Exponential stability in terms of two measures of impulsive stochastic functional differential systems via comparison principle

Statistics and Probability Letters 82 (2012) 1151–1159

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Exponential stability in terms of two measures of impulsive stochastic functional differential systems via comparison principle✩ Fengqi Yao a,b,∗ , Feiqi Deng a a

The Institute of System Engineering, South China University of Technology, Guangzhou 510640, PR China

b

School of Electrical Engineering & Information, Anhui University of Technology, Maanshan 243000, PR China

article

info

Article history: Received 27 March 2011 Received in revised form 6 January 2012 Accepted 8 January 2012 Available online 20 February 2012 Keywords: Impulsive stochastic functional differential systems Exponential stability Stability in terms of two measures Comparison principle

abstract In this paper, based on like-Lyapunov functions and comparison principles, several criteria on the exponential stability in terms of two measures of impulsive stochastic functional differential systems with infinite or finite delays are obtained. The results improve and complement those in earlier publications. Two illustrative examples are also discussed to show the effectiveness and generality of our theorems. © 2012 Published by Elsevier B.V.

1. Introduction Recently, impulsive stochastic differential systems (ISDSs), which are subject to both impulsive effects and stochastic perturbations, have attracted considerable attention. As a result, some theories and methods for impulsive differential systems (IDSs) have been generalized from deterministic systems to stochastic systems, such as Lyapunov function methods (e.g. Wu and Sun, 2006 and Li and Fu, 2011), Lyapunov–Razumikhin techniques (e.g. Peng and Jia, 2010, Cheng and Deng, 2010, Peng and Zhang, 2010 and Liu et al., 2011), Lyapunov–Krasovskii functional (e.g. Yang et al., 2008, Zhu and Cao, 2010, Fu and Li, 2011 and Gu, 2011), fixed point approach (e.g. Sakthivel and Luo, 2009a, Sakthivel and Luo, 2009b, Sakthivel et al., 2010 and Shen et al., 2010) as well as comparison principles (e.g. Caro and Rao, 1996, Liu, 2008, Yao and Deng, 2010 and Li et al., 2008). In particular, the stability of ISDSs without delay was investigated in Caro and Rao (1996), Liu (2008) and Li et al. (2008) and the results showed that by employing comparison principles, the stability properties of an n-dimensional ISDSs could be concluded from the corresponding ones of a scalar and deterministic IDSs. These contributions are of great importance since the stability theory for IDSs has been developed very well (see Lakshmikantham et al., 1989) while the stability results for ISDSs are scarce. Later, in view of the fact that significant progress has also been made in the stability theory of impulsive functional differential systems (IFDSs) (see e.g. Chen and Zheng, 2009, Wu et al., 2010 and Li, 2010), Yao and Deng (2010) extended comparison principle to impulsive stochastic functional differential systems (ISFDSs) and reduced the stability problems of an n-dimensional ISFDSs to the corresponding problems of a scalar IFDSs. On the other hand, rather than the stability of the trivial solution, Caro and Rao (1996) and Yao and Deng (2010) studied the stability in terms of two measures of

✩ This work was supported by the National Natural Science Foundation of China under Grant 60874114.

∗

Corresponding author at: The Institute of System Engineering, South China University of Technology, Guangzhou 510640, PR China. E-mail addresses: [email protected], [email protected] (F. Yao), [email protected] (F. Deng).

0167-7152/$ – see front matter © 2012 Published by Elsevier B.V. doi:10.1016/j.spl.2012.01.005

