Set-stabilization of discrete chaotic systems via impulsive control

Accepted Manuscript Set-stabilization of discrete chaotic systems via impulsive control Liguang Xu, Shuzhi Sam Ge PII: DOI: Reference:

S0893-9659(15)00290-6 http://dx.doi.org/10.1016/j.aml.2015.10.002 AML 4872

To appear in:

Applied Mathematics Letters

Received date: 18 August 2015 Revised date: 4 October 2015 Accepted date: 4 October 2015 Please cite this article as: L. Xu, S.S. Ge, Set-stabilization of discrete chaotic systems via impulsive control, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.10.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Manuscript Click here to view linked References

Set-stabilization of discrete chaotic systems via impulsive control∗ Liguang Xua ,† Shuzhi Sam Geb a b

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, PR China

Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576

Abstract In this letter, for the first time, set-stabilization is addressed for a class of discrete chaotic systems by using impulsive control. By using the Lyapunov stability theory and algebraic inequality techniques, some sufficient conditions for global exponential set-stability of the impulsive controlled discrete chaotic systems are obtained and the attracting set of the systems is also given. It is shown that not only a discrete chaotic system but also an unbounded discrete system can be successfully set-stabilized by impulses. The numerical simulation on the Lozi discrete chaotic system is presented to illustrate the effectiveness of the obtained results. Keywords: Chaotic systems; Impulsive control; Set-stabilization; Attracting set

1

Introduction

Over the past couple of decades, impulsive control has attracted more and more attention from many researchers because impulsive control may be simpler to implement, involve cheaper control mechanisms [1] and more efficient than other control methods (such as pinning control technique [2], adaptive control [3-6], state feedback control [7], anti-windup design [8], etc.) in many cases. Many interesting results about the stability, setstability and other behaviors of impulsive control systems, in which impulses occur in a continuous systems at some instances, have appeared in the literature [9-21]. On the other hand, the interest of researchers on the impulsive control of discrete-time dynamic systems, particularly discrete chaotic systems, has grown very fast and many interesting results have been reported in recent years, see [22-25] and references therein. More recently, Xu and Teo [26] investigated the asymptotical stabilization problem of discrete chaotic systems by using a novel unified impulsive control scheme and obtained some sufficient conditions for asymptotical stability of the impulsive controlled discrete chaotic systems by employing the Lyapunov stability theory. However, the equilibrium point sometimes does not exist in many real physical systems, especially in impulsive controlled chaotic systems. For this reason, it is more important for impulsive controlled chaotic systems to study set-stability than to study stability. However, to our knowledge, few work has been reported on the set-stability of impulsive controlled discrete chaotic system. Motivated by this lack, the present paper is focused on the set-stability of impulsive controlled discrete chaotic systems. By using the Lyapunov stability theory and algebraic inequality techniques, some sufficient conditions for global exponential set-stability are obtained. It is shown that not only a discrete chaotic system but also an unbounded discrete system can be successfully set-stabilized by impulses.

2

Model and preliminaries

Throughout this letter, unless otherwise specified, we use the following notations. Let Z+ = {0, 1, 2, ...}, N = {1, 2, 3, ...}. For x = (x1 , x2 , ..., xn )T ∈ Rn , A = (aij )n×n ∈ Rn×n , let Pnkxk be any vector norm, and kAk Pn be the induced matrix norm, ρ(A) be the spectral radius of A; let kxk1 = i=1 |xi | and kAk1 = max1≤j≤n i=1 |aij |. ∗ The

work is supported by National Natural Science Foundation of China (Grant No. 11501518) and the China Scholarship Council (Grant No. 201208330001). † Corresponding author. E-mail address: [email protected] (L. Xu).

