# Boundedness of Stochastic Delay Differential Systems with Impulsive Control and Impulsive Disturbance

Authors: Liming Wang; Baoqing Yang; Xiaohua Ding; Kai-Ning Wu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper considers thep-moment boundedness of nonlinear impulsive stochastic delay differential systems (ISDDSs). Using the Lyapunov-Razumikhin method and stochastic analysis techniques, we obtain sufficient conditions which guarantee the p-moment boundedness of ISDDSs. Two cases are considered, one is that the stochastic delay differential system (SDDS) may not be bounded, and how an impulsive strategy should be taken to make the SDDS be bounded. The other is that the SDDS is bounded, and an impulsive disturbance appears in this SDDS, then what restrictions on the impulsive disturbance should be adopted to maintain the boundedness of the SDDS. Our results provide sufficient criteria for these two cases. At last, two examples are given to illustrate the correctness of our results.


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## Body

## 1. Introduction

Boundedness is an important property of a given system; for example, in the population models, the boundedness of a biological population is strongly connected with the persistence and extinction [1]. Another important application is on the stability; the practical stability actually is of a kind of boundedness [2]. Impulsive phenomena widely exist in the real world, and known, impulsive effects can change the properties of a given system; for example, given an unstable system, if a suitable impulsive strategy, including the impulsive strength and impulsive moments, is adopted, this system can be stabilized [3]. It is easy to understand that the impulsive effects can destroy the boundedness of a given system when the impulsive strength is large enough and the impulsive interval is small enough. Time delay is extensive in the engineering and applications and impulsive delay differential systems were considered in lots of papers [3–9]. The boundedness of impulsive delay differential systems has also been paid considerable attentions in the past decades. In [10], the authors presented sufficient conditions for uniform ultimate boundedness by virtue of the Lyapunov functional method. The boundedness of variable impulsive perturbations system was considered in [11] and the eventual boundedness was studied in [12]. Recently, the perturbing Lyapunov function method was also used in the study of boundedness [13].Stochastic noise is ubiquitous [14–16] and stochastic delay differential systems (SDDSs) have been one of the focuses of scientific research for many years. Many properties of SDDSs have been studied and lots of papers were published; see [17, 18] and the references therein. Being the wide existence of stochastic delay and impulsive effects, it is a natural task to consider the stochastic delay differential systems with impulsive effects. These systems are described by impulsive stochastic delay differential systems (ISDDSs). In the past ten years, the stability of ISDDSs has attracted a lot of researchers, and a great deal of results on the stability of ISDDSs have been reported; see [19–24] and the references therein.However, little attention has been paid to the boundedness of ISDDSs. In this paper, the boundedness of ISDDSs is considered under two cases. The first case is that the SDDSs may be unbounded, then what kind of impulsive strategy should be taken to make the system be bounded. The second case is that the SDDSs are bounded, then this system can tolerate what kind of impulsive effect to maintain the boundedness.In this paper, sufficient conditions are presented to guarantee the boundedness of ISDDSs; these conditions also admit the global existence of solutions for ISDDSs, which usually was a standard assumption in many papers [25–27]. Making use of the Lyapunov-Razumikhin method, we generalize the results of [10] to the stochastic situation. At last, two examples are given to illustrate the correctness of our results.


