulsive stochastic delay differential systems (see e.g., [9–15]). Particularly, under condition EV(φ(0)+Ik(φ,t),t)≤ρ1kEV(x,t-), t=tk, the pth moment exponential and almost sure exponential stability were investigated in [12–14]. In [12, 13], the authors show that unstable continuous dynamic systems can be stabilized by impulses. The condition ρ1k<1 is assumed in [12] for any k∈ℕ, which is loosen in [13]. More recently, the condition ρ1k<1 is proved unnecessary when continuous dynamic systems are stable in [14].
In most of recent research results, the impulses are usually assumed to take the following form: ΔX(tk)=X(tk+)-X(tk-)=Ik(X(tk-),tk), which indicates the state jump at the impulse time. However, time delays inevitably occurred in the transmission of the impulsive information. Hence, input delays should be considered (see e.g., [5, 16]). In the context of stability of deterministic differential equations with delayed impulses, there have appeared several results in the literature (see e.g., [17–19]). For example, in [17], the asymptotic stability is investigated for a class of delay-free autonomous systems with the impulses of ΔX(tk+)=C1kX((tk-dk)-), and a sufficient asymptotic stability condition is proposed involving the sizes of impulse input delays. In [19], Chen and Zheng considered more general impulses taking the form ΔX(tk+)=Ik(X(tk-),X((tk-dk)-)) and obtained some criteria of exponential stability for nonlinear time-delay systems with delayed impulse effects.
However, most of the existing results of the stability for systems with delayed impulses were considered for the deterministic differential systems. It is noticed that many real world systems are disturbed by stochastic factors. Therefore, it seems interesting to study the stability of stochastic delay differential systems with delayed impulses. Recently, the exponential stability is investigated for impulsive stochastic functional differential system in [20], and exponential stability and uniform stability in terms of two measures were obtained for stochastic differential systems with delayed impulses. Motivated by the above works, the aim of this paper is to study pth moment and almost sure exponential stability of a stochastic delay differential system with delayed impulses. It is shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. Moreover, it is also shown that if a continuous dynamic system is stable, then, under some conditions, the delayed impulses do not destroy the stability of the systems. Our results can generalize some existing results in [20, 21].
The paper is organized as follows. In Section 2, we introduce the notations and definitions. We establish several stability criteria for stochastic differential delay systems with delayed impulses in Section 3. In Section 4, two examples are given to illustrate the effectiveness of our results.
2. Preliminaries
Throughout this paper, let (Ω,ℱ,P) be a complete probability space with some filtration {ℱt}t⩾0 satisfying the usual conditions (i.e., the filtration is increasing and right continuous while ℱ0 contains all P null sets). Let B=(B(t),t≥0) be an m-dimensional ℱt-adapted Brownian motion.
For x∈ℝd, |x| denotes the Euclidean norm of x. For -∞<a<b<∞, we say that a function from [a,b] to ℝd is piecewise continuous, if the function has at most a finite number of jumps discontinuous on (a,b] and are continuous from the right for all points in [a,b). Given r>0, PC([-r,0];ℝd) denotes the family of piecewise continuous functions from [-r,0] to ℝd with norm ∥φ∥r=sup-r≤θ≤0φ(θ). For p≥1 and t≥t0, let Lℱtp([-r,0];ℝd) be the family of ℱt-adapted and PC([-r,0];ℝd)-valued random variables φ such that E∥φ∥rp<∞. Let ℕ=1,2,… and ℝ+=[0,+∞).
In this paper, we consider the following stochastic delay differential systems with delayed impulses:
(1)dX(t)=f(Xt,t)dt+g(Xt,t)dB(t),hhhhhhhhhhhhhht≠tk,t≥t0;ΔX(tk)=X(tk)-X(tk-)=Ik(X(t-),X(t-dk)-),hhhhhhhhhhhhhhhhhhhhhhhhhhhhk∈ℕ;Xt0=ξ(t0+θ),-τ≤θ≤0,
where {tk,k∈ℕ} is a strictly increasing sequence such that tk→∞ as k→∞; {dk≥0,k∈ℕ} are the impulsive input delays satisfying d=maxkdk and τ=max{r,d}. Xt is defined by Xt(θ)=X(t+θ), -r≤θ≤0. Let Xt-(θ)=X((t+θ)-),-r≤θ≤0, where X(t-)=lims→t-X(s). The mappings I:ℝd×PC([-r,0];ℝd)→ℝd, f:PC([-r,0];ℝd)×ℝ+→ℝd, and g:PC([-r,0];ℝd)×ℝ+→ℝd×m are all Borel-measurable functions. For simplicity, denote V(x(t),t) by V(t).
