Exponential stability theorems for discrete-time impulsive stochastic systems with delayed impulses

Exponential stability theorems for discrete-time impulsive stochastic systems with delayed impulses

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Exponential stability theorems for discrete-time impulsive stochastic systems with delayed impulses Ting Cai, Pei Cheng PII: DOI: Reference:

S0016-0032(19)30892-0 https://doi.org/10.1016/j.jfranklin.2019.12.005 FI 4322

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Journal of the Franklin Institute

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Please cite this article as: Ting Cai, Pei Cheng, Exponential stability theorems for discrete-time impulsive stochastic systems with delayed impulses, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.12.005

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Exponential stability theorems for discrete-time impulsive stochastic systems with delayed impulses Ting Caia , Pei Chenga,∗ a

School of Mathematical Sciences, Anhui University, Hefei 230601, China

Abstract This paper investigates the pth moment exponential stability and almost sure exponential stability of discrete-time impulsive stochastic systems with delayed impulses. By using Razumikhin technique, several sufficient conditions for the exponential stability of discrete-time impulsive stochastic systems are derived, which extends the corresponding results for discrete-time stochastic systems without impulses. Both the stability results that impulses act as perturbation and the stability results that impulses act as stabilizer are obtained. Three examples and simulations are also presented to illustrate the effectiveness of the obtained results. Keywords: Discrete-time stochastic systems, Delayed impulses, Exponential stability, Razumikhin technique

1. Introduction In recent years, stochastic systems have aroused widespread interests because of their successful applications to various fields, such as biochemical systems, physical systems, control technology, electrical engineering and so on (see, e.g. [1-4]). Stability and stabilization are two of the most essential and important concepts in modern control theory, especially, in stochastic system analysis and synthesis. It is well known that most stochastic systems involve time delay, which significantly affect the behaviour of the systems in

∗

Corresponding author. Email address: chengpei [email protected] (P.Cheng).

Preprint submitted to The Journal of The Franklin Institute

December 11, 2019

some circumstances. There are extensive literatures on stochastic systems of delay (see, e.g. [5-9]). Besides time delays, impulsive effects may exist widely in stochastic systems. Sometimes, impulses can affect the dynamical behaviors of the systems and even make stable systems become unstable. However, it can also stabilize the original unstable systems. Therefore, it is necessary to consider both time delays and impulsive effects when investigating the stability of stochastic systems. There are many results of stochastic systems with delayed impulses are provided (see, e.g. [10-13]). For instance, a class of time-delays system subject to delayed impulses was studied in [10] by using impulsive delay inequality. Applying the average impulsive approach, the exponential stability criteria of stochastic systems with delayed impulses was derived in [11]. In [12], the global exponential stability was discussed about delay difference equations involving delayed impulses. By using LyapunovKrasovskii functional and the average impulsive interval method, Wang et al. [13] discussed the problem of exponential stability for a class of impulsive positive systems with mixed time-varying delays. For more information about stochastic systems with delay impulses (see, e.g. [14-23]). Along with the development of computer technology, the theory of discretetime systems or control of discrete-time systems have been developed intensively, see [24-26]. The reason is twofold: on the one hand, discrete-time systems are ideal mathematical models in the study of many practical problem; on the other hand, the calculation of discrete-time system has comparative advantages over continuous system. As we know, Razumikhin technique (see [27, 28]) is one of main methods to study the stability of delay systems. By using this kind of technique, the Lyapunov function does not need to be decreasing at all time. It has been successfully applied to investigate the stability of continuous stochastic systems (see [29-33]). In [29], Mao and colleagues established some Razumikhin-type theorems on exponential stability for stochastic functional differential system. The exponential stability of stochastic functional differential systems with delayed impulses was studied by applying the Lyapunov functions and Razumikhin techniques in [30]. For continuous stochastic systems, the continuity and Itˆo’s formula are two important keys to establish corresponding Razumikhin theorems. But for the discrete-time systems, Razumikhin type stability theorems cannot be obtained directly without continuity and Itˆo’s formula. This bring difficulties in using Razumikhin technique to investigate the exponential stability problem for discrete-time delay system. Liu et 2

al. [34] used Razumikhin-type technique to deal with exponential stability of discrete delay systems. Wu et al. [35] investigated pth moment exponential stability of stochastic nonlinear systems by utilizing the Euler-Maruyama method. More recently, Zong et al. [36] extended Razumikhin technique into discrete-time stochastic delay systems, but they did not take impulses into account. In fact, as far as we know, only few Razumikin-type exponential stability theorems have been reported for discrete-time stochastic systems with delay impulse. Hence, it is necessary to investigate the exponential stability for discrete-time stochastic systems with delayed impulses. Motivated by the above discussions, we will consider the exponential stability of the discrete-time stochastic systems with delayed impulses via Razumikhin technique. In this paper, we will establish the Razumikhin-type exponential stability theorems in the main part. When the original impulsefree system is stable, we find the feasible interval of the impulsive to preserve the stability property. While the discrete-time stochastic system without impulsive are unstable, appropriate impulses do contribute the stability of the stochastic system. In particular, the approach and analysis technique provided here are rather different from those used in [36] for discrete-time stochastic systems without impulsive effects, and those given in [12] for impulsive discrete systems with time delays. Finally, some examples are given to illustrate the effectiveness and the advantage of the obtained results. Consequently, comparing with other previously results, our results are much less conservative and more effective than some existing ones. The rest of the paper is organized as follows. Notation and preliminaries are introduced in Section 2. In Section 3, some Razumikhin-type exponential stability and almost sure exponential stability criteria for stochastic systems are established. Their applications in nonlinear and linear discrete-time delay stochastic system are further investigated in Section 4. In Section 5, three examples and simulations are given to show the effectiveness of the main results. Finally, some summaries are mentioned in Section 6. 2. Preliminaries Throughout this paper, let (Ω, F, {Fn }n>0 , P) be a complete probability space with a filtration {Fn }n>0 satisfying the usual conditions (F0 contains all P-null sets). R denotes the field of real numbers, Rd the d-dimensional Euclidean space, and Rd×r the space of d × r real matrices. For a set A ⊂ R, we denote IA be the family of all integer in A. Denote N = I[0, ∞) = {0, 1, 2, . . .} 3

and N+ = I[1, ∞). Let | · | be the Euclidean norm in Rd . If A is a vector T or matrix, its transpose p is denoted by A . If A is a matrix, its trance norm is denoted by |A| = trace(AT A). If A is a symmetric matrix, ∗ represent the elements below the main diagonal of a symmetric matrix. I represents the identity matrix. Let G(N−m , Rd ) = {ϕ : N−m → Rn |ϕ(s) is continuous for all but at most countable points s ∈ N−m and at these points}, where N−m = I[−τ, 0] and τ ∈ N+ . For p > 0, we denote by GbFn (N−m ; Rd ) the family of all Fn -measurable G-valued functions ξ = {ξ(s); s ∈ N−m } satisfying kξkpGb = sups∈N−m E|ξ(s)|p < ∞, where E stands for the mathematical expectation operator with respect to the given probability measure P. And GbFn0 (N−m ; Rd ) denote the family of all Fn0 measurable bounded G(N−m ; Rd )valued functions. Consider the following discrete-time stochastic systems with delayed impulses ( x(n + 1) = f (n, xn ) + g(n, xn )ω(n), n 6= nk − 1, n ∈ N, (1) x(nk ) = Ik (x(nk − 1), x(nk − θk )), k ∈ N+ with initial value x0 = ξ = {ξ(s) : s ∈ N−m } ∈ GbF0 (N−m ; Rd ), where x(n) = (x1 (n), · · · , xd (n))T , xn =: xn (θ) = {x(n + θ) : θ ∈ N−m } ∈ GbFn (N−m ; Rd ). f : N×GbFn (N−m ; Rd ) → Rd and g : N×GbFn (N−m ; Rd ) → Rd×r are Borel measurable. ω(n) = (ω1 (n), · · · , ωr (n))T be r-dimensional mutually independent stochastic sequence defined on the complete probability space (Ω, F, P) satisfying Eω(n) = 0, E|ω(n)|2 = 1 and Eω(i)ω(j) = 0 for i 6= j. For θk ∈ I[2, τ ], the impulsive functions Ik (x(nk − 1), x(nk − θk )) : N+ × GbFn (N−m ; Rd ) → Rd , and the impulsive perturbation at moments nk satisfy 0 = n0 < n1 < · · · < nk < · · · , nk → ∞ (as k → ∞). For the purpose of stability, we assume f (0, n) ≡ 0, g(0, n) ≡ 0 and Ik (0) ≡ 0 for any n ≥ 0, k ∈ N+ , then system (1) admits a trivial solution x(n) ≡ 0 and write x(n, n0 , ξ) = x(n) simply. Definition 1. The trivial solution of system (1) is said to be pth(p > 0) moment exponentially stable if there is a pair of positive constants λ and C such that E|x(n)|p ≤ CEkξkp e−λ(n−n0 ) , n ≥ n0 for any initial value ξ ∈ GbF0 (N−m ; Rd ). When p = 2, it is usually said to be exponentially stable in mean square. Definition 2. The trivial solution of system (1) is said to be almost surely 4

exponentially stable if lim sup n→∞

1 log |x(n)| < 0, n

a.s.