## 2. Preliminaries and Model Description

Let(Ω,F,{Ft}t⩾0,P) be a complete probability space with a filtration {Ft}t⩾0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). Let R=(-∞,+∞), R+=[0,+∞), and N={1,2,…}. If A is a vector or a matrix, its transpose is denoted by AT. Consider PC(J;Rn)={φ:J→Rn, φ(s) is continuous for all but at most countable points s∈J and at these points, φ(s+) and φ(s-) exist and φ(s+)=φ(s)}, where J⊂R is an interval and φ(s+) and φ(s-) denote the right-hand and left-hand limits of the function φ(s) at time s, respectively. Consider PC1,2={φ(t,x):φ(·,x)∈PC and φ(t,x)∈C1,2 if t is not at the uncontinuous points s}. Let PCF0b([-τ,0];Rn)(PCFtb([-τ,0];Rn)) denote the family of all bounded F0(Ft)-measurable, PC-valued random variables. Let |·| be the Euclidean norm in Rn and φτ=sup⁡-τ⩽θ⩽0|φ(t+θ)|.Consider the following nonlinear impulsive stochastic delay differential system:(1)dxt=ft,xtdt+gt,xtdBt,t>t0,t≠tk,k∈N,xtk=xtk-+Itk,xtk-,k∈N,xt0+s=φs,s∈-τ,0,where xt(s)=x(t+s),s∈[-τ,0], f:R+×PC([-τ,0],Rn)→Rn, g:R+×PC([-τ,0],Rn)→Rn×m, I:R+×Rn→Rn and satisfies global Lipschitz condition, τ represents the delay in system (1), impulsive moment tk satisfies 0<t1<t2<⋯<tn<⋯, and tk→∞ as k→∞. B(t) is an m-dimensenal Brownian motion and φ(s)∈PCF0b([-τ,0],Rn).Given a functionV∈PC1,2:R+×Rn→R+, the operator L of V(t,x) with respect to system (1) is defined by(2)LVt,x=Vt+Vxft,xt+12tracegTt,xtVxxgt,xt,where(3)Vt=∂Vt,x∂t,Vx=∂Vt,x∂x1,∂Vt,x∂x2,…,∂Vt,x∂xnT,Vxx=∂2Vt,x∂xi∂xjn×n.Definition 1.
System (1) is said to be (1)
p-moment bounded if, for every B1>0 and t0∈R+, there exists B2=B2(t0,B1) such that if φ∈PCF0b([-τ,0],Rn) with Eφτp⩽B1 and x=x(t,t0,φ) is a solution of (1), then E|x(t,t0,φ)|p⩽B2 for all t⩾t0;(2)
p-moment uniformly bounded if the system (1) is p-moment bounded and B2 is independent of t0;(3)
p-moment ultimately bounded if the system (1) is p-moment bounded and there exists a positive constant B such that for every B3>0 and t0∈R+ there exists some T=T(t0,B3)>0; if φ∈PCF0b([-τ,0],Rn) with Eφτp⩽B3, then E|x(t,t0,φ)|p⩽B for t⩾t0+T;(4)
p-moment uniformly ultimately bounded, if the system (1) is p-moment ultimately bounded and T is independent of t0.


## 3. Boundedness with Impulsive Control

In this section, we consider the first case: when the given SDDS may not be bounded, we adopt an impulsive strategy to get the boundedness. The main result is stated as follows.Theorem 2.
Assume there exist a positive functionV(t,x)∈PC1,2 and positive constants ρ,p,a,b,γ,λ, where 0<λ<1 and 1-λ-γτ>0, such that (1)
a|x|p⩽V(t,x)⩽b|x|p for any (t,x);(2)
fort≠tk, any s∈[-τ,0], and ϕ(t)∈PC([-τ,0],Rn), LV(t,ϕ(0))⩽γV(t,ϕ(0)) whenever V(t,ϕ(0))⩾λV(t+s,ϕ(s)) and |ϕ(0)|p⩾ρ;(3)
V(tk,ϕ(0)+I(tk,ϕ(0)))⩽λV(tk-,ϕ(0)) for all |ϕ(0)|p⩾ρ;(4)
there exists a positive constantρ1⩾ρ such that if |ϕ(0)|p⩽ρ, then |ϕ(0)+I(tk,ϕ(0))|p⩽ρ1;(5)
α=sup⁡k∈Z{tk-tk-1}<∞, αγ<1-λ.
Then the system (1) is p-moment uniformly ultimately bounded.Proof.
We separate the proof into two parts. First, we show thep-moment uniform boundedness and then we give the ultimate uniform boundedness.
Step
1. Let B1>0. Without loss of generality, we assume B1⩾ρ1⩾ρ. Choose B2=B2(B1) such that bB1<λaB2; then we can see B2>B1.
LetEφτp<B1 and t0∈[tl-1,tl) for some positive integer l. Suppose x(t)=x(t,t0,φ) is a solution of system (1) with initial value φ and its maximal interval of existence is [t0-τ,t0+β) for some positive constant β. We will show that, for any t∈[t0-τ,t0+β), E|x(t)|p⩽B2. By the way, if this statement is true, we know that the solution of system (1) is not explored in [t0,t0+β), and the global existence of the solution follows.
