Practical exponential stability of impulsive stochastic functional differential equations

Practical exponential stability of impulsive stochastic functional differential equations Tom´ as Caraballoa, Mohamed Ali Hammamib, Lassaad Mchirib aUniversidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, C/Tarfia s/n, 41012-Sevilla (Spain). E-mail: [email protected] bUniversity of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia. E-mail: [email protected] Abstract This paper is devoted to the investigation of the practical exponential stability of impulsive stochastic functional differential equations. The main tool used to prove the results is the Lyapunov-Razumikhin method which has proven very useful in dealing with stability problems for differential systems when the delays involved in the equations are not differentiable but only continuous. An illustrative example is also analyzed to show the applicability and interest of the main results. Key words: Impulsive stochastic functional differential equations, almost sure practical exponential stability, Lyapunov-Razumikhin method. 2000 MSC: 93E03, 60H10. 1. Introduction Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which some parameters may undergo relatively long periods of smooth variation followed by a short-term rapid change (that is, jumps) in their values. Processes of this type are often investigated in various fields of science and technology (see [11]). Needless to say that, in general, the evolution of a dynamical system may not only be determined by its current state. It is sensible to admit that its past or history may also have some influence in its future evolution. This fact justifies the appearance of several types of delay or memory in the models. Stochastic functional differential equations are often used as appropriate tools for the description and analysis of such systems, see [16]-[17]. Nowadays, impulsive deterministic and stochastic functional differential equations represent an emerging field drawing attention from both theoretical and applied disciplines, and which has been successfully applied to problems in mechanics, electricity, economics, physics and several fields in engineering (see, e.g. [5]- [12] for more details). Over the last decades, many interesting results concerning stability of stochastic functional (and evolution) differential equations have been obtained (see [7]- [8], [15]-[17]). In [14], Mao developed the Razumikhin method on this aspect, and established some Razumikhin-type theorems on exponential stability of stochastic functional differential equations for finitedimensional stochastic