1

Consider a general discrete chaotic system as follows: x(m + 1) = f (x(m)), m ∈ Z+ ,

(1)

where x(m) ∈ Rn represents the state variable, f : Rn → Rn is a continuous function and x(0) = x0 is the initial condition. Let the plant P describe the evolution process (m, x(m)), where x(m) ∈ Rn is the state variable of system (1). Definition 2.1. [27] Sequence {mk , U (k, x(mk ))}, mk ∈ Z+ is said to be an impulsive control law of P if ∆x(mk ) = x(mk + 1) − x(mk ) = U (k, x(mk )), k ∈ N, m0 = 0 < m1 < m2 < · · · < mk < mk+1 < · · · , lim mk = ∞. k→∞

Our main aim is to design an impulsive control law {mk , U (k, x(mk ))} such that system (1) under the control law is exponentially set-stable. Let U (k, x(mk )) = φk (x(mk )), φk : Rn → Rn is a continuous function. Then we obtain the general impulsive discrete chaotic system as follows:   x(m + 1) = f (x(m)), m 6= mk , m ∈ Z+ , ∆x(m) = x(m + 1) − x(m) = φk (x(m)), m = mk , k ∈ N, (2)  x(0) = x0 . Furthermore, we suppose that system (2) satisfies the following hypotheses.

Assumption 2.1. There exist constants s > 0 and ϑ ≥ 0 such that kf (x(m))k ≤ skx(m)k + ϑ, ∀m ∈ Z+ , ∀x ∈ Rn .

(3)

Assumption 2.2. There exist constants lk > 0 such that kx(m) + φk (x(m))k ≤ lk kx(m)k, ∀k ∈ N, ∀x ∈ Rn .

(4)

Assumption 2.3. The impulsive control instants {mk }, mk ∈ Z+ , are such that mk − mk−1 > 1, k ∈ N.

(5)

Remark 2.1. In the study of chaotic systems, Lipschitz condition is usually assumed in literature; see, for instance [22, 24, 25]. However, there is no harm in using Assumption 2.1 in this paper because Lipschitz condition implies Assumption 2.1. In fact, f (x) satisfies Lipschitz condition⇒ kf (x) − f (0)k ≤ skxk ⇒ kf (x)k ≤ skxk + kf (0)k. Remark 2.2. Assumption 2.2 is the most commonly used assumption for impulsive control systems; see, for instance, [16, 17, 26, 28]. Obviously, if φk (0) = 0, then Lipschitz condition is a sufficient condition for Assumption 2.2. Remark 2.3. In general, the jumping impulses cannot happen continuously. So Assumption 2.3 is usually assumed in literature; see, for instance, [22, 26, 29]. Definition 2.2. Set S ⊂ Rn is said to be a globally exponentially attracting set of system (2) if there exist constants K > 0 and β ∈ [0, 1) such that for any solution x(m, 0, x0 ) with the initial value x0 , dist(x(m), S) ≤ Kkx0 kβ k where k ∈ N, k → ∞ when m → ∞, and dist(x(m), S) = inf d(x(m), y), y∈S

where d(·, ·) is any distance in Rn . Definition 2.3. System (2) is said to be globally exponentially set-stable if there exists a globally exponentially attracting set S of system (2). Definition 2.4. System (2) is said to be globally exponentially stable if S = {0} is a globally exponentially attracting set of system (2).

2

3

Main results

In this section, we will derive some sufficient conditions of global exponential set-stability for the impulsive controlled discrete chaotic system (2) by employing the Lyapunov stability theory and algebraic inequality techniques. Theorem 3.1. Suppose that Assumptions 2.1 and 2.2 hold. Then the solution of system (2) satisfies the following estimate kx(m)k ≤ sm−k−1 Πki=0 li kx0 k + Υ,

(6)

where Υ=

m−m Xk −2

sι ϑ +

ι=0

k X j=1

Ãmj −mj−1 −2 ! X sm−mj −(k−j)−1 Πki=j li sι ϑ.

(7)

ι=0

Proof. For m ∈ (mk , mk+1 ], we have kx(m)k ≤sm−mk −1 kx(mk + 1)k + (1 + s + s2 + · · · + sm−mk −2 )ϑ Ãm−m −2 ! Xk m−mk −1 =s kx(mk ) + φk (x(mk ))k + sι ϑ ι=0

Ãm−m −2 ! Xk ≤sm−mk −1 lk kx(mk )k + sι ϑ.

(8)

ι=0

It is clear from (8) that kx(mk+1 )k ≤s

mk+1 −mk −1

Ãmk+1 −mk −2 ! X lk kx(mk )k + sι ϑ.