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ISDSs and ISFDSs, respectively. The notion of stability in terms of two measures has been proven to be very useful which not only unifies a variety of stability concepts in the literature but also offers a general framework for investigation of stability theory (see Lakshmikantham and Liu, 1993 and Liu and Wang, 2007). However, to the best of our knowledge, little work has been done on the exponential stability of ISFDSs in terms of two measures. In this paper, we aim to study this problem by employing comparison principles. Some exponential stability results for both ISFDSs with finite delays (ISFDSs-F) and infinite delays (ISFDSs-I) are obtained. It is worth mentioning that although Cheng and Deng (2010), Peng and Zhang (2010) and Liu et al. (2011) have also investigated the exponential stability of ISFDSs, several aspects make our results novel and meaningful. First of all, instead of the Razumikhin technique, which were used in the papers mentioned above, comparison principles are used in this paper. Secondly, rather than the exponential stability of the trivial solution, a more general stability notion—the exponential stability in terms of two measures is considered. Finally, the results we obtained in this paper can be applied to both ISFDSs-F and ISFDSs-I, while the results in Cheng and Deng (2010), Peng and Zhang (2010) and Liu et al. (2011) are only suitable for ISFDSs-F. The rest of the paper is organized as follows. In Section 2, we introduce some preliminary notes. And Section 3 devotes to the main stability results, followed by two illustrative examples in Section 4. Finally, the paper is closed with some concluding remarks in Section 5. 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. Let B(t ) = (B1 (t ), . . . , Bm (t ))T be an m-dimensional Brownian motion defined on the probability space. R = (−∞, +∞), R+ = [0, +∞), and N = {1, 2, 3, . . .}. Rn denotes the n-dimensional Euclidean space equipped with the Euclidean norm | · |. Denote by LFt (Ω ; Rn ) the family of Ft -measurable Rn -valued random variables. For a, b ∈ R, a < b, let PC ([a, b]; D) = {ϕ : [a, b] → D|ϕ(t ) is continuous for all but at most a finite number of points t¯ at which ϕ(t¯+ ) and ϕ(t¯− ) exist with ϕ(t¯+ ) = ϕ(t¯)}, where ϕ(t¯+ ) and ϕ(t¯− ) denote the right-hand and left-hand limits of ϕ(t ) at t¯, respectively. PC ((−∞, b]; D) = {ϕ : (−∞, b] → D|∀a < b, ϕ|[a,b] ∈ PC ([a, b]; D)}. For 0 6 τ 6 ∞, we equip the linear space PC ([−τ , 0]; D) with the norm ∥ · ∥ defined by ∥ϕ∥ = sup−τ 6θ 60 |ϕ(θ )|. In the case of τ = ∞, the interval [−τ , 0] is understood as (−∞, 0], and θ > −τ is understood as θ > −∞. In this paper, we consider the following ISFDSs

 dx(t ) = f (t , xt )dt + σ (t , xt )dB(t ), x(tk ) = Ik (tk , x(tk− )), x (θ ) = ξ (θ ), t0

t > t0 , t ̸= tk , k ∈ N, θ ∈ [−τ , 0],

(1)

where x ∈ Rn ; f : R+ × PC ([−τ , 0]; Rn ) → Rn ; σ : R+ × PC ([−τ , 0]; Rn ) → Rn×m ; Ik (tk , x(tk− )) : R+ × Rn → Rn represents the impulsive perturbation of x at time tk ; xt ∈ PC Ft ([−τ , 0]; Rn ) is defined as xt (s) = x(t + s), where PC Ft ([−τ , 0]; Rn ) denotes the family of all Ft measurable PC ([−τ , 0]; Rn )-valued random variables; ξ ∈ PC bFt ([−τ , 0]; Rn ) is the initial 0

value, where PC bFt ([−τ , 0]; Rn ) is the family of all bounded and Ft0 measurable PC ([−τ , 0]; Rn )-valued random variables. 0

The fixed moments of time tk satisfy 0 6 t0 < t1 < · · · < tk < · · · , limk→∞ tk = ∞. As a standing hypothesis, we assume that for any ξ ∈ PC bFt ([−τ , 0]; Rn ), there exists a unique solution to system (1) 0

denoted by x(t ; t0 , ξ ) (see Alwan et al., 2010), which is almost surely continuous except at t = tk , k ∈ N, at which it is continuous from right, i.e., x(tk+ ) = x(tk ) with probability one. 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. We introduce the following scalar IFDSs as the comparison system

 u˙ (t ) = g (t , u(t ), ut ), u(tk ) = Ψk (u(tk− )), u (θ ) = ζ (θ ), t0

t > t0 , t ̸= tk , k ∈ N, θ ∈ [−τ , 0],

(2)