ordinary differential systems with delays. The advantage of this method, in contrast to the classical Lyapunov one, is that it is not necessary to use functionals to analyze the stability. Instead, one can consider Lyapunov functions and, additionally, the type of delays that can be handled is more general. For instance, if the problem contains variable delays, these only need to be continuous while to apply the classical Lyapunov theory (see [4]), the variable delay function is required to be continuously differentiable. Some results concerning stability of impulsive stochastic functional differential equations (using the Razumikhin method) have been proved in [5], [9]-[10], [18]-[19] and [12]. However, when the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability of solutions with respect to a small neighborhood of the origin, which yields the concept of practical stability (see [6] and the references therein). It is worth mentioning some previous works on practical stability in the deterministic and stochastic frameworks (see [1, 2, 3, 6]). In particular, we initiated very recently the study of this practical stability for non-delay Preprint submitted to Systems and Control Letters August 29, 2017 stochastic partial differential equations in [1], [2]. Notice that inequality (2.2) (see Section 2 below) implies that x(t) is ultimately bounded by a small bound r>0, that is, |x(t)|is small for sufficiently large t. This can be viewed as a robustness property of almost sure convergence to the origin provided that fand gsatisfy f(0,t)=0 and g(0,t)=0, for all t≥0. In this case the origin becomes an equilibrium point. The content of the paper is as follows. In Section 2 we introduce the basic notations and assumptions. Section 3 is devoted to prove some sufficient conditions ensuring practical p-th moment exponential and almost sure exponential stability of solutions to impulsive stochastic functional differential equations. We prove a sufficient condition ensuring the convergence of solutions to a ball with radius η > 0, even in the case that zero is not an equilibrium point. Finally, two examples to illustrate our abstract theory are included in Section 4. 2. Preliminaries For x∈Rn,|x|denotes the Euclidian norm of x and [x] the biggest integer less than or equal to x. For −∞ <a<b<∞, we say that a function from [a,b] to Rnis piecewise continuous, if the function has at most a finite number of jump discontinuities on (a,b] and 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≤θ≤0|ϕ(θ)|for ϕ∈ PC([−τ, 0]; Rn). Let (Ω,F,P) be a complete probability space with a filtration {Ft}t≥0satisfying the