(9)

ι=0

By induction, we have kx(mk )k ≤ s

mk −k

Πk−1 i=0 li kx0 k

Ãmk −mk−1 −2 ! Ãmj −mj−1 −2 ! k−1 X X X ι mk −mj −(k−j) k−1 + s ϑ+ s Πi=j li sι ϑ. ι=0

(10)

ι=0

j=1

It follows from (8) and (10) that m−k−1

kx(m)k ≤s

Πki=0 li kx0 k

Ãm−m −2 ! Xk + sι ϑ ι=0

Ãmk −mk−1 −2 ! Ãmj −mj−1 −2 ! k−1 X X X m−mk −1 ι m−mj −(k−j)−1 k +s lk s ϑ+ s Πi=j li sι ϑ ι=0

ι=0

j=1

Ãm−m −2 ! Ãmj −mj−1 −2 ! k Xk X X m−k−1 k ι m−mj −(k−j)−1 k =s Πi=0 li kx0 k + s ϑ+ s Πi=j li sι ϑ. ι=0

The proof is completed.

j=1

(11)

ι=0

2

Theorem 3.2. Suppose that Assumptions 2.1 and 2.2 hold. Then the set ¯ ½ ¾ ¯ sT −1 − 1 S = x ∈ Rn ¯¯kxk ≤ ϑ (1 − sT −1 l)(s − 1)

is a globally exponentially attracting set of system (2), and system (2) is globally exponentially set-stable provided the following conditions hold: (i) s > 1 and T = supk∈N {mk − mk−1 } < ∞. (ii) sT −1 l < 1, l = supk∈N {lk }.

3

Proof. By (7), and s > 1 and sT −1 l < 1, we have   ÃT −2 ! ÃT −2 ! k k X X X X sT −1 − 1 ϑ Υ≤ sι ϑ + sm−mj −(k−j)−1 Πki=j li sι ϑ = 1 + sm−mj −(k−j)−1 Πki=j li  s−1 ι=0 ι=0 j=1 j=1     k k T −1 X X s − 1 sT −1 − 1 sm−mk −1 smk −mj −(k−j) lk−j+1  ≤ 1 + ϑ ≤ 1 + sT −1 sT (k−j)−k−j lk−j+1  ϑ s−1 s−1 j=1 j=1   µ ¶ T −1 k T −1 X −1 sT −1 l s −1 sT −1 − 1 T −1 k−j+1  s  (s l) = 1+ ϑ≤ 1+ ϑ = ϑ. (12) s−1 1 − sT −1 l s−1 (1 − sT −1 l)(s − 1) j=1

Since

m = (m − mk ) + (mk − mk−1 ) + · · · + (m1 − m0 ) ≤ m − mk + kT ≤ (k + 1)T,

(13)

we have m → ∞ =⇒ k → ∞ and sm−k−1 Πki=0 li ≤ s(k+1)T −k−1 lk+1 = (sT −1 l)k+1 .

(14)

sT −1 − 1 ϑ. (1 − sT −1 l)(s − 1)

(15)

It from (6), (12) and (14) that kx(m)k ≤(sT −1 l)k+1 kx0 k +

2

The proof is completed.

Theorem 3.3. Suppose that Assumptions 2.1 and 2.2 hold. Then the set ¯ ½ µ ¶ ¾ ¯ l 1 n¯ S = x ∈ R ¯kxk ≤ 1 + ϑ 1 − sT −1 l 1 − s

is a globally exponentially attracting set of system (2), and system (2) is globally exponentially set-stable provided the following conditions hold: (i) s < 1 and T = inf k∈N {mk − mk−1 } > 0. (ii) sT −1 l < 1. Proof. By (7), and s < 1 and sT −1 l < 1, we have   k k X X 1 1 1 sm−mj −(k−j)−1 Πki=j li sm−mk −1 smk −mj −(k−j) Πki=j li  Υ≤ ϑ+ ϑ = 1 + ϑ 1−s 1 − s 1 − s j=1 j=1     k k X X 1 1 ≤ 1 + sT (k−j)−(k−j) Πki=j li  ϑ ≤ 1 + (sT −1 l)k−j l ϑ 1 − s 1 − s j=1 j=1 µ ¶ l 1 ≤ 1+ ϑ T −1 1−s l 1−s

(16)

On the other hand, (m−mk )+

sm−k−1 Πki=0 li ≤ sm−k−1 lk+1 = s

k P

(mi −mi−1 )−k−1

i=1

lk+1 = s(m−mk )+kT −k−1 lk+1 ≤ (sT −1 l)k l.