where ut ∈ PC ([−τ , 0]; R+ ) is defined as ut (θ ) = u(t + θ ), θ ∈ [−τ , 0]; the initial value ζ ∈ PC ([−τ , 0]; R+ ) is bounded; g : R+ × R+ × PC ([−τ , 0]; R+ ) → R is continuous, Lebesgue measurable and nondecreasing with respect to the last argument; Ψk : R+ → R+ is continuous and nondecreasing. Assume that g (t , 0, 0) ≡ 0, Ψk (0) ≡ 0, then system (2) admits a trivial solution u(t ) ≡ 0. We further assume that for any ζ ∈ PC ([−τ , 0]; R+ ), there exists at least one solution to system (2) (see Luo and Shen, 2006 and Ouahab, 2007), and denote by u¯ (t ; t0 , ζ ) the maximal one through (t0 , ζ ). For convenience, we introduce the following function classes: (1) Rτ = [t0 − τ , ∞); (2) K = {b : R+ → R+ is continuous and strictly increasing, b(0) = 0}; (3) Γ = {h : Rτ × Rn → R+ , inf(t ,x) h(t , x) = 0, h(·, x) ∈ PC (Rτ ; R+ ) for each x ∈ Rn , and h(t , ·) ∈ C (Rn ; R+ ) for each t ∈ Rτ };

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(4) S (h, ρ) = {(t , x) ∈ Rτ × LFt (Ω ; Rn ) : Eh(t , x) < ρ, h ∈ Γ , ρ ∈ R+ }, where E stands for the mathematical expectation operator with respect to the probability measure P; (5) Sk (h, ρ) = {(t , x) ∈ [tk−1 , tk ) × LFt (Ω ; Rn ) : Eh(t , x) < ρ, h ∈ Γ , ρ ∈ R+ }, k ∈ N. Definition 1. A function V (t , x) : Rτ × Rn → R+ is said to belong to class v0 if (i) V (t , 0) ≡ 0 for all t ∈ Rτ ; (ii) for k ∈ N, V is continuous on each of the sets [tk−1 , tk ) × Rn , and for all x, y ∈ Rn , lim(t ,y)→(t − ,x) V (t , y) = V (tk− , x) k

exists; (iii) V is continuously once differentiable in t and twice in x in each of the sets (tk−1 , tk ) × Rn , k ∈ N. For each V (t , x) ∈ v0 and t > t0 , t ̸= tk , k ∈ N, define the Kolmogorov operator LV from [t0 , ∞) × PC ([−τ , 0]; Rn ) to R by

LV (t , xt ) = Vt (t , x) + Vx (t , x)f (t , xt ) +

1 2

trace[σ T (t , xt )Vxx (t , x)σ (t , xt )],

(3)

where



∂ V (t , x) Vt ( t , x ) = , ∂t

Vx (t , x) =

 ∂ V (t , x) ∂ V (t , x) ,..., , ∂ x1 ∂ xn

and

 Vxx (t , x) =

∂ 2 V ( t , x) ∂ xi ∂ xj

 . n×n

Definition 2. Let h0 , h ∈ Γ . Then we say that h0 is finer than h, if there exists a constant ρ > 0 and a function γ ∈ K such that

Eh(t , x) 6 γ (Eh0 (t , x)),

∀(t , x) ∈ S (h0 , ρ).

Definition 3. Let V ∈ v0 , h0 ∈ Γ . Then V is said to be h0 -decrescent, if there exists a constant ρ > 0 and a function c ∈ K such that

EV (t , x) 6 c (Eh0 (t , x)),

∀(t , x) ∈ S (h0 , ρ).

Definition 4. Let h0 ∈ Γ , ϕ ∈ PC ([−τ , 0]; Rn ). We define h0 (t , ϕ) = sup h0 (t + s, ϕ(s)),

t > t0 .

−τ 6s60

Definition 5. Let h0 , h ∈ Γ . Then system (1) is said to be (S1) (h0 , h)-exponentially stable, if there exists an α > 0, and for every ε > 0 and t0 ∈ R+ , there exists a δ = δ(t0 , ε) > 0 such that

Eh(t , x(t ; t0 , ξ )) < ε e−α(t −t0 ) ,

∀t > t0

whenever Eh0 (t0 , ξ ) < δ ; (S2) (h0 , h)-uniformly exponentially stable, if the δ in (S1) is independent of t0 ; (S3) (h0 , h)-globally exponentially stable, if there exists a pair of positive constants α and M such that

Eh(t , x(t ; t0 , ξ )) < M Eh0 (t0 , ξ )e−α(t −t0 ) ,

∀t > t 0

for all ξ ∈ PC bFt ([−τ , 0]; Rn ). 0

(h0 , h)-stability notions allow us to unify a variety of stability concepts found in the literature such as partial stability, conditional stability, stability of the trivial solution and stability of invariant sets, etc. For instance, we can immediately get the well-known pth moment exponential stability of the trivial solution or equivalently, of the invariant set {0}, if h0 (t , x) = h(t , x) = |x|p . The two measures h0 , h chosen in the illustrative examples later will demonstrate the generality of Definition 5. Lemma 1 (Yao and Deng (2010)). Assume that there exists a function V ∈ v0 such that (i) for any ϕ ∈ PC Ft ([−τ , 0]; Rn )