usual conditions and W(t)= (W1(t),W2(t), ..., Wm(t)) be an m-dimensional standard Wiener process defined on (Ω,F,P) and adapted to {Ft}t≥0. For p>0 and t≥0, let Lp Ft([−τ, 0],Rn) denote the family of all Ft-measurable PC([−τ, 0]; Rn)-valued random processes ϕ={ϕ(−θ) : −τ≤θ≤0}such that sup−τ≤θ≤0E|ϕ(θ)|p<∞. And let Lp Ft(Ω,Rn) denote the family of all Ft-measurable Rn-valued random variables X, such that E|X|p<∞. Finally, let PCb([−τ, 0]; Rn) denote the family of all bounded PC([−τ, 0]; Rn)-valued functions. In this paper, we consider the following impulsive stochastic functional differential equation:          dx(t)=f(xt,t)dt +g(xt,t)dW(t),t,tk,t≥t0, ∆x(tk)=Ik(x(t− k),tk),k∈N, xt0=ξ, i.e. x(t0+θ)=ξ(θ), θ ∈[−τ, 0], (2.1) where the initial value ξ∈ PCb([−τ, 0]; Rn), x(t)=(x1(t), ..., xn(t))T,xt∈Lp Ft([−τ, 0],Rn), f: Lp Ft([−τ, 0],Rn)×R+−→ Rn,g:Lp Ft([−τ, 0],Rn)× R+−→ Rn×m,Ik(·,·) : Rn×R+−→ Rn, where Ik(x(t− k),tk) represents the impulsive perturbation of xat time tk, and we suppose that there exists t≥t0such that f(0,t),0, g(0,t),0 and Ik(0,t),0, k∈N. The fixed moments of impulse times tksatisfy 0 ≤ t0<t1< ... < tk< ..., tk→ ∞ (as k→ ∞), ∆x(tk)= x(tk)−x(t− k). As standing hypotheses, we assume that f(ϕ, t) and g(ϕ, t) are continuous for almost all t∈[t0,∞) and are Lipschitz in ϕin each compact set in Lp Ft([−τ, 0],Rn); Ik(x,t) is Lipschitz in xfor all x∈Rn,k∈N. As we are interested in the asymptotic behavior of the solutions, we assume that for any ξ∈ PCb([−τ, 0]; Rn), there exists a unique stochastic process, denoted by x(t,t0, ξ), satisfying system (2.1), which is continuous on the right-hand side and possessing limit on the lefthand side. For some results on the existence and uniqueness of solutions of (2.1) the reader is referred to [13]. Now we will establish the definition of p-moment and almost sure practical stability and recall the definition of generating operator for our equation. Definition 2.1. For p>0, system (2.1) is said to be pth moment practically exponentially stable if there exist positive constants λ,Cand ηsuch that E|x(t,t0, ξ)|p≤C||ξ||pe−λ(t−t0)+η, t≥t0, for all ξ∈ PCb([−τ, 0]; Rn). Definition 2.2. The ball Bη:={x∈Rn:|x| ≤ η},η > 0, is said to be almost surely practically exponentially stable if, for all ξ∈ PCb([−τ, 0]; Rn) and every t≥t0, |x(t,t0, ξ)| ≤ C||ξ||pe−λ(t−t0)+η, a.s. (2.2) System (2.1) is said to be almost surely practically exponentially stable if there exists η > 0 such that Bηis almost surely practically exponentially stable. Definition 2.3. Let C2,1(Rn×[t0−τ, ∞); R+) denote the family of all nonnegative functions V(x,t) on Rn×[t0−τ, ∞) that are continuously twice