(17)

It from (6), (16) and (17) that

The proof is completed.

2

µ kx(m)k ≤(sT −1 l)k lkx0 k + 1 +

l 1 − sT −1 l

¶

Theorem 3.4. Suppose that Assumptions 2.1 and 2.2 hold. Then the set ¯ ¾ ½ ¯ 1 (T − 1)ϑ S = x ∈ Rn ¯¯kxk ≤ 1−l 4

1 ϑ. 1−s

(18)

is a globally exponentially attracting set of system (2), and system (2) is globally exponentially set-stable provided the following conditions hold: (i) s = 1 and T = supk∈N {mk − mk−1 } < ∞. (ii) l < 1. Proof. By (7), and s = 1 and l < 1, we have Υ ≤ (m − mk − 1)ϑ +

k X j=1

Πki=j li (mj − mj−1 − 1)ϑ ≤ (1 +

k X j=1

Πki=j li )(T − 1)ϑ ≤

1 (T − 1)ϑ 1−l

(19)

It from (6) and (19) that kx(m)k ≤sm−k−1 Πki=0 li kx0 k + Υ ≤ lk kx0 k + The proof is completed.

2

1 (T − 1)ϑ. 1−l

(20)

For the case ϑ ≡ 0, we easily observe x(m) = 0 is a solution of (2) from Assumptions 2.1 and 2.2. In the following, we give the attractivity of the zero solution and the proof is similar to that of Theorems 3.2-3.4. Corollary 3.1. Suppose that Assumptions 2.1 and 2.2 with ϑ ≡ 0 hold. Then system (2) is globally exponentially stable provided one of the following conditions hold: (i) s > 1, T = supk∈N {mk − mk−1 } < ∞ and sT −1 l < 1. (ii) s < 1, T = inf k∈N {mk − mk−1 } > 0 and sT −1 l < 1. (iii) s = 1 and l < 1. Remark 3.1. Obviously, we can merge the conditions (i)-(iii) of Corollary 3.1, and we induce the following general result by Corollary 3.1. Corollary 3.2. Suppose that Assumptions 2.1 and 2.2 with ϑ ≡ 0 hold. If 0 < T = inf k∈N {mk − mk−1 } and T = supk∈N {mk −mk−1 } < ∞ and maxα∈{T,T } {sα−1 }l < 1, then system (2) is globally exponentially stable. Remark 3.2. Under the assumptions that φk (x) = φ(x), and mk + 2 ≤ mk+1 ≤ mk + 4, Xu and Teo [26] established the following Proposition: Proposition 3.1. ([26, Theorem 2]) Suppose that kx(m) + φ(x(m))k ≤ lkx(m)k, where 0 < l < e, and kf (x(m))k ≤ skx(m)k. Then, system (2) is asymptotically stable if ln s + 31 ln l < 0. Compared to Proposition 3.1 ([26, Theorem 2]), we see our results have wider range of application resulting from the advantage that we do not need the condition 0 < l < e, which can be shown in the following simple example. Example 3.1. Consider the following simple scalar impulsive discrete system:  1 x(m), m 6= 3k, m ∈ Z+ ,  x(m + 1) = 2e 2 ∆x(m) = (e − 1)x(m), m = 3k,  x(0) = x0 .

(21)

1 It is easy to check that the conditions of Corollary 3.2 are satisfied by taking T = T = 3, s = 2e , l = e2 . So system (21) is globally exponentially stable. However, Theorem 2 in [26] is invalid for system (21) since l = e2 > e.

