Impulsive Control for the Synchronization of Chaotic Systems with Time Delay

Abstract

This paper considers impulsive control for the synchronization of chaotic systems with time delays. Based on the Lyapunov functions and the Razumikhin technique, some new synchronization criteria with an exponential convergence rate are derived. Our results show that impulses do contribute to globally exponential synchronization of dynamical systems. Besides, the impulsive moments are independent of the upper bound of time delays. Furthermore, a bigger upper bound of impulsive intervals for the synchronization of chaotic systems can be obtained when compared with many previous studies. Hence, our results are less conservative and more effective for the synchronization analysis. A numerical example is given to show the validity and potential of the developed results.

1. Introduction

In the last two decades, control and synchronization problems of chaotic systems have been extensively studied, due to their potential applications in many areas. For instance, they are used to understand self-organizational behavior in the brain as well as in ecological systems, and they also have been applied to produce secure message communication between a sender and a receiver. So far, different synchronization techniques have been proposed and implemented in practice, such as PC (Pecora and Carroll) method [1], active control [2], adaptive control [3, 4], sliding mode control [5], and impulsive method [6–12].

Impulsive phenomena exist in many biological systems and mechanics fields. Impulsive control has been widely studied and gradually become an interesting and useful synchronization approach [13–16]. Impulsive control can provide an efficient method for some cases in which the systems cannot endure continuous disturbance. The main idea of the impulsive control strategy is to change the states of a system by some sudden jumps instantaneously. Furthermore, using the impulsive control method, the response system needs to receive the information from the drive system only at some discrete instants which means that impulsive control is easier to be implemented to some extent. The asymptotical synchronization of chaotic systems without time delay [17–20] and with time delay [21, 22] has been widely investigated by using impulsive control. However, most of the previous impulsive stability criteria are only valid for some specific systems with small delays due to the restrictive requirement that the time delays are required to be smaller than the length of impulsive interval [23–25].

In this paper, we will investigate globally exponential synchronization of coupled chaotic systems with time delay by using impulsive control. The main contributions of this paper include the following. (i) Our results show that impulses do contribute to globally exponential synchronization of dynamical systems and the time delays in systems not to be smaller than the length of impulsive intervals. Therefore, they can be usually used as an effective control strategy to synchronize the underlying delayed dynamical systems in more practical application. (ii) A new approach for the exponential synchronization of impulsive chaotic systems with any finite delay is given. (iii) In order to deal with uncertainty and/or measurement noise effectively in nominal case for identical chaotic and hyperchaotic systems, the impulse distance should increases which in turn decreases the control cost in accordance with [19]. Thus, having a minimum level of synchronization error with the largest impulsive intervals is generally desired [26]. In our proposed results, a bigger upper bound of impulsive intervals for the synchronization of chaotic systems can be obtained by comparing with the results in [13–16]. It should also be noticed that our paper was inspired in part by the work of Zhou and Wu in [27] for delayed linear differential equations.

The rest of this paper is organized as follows. In Section 2, the problem and some preliminaries are presented. The main result on exponential synchronization is given in Section 3. Section 4 gives an example for illustration, and some conclusions are finally drawn in Section 5.

2. Preliminaries

Denote that 𝑅 is the set of real numbers, 𝑅⁺ is the set of nonnegative real numbers, 𝑅^𝑛 is the 𝑛-dimensional real space, and 𝑁 is the set of positive integers. Let Ψ(𝑡⁺)=lim_𝑠→𝑡+Ψ(𝑠) and Ψ(𝑡⁻)=lim_{𝑠→𝑡−}Ψ(𝑠). For 𝑎,𝑏∈𝑅 with 𝑎<𝑏 and for 𝑆⊂𝑅^𝑛, we define the following function: PC([𝑎,𝑏],𝑆)={Ψ∶[𝑎,𝑏]→𝑆∣Ψ(𝑡)=Ψ(𝑡⁺), for all 𝑡∈[𝑎,𝑏);Ψ(𝑡⁻)existsin𝑆, for all 𝑡∈(𝑎,𝑏] and Ψ(𝑡⁻)=Ψ(𝑡), for all but at most a finite number of points 𝑡∈(𝑎,𝑏]}. For Ψ∈PC([−𝜏,0],𝑅^𝑛), the norm of Ψ is, respectively, defined by ‖Ψ‖=sup_{−𝜏≤𝑠≤0}‖Ψ(𝑠)‖.𝑒_𝑡,𝑒_𝑡−∈PC([−𝜏,0],𝑅^𝑛) are defined by 𝑒_𝑡(𝑠)=𝑒(𝑡+𝑠) and 𝑒_𝑡−(𝑠)=𝑒(𝑡⁻+𝑠) for 𝑠∈[−𝜏,0].