differentiable in xand once in t. For a function V∈ C2,1(Rn×[t0−τ, ∞); R+) we define the operator LV:Lp Ft([−τ, 0],Rn)×[t0,∞)−→ Rfor system (2.1) by LV(ϕ, t)=Vt(ϕ(0),t)+Vx(ϕ(0),t)f(ϕ, t) (2.3) +1 2tracegT(ϕ, t)Vxx(ϕ(0),t)g(ϕ, t), where ϕ∈Lp Ft([−τ, 0],Rn) and 2 Vt(x,t)=∂V ∂t(x,t); Vx(x,t)=(∂V ∂x1 (x,t), ..., ∂V ∂xn (x,t)); Vxx(x,t)=∂2V ∂xi∂xj (x,t)n×n. If we replace ϕin the expression (2.3) by a segment solution xtof system (2.1), then it becomes LV(xt,t)=Vt(x(t),t)+Vx(x(t),t)f(xt,t) (2.4) +1 2tracegT(xt,t)Vxx(x(t),t)g(xt,t). 3. Main results In this section, we first investigate the p-th moment practical exponential stability and the almost sure practical exponential stability of system (2.1) by using the Razumikhin method. The results show that the impulsive effects can produce some kind of stabilization on the system and achieve p-th moment practical exponential stability. Theorem 3.1. Assume that there exist a function V ∈ C2,1(Rn×[t0−τ, ∞); R+)and positive constants p, c, c1, c2, c3,λ, h, ψ,ρk, k ∈N,δ≥0and γ≥1such that (i) c1|x|p≤V(x,t)≤c2|x|p+c3,∀(x,t)∈Rn×[t0− τ, ∞); (ii) EV(x+Ik(x,tk),tk)≤dkEV(x,t− k)+ρk,∀k∈ N,x∈Lp Ft(Ω,Rn),where 0<dk≤1,0< ρk≤d with d =sup k∈N ρk<∞. (iii) ELV(ϕ, t)≤cEV(ϕ(0),t)+ψ, ∀t≥t0,t,tk,k∈ Nand ϕ∈Lp Ft([−τ, 0],Rn),whenever EV(ϕ, t+θ)<qEV(ϕ(0),t)+δ, θ ∈[−τ, 0], where q ≥γeλτ is a constant. (iv) γ≥max 1 dk,e(λ+c)(t1−t0),ln dk≤ −(λ+c)(tk+1− tk),k∈N,and ∆sup =sup k∈Ntk+1−tk≤h<∞,ech ≤c3γ ψh+c3 eλτ. Then, system (2.1) is p-th moment practically exponentially stable. Proof. Given an initial datum ξ∈ PCb([−τ, 0]; Rn), we write x(t,t0, ξ)=x(t) for short. Choose M>0 such that 0<c2e(λ+c)(t1−t0)≤M≤c2γeλτ.(3.1) Then, by condition (i), we have for t∈[t0−τ, t0] EV(x(t),t)≤c2||ξ||p+c3<M||ξ||pe−λ(t1−t0)+r,(3.2) where (ψh+c3)ech ≤r≤c3γeλτ. Next, we shall prove that, for any k∈Nand t∈[tk−1,tk), EV(x(t),t)≤M||ξ||pe−λ(tk−t0)+r.(3.3) We proceed by induction and split our proof into several steps. Step 1: Arguing by contradiction, we first prove that EV(x(t),t)≤M||ξ||pe−λ(t1−t0)+r,t∈[t0,t1).(3.4) Indeed, let us assume that inequality (3.4) is not true. Then, there exists some t∈[t0,t1) such that EV(x(t),t)>M||ξ||pe−λ(t1−t0)+r. Define t∗=inf t∈[t0,t1); EV(x(t),t)>M||ξ||pe−λ(t1−t0)+r. Thanks to (i) and the Lebesgue dominated convergence theorem, the mapping t∈[t0,t1)7→ EV(x(t),t) is continuous. Then lim t→t− 0 EV(x(t),t)=lim t→t+ 0 EV(x(t),t) =EV(x(t0),t0) <M||ξ||pe−λ(t1−t0)+r. Therefore, t∗∈(t0,t1) and EV(x(t∗),t∗)=M||ξ||pe−λ(t1−t0)+r,(3.5) EV(x(t),t)<M||ξ||pe−λ(t1−t0)+r,t∈[t0−τ, t∗).(3.6) Define t∗=sup t∈[t0,t∗]; EV(x(t),t)≤c2||ξ||p+c3, then t∗∈[t0,t∗) and by the continuity of EV(x(·),·), EV(x(t∗),t∗)=c2||ξ||p+c3,(3.7) EV(x(t),t)>c2||ξ||p+c3,t∈(t∗,t∗].