for any initial value ξ ∈ GbF0 (N−m ; Rd ). 3. Main results 3.1. Stability analysis for impulses act as stabilizer In this subsection, we will consider the exponential stability for discretetime impulsive stochastic systems (1) by using the Razumikhin techniques. Theorem 1. Assume that there exists a function V (n, x) and constants p > 0, c1 > 0, c2 > 0, q > 1, µ > 0, and nonnegative constants ρ1 , ρ2 with ρ1 + ρ2 < 1 such that (i) for any x ∈ Rd and n ∈ N, c1 |x|p ≤ V (n, x) ≤ c2 |x|p ; (ii) for any n 6= nk −1, k ∈ N+ , if EV (n+s, x(n+s)) ≤ qEV (n+1, x(n+1)) for all s ∈ N−m , then EV (n + 1, x(n + 1)) ≤ (1 + µ)EV (n, x(n)); (iii) for any x, y ∈ Rd , θk ∈ I[2, τ ], k ∈ N+ , EV (nk , I(x, y)) ≤ ρ1 (1 + ∞ P bk )EV (nk − 1, x) + ρ2 (1 + bk )EV (nk − θk , y), where bk ≥ 0 with bk < k=1 ∞; (iv) q >

1 ρ1 +ρ2

> eαµ , where α = supk∈N+ {nk+1 − nk } < ∞.

Then the trivial solution of system (1) is pth moment exponentially stable. Proof. By condition (iv), choose a sufficiently small number λ > 0 such that qe−λ(τ +1) >

ρ1

eλ

1 > eα(λ+µ) . + ρ2 eλτ

(2)

Set q˜ =: qe−λ(τ +1) > 1. Choose M0 > 0, such that q˜c2 < M0 . Define W (n) = eλn V (n) for any n ∈ I[−τ, ∞). By condition (i), 1 EW (n, x(n)) ≤ c2 Ekξkp < M0 Ekξkp < M0 Ekξkp , q˜

n ∈ I[−τ, n0 ].

(3)

In the following, we will prove that for any n ∈ I[n0 , nk ), EW (n, x(n)) ≤ Mk−1 Ekξkp , 5

(4)

where Mk = M0 Π1≤i≤k (1 + bi ) for k ∈ N+ . We first prove that for n ∈ I[n0 , n1 ), EW (n, x(n)) ≤ M0 Ekξkp .

(5)

If n1 − n0 = 1, (5) obviously holds. We will verify that (5) also holds for n1 − n0 ≥ 2. We assume, on the contrary, there exists n ¯ ∈ I[n0 , n1 − 1) such that EW (¯ n + 1, x(¯ n + 1)) > M0 Ekξkp

(6)

and EW (n, x(n)) ≤ M0 Ekξkp ,

n ∈ I[n0 − τ, n ¯ ].

(7)

By inequalities (6) and (7), we obtain for all s ∈ N−m , EW (¯ n + s, x(¯ n + s)) ≤ M0 Ekξkp < EW (¯ n + 1, x(¯ n + 1)), which implies that EV (¯ n + s, x(¯ n + s)) ≤ e−λ(s−1) EV (¯ n + 1, x(¯ n + 1)) < qEV (¯ n + 1, x(¯ n + 1)).

(8)

Thus, combining (8) and condition (ii), we get that EV (¯ n + 1, x(¯ n + 1)) ≤ (1 + µ)EV (¯ n, x(¯ n)). Together with (6) and (9), we have e−λ EW (¯ n + 1, x(¯ n + 1)) EW (¯ n, x(¯ n)) ≥ 1+µ eαµ −λα−αµ > e EW (¯ n + 1, x(¯ n + 1)) 1+µ > e−α(λ+µ) EW (¯ n + 1, x(¯ n + 1)) 1 > M0 Ekξkp , q˜ where we use the fact eαµ > (1 + µ).

6

(9)

For EW (n0 , x(n0 )) <

M0 Ekξkp , q˜

there exists an n ¯ ∗ ∈ I[n0 , n ¯ ) such that

1 EW (¯ n∗ , x(¯ n∗ )) ≤ M0 Ekξkp q˜ and 1 EW (n, x(n)) > M0 Ekξkp , q˜

n ∈ I(¯ n∗ , n ¯ ].

(10)

By (7) and (10), we obtain for n ∈ I(¯ n∗ , n ¯ ], EW (n − 1 + s, x(n − 1 + s)) ≤ M0 Ekξkp < q˜EW (n, x(n)),

s ∈ N−m ,

which together with the definition of EW (n, x(n)) implies EV (n − 1 + s, x(n − 1 + s)) = e−λ(n−1+s−n0 ) EW (n − 1 + s, x(n − 1 + s)) < e−λ(n−1+s−n0 ) q˜EW (n, x(n)) ≤ q˜eλ(τ +1) EV (n, x(n)) = qEV (n, x(n)). Substituting condition (ii) into this yields that EV (n, x(n)) ≤ (1 + µ)EV (n − 1, x(n − 1)),

n ∈ I(¯ n∗ , n ¯ ].

(11)

From (9) and (11), we further calculate that ∗

EV (¯ n + 1, x(¯ n + 1)) ≤ (1 + µ)n¯ +1−¯n EV (¯ n∗ , x(¯ n∗ )).

(12)

Then by (3) and (12), we have ∗

∗

EW (¯ n + 1, x(¯ n + 1)) ≤ (1 + µ)n¯ +1−¯n eλ(¯n+1−¯n ) EW (¯ n∗ , x(¯ n∗ )). ≤ eαµ eλα EW (¯ n∗ , x(¯ n∗ )) 1 ≤ eα(µ+λ) M0 Ekξkp < M0 Ekξkp . q˜ ∗

Note that (1 + µ)n¯ +1−¯n ≤ (1 + µ)α ≤ eαµ , which contradicts with (6). Therefore, (5) holds. By the induction principle, assume (3) holds for l ∈ N+ , namely, EW (n, x(n)) ≤ Ml−1 Ekξkp , 7

n ∈ I[n0 , nl ).

(13)

Form (13) and condition (iii), we have EW (nl , x(nl )) ≤ ρ1 (1 + bl )eλ EW (nl − 1, x(nl − 1)) +ρ2 (1 + bl )eλθl EW (nl − θl , x(nl − θl )) < (ρ1 eλ + ρ2 eλτ )Ml Ekξkp . Next we claim that, for any n ∈ I[nl , nl+1 ), EW (n, x(n)) ≤ Ml Ekξkp .

(14)

If nl+1 − nl = 1, obviously (14) holds. We claim that (14) also holds for nl+1 − nl ≥ 2. We assume, on the contrary, there exists an n ˆ ∈ I[nl , nl+1 − 1) such that EW (ˆ n + 1, x(ˆ n + 1)) > Ml Ekξkp

(15)

and EW (n, x(n)) ≤ Ml Ekξkp ,

n ∈ I[nl , n ˆ ].

(16)

By (2) and (16), for any s ∈ N−m , EW (ˆ n + s, x(ˆ n + s)) ≤ Ml Ekξkp < EW (ˆ n + 1, x(ˆ n + 1)), then by the definition of EW (n, x(n)), EV (ˆ n + s, x(ˆ n + s)) < eλ(1−s) EV (ˆ n + 1, x(ˆ n + 1)) λ(τ +1)
(17)

This together with condition (ii) and (17) implies EV (ˆ n + 1, x(ˆ n + 1)) ≤ (1 + µ)EV (ˆ n, x(ˆ n)). We derive that EV (ˆ n, x(ˆ n)) ≥

1 EV (ˆ n + 1, x(ˆ n + 1)), (1 + µ)

8

(18)

which implies that 1 e−λ EW (ˆ n + 1, x(ˆ n + 1)) (1 + µ) eαµ −λα−αµ > e EW (ˆ n + 1, x(ˆ n + 1)) (1 + µ) > e−α(λ+µ) EW (ˆ n + 1, x(ˆ n + 1)) λ λτ > (ρ1 e + ρ2 e )Ml Ekξkp .