ELV (t , ϕ) 6 g (t , EV (t , ϕ(0)), EVt ),

t > t0 , t ̸= tk ,

where, and in the sequel, Vt = V (t + s, ϕ(s)), s ∈ [−τ , 0];

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(ii) for any x ∈ LFt (Ω ; Rn )

EV (tk , Ik (tk , x)) 6 Ψk (EV (tk− , x)),

k ∈ N.

Then

EV (t , x(t )) 6 u¯ (t ; t0 , ζ ),

t > t0

provided EV (t0 + s, x(t0 + s)) 6 ζ (s), s ∈ [−τ , 0], where x(t ) is the solution process to system (1). Remark 1. Although τ is a constant in Theorem 1 in Yao and Deng (2010), we can prove the lemma in the case of τ = ∞ without any difficulty. 3. Main results In this section, we shall establish some criteria on the (h0 , h)-exponential stability and (h0 , h)-globally exponential stability for ISFDSs-I or ISFDSs-F by employing comparison principles. Theorem 1. Suppose that system (2) is (uniformly) exponentially stable with convergence rate α > 0, and assume that (i) h0 , h ∈ Γ , h0 is finer than h; (ii) V ∈ v0 , V is h0 -decrescent, and there exist a pair of constants b, ρ > 0 such that bEh(t , x) 6 EV (t , x),

∀(t , x) ∈ Sk (h, ρ dk−1 ),

where and in the sequel, dk−1 = e−α(tk−1 −t0 ) ; (iii) there exists a constant r > 0 such that

∀(tk− , x) ∈ S (h, ρ0 dk ), k ∈ N,

Eh(tk , Ik (tk , x)) < r Eh(tk− , x),

where ρ0 = min{ρ, ρ/r }; (iv) for any ϕ ∈ PC Ft ([−τ , 0]; Rn ) and (t , ϕ(0)) ∈ Sk (h, ρ dk−1 )

ELV (t , ϕ) 6 g (t , EV (t , ϕ(0)), EVt ), and for (tk− , x) ∈ S (h, ρ0 dk )

EV (tk , Ik (tk , x)) 6 Ψk (EV (tk− , x)). Then system (1) is (h0 , h)-(uniformly) exponentially stable with convergence rate α . Proof. Fix any initial data ξ ∈ PC bFt ([−τ , 0]; Rn ) and write x(t ; t0 , ξ ) = x(t ) simply. Furthermore, we write V (t , x(t )) = 0

V (t ), h(t , x(t )) = h(t ), h0 (t , x(t )) = h0 (t ). Since system (2) is uniformly exponentially stable with convergence rate α , for any given ε ∈ (0, ρ0 ) and t0 ∈ R+ , there exists a δ1 = δ1 (ε) > 0 such that u(t ; t0 , ζ ) < bε e−α(t −t0 ) ,

t > t0

(4)

whenever ∥ζ ∥ < δ1 , where u(t ; t0 , ζ ) is any solution to system (2). We shall show that there exists a δ = δ(ε) > 0 such that

Eh(t ) < ε e−α(t −t0 ) ,

t > t0

(5)

whenever Eh0 (t0 , ξ ) < δ . Since V (t ) is h0 -decrescent and h0 is finer than h, there exist positive constants λ, λ0 and functions c , γ ∈ K such that

EV (t ) 6 c (Eh0 (t )),

whenever Eh0 (t ) < λ,

Eh(t ) 6 γ (Eh (t )),

whenever Eh (t ) < λ0 .

0

(6)

0

Choose a δ = δ(ε) > 0 satisfying δ < min{λ0 , λ, γ of h0 that

Eh0 (t0 + s) 6 Eh0 (t0 , ξ ) < δ,

−1

(7)

(ε), c

∀s ∈ [−τ , 0],

−1

(δ1 )} and let Eh0 (t0 , ξ ) < δ . First, we get from the definition (8)

which, together with (6) gives

EV (t0 + s) 6 c (Eh0 (t0 + s)) < c (δ),

∀s ∈ [−τ , 0].