As a standing hypothesis, we assume that f,g, and I are assumed to satisfy necessary assumptions so that, for any ξ∈Lℱtp([-τ,0];ℝd), system (1) has a unique global solution, denoted by X(t;ξ), and, moreover, X(t;ξ)∈Lℱtp([-r,0];ℝd). In addition, we assume that f(0,t)≡0,g(0,t)≡0 and Ik(0,0)≡0, for all t≥t0,k∈ℕ; then system (1) admits a trivial solution X(t)≡0. Moreover, we make the following assumptions on system (1).
There is a constant L>0, such that
(2)E(|f(Xt,t)|p+|g(Xt,t)|p)<Lsup-r≤θ≤0E|X(t)|p,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhht≥t0.
There exist nonnegative bounded sequences {h1k} and {h2k} such that
(3)|Ik(x,y)|≤h1k|x|+h2k|y|,k∈ℕ.
Set h¯=supk(h1k+h2k).
Let C2,1(ℝd×[t0-r,∞);ℝ+) denote the family of all nonnegative functions V(x,t) on ℝd×[t0-r,∞) that are continuously twice differentiable in x and once in t. For each V∈C2,1(ℝd×[t0-r,∞);ℝ+), define an operator ℒV:PC([-r,0];ℝd)×ℝ+→ℝd for system (1) by
(4)ℒV(Xt,t)=Vt(x,t)+Vx(x,t)f(Xt,t)+12trace[gT(Xt,t)Vxx(x,t)g(Xt,t)],
where
(5)Vt(x,t)=∂V(x,t)∂t,Vx(x,t)=(∂V(x,t)∂x1,…,∂V(x,t)∂xd),Vxx(x,t)=(∂2V(x,t)∂xi∂xj)d×d.
The purpose of this paper is to discuss the stability of system (1). Let us begin with the following definition.
Definition 1.
The trivial solution of system (1) is said to be as follows.
(1) pth moment exponentially stable, if, for any initial data ξ∈Lℱt0p([-r,0];ℝd), the solution X(t) satisfies
(6)E|X(t)|p≤CE∥ξ∥pe-λ(t-t0),
or, equivalently,
(7)limsupt→∞1tlogE|X(t)|p≤-λ,
where λ and C are positive constants independent of.
(2) Almost sure exponentially stable, if the solution X(t) satisfies
(8)limsupt→∞1tlog|X(t)|<-λ,
for any initial data ξ∈Lℱt0p([-r,0];ℝd) and λ>0.
3. Main Results
Before establishing the main results, we derive the following lemma, which is useful to present the main results.
Lemma 2.
Let assumptions (A1) and (A2) hold. Suppose that infk∈ℕ{tk-tk-1}=β1 and (l1-1)β1<d≤l1β1 for some positive integer l1. Then
(9)E|X(t)|p≤K1E∥ξ∥τp,t∈[t0-τ,t0+d],
where K1=3l1(p-1)(1+h¯)l1e3p-1L(dp+dp/2).
Proof.