EW (ˆ n, x(ˆ n)) ≥

Noting that EW (nl , x(nl )) ≤ (ρ1 eλ + ρ2 eλτ )Ml Ekξkp , there exists an n ˆ∗ ∈ I[nl , n ˆ ), such that EW (ˆ n∗ , x(ˆ n∗ )) ≤ (ρ1 eλ + ρ2 eλτ )Ml Ekξkp and EW (n, x(n)) > (ρ1 eλ + ρ2 eλτ )Ml Ekξkp ,

n ∈ I(ˆ n∗ , n ˆ ].

Therefore, for all n ∈ I(ˆ n∗ , n ˆ ], EW (n − 1 + s, x(n − 1 + s)) < Ml Ekξkp 1 < EW (n, x(n)) λ ρ1 e + ρ2 eλτ < q˜EW (n, x(n)). Hence for s ∈ N−m , EV (n − 1 + s, x(n − 1 + s)) = e−λ(n−1+s−n0 ) EW (n − 1 + s, x(n − 1 + s)) < e−λ(n−1+s−n0 ) q˜EW (n, x(n)) < eλ(τ +1) q˜EV (n, x(n)) < qEV (n, x(n)). Considering condition (ii), we therefore have that for any n ∈ I(ˆ n∗ , n ˆ ], EV (n, x(n)) ≤ (1 + µ)EV (n − 1, x(n − 1)). Combining with (18) and (19), we obtain EV (ˆ n + 1, x(ˆ n + 1)) ≤ (1 + µ)EV (ˆ n, x(ˆ n)) n ˆ +1−ˆ n∗ ≤ (1 + µ) EV (ˆ n∗ , x(ˆ n∗ )). 9

(19)

∗

Noting that (1 + µ)nˆ +1−ˆn ≤ (1 + µ)α ≤ eαµ , then by (2) and (19), we have ∗

∗

EW (ˆ n + 1, x(ˆ n + 1)) ≤ (1 + µ)nˆ +1−ˆn eλ(ˆn+1−ˆn ) EW (ˆ n∗ , x(ˆ n∗ )) ≤ eαµ eλα EW (ˆ n∗ , x(ˆ n∗ )) ≤ eα(µ+λ) (ρ1 eλ + ρ2 eλτ )Ml Ekξkp < Ml Ekξkp . This contradicts (15). Thus, (14) holds. Q∞By mathematical induction, (4) holds. k=1 (1 + bk ), by (4), we obtain

EW (n, x(n)) ≤ M LEkξkp ,

Set M = M0 L, where L = n ∈ N.

Then, from condition (i) and the definition of W (n, x(n)), we have   ML p E|x(n)| ≤ Ekξkp e−λn , n ∈ N, c1 which implies the trivial solution of system (1) is pth moment exponentially stable. The proof is complete.  Remark 1. From the condition (ii) of Theorem 1, we find that the original system without impulses may be unstable. Theorem 1 shows that the impulses can be regarded as impulsive controllers that would stabilize an unstable discrete-time stochastic system. To take advantage of their positive role, stabilizing impulses should act frequently. Remark 2. The exponential stability for the special case x(nk ) = Ik (x(nk − θ(k)) has been considered in [12]. If let g ≡ 0 and ρ2 = 0, then system (1) becomes the discrete-time system which has been discussed in Zhang [12]. In this sense, Theorem 1 can be reduced to Theorem 3.3 in [12] for impulsive delay systems without stochastic noises. 3.2. Stability analysis for impulses act as destabilizing In Theorem 1, impulses are beneficial to the stability of the stochastic system, while in the following theorem, impulses are potentially harmful to the stability of the impulse-free system. Theorem 2. Assume that there exists a positive definite function V (n, x) and constants p > 0, c1 > 0, c2 > 0, q > 1, 0 < µ < 1, and nonnegative constants ρ1 , ρ2 with ρ1 + ρ2 > 1 such that 10

(i) for any x ∈ Rd and n ∈ N, c1 |x|p ≤ V (n, x) ≤ c2 |x|p ; (ii) for any n 6= nk −1, k ∈ N+ , if EV (n+s, x(n+s)) ≤ qEV (n+1, x(n+1)) for all s ∈ N−m , then EV (n + 1, x(n + 1)) ≤ (1 − µ)EV (n, x(n)); (iii) for any x, y ∈ Rd , θk ∈ I[2, τ ], k ∈ N+ , EV (nk , I(x, y)) ≤ ρ1 (1 + ∞ P bk < bk )EV (nk − 1, x) + ρ2 (1 + bk )EV (nk − θk , y), where bk ≥ 0 with k=1 ∞; (iv) ρ1 + ρ2 < q < eµα , where α = inf k∈N+ {nk+1 − nk } ≥ 1. Then the trivial solution of system (1) is pth moment exponentially stable. Proof. By condition (iv), choose a sufficiently small number λ ∈ (0, µ) such that (ρ1 eλ + ρ2 eλτ ) < qe−λ(τ +1) < e(µ−λ)α .

(20)

Set q˜ =: qe−λ(τ +1) > 1. Choose M0 > 0 such that q˜c2 < M0 . Define W (n) = eλn V (n) for any n ∈ I[−τ, ∞). By condition (i), for any n ∈ I[−τ, n0 ], we have 1 EW (n, x(n)) ≤ c2 Ekξkp < M0 Ekξkp < M0 Ekξkp . q˜ Next, we will prove that for any n ∈ I[n0 , nk ), EW (n, x(n)) ≤ Mk−1 Ekξkp ,

(21)

where Mk = M0 Π1≤i≤k (1 + bi ) for k ∈ N+ . We first prove that for any n ∈ I[n0 , n1 ), EW (n, x(n)) ≤ M0 Ekξkp .

(22)

If n1 − n0 = 1, (22) obviously holds. We will verify that (22) also holds for n1 − n0 ≥ 2. We assume, there exists an n ¯ ∈ I[n0 , n1 − 1) such that EW (¯ n + 1, x(¯ n + 1)) > M0 Ekξkp

(23)

and EW (n, x(n)) ≤ M0 Ekξkp , 11

n ∈ I[n0 − τ, n ¯ ].

(24)

By inequalities (23) and (24), we obtain for all s ∈ N−m , EW (¯ n + s, x(¯ n + s)) < EW (¯ n + 1, x(¯ n + 1)), which implies that EV (¯ n + s, x(¯ n + s)) < e−λ(s−1) EV (¯ n + 1, x(¯ n + 1)) < qEV (¯ n + 1, x(¯ n + 1)).

(25)

Thus, combining (25) and condition (ii), we get EV (¯ n + 1, x(¯ n + 1)) ≤ (1 − µ)EV (¯ n, x(¯ n)).

(26)

By virtue of (20), (23) and (26), one can derive that e−λ EW (¯ n + 1, x(¯ n + 1)) 1−µ > e−λ EW (¯ n + 1, x(¯ n + 1)) 1 > M0 Ekξkp . q˜

EW (¯ n, x(¯ n)) ≥

Note that EW (n0 , x(n0 )) < 1q˜ M0 Ekξkp , then exists an n ¯ ∗ ∈ I[n0 , n ¯ ) such that 1 EW (¯ n∗ , x(¯ n∗ )) ≤ M0 Ekξkp q˜ and 1 EW (n, x(n)) > M0 Ekξkp , q˜

n ∈ I(¯ n∗ , n ¯ ].

(27)

By (24) and (27), we obtain that for any n ∈ I(¯ n∗ , n ¯ ], EW (n − 1 + s, x(n − 1 + s)) ≤ M0 Ekξkp < q˜EW (n, x(n)),

s ∈ N−m ,

which further implies EV (n − 1 + s, x(n − 1 + s)) < e−λ(s−1) q˜EV (n, x(n)) ≤ q˜eλ(τ +1) EV (n, x(n)) = qEV (n, x(n)). 12

It then follows from condition (ii) that EV (n, x(n)) ≤ (1 − µ)EV (n − 1, x(n − 1)),

n ∈ I(¯ n∗ , n ¯ ].

(28)

By using simple induction, we obtain form (26) and (28) that ∗

EV (¯ n + 1, x(¯ n + 1)) ≤ (1 − µ)(¯n+1−¯n ) EV (¯ n∗ , x(¯ n∗ )). Following (20) and (26), we further calculate that ∗

∗

EW (¯ n + 1, x(¯ n + 1)) ≤ (1 − µ)(¯n+1−¯n ) eλ(¯n+1−¯n ) EW (¯ n∗ , x(¯ n∗ )) ∗ ∗ < e−µ(¯n+1−¯n ) eλ(¯n+1−¯n ) EW (¯ n∗ , x(¯ n∗ )) < EW (¯ n∗ , x(¯ n∗ )) < M0 Ekξkp , ∗

∗

here we use the fact that (1 − µ)(¯n+1−¯n ) < e−µ(¯n+1−¯n ) . This contracts with (23). Therefore, (22) holds. To apply the mathematical induction, assume (21) holds for l ∈ N+ , EW (n, x(n)) ≤ Ml−1 Ekξkp ,

n ∈ I[n0 , nl ).