Let ζ (s) = EV (t0 + s), s ∈ [−τ , 0]. Then

∥ζ ∥ = sup ζ (s) = sup EV (t0 + s) 6 c (δ) < δ1 , −τ 6s60

hence (4) holds.

−τ 6s60

(9)

F. Yao, F. Deng / Statistics and Probability Letters 82 (2012) 1151–1159

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Define L(t ) = h(t )eα(t −t0 ) , t ∈ Rτ . Then to prove (5), we only need to show

EL(t ) < ε,

∀t > t0 .

(10)

First, from (7) and (8) we obtain

EL(t0 + s) = Eh(t0 + s)eα s 6 Eh(t0 + s) 6 γ (Eh0 (t0 + s)) < γ (δ) < ε,

∀s ∈ [−τ , 0].

(11)

If (10) were not true, there would exist a solution process x(t ) to system (1) with Eh0 (t0 , ξ ) < δ and some t > t0 such that EL(t ) > ε . Set t ∗ = inf{t > t0 : EL(t ) > ε}. Suppose t ∗ ∈ [tk−1 , tk ), k ∈ N, then

EL(t ∗ ) > ε, and

EL(t ) < ε,

t0 − τ 6 t < tk−1 .

(12)

If k = 1, i.e. t ∈ [t0 , t1 ), then EL(t0 ) < ε < ρ in view of (11). If k > 2, from (12) we have ∗

Eh(tk−−1 ) = EL(tk−−1 )e−α(tk−1 −t0 ) 6 ε e−α(tk−1 −t0 ) < ρ0 dk−1 , then it follows from condition (iii) and the above inequality that

EL(tk−1 ) = Eh(tk−1 )eα(tk−1 −t0 ) < r Eh(tk−−1 )eα(tk−1 −t0 ) < r ρ0 6 ρ. So, for any k ∈ N, we have EL(tk−1 ) < ρ . Thus, we can find a t 0 ∈ [tk−1 , t ∗ ] such that

ε 6 EL(t 0 ) < ρ, EL(t ) < ρ,

(13)

t ∈ [t0 − τ , t ]. 0

(14)

From (12) and (14), we know that − Eh(tm ) = EL(tm− )e−α(tm −t0 ) 6 ε e−α(tm −t0 ) < ρ0 dm ,

Eh(t ) = EL(t )e−α(t −t0 ) < ε e−α(tj−1 −t0 ) = ρ dj−1 , Eh(t ) = EL(t )e−α(t −t0 ) < ρ e−α(tk−1 −t0 ) = ρ dk−1 ,

m = 1, . . . , k − 1, t ∈ [tj−1 , tj ), j = 1, . . . , k − 1, t ∈ [tk−1 , t 0 ].

Consequently, by conditions (ii), (iv) and applying Lemma 1, together with (4), we derive bEh(t ) 6 EV (t ) 6 u¯ (t ; t0 , ξ ) < bε e−α(t −t0 ) ,

t ∈ [t0 , t 0 ].

In particular,

Eh(t 0 ) < ε e−α(t

0 −t

0)

,

which contradicts (13). Therefore, (10) is true and system (1) is (h0 , h)-uniformly exponentially stable. If system (2) is exponentially stable but not uniformly, then the choice of δ1 is dependent on t0 , which leads to the dependence on t0 of δ , and we can similarly achieve the nonuniform (h0 , h)-exponential stability of system (1). The proof is complete.  Remark 2. Since Sk (h, ρ dk−1 ) ⊂ S (h, ρ), S (h, ρ0 dk ) ⊂ S (h, ρ0 ), so conditions (ii)–(iv) of Theorem 1 are weaker than conditions (ii)–(iv) of Theorem 2 in Yao and Deng (2010). In fact, we assume that the comparison system (2) is exponentially stable in this paper, and only asymptotically stable in Yao and Deng (2010). That is to say, the comparison system in this paper has been burden stronger restrictions. Now, we turn to investigate the (h0 , h)-globally exponentially stable of system (1). Theorem 2. Assume that there exists a function V ∈ v0 and a couple of positive constants c1 , c2 such that (i) h0 , h ∈ Γ , c1 h(t , x) 6 V (t , x) 6 c2 h0 (t , x), ∀(t , x) ∈ Rτ × Rn ; (ii) for t > t0 , t ̸= tk and ϕ ∈ PC Ft ([−τ , 0]; Rn )

ELV (t , ϕ) 6 g (t , EV (t , ϕ(0)), EVt ), for k ∈ N and x ∈ LFt (Ω ; Rn )

EV (tk , Ik (tk , x)) 6 Ψk (EV (tk , x)). Then system (1) is (h0 , h)-globally exponentially stable if system (2) is globally exponentially stable.