(3.8) Consequently, for all t∈[t∗,t∗], we have EV(x(t+θ),t+θ)≤M||ξ||pe−λ(t1−t0)+r ≤c2γeλτ||ξ||pe−λ(t1−t0)+r < γeλτc2||ξ||p+r γe−λτ ≤γeλτEV(x(t∗),t∗) ≤qEV(x(t∗),t∗). Then, for all θ∈[−τ, 0], EV(x(t+θ),t+θ)<qEV(x(t),t)+δ. (3.9) By the Razumikhin-type condition (iii), ELV(xt,t)≤cEV(x(t),t)+ψ, t∈[t∗,t∗].(3.10) 3 Applying the Itˆ o formula on [t∗,t∗] EV(x(t∗),t∗) =EV(x(t∗),t∗)+Zt∗ t∗ ELV(xs,s)ds ≤EV(x(t∗),t∗)+cZt∗ t∗ EV(x(s),s)ds +ψ(t∗−t∗) ≤EV(x(t∗),t∗)+cZt∗ t∗ EV(x(s),s)ds +ψ∆sup, and thus, EV(x(t∗),t∗)≤EV(x(t∗),t∗) +cZt∗ t∗ EV(x(s),s)ds +ψh. By (3.1), (3.5), (3.7) and the Gronwall inequality, EV(x(t∗),t∗)≤EV(x(t∗),t∗)+ψhec(t∗−t∗) <c2||ξ||p+c3+ψhec(t1−t0) ≤M||ξ||pe−λ(t1−t0)+(c3+ψhech ≤M||ξ||pe−λ(t1−t0)+r =EV(x(t∗),t∗), which is a contradiction. Therefore, (3.4) holds and (3.3) must be true for k=1. Step 2: Now, we assume that (3.3) holds for all k≤m,m∈N∗, i.e. for k=1,2, .., m EV(x(t),t)≤M||ξ||pe−λ(tk−t0)+r,t∈[tk−1,tk).(3.11) We will show that EV(x(t),t)≤M||ξ||pe−λ(tm+1−t0)+r,t∈[tm,tm+1).(3.12) Suppose (3.12) does not hold true and define t∗=inf t∈[tm,tm+1); EV(x(t),t)>M||ξ||pe−λ(tm+1−t0)+α where α=(R+ψh)ech with R=r+d. Thanks to conditions (ii), (iv) and inequality (3.11), EV(x(tm),tm)≤dmEV(x(t− m),t− m)+ρm, ≤dmM||ξ||pe−λ(tm−t0)+dmr+ρm, ≤M||ξ||pe−λ(tm−t0)e−(λ+c)(tm+1−tm)+dmr+ρm, <M||ξ||pe−λ(tm+1−t0)+dmr+ρm, ≤M||ξ||pe−λ(tm+1−t0)+r+d. Thus, EV(x(tm),tm)<M||ξ||pe−λ(tm+1−t0)+R.(3.13) Now, (3.13) and the continuity of the mapping t∈ [tm,tm+1)7→ EV(x(t),t) ensure the existence of t∗∈ (tm,tm+1) such that EV(x(t∗),t∗)=M||ξ||pe−λ(tm+1−t0)+α, (3.14) EV(x(t),t)<M||ξ||pe−λ(tm+1−t0)+α, t∈[tm,t∗).(3.15) Define t∗=sup t∈[t0−τ, t∗]; EV(x(t),t)≤dmM||ξ||pe−λ(tm−t0)+R. Then t∗∈[tm,t∗) and EV(x(t∗),t∗)=dmM||ξ||pe−λ(tm−t0)+R,(3.16) EV(x(t),t)>dmM||ξ||pe−λ(tm−t0)+R,t∈(t∗,t∗].(3.17) Fix any t∈[t∗,t∗], and assume that t+θ≥tm, for all θ∈[−τ, 0], then (3.14)-(3.17) imply that EV(x(t+θ),t+θ)≤M||ξ||pe−λ(tm+1−t0)+α, <M||ξ||pe−λ(t+θ−t0)+α, ≤γeλτdmM||ξ||pe−λ(tm−t0)+α =γeλτdmM||ξ||pe−λ(tm−t0)+α γe−λτ, ≤γeλτEV(x(t∗),t∗), ≤qEV(x(t∗),t∗). Then, EV(x(t+θ),t+θ)<qEV(x(t),t)+δ, (3.18) for t∈[t∗,t∗], θ ∈[−τ, 0]. Applying again the Itˆ o formula on [t∗,t∗] and taking into account the Razumikhintype condition (iii), we obtain EV(x(t∗),t∗)=EV(x(t∗),t∗)+Zt∗ t∗ ELV(xs,s)ds, ≤EV(x(t∗),t∗)+cZt∗ t∗ EV(x(s),s)ds +ψ∆sup. Thus, EV(x(t∗),t∗)≤EV(x(t∗),t∗) +cZt∗ t∗ EV(x(s∗),s∗)ds +ψh. 4 Condition (iv), (3.14) and (3.16) and the Gronwall inequality imply EV(x(t∗),t∗) ≤EV(x(t∗),t∗)+ψhec(t∗−t∗) =dmM||ξ||pe−λ(tm−t0)+R+ψhec(t∗−t∗), <dmM||ξ||pe−λ(tm−t0)+R+ψhec(tm+1−tm), ≤M||ξ||pe−λ(tm−t0)e−(λ+c)(tm+1−tm)+R+ψhec(tm+1−tm), ≤M||ξ||pe−λ(tm+1−t0)+R+ψhech. Then, EV(x(t∗),t∗)<M||ξ||pe−λ(tm+1−t0)+α, (3.19) which