Since (l1-1)β1<d≤l1β1, the maximum number of impulsive times on the interval (t0,t0+d] is l1. Suppose that the impulsive instants on (t0,t0+d] are ti,1≤i≤m≤l1. For t∈(t0,t1), using (A1), we have
(10)E|X(t)|p=E|ξ(0)+∫t0tf(Xs,s)ds+∫t0tg(Xs,s)dB(s)|p≤3p-1[E∥ξ∥τp+(t-t0)p-1E∫t0t|f(Xs,s)|pdshhhhhhh+(t-t0)(p-2)/2E∫t0t|g(Xs,s)|pds]≤3p-1[E∥ξ∥τp+L(t-t0)p-1∫t0tEsupt0-r≤u≤s|X(u)|pdshhhhhhh+L(t-t0)(p-2)/2∫t0tEsupt0-r≤u≤s|X(u)|pds]≤3p-1E∥ξ∥τp+3p-1L×[(t-t0)p-1+(t-t0)(p-2)/2]×∫t0tEsupt0-r≤u≤s|X(u)|pds,
which implies
(11)Esupt0-r≤s≤t|X(s)|p≤3p-1E∥ξ∥τp+3p-1L(dp-1+d(p-2)/2)×∫t0tEsupt0-r≤u≤s|X(u)|pds.
Using the Gronwall inequality, it follows that
(12)Esupt0-r≤s≤t|X(s)|p≤3p-1E∥ξ∥τpe3p-1L(dp-1+d(p-2)/2)(t-t0),hhhhhhhhhhhhhhhhhht∈(t0,t1).
According to (A2), we get
(13)|X(t1)|=|X(t1-)+Ik(X(t1-),X((t1-d1)-))|≤|X(t1-)|+h11|X(t1-)|+h12|X((t1-d1)-)|.
It follows that
(14)E|X(t1)|p≤3p-1(1+h¯)E∥ξ∥τpe3p-1L(dp-1+d(p-2)/2)(t-t0).
Hence,
(15)E|X(t)|p≤3p-1(1+h¯)E∥ξ∥τpe3p-1L(dp-1+d(p-2)/2)(t1-t0),hhhhhhhhhhhhhhhhhhhhht∈[t0-τ,t1].
Repeating the above argument gives that, for t∈[t0-τ,tm],
(16)E|X(t)|p≤3l1(p-1)(1+h¯)l1E∥ξ∥τpe3p-1L(dp-1+d(p-2)/2)(tm-t0).
Since there are no impulses on (tm,t0+d], we obtain
(17)E|X(t)|p≤3l1(p-1)(1+h¯)l1E∥ξ∥τpe3p-1L(dp-1+d(p-2)/2)d,hhhhhhhhhhhhhhhhhhht∈[t0-τ,t0+d].
This completes the proof.
When the continuous dynamics in system (1) is unstable, the following theorem shows that the system (1) can be stabilized by the delayed impulses.
Theorem 3.
Let the assumptions in Lemma 2 hold. Assume that there exist positive constants c1, c2, γ1, and λ and p≥1 such that
c1|x|p≤V(x,t)≤c2|x|p;
for t∈[tk-1,tk),k∈ℕ,
(18)EℒV(φ(θ),t)≤γ1EV(φ(0),t),
provided that φ∈Lℱtp([-r,0];ℝd) satisfies EV(φ(θ),t+θ)≤qEV(φ(0),t), θ∈[-r,0];
there exist nonnegative constant sequences ρ1k,ρ2k, and ηk such that
(19)EV(t,x+Ik(x,y))≤ρ1kηkEV(x,t-)+ρ2kηkEV(y,(t-dk)-),t=tk,
where ∏k=1∞ηk<∞;
let γ=supm∈ℕ{ρ1m+ρ2meλd},q≥(eλr/γ) and γ<e-(γ1+λ)β2, where β2=supk∈ℕ{tk-tk-1}<∞.
Then the trivial solution of system (1) is pth moment exponentially stable.
Proof.
Define W(t)=eλ(t-t0-d)V(t). From Itô’s differential formula, we have
(20)dW(t)=ℒW(t)+eλ(t-t0-d)Vx(Xt,t)g(Xt,t)dB(t),
for t∈[tk-1,tk), k∈ℕ. It is easy to calculate that
(21)ℒW(t)=λeλ(t-t0-d)V(t)+eλ(t-t0-d)ℒV(t).
Let Δt>0 be small enough such that t+Δt∈(tk-1,tk), then
(22)EW(t+Δt)-EW(t)=∫tt+ΔtEℒW(s)ds,
which implies that
(23)D+EW(t)=EℒW(t),t∈[tk-1,tk),k∈ℕ.