Now we introduce the coupled chaotic systems that will be studied. It usually consists of two chaotic systems at the transmitter and the receiver ends.

At the transmitter end, we have 𝑥̇(𝑡)=𝐴𝑥(𝑡)+Φ₁(𝑥(𝑡)), (2.1) and, at the receiver end, we have 𝑦̇(𝑡)=𝐴𝑦(𝑡)+Φ₂𝑦(𝑡),𝑥(𝑡−𝜏),𝑦(𝑡−𝜏),𝑡≠𝑡_𝑘,Δ𝑦(𝑡)=−𝐵_𝑘𝑒(𝑡),𝑡=𝑡_𝑘,𝑦𝑡₀−𝑠=𝜑(𝑠),𝑠∈[0,𝜏], (2.2) where the matrices 𝐴,𝐵_𝑘∈𝑅^𝑛×𝑛,  𝑒(𝑡)=𝑥(𝑡)−𝑦(𝑡) are the synchronization error states between the states of system (2.1) and system (2.2), and the functions Φ₁,Φ₂ are the continuous functions in their respective domain of definition. The time sequence {𝑡_𝑘}^+∞_𝑘=1 satisfies 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯,  lim_𝑘→∞𝑡_𝑘=+∞, the time delay 𝜏>0, and Δ𝑦(𝑡_𝑘)=𝑦(𝑡⁺_𝑘)−𝑦(𝑡⁻_𝑘), in which 𝑦(𝑡⁺_𝑘)=lim_𝑠→0⁺𝑦(𝑡_𝑘+𝑠),𝑦(𝑡⁻_𝑘)=lim_𝑠→0⁻𝑦(𝑡_𝑘+𝑠) for 𝑘=1,2,…. A typical form of the function Φ₂ is given as follows: Φ₂𝑦(𝑡),𝑥(𝑡−𝜏),𝑦(𝑡−𝜏)=Φ₁𝑦(𝑡)+𝐹𝐾𝑥(𝑡−𝜏)−𝑦(𝑡−𝜏), (2.3) where 𝐹∈𝑅^𝑛×𝑚,𝐾∈𝑅^𝑚×𝑛 are constant matrices. The second term is known as the delayed feedback controller, which is applied to the input state and then influences the system function.

Then, we can get the following error dynamical system: 𝑒̇(𝑡)=𝐴𝑒(𝑡)+Ψ𝑥(𝑡),𝑦(𝑡),𝑥(𝑡−𝜏),𝑦(𝑡−𝜏),𝑡≠𝑡_𝑘,Δ𝑒(𝑡)=𝐵_𝑘𝑒(𝑡),𝑡=𝑡_𝑘,𝑘=1,2,…, (2.4) where Ψ(𝑥(𝑡),𝑦(𝑡),𝑥(𝑡−𝜏),𝑦(𝑡−𝜏))=Φ₁(𝑥(𝑡))−Φ₂(𝑦(𝑡),𝑥(𝑡−𝜏),𝑦(𝑡−𝜏)).

Obviously, 𝑒(𝑡)=0 is a trivial solution of system (2.4). We shall analyze the dynamics of system (2.4) and drive criteria under which its trivial solution is globally exponentially stable. It is clear that the globally and exponential stability of trivial solution of (2.4) implies the global synchronization of systems (2.1) and (2.2).

For a given 𝑡>𝑡₀ and 𝜑∈PC([−𝜏,0],𝑅^𝑛), the initial value problem of (2.4) is 𝑒̇(𝑡)=𝐴𝑒(𝑡)+Ψ𝑥(𝑡),𝑦(𝑡),𝑥(𝑡−𝜏),𝑦(𝑡−𝜏),𝑡≠𝑡_𝑘,Δ𝑒(𝑡)=𝐵_𝑘𝑒(𝑡),𝑡=𝑡_𝑘𝑒𝑡,𝑡∈𝑁₀−𝑠=𝜑(𝑠),𝑠∈(0,𝜏). (2.5)

We assume that (2.4) has a unique solution with respect to initial conditions. Denote by 𝑒(𝑡)=𝑒(𝑡,𝑡₀,𝜑) the solution of (2.4) such that 𝑒(𝑡₀−𝑠)=𝜑(𝑠),  𝑠∈(0,𝜏). Also we assume that the function Ψ satisfies the following assumption: for certain positive definite matrix 𝑃, there exist constant matrices 𝐷_𝑖∈𝑅^𝑛×𝑛,  𝑖=1,2, such that 𝑥−𝑦^𝑇𝑃Ψ𝑥,𝑦,𝑢,𝑣≤𝑥−𝑦^𝑇𝑃𝐷₁𝑥−𝑦+𝑥−𝑦^𝑇𝑃𝐷₂(𝑢−𝑣). (2.6) Now we have the following definitions.