is a contradiction. Therefore (3.12) must hold true. Thus, by induction, we have obtained that (3.3) holds for all k∈N. Step 3: Now, we can finish our proof. Thanks to condition (i) and inequality (3.3), E|x(t)|p≤M||ξ||p c1 e−λ(tk−t0)+r c1 ≤M||ξ||p c1 e−λ(t−t0)+r c1 ,(3.20) for t∈[tk−1,tk),k∈N. Therefore, system (2.1) is p-th moment practically exponentially stable with η=r c1 .2 Now, before proving our main result on almost sure practical exponential stability, we recall a result which will be crucial for our analysis and whose proof can be found in [16]. Lemma 3.2. Let g =(g1, ...., gm)∈L2(R+,Rm), and let T, α,βbe positive numbers. Then Psup 0≤t≤TZt 0 g(s)dWs−α 2Zt 0 |g(s)|2ds> β≤exp(−αβ). Theorem 3.3. Assume conditions of Theorem 3.1 with p≥1,∆sup =supk∈Ntk−tk−1≤h<∞and ∆inf = infk∈Ntk−tk−1>0. Assume that there exist positive constants β, L and µ≥1 such that, for all (ϕ, t)∈Lp Ft([−τ, 0],Rn)×[t0,∞), E|f(ϕ, t)|p+|g(ϕ, t)|p≤Lsup −τ≤θ≤0 E|ϕ(θ)|p+β. (3.21) Then (3.20) implies |x(t,t0, ξ)| ≤ C||ξ||pe−λ(t−t0)+µ1 p,∀t≥t0,a.s. (3.22) In other words, the p-th moment practical exponential stability implies the almost sure practical exponential stability. Proof. Take 0 < δ < ∆inf sufficiently small. For such a fixed δ > 0, set kδ=htk−tk−1 δi∈N. Then kδ≤∆sup δ< ∞and for any t∈[tk−1,tk), there exists some iwith 1≤i≤kδ+1 such that tk−1+(i−1)δ≤t<tk−1+iδ. Thus, for any t∈[tk−1,tk), k∈N, one has Ehsup tk−1≤t<tk |x(t)|pi≤ kδ+1 X i=1 Ehsup tk−1+(i−1)δ≤t<tk−1+iδ |x(t)|pi. (3.23) From [16, page 178] we have for any a,b,c≥0, (a+b+c)p≤3pap+bp+cp. For each isatisfying 1 ≤i≤kδ+1, k∈N, Zt tk−1+(i−1)δ dx(s)=Zt tk−1+(i−1)δ f(xs,s)ds +Zt tk−1+(i−1)δ g(xs,s)dW(s). Thus, x(t)=x(tk−1+(i−1)δ)+Zt tk−1+(i−1)δ f(xs,s)ds +Zt tk−1+(i−1)δ g(xs,s)dW(s), and |x(t)|≤|x(tk−1+(i−1)δ)|+Ztk−1+iδ tk−1+(i−1)δ |f(xs,s)|ds +Zt tk−1+(i−1)δ g(xs,s)dW(s). Then, |x(t)|p≤3p|x(tk−1+(i−1)δ)|p +3pZtk−1+iδ tk−1+(i−1)δ |f(xs,s)|dsp +3pZt tk−1+(i−1)δ g(xs,s)dW(s) p. Therefore, sup tk−1+(i−1)δ≤t<tk−1+iδ |x(t)|p≤3p|x(tk−1+(i−1)δ)|p(3.24) +3pZtk−1+iδ tk−1+(i−1)δ |f(xs,s)|dsp, +3psup tk−1+(i−1)δ≤t<tk−1+iδZt tk−1+(i−1)δ g(xs,s)dW(s) p. 5 Taking expectation in (3.24), Ehsup tk−1+(i−1)δ≤t<tk−1+iδ |x(t)|pi(3.25) ≤3pE|x(tk−1+(i−1)δ)|p +3pEZtk−1+iδ tk−1+(i−1)δ |f(xs,s)|dsp +3pEsup tk−1+(i−1)δ≤t<tk−1+iδZt tk−1+(i−1)δ g(xs,s)dW(s) p. In view of (3.20), we have E|x(tk−1+(i−1)δ)|p≤M||ξ||p c1 e−λ(tk−1−t0)+r c1 .