(29)

We proceed to prove that EW (n, x(n)) ≤ Ml Ekξkp ,

n ∈ I[nl , nl+1 ).

To do this, we first claim EW (nl − θl , x(nl − θl )) ≤

e(µ−λ)τ Ml−1 Ekξkp , q˜

θl ∈ I[2, τ ].

(30)

Suppose not, then there exists θ0 ∈ I[2, τ ], such that EW (nl −θ0 , x(nl −θ0 )) > e(µ−λ)τ Ml−1 Ekξkp . Without lose of generality, we assume nl − θ0 ∈ I(ni−1 , ni ], q˜ i ≤ l. In what follows, two cases are considered. (µ−λ)τ Case I. EW (n, x(n)) > e q˜ Ml−1 Ekξkp for all n ∈ I[ni−1 , nl − θ0 ]. By (29), for all n ∈ I[ni−1 , nl − θ0 ], we get EW (n − 1 + s, x(n − 1 + s)) < Ml−1 Ekξkp < e(µ−λ)τ Ml−1 Ekξkp < q˜EW (n, x(n)), s ∈ N−m , 13

which implies, for any n ∈ I[ni−1 , nl − θ0 ], EV (n, x(n)) ≤ (1 − µ)EV (n − 1, x(n − 1)). This leads to EV (nl − θ0 , x(nl − θ0 )) ≤ (1 − µ)nl −θ0 −ni−1 EV (ni−1 , x(ni−1 )). Form (20) and (29), we have EW (nl − θ0 , x(nl − θ0 )) ≤ (1 − µ)nl −θ0 −ni−1 eλ(nl −θ0 −ni−1 ) EW (ni−1 , x(ni−1 )) ≤ e(λ−µ)(nl −θ0 −ni−1 ) EW (ni−1 , x(ni−1 )) < e(µ−λ)τ e(λ−µ)(nl −ni−1 ) Ml−1 Ekξkp < e(µ−λ)τ e(λ−µ)α(l−i+1) Ml−1 Ekξkp e(µ−λ)τ < (l−i+1) Ml−1 Ekξkp q˜ e(µ−λ)τ < Ml−1 Ekξkp , q˜ which leads a contradiction. Therefore, (30) holds. Case II. There exists an n ˜ ∈ I(ni−1 , nl − θ0 ], such that EW (˜ n, x(˜ n)) >

e(µ−λ)τ Ml−1 Ekξkp q˜

(31)

and EW (n, x(n)) ≤

e(µ−λ)τ Ml−1 Ekξkp , q˜

n ∈ I[ni−1 , n ˜ ).

Then, follows from condition (ii), (29) and (31), we get that EW (˜ n − 1 + s, x(˜ n − 1 + s)) < e(µ−λ)τ Ml−1 Ekξkp < q˜EW (˜ n, x(˜ n)), s ∈ N−m , from condition (ii), which implies EV (˜ n, x(˜ n)) ≤ (1 − µ)EV (˜ n − 1, x(˜ n − 1)). By the definition of EW (n, x(n)), we have EW (˜ n − 1, x(˜ n − 1)) ≥

e−λ 1 EW (˜ n, x(˜ n)) > Ml−1 Ekξkp . 1−µ q˜ 14

(32)

˜ ), then set n ˜ ∗ = ni−1 ; If EW (n, x(n)) > 1q˜ Ml−1 Ekξkp for all n ∈ I[ni−1 , n otherwise, there exists an n ˜ ∗ ∈ I[ni−1 , n ˜ ), such that 1 EW (˜ n∗ , x(˜ n∗ )) ≤ Ml−1 Ekξkp q˜ and 1 EW (n, x(n)) > Ml−1 Ekξkp , q˜

n ∈ I(˜ n∗ , n ˜ ).

Thus for all n ∈ I(˜ n∗ , n ˜ ) and s ∈ N−m , we have EW (n − 1 + s, x(n − 1 + s)) < Ml−1 Ekξkp < q˜EW (n, x(n)), which implies that EV (n, x(n)) < (1 − µ)EV (n − 1, x(n − 1)),

n ∈ I(˜ n∗ , n ˜ ).

It follows from (32) and (33), we obtain EV (˜ n, x(˜ n)) ≤ (1 − µ)EV (˜ n − 1, x(˜ n − 1)) n ˜ −˜ n∗ ∗ ≤ (1 − µ) EV (˜ n , x(˜ n∗ )). Therefore, by (20), (29) and (32), we see that for n ∈ I(˜ n∗ , n ˜ ], ∗

∗

EW (˜ n, x(˜ n)) ≤ (1 − µ)(˜n−˜n ) eλ(˜n−˜n ) EW (˜ n∗ , x(˜ n∗ )) ∗ ∗ < e−µ(˜n−˜n ) eλ(˜n−˜n ) EW (˜ n∗ , x(˜ n∗ )) ∗ < e(λ−µ)(˜n−˜n ) EW (˜ n∗ , x(˜ n∗ )) ∗ e(λ−µ)(˜n−˜n ) < Ml−1 Ekξkp q˜ ∗ e(λ−µ)α(˜n−˜n ) < Ml−1 Ekξkp q˜ 1 < (˜n−˜n∗ +1) Ml−1 Ekξkp q˜ e(µ−λ)τ < (˜n−˜n∗ +1) Ml−1 Ekξkp q˜ e(µ−λ)τ < Ml−1 Ekξkp , q˜ 15

(33)

which contradicts (31). Hence, (30) holds. Similarly, we can prove EV (nl − 1, x(nl − 1)) <

e(µ−λ)τ Ml−1 Ekξkp . q˜

(34)

Then it is follows from (30), (34) and conditions (iii) and (iv), we obtain EW (nl , x(nl )) ≤ ρ1 (1 + bl )eλ EW (nl − 1, x(nl − 1)) +ρ2 (1 + bl )eλτ EW (nl − θl , x(nl − θl )) (ρ1 eλ + ρ2 eλτ ) < (1 + bl )Ml−1 Ekξkp q˜ < Ml Ekξkp . Now we show that, for any n ∈ I[nl , nl+1 ), EW (n, x(n)) ≤ Ml Ekξkp .

(35)

If nl+1 − nl = 1, (35) obviously holds. We claim that (35) also holds for nl+1 − nl ≥ 2. Assume exists an n ˆ ∈ I[nl , nl+1 − 1) such that EW (ˆ n + 1, x(ˆ n + 1)) > Ml Ekξkp

(36)

and EW (n, x(n)) ≤ Ml Ekξkp ,

n ∈ I[nl , n ˆ ].

(37)

It can be deduced that, by (36) and (37), EW (ˆ n + s, x(ˆ n + s)) < EW (ˆ n + 1, x(ˆ n + 1)),

s ∈ N−m .

It follows immediately that for s ∈ N−m , EV (ˆ n + s, x(ˆ n + s)) ≤ eλ(1−s) EV (ˆ n + 1, x(ˆ n + 1)) λ(τ +1)
(38)

This, together with condition (ii) and (38) yields EV (ˆ n + 1, x(ˆ n + 1)) < (1 − µ)EV (ˆ n, x(ˆ n)), 16

(39)

by (36) and (39), we have EW (ˆ n, x(ˆ n)) ≥

e−λ 1 EW (ˆ n + 1, x(ˆ n + 1)) > Ml Ekξkp . 1−µ q˜

If EW (n, x(n)) > 1q˜ Ml Ekξkp for n ∈ I[nl , n ˆ ), then set n ˜ ∗ = nl ; otherwise, there exists an n ˆ ∗ ∈ I[nl , n ˆ ), such that 1 EW (ˆ n∗ , x(ˆ n∗ )) ≤ Ml Ekξkp q˜ and 1 EW (n, x(n)) > Ml Ekξkp , q˜

n ∈ I(ˆ n∗ , n ˆ ].

(40)

Thus for all n ∈ I(ˆ n∗ , n ˆ ] and s ∈ N−m , by (37) and (40), we have EW (n − 1 + s, x(n − 1 + s)) ≤ Ml Ekξkp < q˜EW (n, x(n)), which together with the definition of EW (n, x(n)) implies EV (n, x(n)) < (1 − µ)EV (n − 1, x(n − 1)),

n ∈ I(ˆ n∗ , n ˆ ].