Remark 3.3. It is worth mentioning that our results not only can be applied to set-stabilize the discrete chaotic system by utilizing impulsive control, but also can be applied to set-stabilize an unbounded discrete system by utilizing impulsive control. For example, consider the following simple discrete linear systems: µ ¶ 0.4 0.5 x(m + 1) = Ax(m), A = , (22) 0 1.1 the initial value is given as x(0) = (1, 1)T . Obviously, (22) is unbounded system since ρ(A) = 1.1 > 1, which is shown in Fig. 1. However, if we design the impulsive controller as {mk , U (k, x) = φk (x)}, where mk = 2k U (k, x(m)) = φk (x) = (−0.4)x. 5

(23)

Then, it follows from Corollary 3.1 that the trivial solution of (22) under this impulsive control is globally exponentially stable, which is shown in Fig. 2. Remark 3.4. In [28, 30], the authors have developed some interesting methods for investigating the stability and set-stability of impulsive discrete systems. However, these methods all effectiveless for studying the stability of the systems (22)-(23) since ρ(A) = 1.1 > 1. 14000

1.4

12000

1.2

x1(m)

x1(m)

x2(m)

8000

0.8

x2(m)

x(m)

1

x(m)

10000

6000

0.6

4000

0.4

2000

0.2

0

0

20

40

60

80

0

100

0

20

40

m

Figure 1:

4

60

80

100

m

Figure 2:

State trajectories of the impulses-free system (22).

State trajectories of the impulsive system (22)-(23).

Example

The following illustrative example will demonstrate the effectiveness of our results. Example 4.1. Consider the following generalized Lozi map: ½ x1 (m + 1) = −1.7|x1 (m)| + x2 (m) + 2, x2 (m + 1) = 0.4x1 (m) + 3,

(24)

the initial value is given as x(0) = (1, 1)T . It is well known that system (24) exhibits chaotic behaviors which are presented in Figs. 3-4. By some simple computations, we have kf (x(m))k1 ≤ 2.1kx(m)k1 + 5. So s = 2.1 > 1, and we can devise a simple linear impulsive controller as {mk , U (k, x) = φk (x)}, where mk = 2k and U (k, x) = φk (x) = (e−2 − 1)x.

(25)

We have T = 2, l = e−2 , and sT −1 l = 2.1e−2 = 0.2842 < 1. All the conditions of Theorem 3.2 are satisfied, therefore by Theorem 3.2, we obtain that system (24)-(25) is exponentially set-stable, and the set ¯ ½ ¾ ¯ (sT −1 − 1)ϑ 2¯ S = x ∈ R ¯kxk1 ≤ = 6.9852 (26) (1 − sT −1 l)(s − 1)

is a globally exponentially attracting set of system (24)-(25). Simulation results are shown in Figs. 5-6. From Figs. 5-6, it is shown that chaotic behaviors of system (24) had be removed law (25), and the trajectories of system (24)-(25) are all attracted by the ¯by impulsive control © ª set S = x ∈ R2 ¯kxk1 ≤ 6.9852 .

5

Conclusions

We have investigated the set-stabilization problem for a class of impulsive controlled discrete chaotic systems. Sufficient conditions for global exponential set-stability of the impulsive controlled discrete chaotic systems have been presented based on the Lyapunov stability theory and algebraic inequality techniques. The obtained results shown that not only a discrete chaotic system but also an unbounded discrete system can be successfully set-stabilized by impulses.

6

12

12

10

x1(m)

8

x2(m)

10

6

8

1

||x||

x(m)

4 2

6

0 4

−2 −4

2

−6 −8

0

100

200

300

400

0

500

0

100

200

m

Figure 3:

300

400

500

m

State trajectories of the impulses-free system (24).

Figure 4:

3.5

Behavior of kx(t)k1 of the impulses-free system (24).

12

3

10 x1(m)

2.5

x2(m)

8 6.9852 ||x||

1

x(m)

2

6

1.5 4 1 2

0.5

0

0

100

200

300

400

0

500

m

Figure 5:

State trajectories of the impulsive system (24)-(25).

0

100

200

300

400

500

m

Figure 6:

7

Behavior of kx(t)k1 of the impulsive system (24)-(25).