(3.26) Thanks now to (3.25), (3.26), (5.1)-(5.2), we deduce Ehsup tk−1+(i−1)δ≤t<tk−1+iδ |x(t)|pi(3.27) ≤3phδp+Cpδp 2Leλτ +1iM||ξ||p c1 e−λ(tk−1−t0), +3phr c1 +βδp+Cpβδ p 2+Lr c1Cpδp 2+δpi. Then, from (3.23) and (3.27), Ehsup tk−1≤t<tk |x(t)|pi ≤3pkδ+1hδp+Cpδp 2Leλτ +1iM||ξ||p c1 e−λ(tk−1−t0), +3pkδ+1hr c1 +βδp+Cpβδ p 2+Lr c1Cpδp 2+δpi. Therefore, Ehsup tk−1≤t<tk |x(t)|pi≤Dp||ξ||pe−λ(tk−1−t0)+µ, (3.28) where Dp=3ph δ+1M c1hδp+Cpδp 2Leλτ +1i, and µ=3ph δ+1hr c1 +βδp+Cpβδ p 2+Lr c1Cpδp 2+δpi. We will assume that 3pr c1 ≥1. Notice that this assumption guarantees that µ≥1. Let ∈(0, λ) be arbitrary. Then Lemma 3.2 yields Pnsup tk−1≤t<tk |x(t)|p>Dp||ξ||pe−(λ−)(tk−1−t0)o≤e−(tk−1−t0). By the Borel-Cantelli lemma we obtain sup tk−1≤t<tk |x(t)|p≤Dp||ξ||pe−(λ−)(tk−1−t0) ≤Dp||ξ||pe−(λ−)(tk−1−t0)+µ, a.s., which implies |x(t)|p≤Dp||ξ||pe−(λ−)(tk−1−t0)+µ, a.s, for t∈[tk−1,tk), k∈N. Letting →0 we deduce that, for any t∈[tk−1,tk), k∈N, |x(t)|p−µ≤Dp||ξ||pe−λ(t−t0),a.s. We know that |x(t)|p−µ=|x(t)| − µ1 p p X n=1 |x(t)|p−nµn−1 p. Thanks to condition µ≥1, |x(t)|p−µ≥|x(t)| − µ1 p. Then, |x(t)| − µ1 p≤Dp||ξ||pe−λ(t−t0),∀t≥t0,a.s. It thus follows |x(t)| ≤ Dp||ξ||pe−λ(t−t0)+µ1 p,∀t≥t0,a.s.(3.29) Therefore, system (2.1) is almost surely practically exponentially stable with η=µ1 p.2 Remark 3.4. Observe that theorems 3.1 and 3.3 provide sufficient conditions for the practical stability of solutions both in the pth-moment and pathwise. These can be interpreted as some kind of stabilization results because the problem without impulses may be unstable while the impulsive one may become practically stable. Also, when the problem is already practically stable before the impulses are taken into account, the precedent results provide sufficient conditions for the system to remain practically stable, which is a proof for the robustness of the problem. 4. An application examples Example 4.1. Consider a scalar ISDDS of the form: (dx(t)=x(t)dt +α(x(t),x(t−0.1),t)dW(t),t,tk,t≥0, ∆x(tk)=−0.5x(t− k),k∈N, (4.1) Then, f(xt,t)=x(t), g(xt,t)=α(xt(0),xt(−0.1),t)=α(x(t),x(t−0.1),t), We assume that there exists σ > 0such that |α(x,y,t)| ≤ σ. Then Ef(xt,t) p+Eg(xt,t) p≤Ex(t) p+σp, 6 Thus, condition (3.21) holds for system (4.1) with L =1 and β=σp. Let p =2and V(x,t)=x2. Therefore, the constants in Theorem 3.1 are c1=c2=1, c3=0.1, q=5,δ=0. Then, we have EV(x(t− k)+Ik(x(t− k),tk),tk)≤0.45EV(x(t− k),tk)+0.1 and ELV(xt,t)≤2Ex(t) 2+σ2. Thus, c =2, dk=0.45, d =0.1=ρkand ψ=σ2. We choose tk−tk−1=h=0.2,λ=0.01 and γ=2.5. Then, c3γeλτ =0.25, and we choose ψ=σ2=0.3, ψh+c3ech =0.238. Thus, we can pick r =0.24. Then, all assumptions in the previous two theorems are fulfilled. Therefore, system (4.1) is mean square (i.e. second moment) practically exponentially stable with η=r c1 =0.24. Now, we exhibit the following