Therefore, by (20), (39) and (41) for any n ∈ I(ˆ n∗ , n ˜ ], ∗

∗

EW (ˆ n + 1, x(ˆ n + 1)) ≤ (1 − µ)(ˆn+1−ˆn ) eλ(ˆn+1−ˆn ) EW (ˆ n∗ , x(ˆ n∗ )) ∗ ∗ < e−µ(ˆn+1−ˆn ) eλ(ˆn+1−ˆn ) EW (ˆ n∗ , x(ˆ n∗ )) ∗ e(λ−µ)(ˆn+1−ˆn ) < Ml Ekξkp q˜ ∗ e(λ−µ)α(˜n+1−˜n ) < Ml Ekξkp q˜ 1 < (˜n−˜n∗ +2) Ml Ekξkp q˜ < Ml Ekξkp . This contradicts with (36). Therefore, (35) holds. Q∞By mathematical induction, (21) holds. Set M = M0 L, where L = k=1 (1 + bk ), by (21), we obtain EW (n, x(n)) ≤ M LEkξkp , 17

n ∈ N.

Then, by the definition of W (n, x(n)) and condition (i), we get that   ML p Ekξkp e−λn , n ∈ N. E|x(n)| ≤ c1

(41)

The proof is complete.  Remark 3. When ρ1 + ρ2 > 1, the impulse may be viewed as a destabilizing factor and in order to make the underlying system stable, the impulse should not occur too frequently. Remark 4. When there do not exist impulses in (1), the result of Theorem 2 can be reduced to Theorem 3.4 in [36]. However, when condition (iii) of Theorem 2 holds, the condition (iii) of Theorem 3.4 in [36] may not hold, thus Theorem 2 has a wider range of applications than the results in [36]. 3.3. Almost sure exponential stability At the end of this section, under an irrestrictive condition, we shall establish a theorem about the almost exponentially stable of system (1). Theorem 3. For any p > 1, under the conditions of Theorem 2, (41) implies lim sup n→∞

λ 1 log |x(n)| ≤ − , n p

a.s.

(42)

namely, the pth moment exponential stability of the trivial solution of system (1) implies the almost sure exponential stability. Proof. By the Chebyshev inequality and (41), for n ∈ N, P{|x(n)|p > n2 e−λn } ≤

M LEkξkp eλn p E|x(n)| ≤ . n2 c1 n2

In view of the Borel-Cantelli lemma we see that for almost all w ∈ Ω, |x(n)|p ≤ n2 e−λn

(43)

hold for all but finitely many n. Hence, there exists an n0 = n0 (ω), for all w ∈ Ω excluding a P -null set, for which (43) holds whenever n ≥ n0 . Consequently, for almost all w ∈ Ω, 1 λ 2 log n log |x(n)| ≤ − + , n p pn

whenever n ≥ n0 . Letting n → ∞ yields (42), as required.  Remark 5. Generally speaking, the pth moment exponential stability and almost sure exponential stability do not imply each other and additional conditions are required in order to deduce one from the other (cf. [35]). 18

4. Applications In this section, using the established theorems in the previous section, we shall consider the stability of the following discrete-time delay stochastic systems with delayed impulses   x(n + 1) = f (n, x(n), x(n − δ(n))) +g(n, x(n), x(n − δ(n)))ω(n), n 6= nk − 1, n ∈ N,   x(nk ) = Ik (x(nk − 1), x(nk − θk )), k ∈ N+ (44) with initial value x0 = ξ = {ξ(s) : s ∈ N−m } ∈ GbF0 (N−m ; Rd ), where f : N × GbF0 (N−m ; Rd ) × GbF0 (N−m ; Rd ) → Rd and g : N × GbF0 (N−m ; Rd ) × GbF0 (N−m ; Rd ) → Rd×r , δ : N → I[0, τ ] and θk ∈ I[2, τ ]. Corollary 1. Let λ, a, b all be positive number, assume that the system (44) satisfies the following: (i) E|f (n, x(n), x(n − δ(n))|2 + E|g(n, x(n), x(n − δ(n)))ω(n)|2 < 21 (aE|x(n)|2 + bE|x(n − δ(n))|2 ); (ii) E|x(nk )|2 ≤ β1 E|x(nk − 1)|2 + β2 E|x(nk − θk )|2 , where β1 , β2 > 0, and β1 + β2 < 1; (iii)

1 β1 +β2

> eα(a+b) , where α = supk∈N+ {nk+1 − nk } < ∞.

Then the trivial solution of system (44) is mean square exponentially stable. Proof. By condition (i), we have E|x(n + 1)|2 = E|f (x, n(x), x(n − δ(n))) +g(n, x(n), x(n − δ(n)))ω(n)|2 ≤ 2E|f (x, n(x), x(n − δ(n)))|2 +2E(|g(n, x(n), x(n − δ(n)))|2 ω 2 (n)) < aE|x(n)|2 + bE|x(n − δ(n))|2 . Hence, for −δ(n) ∈ N−m , if E|x(n − δ(n))|2 ≤ qE|x(n + 1)|2 hold, then E|x(n + 1)|2 ≤

a E|x(n)|2 . 1 − bq 19

This implies that the condition (ii) in Theorem 1 holds with µ =

a+bq−1 . 1−bq

λ(τ +1)

Choose a small enough constant λ > 0, such that q > β1 eeλ +β2 eλτ , then the conclusion follows from Theorem 1 directly. Corollary 2. Let λ, a, b all be positive number, assume that system (44) satisfies the following: (i) E|f (n, x(n), x(n − δ(n)))|2 + E|g(n, x(n), x(n − δ(n)))ω(n)|2 < 21 (aE|x(n)|2 + bE|x(n − δ(n))|2 ); (ii) E|x(nk )|2 ≤ β1 E|x(nk − 1)|2 + β2 E|x(nk − θk )|2 , where β1 , β2 > 0, and β1 + β2 > 1; (iii) β1 + β2 < eα(a+b) , where α = inf k∈N+ {nk+1 − nk } > 1. Then, for any initial data x0 = ξ(s) ∈ GbF0 (N−m ; Rd ) the solution x(n) of system (44) satisfies Ekx(n)k2 ≤ Ekξk2 e−(λ/2)n . Therefore, the trivial solution of system (44) is mean square exponentially stable. Proof. By condition (i), it follows immediately that E|x(n + 1)|2 ≤ aE|x(n)|2 + bE|x(n − δ(n))|2 . Hence, for −δ(n) ∈ N−m , if E|x(n − δ(n))|2 ≤ qE|x(n + 1)|2 hold, then E|x(n + 1)|2 ≤

a E|x(n)|2 . 1 − bq

(45)

Let µ = 1−(a+bq) , using inequality (45), we get condition (ii) of Theorem 1−bq 2. Choose a small enough constant λ > 0, such that q > eλ(τ +1) (β1 eλ +β2 eλτ ), then the other conditions of the Theorem 2 are easy to be verified. Next, we present the stability of discrete-time linear delay stochastic systems with impulses   x(n + 1) = A0 x(n) + A1 x(n − τ ) (46) +[B0 x(n) + B1 x(n − τ )]ω(n), n 6= nk − 1, n ∈ N,   x(nk ) = C0 x(nk − 1) + C1 x(nk − θk ), k ∈ N+ 20

with initial value x0 = ξ = {ξ(s) : s ∈ N−m } ∈ GbF0 (N−m ; Rd ), where A0 , A1 , B0 , B1 , C0 and C1 ∈ Rd×d and θk is a time-varying delay satisfying θk ∈ I[2, τ ]. Corollary 3. Assume that for any scalar α > 0, choose positive scalars µ1 , µ2 such that µ1 + µ2 < 1. If there exist a positive definite matrix X, positive scalars λ1 and λ2 with λ1 + λ2 > 21 , such that the following matrix inequalities holds   −µ1 X 0 XC0T  ∗ −µ2 X XC1T  ≤ 0, (47) ∗ ∗ −X 



 −λ2 X B1 X A1 X  ∗ −X 0  < 0. ∗ ∗ −X

 −λ1 X B0 X A0 X  ∗ −X 0  < 0, ∗ ∗ −X

(48)