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Proof. Suppose that system (2) is globally exponentially stable, then there exists a pair of positive constants α and M such that u(t ; t0 , ζ ) < M ∥ζ ∥e−α(t −t0 ) ,

t > t0 ,

(15)

where u(t ; t0 , ζ ) is any solution to system (2) through (t0 , ζ ). Let ζ (s) = EV (t0 + s), s ∈ [−τ , 0]. It follows from condition (ii) and Lemma 1 that

EV (t ) 6 u¯ (t ; t0 , ζ ),

t > t0 .

(16)

And by condition (i), we have

∥ζ ∥ = sup EV (t0 + s) 6 c2 sup Eh0 (t0 + s) 6 c2 Eh0 (t0 , ξ ). −τ 6s60

(17)

−τ 6s60

Consequently, by condition(i) and (15)–(17), we get

 Eh0 (t0 , ξ )e−α(t −t0 ) , Eh(t ) < M

t > t0 ,  where M = c2 /c1 M. Therefore, system (1) is (h0 , h)-globally exponentially stable and the proof is complete.



Corollary 1. If condition (ii) of Theorem 2 holds while condition (i) is replaced by

(i)∗ there exists a constant p > 0 such that c1 |x|p 6 V (t , x) 6 c2 |x|p ,

∀(t , x) ∈ Rτ × Rn .

Then system (1) is pth moment globally exponentially stable if system (2) is globally exponentially stable. Proof. It is a direct consequence of Theorem 2 taking h(t , x) = h0 (t , x) = |x|p .



If τ > 0 is a constant, we have the following result for ISFDS-F. Theorem 3. Assume that there exists a constant c1 > 0 and functions V ∈ v0 , c2 ∈ C (Rτ ; R+ ) such that condition (ii) of Theorem 2 holds while condition (i) is replaced by

(i)∗ h0 , h ∈ Γ , c1 h(t , x) 6 V (t , x) 6 c2 (t )h0 (t , x), ∀(t , x) ∈ Rτ × Rn . Then system (1) is (h0 , h)-globally exponentially stable if system (2) is globally exponentially stable. Proof. The proof is similar to that of Theorem 2, only replacing (17) with

∥ζ ∥ = sup EV (t0 + s) −τ 6s60

6

sup c2 (t0 + s) · sup Eh0 (t0 + s) 6 c˜2 Eh0 (t0 , ξ ), −τ 6s60

−τ 6s60

where c˜2 = sup−τ 6s60 c2 (t0 + s) > 0 is a constant because function c2 is continuous and the interval [−τ , 0] is compact.



Remark 3. In the literature, condition c1 |x|p 6 V (t , x) 6 c2 |x|p is often required to guarantee the exponential stability (see e.g. Cheng and Deng, 2010, Peng and Zhang, 2010 and Liu et al., 2011). Obviously, condition (i)∗ of Theorem 3 is much weaker and enlarges the scope of Lyapunov function candidates. We will see this point in Example 1. In the following, we discuss the special case τ = 0. In this case, system (1) reduces to the following ISDS

 dx(t ) = f1 (t , x(t ))dt + σ1 (t , x(t ))dB(t ), x(t ) = I (t , x(tk− )), x(tk ) = xk , k 0 0

t > t0 , t ̸= tk , k ∈ N,

(18)

where f1 : R+ × Rn → Rn ; σ1 : R+ × Rn → Rn×m . For any x0 ∈ Rn , assume that there exists a unique solution to system (18). We note that the Kolmogorov operator (3) with respect to system (1) is a functional of xt , but only a function of t and x(t ) with respect to system (18). So, to study the stability of system (18), the comparison system should be an IDS as follows:

 u˙ (t ) = g1 (t , u(t )), u(t ) = Ψ (u(t − )), u(tk ) = u k, k 0 0

t > t0 , t ̸= tk , k ∈ N,

(19)

where g1 : R+ × R+ → R is continuous and Lebesgue measurable, g (t , 0) = 0, ∀t > t0 ; u0 ∈ R+ ; Ψk is the same as in system (2). Assume that for any u0 ∈ R+ , there exists at least one solution to system (18) and denote by uˆ (t ; t0 , u0 ) the maximal one through (t0 , u0 ). As a special case of Lemma 1, we give the following lemma, which is the comparison result for ISDS without delay as well as the ground of Caro and Rao (1996), Liu (2008) and Li et al. (2008).