For the sake of contradiction, supposeE|x(t)|p⩾B2 for some t∈[t0,t0+β). Then there exists t^=inf⁡{t∈[t0-τ,t0+β)∣E|x(t)|p>B2}. Note that E|x(t)|p⩽Eφτp⩽B1<B2 for t∈[t0-τ,t0]; we see that t^∈(t0,t0+β) and E|x(t)|p⩽B2 for t∈[t0-τ,t^) and E|x(t^)|p⩾B2.
WriteV(t,x(t))=V(t). For t∈[t0-τ,t0], we have EV(t)⩽bE|x(t)|p⩽bEφτp⩽bB1<λaB2<aB2, and EV(t^)⩾aE|x(t^)|p⩾aB2. Define t∗=inf⁡{t∈[t0,t^]∣EV(t)⩾aB2} and then t∗∈(t0,t^] and EV(t)<aB2 for t∈[t0-τ,t∗) and EV(t∗)⩾aB2.
We claim thatt∗≠tk for any k∈N and then EV(t∗)=aB2.
If it is not true, supposet∗=tk for some k. If E|x(tk-)|p⩾ρ, then aB2⩽EV(tk)⩽λEV(tk-)<λaB2<aB2, which is a contradiction. If E|x(tk-)|p<ρ, then Extk|p=Extk-+I(tk,x(tk-))|p<ρ1<B1. Then aB2<EV(tk)<bB1<λaB2<aB2, which is a contradiction.
Now we will proceed under two cases.
Case
1.Consider tl-1⩽t0<t∗<tl.
Lett-=sup⁡{t∈[t0,t∗]∣EV(t)⩽λaB2}. Since EV(t0)<bB1<λaB2, EV(t∗)=aB2>λaB2, and EV(t) is continuous on [t0,t∗], then t-∈(t0,t∗) and EV(t-)=λaB2 and, when t∈[t-,t∗], EV(t)⩾λaB2. Hence, for t∈[t-,t∗] and s∈[-τ,0], we have(4)λEVt+s⩽λaB2⩽EV(t),bB1⩽λaB2⩽EVt⩽bExtp,and we can get(5)Extp⩾B1⩾ρ.Then, by virtue of condition (2), for t∈[t-,t∗],(6)ELVt⩽γEVt,EVt∗-EVt-=∫t-t∗‍ELVsds⩽∫t-t∗‍γEVsds<γαaB2.However,(7)EVt∗-EVt-=aB2-λaB2=1-λaB2>γαaB2,which is contradiction. Then we get, in this case,(8)Extp⩽B2.
Case
2.Consider tk<t∗<tk+1 for some k⩾l.
Note thatEV(tk)⩽λaB2. This inequality can be obtained by the following reason: if E|x(tk-)|p⩾ρ, then EV(tk)⩽λEV(tk-)⩽λaB2. If E|x(tk-)|p<ρ, we get E|x(tk)|p<ρ1<B1, and then(9)EVtk<bB1<λaB2.
Definet-=sup⁡{t∈[tk,t∗]∣EV(t)⩽λaB2}, and then t-∈[tk,t∗), EV(t-)=λaB2, and EV(t)⩾λaB2 for t∈[t-,t∗]. The same argument as the one in Case 1 yields a contradiction. Therefore, in this case, we have, for any t∈[t0-τ,∞),(10)Extp⩽B2.
Now we get that, under conditions (1) to condition (5), the solutions of (1) are p-moment uniformly bounded. That is, if Eφτp⩽ρ1, there exists a constant B>0, such that E|x(t,t0,φ)|p⩽B for all t⩾t0-τ, and, from the proof, we have bρ1<λaB.
Step
2. Now, let B3>0 and assume, without loss of generality, that B3>B. Then, from the proof of uniform boundedness, there exists some B2=B2(B3)>B3 for which if Eφτp⩽B3, then E|x(t)|p⩽B2 for t⩾t0-τ.