Then the system (46) is exponentially stable in mean square with impulse 2 ) ln(µ1 +µ2 ) time sequences that satisfy supk∈N+ {nk − nk−1 } ≤ −(1−2qλ . 2λ1 +2qλ2 −1 T d −1 Proof. Define V (n) = x (n)Qx(n) for x ∈ R , where Q = X , using the first equation of system (46), we have EV (n + 1, x(n + 1)) = E[xT (n)AT0 QA0 x(n)] + E[xT (n)B0T QB0 x(n)] +E[xT (n − τ )AT1 QA1 x(n − τ )] +2E[xT (n)AT0 QA1 x(n − τ )] +E[xT (n − τ )B1T QB1 x(n − τ )] +2E[xT (n)B0T QB1 x(n − τ )]. Using the elementary inequality for any vectors a, b ∈ Rd and matrix Z > 0, aT b + bT a ≤ aT Za + bT Z −1 b, we obtain EV (n + 1, x(n + 1)) ≤ 2E[xT (n)M1 x(n)] + 2E[xT (n − τ )M2 x(n − τ )], (49) where M1 = AT0 QA0 + B0T QB0 and M2 = AT1 QA1 + B1T QB1 . 21

By applying the Schur complement techniques [38] and the first matrix inequality (48), we have X T M1 X = (A0 X)T Q(A0 X) + (B0 X)T Q(B0 X) < λ1 X, similarly, by the second matrix inequality (48), we have X T M2 X < λ2 X. Applying this and inequality (49) one obtains that EV (n + 1, x(n + 1)) ≤ 2E[xT (n)M1 x(n)] + 2E[xT (n − τ )M2 x(n − τ )] < 2λ1 EV (n, x(n)) + 2λ2 EV (n − τ, x(n − τ )). Assume that EV (n − τ, x(n − τ )) ≤ qEV (n + 1, x(n + 1)) hold, then EV (n + 1, x(n + 1)) ≤

2λ1 EV (n, x(n)). 1 − 2qλ2

(50)

1 +qλ2 )−1 Set µ = 2(λ1−2qλ > 0, using inequality (50), the condition (ii) of Theorem 2 1 is satisfied. For matrix inequality (47) combining Schur complement yields    T   −µ1 Q 0 + C0 C1 Q C0 C1 ≤ 0. 0 −µ2 Q

Similarly, using the second equation of system (46), one has

V (nk , x(nk )) = xT (nk )Qx(nk )  T   T   x(nk − 1) x(nk − 1)  C0 C1 Q C0 C1 = x(nk − θk ) x(nk − θk )  T    x(nk − 1) µ1 Q 0 x(nk − 1) ≤ x(nk − θk ) 0 µ2 Q x(nk − θk )

= µ1 xT (nk − 1)Qx(nk − 1) + µ2 xT (nk − θk )Qx(nk − θk ) = µ1 V (nk − 1, x(nk − 1)) + µ2 V (nk − θk , x(nk − θk )).

Thus, we have EV (nk , x(nk )) ≤ µ1 EV (nk − 1, x(nk − 1)) + µ2 EV (nk − θk , x(nk − θk )). Choose a small enough constant λ > 0 and constant q > 1, such that q > eλ(τ +1) µ1 eλ +µ2 eλτ

λ(τ +α+1)+α

>e

2(λ1 +qλ2 )−1 1−2qλ2

. Let bk ≡ 0 for each k ∈ N+ in condition 22

(iii) of Theorem 1, the conclusion follows from Theorem 1 immediately. Then the system (46) is exponentially stable in mean square with impulse time 2 ) ln(µ1 +µ2 ) sequences that satisfy supk∈N+ {nk+1 − nk } ≤ −(1−2qλ . 2λ1 +2qλ2 −1 Corollary 4. Assume that for any scalar α > 0, choose positive scalars µ1 , µ2 such that µ1 + µ2 > 1. If there exist a positive-definite matrix X, positive scalars λ1 , λ2 ∈ (0, 1/2) such that the following matrix inequalities hold   −µ1 X 0 XC0T  ∗ −µ2 X XC1T  ≤ 0, (51) ∗ ∗ −X 

 −λ2 X B1 X A1 X  ∗ −X 0  < 0. ∗ ∗ −X



 −λ1 X B0 X A0 X  ∗ −X 0  < 0, ∗ ∗ −X

(52)

Then the system (46) is exponentially stable in mean square with impulse 2 ) ln(µ1 +µ2 ) time sequences that satisfy inf k∈N+ {nk − nk−1 } ≥ (1−2qλ . 1+2λ1 −2qλ2 T d −1 Proof. Define V (n) = x (n)Qx(n) for x ∈ R , where Q = X , then similar to the proof of Corollary 3, we get that EV (n + 1, x(n + 1)) ≤ 2E[xT (n)M1 x(n)] + 2E[xT (n − τ )M2 x(n − τ )] < 2λ1 EV (n, x(n)) + 2λ2 EV (n − τ, x(n − τ )). Assuming that EV (n − τ, x(n − τ )) ≤ qEV (n + 1, x(n + 1)) hold, then EV (n + 1, x(n + 1)) ≤

2λ1 EV (n, x(n)). 1 − 2qλ2

(53)

1 +qλ2 ) Set µ = 1−2(λ , using inequality (53), we get condition (ii) of Theorem 1−2qλ2 2. Similarly to the proof of impulse time nk , for k ∈ N+ , one has

EV (nk , x(nk )) ≤ µ1 EV (nk − 1, x(nk − 1)) + µ2 EV (nk − θk , x(nk − θk )). Choose a small enough constant λ > 0 and constant q > 1, such that λ(τ +1)+(

1+2(λ1 −qλ2 ) −λ)α

1−2qλ2 eλ(τ +1) (µ1 eλ + µ2 eλτ ) < q < e . If µ1 + µ2 > 1, the condition (iii) of Theorem 2 obviously holds, the conclusion follows from Theorem 2 immediately. Then the system (46) is exponentially stable in mean square 2 ) ln(µ1 +µ2 ) with impulses satisfy inf k∈N+ {nk+1 − nk } ≥ (1−2qλ . 1−2λ1 −2qλ2

23

5. Examples In this section, the applicability of the results derived in the preceding section will be illustrated by three examples. Example 1. Consider the following scalar stochastic discrete-time delay system.   x(n + 1) = a0 x(n) + a1 x(n − τ ) (54) +[b0 x(n) + b1 x(n − τ )]w(n), n 6= nk − 1, n ∈ N,   x(nk ) = c0 x(nk − 1) + c1 x(nk − θk ), k ∈ N+ where a0 , a1 , b0 , b1 , c0 , c1 ∈ R+ , θk is a time-varying delay satisfying θk ∈ I[2, τ ]. Let V (n) = |x(n)|2 , by simple calculation, we get that EV (n + 1, x(n + 1)) = E|x(n + 1)|2 = 2(a20 + b20 )EV (n, x(n)) +2(a21 + b21 )EV (n − τ, x(n − τ )) 2(a20 + b20 ) EV (n, x(n)), ≤ 1 − 2q(a21 + b21 ) whenever EV (n − τ, x(n − τ )) ≤ qEV (n + 1, x(n + 1)). One can derive from systems (54) that EV (nk ) ≤ c20 EV (nk − 1) + c21 EV (nk − θk ). 2(a2 +b2 )+2(a2 +b2 )q−1

λ(τ +1)

0 1 1 Case 1. c20 + c21 < 1, given λ > 0, let q = c2 eeλ +c2 eλτ , µ = 0 1−2(a , 2 +b2 )q 0 1 1 1 2 2 ρ1 = c0 , ρ2 = c1 , and bk = 0, it is easy to verify that systems (54) is mean square exponentially stable with impulse time sequences satisfies supk∈N+ {nk+1 −

nk } <

ln(c20 +c21 ) 2 2 2 1−(2(a2 0 +b0 )−2(a1 +b1 )q) 2 )q 1−2(a2 +b 1 1

.