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Lemma 2. Assume that there exists a function V ∈ v0 such that for any x ∈ LFt (Ω ; Rn )

ELV (t , x) 6 g (t , EV (t , x)),

t > t0 , t ̸= tk ,

(20)

and

EV (tk , Ik (tk , x)) 6 Ψk (EV (tk− , x)),

k ∈ N.

(21)

Then

EV (t , x(t ; t0 , x0 )) 6 uˆ (t ; t0 , u0 ),

t > t0

provided EV (t0 , x0 ) 6 u0 , where x(t ; t0 , x0 ) is the solution process to system (18). Corresponding to the above results for system (1), we can derive the criteria on the (h0 , h)-(uniformly) exponentially stable or (h0 , h)-globally exponentially stable of system (18) if we replace ELV (t , ϕ) 6 g (t , EV (t , ϕ(0)), EVt ) by (20). For instance, we present the pth moment globally exponentially stable of system (18) corresponding to Theorem 3 in the next result. Corollary 2. Assume that condition (i)∗ of Theorem 3 holds and for any x ∈ LFt (Ω ; Rn )

(ii)′ ELV (t , x) 6 g (t , EV (t , x)),

t > t0 , t ̸= tk ;

EV (tk , Ik (tk , x)) 6 Ψk (EV (tk , x)),

k ∈ N.

Then system (18) is (h0 , h)-globally exponentially stable if system (19) is globally exponentially stable. Remark 4. If h(t , x) = h0 (t , x) = |x|p , c2 (t ) = c2 , where c2 is a constant, then Corollary 2 reduces to the result of pthmoment exponential stability in Theorem 3.3 in Liu (2008). 4. Illustrative examples In this section, we give two examples to show the effectiveness and advantage of our results. For simplicity, we write xi (t ) = xi , xi (t − τ ) = xiτ , i = 1, 2. Example 1. Consider the following impulsive stochastic functional differential system with finite delays

 √  dx1 = (−x1 − x2 )dt + 2x1τ dB(t ), √   dx2 = [−x2 + 2(t + τ + 3/2)x1 ] dt + 2x2τ dB(t ),  x (t ) = β x1 (tk− ),   1 k x2 (tk ) = β x2 (tk− ),

t > 0, t ̸= tk , t > 0, t ̸= tk , k ∈ N, k ∈ N,

(22)

where τ > 0 is a constant, β ∈ (−1, 1) \ {0}. First, let h01 (t , x) = h1 (t , x) = x21 + x22 and choose V1 (t , x) = (t +τ + 3/2)x21 + 1/2x22 . Obviously, 1/2(x21 + x22 ) 6 V1 (t , x) 6 (t + τ + 3/2)(x21 + x22 ), i.e., condition (i)∗ of Theorem 3 holds with c1 = 1/2, c2 (t ) = t + τ + 3/2. And a simple calculation gives

LV1 = x21 − 2(t + τ + 3/2)x21 − x22 + 2(t + τ + 3/2)x21τ + x22τ

= x21 − 2V1 (t ) + 2V1 (t − τ ) + 2τ x21τ 6 −V1 (t ) + 2(1 + τ )V1 (t − τ ).

V1 (tk ) = β 2 V1 (tk− ). So the comparison system is



u˙ (t ) = −u(t ) + 2(1 + τ )u(t − τ ), u(tk ) = β 2 u(tk− ),

t > 0, t ̸= tk , k ∈ N.

(23)

According to Corollary 3.2 in Li (2010), it is easy to see that system (23) is globally exponentially stable if sup{tk − tk−1 } < − k∈N

2 ln |β| , −1 + 2(1 + τ )β −2 eγ τ

(24)

where γ > 0 is any given constant. Therefore, according to Theorem 3, we conclude that system (22) is (h10 , h1 )-globally exponentially stable under condition (24), i.e., the trivial solution of system (22) is globally exponentially stable in mean square. Next, let h02 (t , x) = x21 + x22 , h2 (t , x) = V2 (t , x) = (t + τ + 3/2)x21 + 1/2x22 . Obviously, condition (i)∗ of Theorem 3 still holds, and the comparison system is still system (23). Therefore, system (22) is also (h20 , h2 )-globally exponentially stable if (24) holds.