Take a constantd satisfying 0<d⩽(1-λ-γτ)aB/(1-γτ); it is easy to verify that 0<d<(1-λ)aB. Let N=N(B3) be the smallest positive integer for which bB2<aB+Nd and T=T(B3)=α+(τ+α)(N-1). Given a solution x(t)=x(t,t0,φ) where Eφτp⩽B3 and t0∈[tl-1,tl), we will show E|x(t)|p⩽B for t⩾t0+T.
Given a constantA satisfying aB⩽A-d⩽bB2 and j>l, we will show that if EV(t)⩽A for t∈[tj-τ,tj), then EV(t)⩽A-d for t⩾tj.
For the sake of contradiction, suppose that there exists somet⩾tj for which EV(t)>A-d and define(11)t∗=inf⁡{t⩾tj∣EVt>A-d},and we suppose t∗∈[tk,tk+1) for some k∈N. We can get EV(t)⩽A-d for t∈[tj-τ,t∗) and EV(t∗)⩾A-d.
We claim thatEV(tk)⩽λA. The fact follows that if E|x(tk-)|p⩾ρ, then EV(tk)⩽λEV(tk-)⩽λA. If E|x(tk-)|p<ρ and we have E|x(tk)|p⩽ρ1, then EV(tk)⩽bρ<bB⩽λaB⩽λA.
Now, sinceaB⩽A, we have λA=A-(1-λ)A<A-(1-λ)aB<A-d and EV(tk)<A-d. This implies that t∗≠tk; that is, t∗∈(tk,tk+1) and EV(t∗)=A-d since EV(t) is continuous at t∗. Also, for t∈[tk,t∗], we have EV(t)⩽A-d. Define(12)t-=sup⁡{t∈tk,t∗∣EV(t)⩽λ(A-d)}.Since EV(t∗)=A-d>λA>λ(A-d), we have t-∈[tk,t∗) and EV(t-)=λ(A-d) and EV(t)⩾λ(A-d) for t∈[t-,t∗]. Then, if t∈[t-,t∗] and s∈[-τ,0],(13)λEV(t+s)⩽λ(A-d)<EV(t),bExtp>EVt>λA-d>λaB>bρ,which yields E|x(t)|p>ρ. Then, in light of condition (2),(14)ELV(t)⩽γEV(t).In terms of Itô formula,(15)EVt∗-EVt-=∫t-t∗‍ELVsds⩽∫t-t∗‍γEVsds⩽γαA-d.But(16)EVt∗-EVt-=A-d-λA-d>γαA-d,and this contradiction proves that EV(t)<A-d for all t⩾tj.Now we define a sequencetk(i)∈{tk,k=l,l+1,…}, satisfying tk(1)=tl and tk(i)-1-τ⩽tk(i-1)⩽tk(i)-τ, and then we have tk(i)⩽tk(i)-1+α⩽tk(i-1)+τ+α. By induction, we get tk(N)⩽t0+α+(τ+α)(N-1)=t0+T. We know that when t∈[t0-τ,tl), that is, t∈[t0-τ,tk(1)), EV(t)⩽bB2; then by induction we get EV(t)⩽bB2-Nd for t∈[tk(N),∞) and then EV(t)⩽aB for t∈[t0+T,∞). Using condition (1), we get that aE|x(t)|p⩽EV(t)⩽aB; that is,(17)Extp⩽B.Remark 3.
Condition (2) means the system without impulse may be unbounded. If the impulsive effects satisfy condition (3) to condition (5), then this system can be bounded.


## 4. Boundedness with Impulsive Disturbance

In this section, we consider the case that the SDDS is bounded, and when the impulsive disturbance appears in the SDDS, then what restrictions should be added to the disturbance to maintain the boundedness. The result is stated as follows.Theorem 4.
Assume that there exist a positive functionV(t,x) and positive constants a,b,c,p,λ1,λ2,γ, where 1⩽λ1<λ2, such that (1)
a|x|p⩽V(t,x)⩽b|x|p for any (t,x);(2)
fort≠tk, any s∈[-τ,0], and ϕ(s)∈PC([-τ,0],Rn), LV(t,ϕ(0))⩽-γV(t,ϕ(0)) whenever λ2V(t,ϕ(0))⩾V(t+s,ϕ(s)) and |ϕ(0)|p⩾ρ;(3)
V(tk,ϕ(0)+I(tk,ϕ(0)))⩽λ1V(tk-,ϕ(0)) for all |ϕ(0)|p⩾ρ;(4)
there exists a positive constantρ1⩾ρ such that if |ϕ(0)|p⩽ρ, then |ϕ(0)+I(τk,ϕ(0))|p⩽ρ1;(5)
there exist positive constantsμ and α, such that μ⩽tk-tk-1⩽α and μγ>λ2-1.