Table 1: The convergence rate λ with different ρ1 + ρ2

ρ1 + ρ2 λ by Th 1

0.10 1.9026

0.20 1.2094

0.35 0.6498

0.50 0.2931

0.65 0.0308

Firstly, supposing µ = 0.2, α = 2, τ = 2, we consider the convergence rate λ is influenced by the parameter ρ1 + ρ2 . The convergence rate λ with 24

different ρ1 + ρ2 solved by (2) are listed in Table 1. One can see from Table 1 that: the convergence rate λ decreases with the parameter ρ1 + ρ2 , which means the larger the elastic number of the impulse is, the smaller of the −λ is permitted. Table 2: The convergence rate λ with different α

α λ by Th 1

1 1.1863

2 0.9863

3 0.7862

4 0.5865

5 0.3864

Secondly, fixing µ = 0.2, ρ1 + ρ2 = 0.25, τ = 2, we study the convergence rate λ of the parameter α. The maximum of λ under different α by Theorem 1 can be obtained from (2) are listed in Table 2. Table 2 shows that: the convergence rate λ decreases with the increase of α. Table 3: The convergence rate λ with different µ

µ λ by Th 1

0.10 1.857

0.15 1.0863

0.25 0.8862

0.50 0.3864

0.65 0.0872

Thirdly, given α = 2, ρ1 + ρ2 = 0.25, τ = 2, the convergence rate λ under different µ are listed in Table 3. Table 3 tells that: the allowable λ decreases with the increase of µ. Case 2. c20 + c21 > 1, given λ > 0, taking q = eλ(τ +1) (c20 eλ + c21 eλτ ), 1−2(a20 +b20 )−2(a21 +b21 )q , ρ1 = c20 , ρ2 = c21 , and bk = 0, then the conditions of µ= 1−2(a21 +b21 )q Theorem 2 are satisfied with impulse time sequences inf k∈N+ {nk+1 − nk } > ln(c20 +c21 ) . our result shows that a stable system, which is subject to 1−2(a2 +b2 )−2(a2 +b2 )q 0 0 1 2 1−2(a2 1 +b1 )q

1

delayed impulses, can still keep stability. Table 4: The convergence rate λ with different ρ1 + ρ2

ρ1 + ρ2 λ by Th 2

1.05 0.3512

1.20 0.2165

1.35 0.0998

1.40 0.0635

1.50 0.0079

For case 2, we study the relationship between the convergence rate λ and the correlation coefficients τ , α, µ, ρ1 + ρ2 . The convergence rate of λ under different parameters by Theorem 2 can be obtained from (20) are listed in Table 4-6. One can see from Table 4 that: choose µ = 0.2, α = 2, τ = 2, then 25

Table 5: The convergence rate λ with different α

α λ by Th 2

1 0.0177

2 0.2165

3 0.4202

4 0.6183

5 0.8225

Table 6: The convergence rate λ with different µ

µ λ by Th 2

0.10 0.0186

0.25 0.3176

0.50 0.8280

0.65 1.1362

0.70 1.2177

the convergence rate λ decreases with the impulses ρ1 + ρ2 . Setting µ = 0.2, ρ1 + ρ2 = 1.2, τ = 2, according to Table 5, we conclude that the convergence rate of λ increases with the increase of α. Fixing α = 2, ρ1 + ρ2 = 1.2, τ = 2, it can be seen from Table 6 that the property of the convergence rate λ increases with the increase of µ. Remark 6. Example 1 has been discussed in [12], the results in Zhang [12] would not be applicable. When c20 + c21 < 1, we obtain that the impulses ln c21 sequence satisfies supk∈N+ {nk+1 − nk } < 1−2a2 −2a 2 q (where b0 = 0, b1 = 0 1 0 1−2a2 1q

and c0 = 0). In this case, we see that the infimum of impulsive intervals is not restricted and only the supremun interval length is constrained. In this sense, our results have a wider adaptive range than those given in [12]. Remark 7. When c20 +c21 > 1, impulses sequence should satisfy inf k∈N+ {nk+1 − ln c21 nk } > 1−(2a2 +2a 2 q) . It should be noted that only the infimum impulsive in0 1 1−2a2 1q

terval is restricted, and the results in [12] still cannot be used to study the stability of (54). Example 2. Consider a two dimensional linear discrete-time stochastic system   x(n + 1) = A0 x(n) + A1 x(n − τ ) (55) +[B0 x(n) + B1 x(n − τ )]w(n), n 6= 3k − 1,   x(nk ) = C0 x(nk − 1) + C1 x(nk − τ ), k ∈ N+ .

In the following, two cases are considered to analyse the results. Case 1: 0 < µ1 + µ2 < 1; where         1 0 0.3 0 0.2 0 1 0.4 A0 = , A1 = , B0 = , B1 = . 0 1 0.2 0.1 0 0.3 0.5 1 26

It is not difficult to find that system (55) is exponentially unstable, the state sample trajectory with τ = 2, initial value ξ = [2, −0.5]T are simulated in Figure 1. By Corollary 3, system (55) with impulsive control x(nk ) = C0 x(nk − 1) + C1 x(nk − 2) is exponentially stabilize, where choose V (n) = kx(n)k2 , C0 = 0.25I, C1 = 0.15I, µ1 = 0.25, µ2 = 0.15, λ1 = 1, λ2 = 0.12, then the matrix inequalities (47) and (48) are feasible with   2.1257 −0.0202 X= . −0.0202 0.1145 1 and µ = 2.125, then the conditions of Corollary 3 are satisfied Setting q > 0.4 with α = supk∈N+ {nk − nk−1 } < 3.054. It can be deduced that the system is exponentially stable in mean square, which can also be further supported by Figure 2.

7

×104

2.5 x1

x1 x2

6

x2

2 5

1.5

4

x(n)

x(n)

3 2 1 0

1

0.5

0

-1

-0.5 -2 -3

0

20

40

60

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100

120

140

160

180

200

time n

-1

0

20

40

60

80

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140

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time n

Figure 1: The system (55) without impulses. Figure 2: The system (55) with impulses when nk − nk−1 = 3.

Case 2: µ1 + µ2 > 1; where         −0.1 1.1 0.4 0.1 0 −0.2 0.3 0.5 A0 = , A1 = , B0 = , B1 = . 0.1 −0.1 1 1.2 −0.3 0.2 0.2 0.3 27

The simulation results of system (55) with τ = 2, initial value ξ = [−3, 0.15]T are shown as in Figure 3. From Figure 3, one can find the system (55) without impulses is mean-square exponentially stable. Now we apply Corollary 4 to design the impulses controller to destroy above system. Choosing impulsive control x(nk ) = C0 x(nk − 1) + C1 x(nk − 2) to destruction of system stability, where choosing V (n) = kx(n)k2 , C0 = 0.75I, C1 = 1.5I, µ1 = 0.75, µ2 = 1.5, λ1 = 0.08, λ2 = 0.2, by solving matrix inequalities (51) and (52), one can obtain the following feasible with   0.8432 −0.0568 X= . −0.0568 0.5961 1 and µ = 0.2, then the conditions of Corollary 4 are satisfied Setting q ≥ 0.5 with α = inf k∈N+ {nk − nk−1 } > 4.054. It can be deduced that the stability of systems (55) will be subjected to delayed impulsive perturbation. The simulation results means that the impulses should not so frequently, the time interval between the nearest two impulses should be large enough, which can also be further supported by Figure 4.

1

12 x1

x1 x2

0.5

x2

10

8

0

6

x(n)

x(n)

-0.5

-1

4

2

-1.5 0 -2

-2

-2.5

-3

-4

0

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time n

-6

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time n

Figure 3: The system (55) without impulses. Figure 4: The system (55) under destabilizing impulsive perturbations.

Remark 8. When µ1 + µ2 < 1, the function V (n) will jump down along the 28

state trajectories of system (55) at impulse times nk . That is, it is possible to make an unstable discrete-time system become stable as long as the impulses happen frequently enough. Remark 9. The parameters µ1 + µ2 = 0.75 + 1.5 > 1 in (55) means that the impulse acts as a destabilizing factor. In this case, destabilizing impulses are expected to happen seldom. Example 3. Consider the following discrete population model with delayed impulses, which has been discussed in [39] y(n) =

αy(n − m) 1 + βy(n − m) + (1 − β)y(n − k)

(56)

where α > 1, β > 0, m as the delay in population reproduction while k as delay in the environment reaction to the growth of population. Here y(n) is the size of the population at time n; α is the rate of the growth of population if the resources were unlimited and the individuals are independent of each other. It is convinient for us to study the stability of the equation, introduce the deviation x(n) = y(n) − y0 , we get linearization of (56) x(n) =

β + α(1 − β) (1 − β)(α − 1) x(n − m) − x(n − k), α α

which has been considered in [39]. By setting α = 4, β = 21 , m = 2, k = 3, we obtain the following linear equation 5 3 x(n) = x(n − 2) − x(n − 3), 8 8

(57)

which has been investigate in [40]. The solution x(n) = 0 of model (57) is unstable, and the state sample trajectories are simulated in Figure 5. In what follows, we shall design a stochastic controller to stabilize this unstable system. In this case, system (57) becomes  3 5  x(n) = 8 x(n − 2) − 8 x(n − 3) (58) +[ 58 x(n − 2) − 83 x(n − 3)]w(n), n 6= 2k,   x(nk ) = 0.03x(nk − 1) + 0.02x(nk − 2), k ∈ N+ ,

where the initial value x(−3) = x(−2) = x(0) = 5. Let V (n) = E|x(n)| + 0.42E|x(n − 1)| + 0.189E|x(n − 2)|, by simple calculation, then when n 6= 2k, EV (n) ≤ 2EV (n−1), EV (nk ) = 0.45EV (nk −1)+0.191EV (nk −2). According 29

4

4

3.5

3.5

3

3

x(n)

x(n)

to Theorem 1, set q = 14 , µ = 1, ρ1 = 0.45, ρ2 = 0.191, bk ≡ 0, condition (ii)-(iii) of Theorem 1 hold, and system (58) is exponential stability in mean square, the state sample trajectories are simulated in Figure 6. Obviously, the maximum impulsive interval α = supk∈N+ {nk − nk−1 } < 4.054, under such circumstances, the results in [40] would not be applicable.