In view of Lemma 2 and (H1), we obtain
(24)EW(t)≤c2K1E∥ξ∥p≤γM,t∈[t0-τ,t0+d],
where M=c2K1E∥ξ∥p/γ. In the following, we will prove
(25)EW(t)≤M,t≥t0+d.
We first show that
(26)EW(t)≤M,t∈[t0+d,t1).
Suppose that it is not true; then there exist some t∈(t0+d,t1) such that EW(t)>M. Set t*=inf{t∈[t0+d,t1):EW(t)>M}; we have t*∈(t0+d,t1) and EW(t*)=M. Let t*=sup{t∈[t0+d,t*):EW(t)≤γM}. For t∈[t*,t*], we see that
(27)EW(t)≥γM≥γEW(t+θ),θ∈[-r,0].
Hence,
(28)EV(X(t),t)≥γe-λrEV(X(t+θ),t+θ)≥1qEV(X(t+θ),t+θ),θ∈[-r,0].
Combining this with (H2), we obtain that, for t∈[t*,t*],
(29)D+EW(t)≤eλ(t-t0-d)(γ1+λ)EV(t)=(γ1+λ)EW(t).
So, we derive that
(30)EW(t*)≤EW(t*)e(γ1+λ)(t*-t*)≤γMe(γ1+λ)β2<M.
It is a contradiction; therefore, (26) holds for t∈(t0+d,t1).
Now, we assume that EW(t)≤∏k=1m-1ηkM,t∈[tm-1,tm), m∈ℕ. We will show that
(31)EW(t)≤∏k=1mηkM,t∈[tm,tm+1).
By (H3), we derive that
(32)EW(tm)=eλ(tm-t0-d)EV(tm)≤eλ(tm-t0-d)×(ρ1mηmEV(tm-)+ρ2mηmEV((tm-dm)-))≤ρ1mηmEW(tm-)+ρ2meλdηmEW((tm-dm)-)≤(ρ1m+ρ2meλd)∏k=1mηkM=γ∏k=1mηkM.
Now, we assume that (31) is not true. Set t*=inf{t∈[tm,tm+1):EW(t)>∏k=1mηkM}; then we have t*∈(tm,tm+1) and EW(t*)=∏k=1mηkM. Let t*=sup{t∈[tm,t*):EW(t)≤γ∏k=1mηkM}. For t∈[t*,t*], we have
(33)EW(t)≥γ∏k=1mηkM≥γEW(t+θ),θ∈[-r,0].
Hence,
(34)EV(X(t),t)≥γe-λrEV(X(t+θ),t+θ)≥1qEV(X(t+θ),t+θ),θ∈[-r,0].
This yields that D+EW(t)≤(γ1+λ)EW(t),t∈[t*,t*]. Therefore,
(35)EW(t*)≤EW(t*)e(γ1+λ)(t*-t*)≤γ∏k=1mηkMe(γ1+λ)β2<∏k=1mηkM,
which leads to a contradiction. Thus, (31) holds.
By mathematical induction, we have
(36)EW(t)<M∏k=1∞ηk,t≥t0.
This implies that
(37)E|X(t)|p<M∏k=1∞ηkc1e-λ(t-t0-d),t≥t0.
This completes the proof.
Remark 4.
In Theorem 3, the positive constant ηk is introduced in (H3), where ηk>1 and ηk≤1 are allowed. As mentioned in [13], the constant ηk is introduced in (H3), which makes it possible to tolerate certain perturbations in the overall impulsive stabilization process; that is, it is not strictly required by Theorem 3 that each impulse contributes to stabilize the system; there can exist some destabilized impulses. Moreover, when η3k-2=1/2,η3k-1=1/2, η3k=4, for k∈ℕ, we have Πk=1∞ηk<5 and ∑k=1∞(ηk-1)=+∞. Then, Theorem 3 can be used, but the results in [20, 21] cannot be applicable to this case.
In the following theorem, we will show that if the continuous dynamics is stable, then, under some condition, the system is still stable with the delayed impulsive effects.
Theorem 5.