Then, the system (1) is p-moment uniformly ultimately bounded.Proof.
Step
1.Let B1>0; without loss of generality, we assume B1⩾ρ1. Choose B2=B2(B1), such that λ2bB1<aB2, and then we get B2>B1. Let Eφτp⩽B1 and assume t0∈[tl-1,tl); moreover, we assume that (1) has a maximal interval of existence, [t0-τ,t0+β).
We will prove thatE|x(t)|p⩽B2 for t∈[t0,t0+β). This will show that β=∞ and that solutions of (1) are uniformly bounded.
For the sake of contradiction, we suppose thatE|x(t)|p>B2 for some t∈[t0,t0+β). Let t^=inf⁡{t∈[t0,t0+β)∣E|x(t)|p>B2}. Note that E|x(t)|p⩽Eφτp<B1<B2 for t∈[t0-τ,t0], and we get t^∈(t0,t0+β), E|x(t)|p⩽B2 for t∈[t0-τ,t^) and E|x(t^)|p⩾B2.
Fort∈[t0-τ,t0], we have EV(t)⩽bE|x(t)|p⩽bEφτp⩽bB1 and then EV(t)⩽λ2EV(t)⩽λ2bB1<aB2. Particularly, EV(t0)⩽λ2EV(t0)<aB2 and EV(t^)⩾aE|x(t^)|p⩾aB2.
Definet∗=inf⁡{t∈[t0,t^]∣EV(t)⩾aB2} and then t∗∈(t0,t^], EV(t∗)⩾aB2, and EV(t)<aB2 for t∈[t0-τ,t∗).
Now we will proceed under two cases.
Case
1.Consider tl-1⩽t0<t∗<tl.
Under this case, we haveEV(t∗)=aB2  because of the continuity of V(t) on (tk,tk+1) and λ2EV(t∗)=λ2aB2>aB2. Define t-=sup⁡{t∈[t0,t∗]∣λ2EV(t)⩽aB2} and then t-≠t∗, λ2EV(t-)=aB2, and λ2EV(t)⩾aB2 for t∈[t-,t∗]. Therefor, for any t∈[t-,t∗] and s∈[-τ,0], we have EV(t+s)⩽aB2<λ2EV(t) and λ2bB1<aB2<λ2EV(t), which yields EV(t)>bB1, and then we have E|x(t)|p>B1⩾ρ. Using condition (2), we have, when t∈[t-,t∗],(18)ELV(t)⩽-γEV(t).By virtue of Itô formula, we have(19)EVt∗-EVt-=∫t-t∗‍ELVsds⩽∫t-t∗‍-γEVsds⩽0.However,(20)EVt∗=aB2>aB2λ2=EVt-.This contradiction gives(21)Extp⩽B2fort∈t0,t0+β.Case2. Consider tk⩽t∗<tk+1 for some k⩾l.
We first showλ2EV(tk-)⩽aB2. We have two situations to contemplate: k=l and k>l.
Ifk=l, we suppose λ2EV(tl-)>aB2. Define t-=sup⁡{t∈[t0,tl)∣λ2EV(t)⩽aB2} and then t-∈(t0,tl) and λ2EV(t-)=aB2. In light of the definition of t-, we have, for t∈[t-,tl) and s∈[-τ,0],(22)λ2EVt⩾aB2⩾EVt+s,and, for t∈[t-,tl),(23)Extp⩾B1⩾ρ.By virtue of condition (2), an analogous calculation of EV(tl-)-EV(t-) yields EV(tl-)⩽EV(t-); then we get(24)aB2<λ2EVtl-⩽λ2EVt-=aB2.
Ifk>l, we suppose λ2EV(tk-)>aB2. We will proceed under two subcases.