2.5

2.5

2

2

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1.5

1

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time n

1

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25

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time n

Figure 5: The system (58) without impulses. Figure 6: The system (58) with impulses when nk − nk−1 = 2.

Remark 10. We find that those results in [38, 39] are invalid for this example because stochastic perturbations w(n) ≡ 0 is not always. Furthermore, it should be pointed out that the impulsive control in (58) make the results in [40] can not be used to investigate the exponential stability of this system. From [40], the impulsive interval depends on the upper bound and the lower bound, in this case, our result is less conservative than [40]. 6. Conclusions This paper has studied the exponential stability for discrete-time impulsive stochastic systems with delay impulses. Using Razumikhin method, several criteria have been obtained to guarantee the pth moment exponential stability and almost sure exponential stability of the discrete-time impulsive 30

stochastic systems. The proposed results improve the related existing results on the same topic by removing some restrictive conditions. Finally, we give there examples to verify the validity of our results. Acknowledgements The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions which improved the quality of the paper. This work was supported by the National Natural Science Foundation of China (No.11771001), the Key Natural Science Project of Anhui Provincial Education Department (No.KJ2018A0027). References References [1] V. Lakshmikantham, D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, Singapore, World Scientific, 1989. [2] X. Mao, Stochastic differential equations and applications, England: Horwood Publishing Limited, Horwood, 1997. [3] S. E. Mohamed, Stochastic Functional Differential Equations, Longman: Scientific and technical, New York, 1986. [4] B. ∅ksendal, Stochastic Differential Equations: An Introduction with Applications, 4th edition, Berlin etc, Springer-Verlag, 1995. [5] X. Zong, G. Yin, L. Wang, T. Li, J. Zhang, Stability of stochastic functional differential systems using degenerate it Lyapunov functionals and applications, Automatica 91 (2018) 197-207. [6] P. Cheng, F. Deng, F. Yao, Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects, Nonlinear Anal. Hybrid Syst. 30 (2018) 106-117. [7] T. Zhang, F. Deng, W. Zhang, Finite-time stability and stabilization of linear discrete time-varying stochastic systems, J. Franklin. Inst. 356 (3) (2019) 1247-1267.

31

[8] M. Hua, D. Zheng, F. Deng, J. Fei, P. Cheng, X. Dai, H ∞ filtering for nonhomogeneous Markovian jump repeated scalar nonlinear systems with multiplicative noises and partially mode-dependent characterization, IEEE T. SYST. MAN CY-S. (2019) DOI: 10.1109/TSMC. 2019. 2919146. [9] S. Peng, F. Deng, New criteria on pth moment input-to-state stability of impulsive stochastic delayed differential systems, IEEE Trans. Autom. Control 62 (7) (2017) 3573-3579. [10] X. Li, S. Song, Stabilization of delay systems: delay-dependent impulsive control, IEEE Trans. Autom. Control 62 (1) (2017) 406-411. [11] X. Fu, Q. Zhu, Y. Guo, Stabilization of stochastic functional differential systems with delayed impulses, Appl. Math. Comput. 346 (1) (2019) 776-789. [12] Y. Zhang, Global exponential stability of delay difference equations with delayed impulses, Math. Comput. Simulation 132 (2017) 183-194. [13] Y. Wang, J. Zhang, L. Meng, Exponential stability of impulsive positive systems with mixed time-varying delays, IET Control Theory Appl. 8 (15) (2014) 1537-1542. [14] X. Li, J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Autom. Control 63 (1) (2018) 306-311. [15] M. Yang, Y. Wang, J. Xiao, Y. Huang, Robust synchronization of singular complex switched networks with parametric uncertainties and unknown coupling topologies via impulsive control, Nonlinear Sci Numer. Simul. 17 (11) (2012) 4404-4416. [16] X. Li, D. O’Regan, H. Akca, Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays, IMA J. Appl. Math. 80 (1) (2015) 85-99. [17] W. Chen, W. Zheng, Exponential stability of nonlinear time-delay systems with delayed impulse effects, Automatica 47 (5) (2011) 1075-1083.

32

[18] D. Li, P. Cheng, S. He, Exponential stability of hybrid stochastic functional differential systems with delayed impulsive effects: average impulsive interval approach, Math. Methods Appl. Sci. 40 (11) (2017) 41974210. [19] X. Yang, X. Li, Q. Xi, P. Duan, Review of stability and stabilization for impulsive delayed systems, Math. Biosci. Eng. 15 (6) (2018) 1495-1515. [20] Q. Zhu, pth moment exponential stability of impulsive stochastic functional differential equations with Markovian switching, J. Franklin Inst. 351 (7) (2014) 3965-3986. [21] B. Li, Stability of stochastic functional differential equations with impulses by an average approach, Nonlinear Anal. Hybrid Syst. 29 (2018) 221-233. [22] K. Wu, X. Ding, L. Wang, Stability and stabilization of impulsive stochastic delay difference equations, Discrete Dyn. Nat. Soc. (2010) Article ID 592036. [23] D. Li, P. Cheng, M. Hua, F. Yao, Robust exponential stability of uncertain impulsive stochastic neural networks with delayed impulses, J. Frankl. Inst. 355 (17) (2018) 8597-8618. [24] L. Ma, W. Zhang, Y. Zhao, Study on stability and stabilizability of discrete-time mean-field stochastic systems, J. Frankl. Inst. 356 (4) (2019) 2153-2171. [25] T. Hou, W. Zhang, B. S. Chen, Study on general stability and stabilizability of linear discrete time stochastic systems, Asian J. Control 13 (6) (2011) 977-987. [26] S. B. Stojanovic, Robust finite-time stability of discrete time systems with interval time-varying delay and nonlinear perturbations, J. Frankl. Inst. 354 (11) (2017) 4549-4572. [27] B. S. Razumikhin, On the stability of systems with a delay, Prikl. Mat. Mekh. 20 (4) (1956) 500-512. [28] B. S. Razumikhin, Applications of Lyapunov’s method to problems in the stability of systems with a delay, Automat. i Telemeh. 21 (1960) 740-749. 33

[29] X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Process. Appl. 65 (2) (1996) 233-250. [30] P. Cheng, F. Deng, F. Yao, Exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses, Commun. Nonlinear Sci Numer. Simul. 19 (6) (2014) 2104-2114. [31] Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with L´ evy noise and Markov switching, Int. J. Control Autom. 90 (8) (2017) 1703-1712. [32] S. Peng, Y. Zhang, Razumikhin-type theorems on pth moment exponential stability of impulsive stochastic delay differential equations, IEEE Trans. Autom. Control 55 (8) (2010) 1917-1922. [33] Y. Li, J. Lu, C. Kou, X. Mao, J. Pan, Robust discrete-state-feedback stabilization of hybrid stochastic systems with time-varying delay based on Razumikhin technique, Stat. Probab. Lett. 139 (2018) 152-161. [34] B. Liu, H. J. Marquez, Razumikhin-type stability theorems for discrete delay systems, Automatica 43 (7) (2007) 1219-1225. [35] F. Wu, X. Mao, P. E. Kloeden, Discrete Razumikhin-type technique and stability of the Euler-Maruyama method to stochastic functional differential equations, Discret Contin. Dyn. Syst. 33 (2) (2013) 885-903. [36] X. Zong, D. Lei, F. Wu, Discrete Razumikhin-type stability theorems for stochastic discrete-time delay systems, J. Frankl. Inst. 355 (17) (2018) 8245-8265. [37] B. Zhou, Improved Razumikhin and Krasovskii approaches for discretetime time-varying time-delay systems, Automatica 91 (2018) 256-269. [38] S. Boyd, L. Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theroy, Philadephia, PA: Siam, 1994. [39] M. Nigmatulin, M. Kipnis. Stability of the discrete population model with two dalays, International Conference on Physics and Control, IEEE, 2003, 313-315.

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[40] Y. Zhang, Exponential stability of impulsive discrete systems with time delays, Appl. Math. Lett. 25 (2012) 2290-2297.

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