Assume that the assumptions in Lemma 2 hold. Suppose that there exist positive constants c1, c2, γ2, and λ and p≥1 such that
c1|x|p≤V(t,x)≤c2|x|p;
for t∈[tk-1,tk), k∈ℕ and V∈C2,1(ℝn×[t0-r,∞);ℝ+),
(38)EℒV(φ(θ),t)≤-γ2EV(φ(0),t),
provided that φ∈Lℱtp([-r,0];ℝd) satisfies EV(φ(θ),t+θ)≤qEV(φ(0),t), θ∈[-r,0];
EV(x+Ik(x,y),t)≤ρ1kEV(x,t-)+ρ2kEV(y,(t-dk)-), for all t=tk;
supk∈ℕ{(ρ1k/min{qe-λr,e(γ2-λ)β1})+ρ2keλd}<1, qe-λr>1 and γ2>λ.
Then the trivial solution of system (1) is pth moment exponentially stable.
Proof.
Since maxk∈ℕ{(ρ1k/min{qe-λr,e(γ2-λ)β1})+ρ2keλd}<1, qe-λr>1 and e(γ2-λ)β1>1, there exists a constant q¯>1 such that
(39)1<q¯<min{qe-λr,e(γ2-λ)β1},ρ1kq¯+ρ2keλd≤1,hhhhhhhhhhhihhhhhhhhhhhhhhhhhhhk∈ℕ.
By Lemma 2 and (H1), we have
(40)EW(t)≤c2K1E∥ξ∥p≤1q¯M,t∈[t0-τ,t0+d],
where M=c2q¯K1E∥ξ∥p. We first show
(41)EW(t)≤M,t∈[t0+d,t1].
This can be verified by a contradiction. Suppose that it is not true, then there exist some t∈[t0+d,t1) such that EW(t)>M. Set t*¯=inf{t∈[t0+d,t1):EW(t)≤M}, then t*¯∈(t0+d,t1). Let t*¯=sup{t∈[t0+d,t*¯):EW(t)≥(1/q¯)M}. For t∈[t*¯,t*¯], we get
(42)q¯EW(t)≥M≥EW(t+θ),-r≤θ≤0.
Hence,
(43)EV(X(t),t)≥1q¯e-λrEV(X(t+θ),t+θ)≥1qEV(X(t+θ),t+θ),-r≤θ≤0.
It follows that, for t∈[t*¯,t*¯],
(44)D+EW(t)≤λeλ(t-t0-d)EV(t)-γ2eλ(t-t0-d)EV(t)=(λ-γ2)EW(t)<0,
which yields that EW(t*¯)≤EW(t*¯)=(1/q¯)M<M. This is a contradiction; therefore, (41) holds for [t0+d,t1).
Now we assume that
(45)EW(t)≤M,t∈[tm-1,tm),m∈ℕ.
We will show that
(46)EW(t)≤M,t∈[tm,tm+1).
In order to do this, we first prove that
(47)EW(tm-)≤1q¯M.
Suppose this is not true, then EW(tm-)>(1/q¯)M. There exist two possible cases as follows.
Case 1. EW(t)>(1/q¯)M, for all t∈[tm-1,tm). Obviously, for t∈[tm-1,tm),
(48)q¯EW(t)≥M≥EW(t+θ),-r≤θ≤0.
Thus, we can get D+EW(t)≤(λ-γ2)EW(t), which implies that
(49)EW(tm-)≤EW(tm-1)e(λ-γ2)(tm-tm-1)≤EW(tm-1)e(λ-γ2)β1<1q¯M.
This is a contradiction.
Case 2. There exist some s∈[tm-1,tm) such that EW(s)≤(1/q¯)M. In this case, set t¯=sup{t∈[tm-1,tm):EW(t)<(1/q¯)M}; then EW(t¯)=(1/q¯)M. Since, for t∈[t¯,tm),
(50)q¯EW(t)≥M≥EW(t+θ),-r≤θ≤0,
it follows that D+EW(t)≤0, which gives EW(tm-)≤EW(t¯)=(1/q¯)M. This is also a contradiction.