References [1] X. Cheng, Z. Guan, X. Liu, Decentralized impulsive control for a class of uncertain interconnected systems, J. Zhejiang Univ. Sci. 5 (2004) 274-282. [2] X. Li, X. Wang, G. Chen, Pinning a complex dynamical network to its equilibrium, IEEE Trans. Circuit Syst. I 51 (2004) 2074-2087. [3] J. Zhou, J. Lu, and J. L¨ u, Adaptive synchronization of an uncertain complex dynamical network, IEEE Trans. Autom. Control, 51 (2006) 652-656. [4] M. Rehan, K. Hong, LMI-based robust adaptive synchronization of FitzHugh-Nagumo neurons with unknown parameters under uncertain external electrical stimulation, Phys. Lett. A 375 (2011) 1666-1670. [5] M. Rehan, K. Hong, M. Aqil, Synchronization of multiple chaotic FitzHugh-Nagumo neurons with gap junctions under external electrical stimulation, Neurocomputing 74 (2011) 3296-3304. [6] M. Iqbal, M. Rehan, K. Hong, et al., Sector-condition-based results for adaptive control and synchronization of chaotic systems under input saturation, Chaos, Solitons & Fractals 77(2015) 158-169. [7] J. Wu, L. Jiao, Synchronization in dynamic networks with nonsymmetrical time-delay coupling based on linear feedback controller, Physica A 387 (2008) 2111-2119. [8] M. Rehan, K. Hong, Decoupled-architecture-based nonlinear anti-windup design for a class of nonlinear systems, Nonlinear Dyn. 73 (2013) 1955-1976. [9] D. Bainov, P. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, Ellis Horwood Limited, Chichester, 1989. [10] D. He, L. Xu, Ultimate boundedness of non-autonomous dynamical complex networks under impulsive control, IEEE Trans. Circuits Syst. II 62 (2015) 997-1001. [11] V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [12] X. Liu, Stability of impulsive control systems with time delay, Math. Comput. Modelling 39 (2004) 511-519. [13] X. Liu, Impulsive synchronization of chaotic systems subject to time delay, Nonlinear Anal. 71 (2009) e1320-e1327. [14] J. Sun, Impulsive control of a new chaotic system, Math. Comput. Simul. 64 (2004) 669-677. [15] S. Xie, Stability of sets of functional differential equations with impulse effect, Appl. Math. Comput. 218 (2011) 592-597. [16] D. Xu, Z. Yang, Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl. 305 (2005) 107-120. [17] L. Xu, D. Xu, Exponential stability of nonlinear impulsive neutral integro-differential equations, Nonlinear Anal. 69 (2008) 2910-2923. [18] L. Xu, S.S. Ge, The pth moment exponential ultimate boundedness of impulsive stochastic differential systems, Appl. Math. Lett. 42 (2015) 22-29. [19] Z. Yang, D. Xu, Stability analysis of delay neural networks with impulsive effects, IEEE Trans. Circuits Syst. II 52 (2005) 517-521. [20] Z. Yang, D. Xu, Stability Analysis and design of impulsive control systems with time delay, IEEE Trans. Automatic Control 52 (2007) 1448-1454. [21] Y. Zhang, Analysis and design of set-stability for impulsive control systems, Phys. Lett. A 370 (2007) 459-464. [22] Y. Gao, X. Zhang, G. Lu, Y. Zheng, Impulsive synchronization of discrete-time chaotic systems under communication constraints, Commun Nonlinear Sci Numer Simulat 16 (2011) 1580-1588. [23] Z. Guan, N. Liu, Generating chaos for discrete time-delayed systems via impulsive control, Chaos 20, 013135 (2010). [24] L. Zhang, H. Jiang, Q. Bi, Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems, Nonlinear Dyn. 59 (2010) 529-534. [25] L. Zhang, H. Jiang, Impulsive generalized synchronization for a class of nonlinear discrete chaotic systems, Commun Nonlinear Sci Numer Simulat 16 (2011) 2027-2032. [26] H. Xu, K. Teo, Stabilizability of discrete chaotic systems via unified impulsive control, Phys. Lett. A 374 (2009) 235-240. [27] T. Yang, Impulsive Systems and Control: Theory and Application. New York: Nova Science, 2001. [28] W. Zhu, D. Xu, Z. Yang, Global exponential stability of impulsive delay difference equation, Appl. Math. Comput. 181 (2006) 65-72. [29] B. Liu, X. Liu, Robust stability of uncertain discrete impulsive systems, IEEE Trans. Circuits Syst. II 54 (2007) 455-459. [30] W. Zhu, Invariant and attracting sets of impulsive delay difference equations with continuous variables, Comput. Math. Appl. 55 (2008) 2732-2739.

8