Subcase
1. Consider λ2EV(t)>aB2 for all t∈[tk-1,tk).
Under this situation, we haveλ2EV(t)>aB2⩾EV(t+s) and E|x(t)|p⩾ρ for all t∈[tk-1,tk) and s∈[-τ,0]. In terms of condition (2), an analogous discussion as done in Case 1 gives(25)EVtk--EVtk-1=∫tk-1tk-‍ELVsds⩽∫tk-1tk-‍-γEVsds⩽-γμaB2λ2.However, by virtue of condition (5),(26)EVtk--EVtk-1⩾aB2λ2-aB2=1λ2-1aB2>-γμaB2λ2.This contradiction implies(27)λ2EVtk-⩽aB2fortk⩽t∗<tk+1,k⩾l.Subcase2. Consider λ2EV(t)⩽aB2 for some t∈[tk-1,tk).
Definet-=sup⁡{t∈[tk-1,tk)∣λ2EV(t)⩽aB2} and then t-∈[tk-1,tk) and λ2EV(t-)=aB2. Using the definition of t-, we get, for t∈[t-,tk) and s∈[-τ,0], λ2EV(t)⩾aB2⩾EV(t+s). Since λ2EV(t)⩾aB2, using the fact ρ1⩾ρ,λ2bB1<aB2 and b|x|p⩾V(t,x), we can get E|x(t)|p⩾ρ. By virtue of condition (2), we get, for t∈[t-,tk), (28)ELV(t)⩽-γEV(t).An analogous discussion as done in the case k=l gives EV(t-)⩾EV(tk-). Then we have(29)aB2<λ2EV(tk-)⩽λ2EV(t-)=aB2.This contradiction gives(30)λ2EVtk-⩽aB2fortk⩽t∗<tk+1,k⩾l.
Now we claimEV(tk)<aB2. If E|x(tk-)|p⩾ρ, we get EV(tk)⩽λ1EV(tk-)<λ2EV(tk-)<aB2. If E|x(tk-)|p<ρ, we get EV(tk)⩽bρ1<bB1<λ2bB1<aB2. That is, the following inequality holds:(31)EV(tk)<aB2.
SinceEV(t∗)⩾aB2, we have t∗≠tk and EV(t∗)=aB2.
Ifλ2EV(t∗)⩾aB2 for all t∈[tk,t∗], then let t-=tk and we have EV(t-)<aB2. Otherwise, let t-=sup⁡{t∈[tk,t∗)∣λ2EV(t)⩽aB2}, and we have EV(t-)<λ2EV(t-)=aB2. Since EV(t∗)=aB2, we get t-∈[tk,t∗). Moreover, for t∈[t-,t∗], we have λ2EV(t)⩾aB2>EV(t+s) and, by virtue of λ2bB1<aB2<λ2EV(t), we obtain EV(t)>bB1 and then E|x(t)|p>B1>ρ. In terms of condition (2) and Itô formula, we can obtain EV(t-)⩾EV(t∗). But EV(t-)<aB2=EV(t∗), which is a contradiction and yields(32)Extp⩽B2fort∈t0,t0+β.
Now we get that, under condition (1) to condition (5), the solutions of (1) are p-moment uniformly bounded. Then we know that if Eφτp⩽ρ1, there exists a constant B>0, such that E|x(t,t0,φ)|p⩽B for all t⩾t0-τ, and, from the above proof, we have λ2bρ1<aB.
Step
2. Now, let B3>0 and assume, without loss of generality, that B3>B. Then, from the proof of uniform boundedness, there exists a constant B2=B2(B3)>B3 for which if Eφτp⩽B3, then E|x(t)|p⩽B2 for t⩾t0-τ.
Take a constantd satisfying 0<d⩽min⁡{aB-bρ1,((λ2-λ1)/λ2)aB}, N=min⁡{n>((bB2-aB)/d)}, and T=α+(2N-1)(α+τ).
Letx(t)=x(t,t0,φ) be a solution of (1) with Eφτp⩽B3, t0∈[tl-1,tl). We will show E|x(t)|p⩽B for t⩾t0+T.
Given a positive numberA satisfying aB⩽A⩽bB2 and j⩾l, we will show that if EV(t)⩽A for t∈[tj-τ,tj) and λ2EV(tj-)⩽A, then EV(t)⩽A for t⩾tj and λ2EV(tj+1-)⩽A.