Hence, (47) holds. In the following equation, we will show that EW(tm)≤M. In view of (H3), we obtain
(51)EW(tm)=eλ(tm-t0-d)EV(tm)≤eλ(tm-t0-d)(ρ1EV(tm-)+ρ2EV((tm-dm)-))≤ρ1EW(tm-)+ρ2eλdEW((tm-dm)-)≤(ρ1q¯+ρ2eλd)M≤M.
We go on proving (46). Suppose that it is not the case; then, there exist some t∈[tm,tm+1). Set t*¯=inf{t∈[tm,tm+1):EW(t)>M}; then, we have t*¯∈(tm,tm+1). If EW(t)≥(1/q¯)M, set t*¯=tm; otherwise, set t*¯=sup{t∈[tm,t*¯):EW(t)≤(1/q¯)M}. For t∈[t*¯,t*¯], we derive
(52)q¯EW(t)≥M≥EW(t+θ),-r≤θ≤0,
which implies that
(53)EV(t)≥1q¯e-λrEV(t+θ,X(t+θ))≥1qEV(t+θ,X(t+θ)),-r≤θ≤0.
It follows that D+EW(t)<0 for t∈[t*¯,t*¯]. Consequently, EW(t*¯)<EW(t*¯). This is a contradiction. Thus, (46) holds. By mathematical induction, we see that
(54)EW(t)≤M,t≥t0-τ.
Then we can get from (H1) that
(55)E|X(t)|p≤Mc1e-λ(t-t0-d),t≥t0-τ.
This completes the proof.
Remark 6.
When the continuous system in system (1) is stable, the system (1) can always be stable with stabilized impulses. Thus, ρ1k+ρ2k<1 is permissible in Theorem 5, and only one constraint qe-λr>1 is assumed for constant q. However, ρ1+ρ2≥1 and ρ1+ρ2ec~τ>q are necessary in Theorem 3.2 of [20]. Thus, in this aspect, Theorem 5 is more general than the results existing in [20].
The following theorem shows that the trivial solution of system (1) is almost sure exponentially stable, under some additional conditions.
Theorem 7.
Suppose that p≥1 and the conditions in Theorem 3 or Theorem 5 hold. Then, the trivial solution of system (1) is almost sure exponentially stable.
Proof.
Using Theorem 3 or Theorem 5, we derive that the trivial solution of system (1) is pth moment exponentially stable. Therefore, there exists a positive constant C1 such that
(56)E|X(t)|p≤C1e-λ(t-t0).
It is obvious that
(57)E(sup0≤s≤r|X(t+s)|p)≤4p-1(|∑t≤tk≤t+rIk(X(t-),X(t-dk)-)|pE|X(t)|p+E(∫tt+r|f(Xs,s)|ds)phhhhhhhhh+E|sup0≤s≤r∫tt+sg(Xu,u)dB(u)|phhhhhhhhh+E|∑t≤tk≤t+rIk(X(t-),X(t-dk)-)|p).
Combining the Hölder inequality with (A1) and (56) implies that
(58)E∫tt+r|f(Xs,s)|pds≤Lrp-1∫tt+rsup-r≤θ≤0E|X(s+θ)|pds≤C1Lrpe-λ(t-r-t0).
By virtue of Burkholder-Davis-Gundy inequality, (A1), and (56), we have
(59)E(sup0≤s≤r∫tt+s|g(Xu,u)|dB(u))≤Lr(p/2)-1∫tt+rsup-r≤θ≤0E|X(s+θ)|pds≤C1C(p)Lrp/2e-λ(t-r-t0),
where C(p) is a positive constant depending on p only. Thanks to (A2) and (56), we see that
(60)E(∑t≤tk≤t+r|Ik(X(t-),X(t-dk)-)|)p≤l1pEsupt≤tk≤t+r|Ik(X(t-),X(t-dk)-)|p≤l1p2p-1C1h¯eλde-λ(t-r-t0).
Substituting (58)–(60) into (57) gives that
(61)E(sup0≤s≤r|X(t+s)|p)≤C2e-λt,
where C2 is a positive constant. Then for all ɛ∈(0,λ) and n∈ℕ, we have
(62)P(ω:sup0≤s≤r|X(nr+s)|p>e-(λ-ɛ)nr)≤C2e-ɛnr.