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We would like to point out that the exponential stability results in Cheng and Deng (2010), Peng and Zhang (2010) and Liu et al. (2011) are invalid because the condition c1 |x|p 6 V (t , x) 6 c2 |x|p is required in their theorems.  Example 2. Consider the following impulsive stochastic functional differential system with finite delays:

   dx = −x1 [x2 − (2 + sin t )x1 ]2 − x1 cos t /(2 + sin t ) dt + (2 + sin(t − τ )) x1τ dB(t ),    1 dx2 = −x2 [x2 − (2 + sin t )x1 ]2 dt + (2 + sin t )x2τ dB(t ), x (t ) = 1/2x1 (tk− ),    1 k x2 (tk ) = 1/2x2 (tk− ),

t > t0 , t ̸= tk , t > t0 , t = ̸ tk , k ∈ N, k ∈ N.

(25)

First, let V1 (t , x) = h01 (t , x) = h1 (t , x) = x21 + x22 . Then condition (i) of Theorem 2 holds with c1 = c2 = 1, and

LV1 6 −2 [x2 − (2 + sin t )x1 ]2 (x21 + x22 ) − 2x21 cos t /(2 + sin t ) + 9(x21τ + x22τ ) 6 2| cos t |/(2 + sin t )(x21 + x22 ) + 9(x21τ + x22τ ) 6 2V1 (t ) + 9V1 (t − τ ),

V1 (tk ) =

1 4

V1 (tk− ).

Thus, the comparison system is u˙ = 2u(t ) + 9u(t − τ ), 1 u(tk ) = u(tk− ), 4



t > t0 , t ̸= tk , k ∈ N.

(26)

According to Corollary 3.2 in Li (2010), we know that system (28) is globally exponentially stable if sup{tk − tk−1 } < ln 2/(1 + 18eγ τ ),

(27)

k∈N

where γ > 0 is an arbitrary constant. Hence, by Theorem 2, we conclude that system (25) is (h10 , h1 )-globally exponentially stable under condition (27), i.e., the trivial solution is globally exponentially stable in mean square. In the following, we choose h02 (t , x) = h2 (t , x) = V (t , x) = [x2 − (2 + sin t )x1 ]2 . Obviously, condition (i) of Theorem 2 holds with c1 = c2 = 1, and a simple calculation yields

LV2 = −2V22 (t ) + (2 + sin t )2 V2 (t − τ ) 6 9V2 (t − τ ), V2 (tk ) =

1 4

V (tk− ).

Thus, the comparison system is u˙ = 9u(t − τ ), 1 u(tk ) = u(tk− ), 4



t > t0 , t ̸= tk , k ∈ N,

(28)

which, according to Corollary 3.2 in Li (2010), is globally exponentially stable if sup{tk − tk−1 } < k∈N

ln 2 18eγ τ

,

(29)

where γ > 0 is an arbitrary constant. Hence, we conclude by Theorem 2 that system (25) is (h20 , h2 )-globally exponentially stable if (29) holds.  5. Conclusion In this paper, some criteria on the (h0 , h)-(uniformly) exponential stability and (h0 , h)-globally exponential stability for both ISFDSs-I and ISFDSs-F have been established by employing like-Lyapunov functions and comparison principle. The results obtained show that the (h0 , h)-exponential stability properties of ISFDSs can be concluded from the exponential stability properties of a scalar IFDSs. Moreover, for ISFDSs-F, the usual requirement c1 |x|p 6 V (t , x) 6 c2 |x|p ensuring the pth-moment exponential stability has been weaken. Finally, some illustrative examples have been given to demonstrate the effectiveness and generality of our results. References Alwan, M.S., Liu, X., Xie, W., 2010. Existence, continuation, and uniqueness problems of stochastic impulsive systems with time delay. J. Franklin Inst. 347, 1317–1333. Caro, E.A., Rao, A., 1996. Stability analysis of impulsive stochastic differential systems in terms of two measures. In: Bringing Together Education, Science and Technology, pp. 132–135. Chen, W., Zheng, W., 2009. Robust stablity and H∞ -control of uncertain impulsive systems with time-delay. Automatica 45, 109–117.

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