numerical example Example 4.2. Consider the linear impulsive stochastic delay system          dx(t)=Ax(t)+Bx(t−1)dt +Cx(t) +Dx(t−1)dW(t),t,tk,t≥0, ∆x(tk)=−0.4x(t− k),k∈N∗, (4.2) where A = −0.4 0 0−0.4!, B = 1 0 0 1 !, C = 0.9 0 0 0.9!, D = 1 0 0 1 !, x(t)=x1(t),x2(t)T. Then, f (x(t),x(t−1),t)=Ax(t)+Bx(t−1), g(x(t),x(t− 1),t)=Cx(t)+Dx(t−1)and Ik(x(t− k),tk)=−0.4x(t− k). We assume that ||A|| =pλmaxATAfor all A ∈ M2(R). Let p =2and V(x,t)=|x|2. Then, Ef(x(t),x(t−1),t) 2+Eg(x(t),x(t−1),t) 2 ≤2||A||2+||C||2Ex(t) 2+2||B||2+||D||2Ex(t−1) 2, ≤1.7Ex(t) 2+4Ex(t−1) 2+0.4, ≤4Ex(t) 2+Ex(t−1) 2+0.4. Therefore, condition (3.21) holds for system (4.1) with L=4and β=0.4. Let p =2and V(x,t)=|x|2. Then, the constants in Theorem 3.1 are c1=c2=1, c3=0.3, q=6.12,δ=0. Consequently, EV(x(t− k)+Ik(x(t− k),tk),tk)≤0.6EV(x(t− k),tk)+0.3 and ELV(xt,t) ≤2Ex(t)Tf(x(t),x(t−1),t)+Eg(x(t),x(t−1),t) 2, ≤2Ex(t)TAx(t)+2Ex(t)TBx(t−1)+Eg(x(t),x(t−1),t) 2, ≤2Ex 2||A||2+||B||2+4Ex 2, ≤6.08EV(x,t)+0.5. Thus, c =6.08, dk=0.6, d =0.5=ρkand ψ=0.5. We choose tk−tk−1=h=0.2,λ=0.02 and γ=6. Then, c3γeλτ =1.83, and ψh+c3ech =1.348. Thus, we can choose r =1.35. Then, all the assumptions of the previous two theorems are fulfilled. Therefore, system (4.2) is mean square practically exponentially stable with η=r c1 =1.35. 5. Appendix Lemma 5.1. Under conditions in Theorem 3.1 and Theorem 3.3, we define J1=EZtk−1+iδ tk−1+(i−1)δ |f(xs,s)|dsp, J2=Esup tk−1+(i−1)δ≤t<tk−1+iδZt tk−1+(i−1)δ g(xs,s)dW(s) p. Then J1≤Lδpeλτ M||ξ||p c1 e−λ(tk−1−t0)+δpβ+Lr c1,(5.1) and J2≤LCpδp 2M||ξ||p c1 eλτe−λ(tk−1−t0)+Cpβδ p 2+Lr c1 Cpδp 2. (5.2) 7 Proof. By (3.20), (3.21) and the H¨ older inequality, J1≤EhZtk−1+iδ tk−1+(i−1)δ dsp−1Ztk−1+iδ tk−1+(i−1)δ |f(xs,s)|pdsi, ≤δp−1EZtk−1+iδ tk−1+(i−1)δ |f(xs,s)|pds, ≤δp−1Ztk−1+iδ tk−1+(i−1)δLsup −τ≤θ≤0 E|x(s+θ)|p+βds, ≤Lδp−1Ztk−1+iδ tk−1+(i−1)δ sup −τ≤θ≤0 E|x(s+θ)|pds +βδp, ≤Lδp−1M||ξ||p c1Ztk−1+iδ tk−1+(i−1)δ e−λ(s−τ−t0)ds +βδp+Lr c1 δp, ≤LδpM||ξ||p c1 e−λ(tk−1+(i−1)δ−τ−t0)+δpβ+Lr c1. Then, J1≤Lδpeλτ M||ξ||p c1 e−λ(tk−1−t0)+δpβ+Lr c1.(5.3) On the other hand, by the Burkholder-Davis-Gundy inequality and H¨ older inequality, we have J2≤CpEZtk−1+iδ tk−1+(i−1)δ |g(xs,s)|2ds p 2 ≤Cpδp 2−1EZtk−1+iδ tk−1+(i−1)δ |g(xs,s)|pds,(5.4) where Cp>0 is a constant which only depends on p. From (3.21), for all (ϕ, t)∈Lp Ft([−τ, 0],Rn)×[t0,∞), we have E|g(ϕ, t)|p≤Lsup −τ≤θ≤0 E|ϕ(θ)|p+β. By inequality (5.4), J2≤Cpδp 2−1Ztk−1+iδ tk−1+(i−1)δLsup −τ≤θ≤0 E|x(s+θ)|p+βds, ≤LCpδp 2−1Ztk−1+iδ tk−1+(i−1)δ sup −τ≤θ≤0 E|x(s+θ)|pds +Cpβδ p 2. Thus, J2≤LCpδp 2M||ξ||p c1 eλτe−λ(tk−1−t0)+Cpβδ p 2+Lr c1 Cpδp 2. 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