For the sake of contradiction, suppose that there exists a constantt∈[tj,tj+1) for which EV(t)>A and define(33)t∗=inf⁡t∈tj,tj+1∣EVt⩾A.Note that EV(tj)<A, and we have that if E|x(tj-)|p⩾ρ, then EV(tk)⩽λ1EV(tj-)<λ2EV(tj-)⩽A. If E|x(tk-)|p<ρ, we have EV(tj)⩽bρ1<λ2bρ1<aB⩽A. Then we get t∗≠tj, EV(t∗)=A, and EV(t)⩽A for t∈(tj,tj+1].
Ifλ2EV(t)>A for all t∈[tj,tj+1), we let t-=tj, and then EV(t-)=EV(tj)<A. Otherwise, let t-=sup⁡{t∈[tj,t∗]∣λ2EV(t)⩽A}, and we get EV(t-)⩽λ2EV(t-)=A. Since λ2EV(t∗)=λ2A>A, t-≠t∗. For t∈[t-,t∗] and s∈[-τ,0], we have λ2EV(t)⩾A⩾EV(t+s). Moreover, for t∈[t-,t∗],(34)λ2EVt⩾A⩾aB>λ2bρ1,and we get E|x(t)|p⩾ρ1⩾ρ. By virtue of condition (2) and Itô formula, we can get EV(t-)⩾EV(t∗). However, EV(t∗)=A>EV(t-).
Now we have provenEV(t)⩽A for t∈[tj,tj+1), and we are on the position to show λ2EV(tj+1-)⩽A. This will follow in the same way as the arguments used in the proof of uniform boundedness, where we show λ2EV(tk-)⩽aB2 for the case k>l; we just need to replace k by j+1 and ab2 by A.
By induction, we get that ifEV(t)⩽A for t∈[tj-τ,tj) and λ2EV(tj-)⩽A, then EV(t)⩽A for all t⩾tj and λ2EV(tk-)⩽A for k⩾j+1.
Next, we will showEV(t)⩽A-d for t∈[tj+1,tj+2), if EV(t)⩽A for all t⩾tj and λ2EV(tk-)⩽A, k⩾j.
We first showEV(tj+1)⩽A-d. This can be easily verified under two situations: iIf E|x(tj+1-)|p⩽ρ, we have EV(tj+1)⩽bρ1⩽aB-d⩽A-d; if E|x(tj+1-)|p>ρ, EV(tj+1)<λ1EV(tj+1)=(λ1/λ2)λ2EV(tj+1-)⩽(λ1/λ2)A<A-d.
In order to verifyEV(t)⩽A-d for all t∈[tj+1,tj+2), suppose that EV(t)>A-d for some t∈[tj+1,tj+2). Let t∗=inf⁡{t∈[tj+1,tj+2)∣EV(t)⩾A-d}; we know t∗≠tj+1 and then EV(t∗)=A-d and λ2EV(t∗)=λ2(A-d)>A.
Ifλ2EV(t)>A for all t∈[tj+1,t∗], let t-=tj+1, EV(t-)=EV(tj+1)<A-d.
Ifλ2EV(t)>A for some t∈(tj+1,t∗], let t-=sup⁡{t∈[tj+1,t∗]∣λ2EV(t)⩽A} and we know t-≠t∗, EV(t-)=A/λ2.
Fort∈[t-,t∗] and s∈[-τ,0], λ2EV(t)⩾A>A-d>EV(t+s) and EV(t)⩾A/λ2>aB/λ2>bρ1, and we get E|x(t)|p>ρ1⩾ρ. In terms of condition (2) and Itô formula, we can get EV(t∗)<EV(t-). However, EV(t∗)=A-d>EV(t-), which yields(35)EV(t)⩽A-d.
Applying our results to successive intervals of the form[tk,tk+1) for k⩾j+1, we can get EV(t)⩽A-d for t⩾tj+1.
Now we need a factλ2EV(tj+2-)⩽A-d. This can be verified just as we did in the proof of uniform boundedness, where we show λ2EV(tk-)⩽aB2 for the case k>l.