Using the Borel-Cantelli Lemma, we see that there exists an n0(ω) such that, for almost all ω∈Ω, n≥n0(ω),
(63)sup0≤s≤r|X(t+s)|p≤e-(λ-ɛ)nr,
where nr≤t≤(n+1)r. It follows that
(64)limsupn→∞logsupnr≤t≤(n+1)r|X(t)|(n+1)r≤-(λ-ɛ)p,a.s.
Consequently,
(65)limsupt→∞log|X(t)|t≤-(λ-ɛ)p,a.s.
Let ɛ→0; then the result follows.
4. Numerical Examples
In this section, two numerical examples are given to show the effectiveness of the main results derived in the preceding section.
Example 1.
Consider a stochastic delay differential system with delayed impulses as follows:
(66)dX(t)=[0.5X(t)+0.125X(t-0.2)]dt+0.5X(t-0.2)dB(t),t≠tk,t≥t0,ΔX(tk)=-0.7X(tk-)+0.2X((tk-0.6)-),hhhhhhhhhhhhhhhhhhhhhk∈ℕ,X(0)=1.2;X(θ)=0,-0.6≤θ<0,
where tk-tk-1=0.3. Let p=2,V(t,x)=x2,c1=c2=1, and q=4. Then
(67)EℒV(t,x)=E|X(t)|2+0.25EX(t)X(t-0.2)+0.25E|X(t-0.2)|2≤1.5E|X(t)|2+0.75qE|X(t-0.2)|2≤4.5E|X(t)|2.
Choose γ1=4.5, γ=0.265, β1=0.2, d=0.6, ρ1k=0.18, ρ2k=0.08, η3k-2=1/2, η3k-1=1/2, η3k=4, λ=0.1, and h¯=L=1. Clearly, (A1) and (A2) hold, and q>(eλr/γ)=2.082, γ=0.265<e-(γ1+λ)β2=0.316. Thus, by Theorems 3 and 7 the trivial solution of system (66) is pth moment and an almost sure exponential stability.
It can be seen in Figures 1 and 2 that unstable continuous dynamics of system (66) can be successfully stabilized by delayed impulses.
System without impulses for Example 1.
System with impulses for Example 1.
Example 2.
Consider a stochastic delay differential system with delayed impulses as follows
(68)dX(t)=[-0.9X(t)+0.125X(t-1)]dt+0.5X(t-1)dB(t),t≠tk,t≥t0,ΔX(tk)=-0.5X(tk-)+0.2X((tk-2)-),k∈ℕ,X(0)=-1;X(θ)=0,-2≤θ<0,
where tk-tk-1=0.5. Let p=2,V(t,x)=x2,c1=c2=1, and q=4/3; then
(69)EℒV(t,x)=-1.8E|X(t)|2+0.25EX(t)X(t-1)+0.25E|X(t-1)|2≤-1.3E|X(t)|2+3q4E|X(t)|2=-0.3E|X(t)|2.
Choose γ2=0.3, β1=0.5, d=0.6, ρ1k=0.5, ρ2k=0.08, λ=0.1, h¯=1, and L=1.2. Therefore, (A1) and (A2) hold, and maxk∈ℕ{(ρ1k/min{qe-λr,e(γ2-λ)β1})+ρ2keλd}=0.651<1 and q=4/3>eλr=1.106. Thus, by Theorems 5 and 7 the trivial solution of system (68) is pth moment and an almost sure exponential stability.
It can be seen from Figures 3 and 4 that the delayed impulses can robust the stability of the system (68).
System without impulses for Example 2.
System with impulses for Example 2.
5. Conclusion
The pth moment and almost sure exponential stability are investigated in this paper. Using Razumikhin methods, several sufficient conditions are established for stability of stochastic delay differential systems with delayed impulses. Finally, two numerical simulation examples are offered to verify the effectiveness of the main results.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The project was sponsored by the Natural Science Foundation of China (no. 11326121), the Anhui Excellent Youth Fund (2013SQRL033ZD), and the Natural Science Foundation of Anhui Province (Grant no. 1408085QA09).
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