Taketk(i)∈{tj,j=l,l+1,…} satisfying tk(i-1)+τ⩽tk(i)⩽tk(i-1)+1+τ. Take A=bB2, when t⩾tk(2N), and we get EV(t)⩽bB2-Nd<aB. Since tk(2N)⩽tk(1)+(2N-1)(α+τ)⩽t0+α+(2N-1)(α+τ)=t0+T, we have EV(t)⩽aB when t>t0+T. By virtue of condition (1), E|x(t)|p⩽B for t⩾t0+T, which completes the proof.Remark 5.
Theorem4 considers that a bounded system without impulse can tolerate what kind of impulsive effects to hold the boundedness. It is not surprising that condition (3) to condition (5) should be satisfied: the interval of impulsive moments (μ) should be large and impulsive strength (λ1) should be small.


## 5. Examples

In this section, we present two examples to illustrate our results.Example 1.
Consider the following impulsive stochastic delay differential system:(36)dxt=12xt+12xtdt+xt-120dBt,t>0,t≠k10,k=1,2,…,xk10=22xk10-,where B(t) is a one-dimension Brownian motion.
DefineV(t,x)=x2; the smoothness requirement is satisfied. Let a=b=1 and p=2; condition (1) of Theorem 2 follows. For any solution x(t) of system (36), we have(37)LVt,x=2x12xt+12xt+x2t-120=x2t+1+x2t-120.
Takeλ=1/2; condition (3) of Theorem 2 is satisfied.
Now letρ=1; then, when |x(t)|2⩾1 and V(t,x)⩾λV(t,x(t-τ)), that is, x2(t)⩾(1/2)x2(t-1/20), we have (38)LVt,x⩽x2t+x2t+2x2t=4x2t=4Vt,x.Then let γ=4; condition (2) of Theorem 2 is verified.
Condition (4) of Theorem2 can be verified by taking ρ1=1.
Takeα=1/10 and then αγ=(1/10)×4=2/5<1/2=1-λ; condition (5) of Theorem 2 is verified.
Therefore, according to Theorem2, solutions of system (36) are mean square uniformly ultimately bounded. The boundedness can be read from Figure 1, where we take initial condition x(t)=1,t∈[-1/20,0].
To see the contribution of impulsive effect on boundedness, we consider the following system:(39)dxt=12xt+12xtdt+xt-120dBt,t>0,which is the situation of system (36) without impulses. It is easy to be verified that system (39) is unbounded; see Figure 2, where we also take initial condition x(t)=1,t∈[-1/20,0].Figure 1
Mean square uniform ultimate boundedness of solution of system (36).Figure 2
Unboundedness of solution of system (39).Now we give another example to illustrate the correctness of Theorem4.Example 2.
Consider(40)dxt=-4xt+12xtdt+xt-12dBt,t>0,t≠2k,k=1,2,…,x2k=2x2k-,where B(t) is a one-dimension Brownian motion.
DefineV(t,x)=x2; the smoothness requirement is satisfied. Let a=b=1 and p=2; condition (1) of Theorem 4 follows. For any solution x(t) of system (40), we have(41)LVt,x=2x-4xt+12xt+x2t-12=-8x2t+1+x2t-120.
Takeλ1=2, condition (3) of Theorem 4 is satisfied.
Now letρ=1 and λ2=3; then, when |x(t)|2⩾1 and V(t,x)⩾λ2V(t,x(t-τ)), that is, 3x2(t)⩾x2(t-1/2), we have(42)LVt,x⩽-8x2t+x2t+3x2t=-4x2t=-4Vt,x.Then, let γ=4; condition (2) of Theorem 2 is verified.
Condition (4) of Theorem2 can be verified by taking ρ1=2.
Takeμ=2 and then μγ=2×8=16>3-1=λ2-1 and condition (5) of Theorem 4 is verified.
Therefore, according to Theorem4, solutions of system (40) are mean square uniformly ultimately bounded. The boundedness can be seen in Figure 3, where we take initial condition x(t)=3,t∈[-1/2,0].
We also present the simulation of system (40) without impulsive effects; that is,(43)dxt=-4xt+12xtdt+xt-12dBt,t>0.The property of system (43) can be read from Figure 4, where we take initial condition x(t)=3,t∈[-1/2,0].Figure 3
Mean square uniform ultimate boundedness of solution of system (40).Figure 4
Simulation of system (43).



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