Definition 2.1. The trivial solution of (2.4) is said to be globally and exponentially stable if for any initial data 𝑒_𝑡0=𝜑∈PC([−𝜏,0],𝑅^𝑛) and ‖Ψ‖_𝜏=sup_{−𝜏≤𝑠≤0}‖Ψ(𝑠)‖, there exist some constants 𝜆>0 and 𝑀≥1 such that 𝑒𝑡,𝑡₀,𝜑≤𝑀𝜑_𝜏𝑒^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀. (2.7)

Definition 2.2. Given a function 𝑉∶𝑅⁺×𝑅^𝑛→𝑅⁺, the upper right-hand derivative of 𝑉 with respect to system (2.4) is defined by 𝐷⁺𝑉(𝑡,𝑒(𝑡))=lim_𝛿→0+1sup𝛿[𝑉(𝑡+𝛿,𝑒(𝑡+𝛿))−𝑉(𝑡,𝑒(𝑡))]. (2.8)

3. Synchronization of Chaotic Systems

In the following, we shall address the exponential stability problem for impulsive delayed nonlinear differential equation (2.4), which implies the global synchronization of two chaotic systems. Our result shows that impulses play an important role in making the delayed nonlinear differential equations globally and exponentially stable.

Theorem 3.1. Let 𝑃∈𝑅^𝑛×𝑛 be a positive definite matrix and 𝜆_𝑘(0<𝜆_𝑘≤1,𝑘∈𝑁) be largest eigenvalue of 𝑃⁻¹(𝐼+𝐵_𝑘)^𝑇𝑃(𝐼+𝐵_𝑘). Assume that there exist constants 𝛼_𝑖,  𝑖=1,2 with 𝛼₂∈𝑅⁺ and 𝜆>0,  𝜎>0, such that for all 𝑘∈𝑁, one has(i)𝐴^𝑇𝑃+𝑃𝐴+𝑃𝐷₁−𝛼₁𝑃𝑃𝐷₂𝐷^𝑇₂𝑃−𝛼₂𝑃≤0, (3.1)(ii)𝜎−𝜆−𝛼₁+𝛼₂𝜆_𝑘−1𝑒^𝜆𝜏≥0, (3.2) where 0<𝜆₀≤1,(iii)ln𝜆_𝑘−1<−(𝜎+𝜆)𝑡_𝑘−𝑡_𝑘−1. (3.3)Then, the trivial solution of system (2.4) is globally and exponentially stable with a convergence rate of 𝜆/2 for any fixed delay 𝜏∈(0,+∞), that is, system (2.1) and system (2.2) are globally and exponentially synchronized.

Proof. Let 𝑒(𝑡)=𝑒(𝑡,𝑡₀,𝜑) be any solution of system (2.5), and consider a Lyapunov function as follows: 𝑉(𝑡,𝑒(𝑡))=𝑒^𝑇(𝑡)𝑃𝑒(𝑡). (3.4) Let 𝜇₁>0 and 𝜇₂>0 be the smallest and largest eigenvalues of positive definite matrix 𝑃, respectively. Then, we can obtain 𝜇₁‖𝑒(𝑡)‖²≤𝑉(𝑡,𝑒(𝑡))≤𝜇₂‖𝑒(𝑡)‖². (3.5) Now, we are in a position to prove that, for any 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑉(𝑡,𝑒(𝑡))≤𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)},𝑘∈𝑁. (3.6) For 𝑡≠𝑡_𝑘(𝑘∈𝑁), it follows from assumption (2.6) and condition (i) that 𝐷⁺𝑉(𝑡,𝑒(𝑡))=𝑒^𝑇̇𝑃𝑒(𝑡)+𝑒^𝑇(𝑡)𝑃𝑒̇(𝑡)=𝑒^𝑇(𝑡)𝐴^𝑇+Ψ^𝑇𝑃𝑒(𝑡)+𝑒^𝑇(𝑡)𝑃[𝐴𝑒(𝑡)+Ψ]=𝑒^𝑇𝐴^𝑇𝑃+𝑃𝐴𝑒(𝑡)+𝑒^𝑇(𝑡)𝑃Ψ+Ψ^𝑇𝑃𝑒(𝑡)≤𝑒^𝑇(𝑡)𝐴^𝑇𝑃+𝑃𝐴+2𝑃𝐷₁𝑒(𝑡)+𝑒^𝑇(𝑡)𝑃𝐷₂×𝑒(𝑡−𝜏)+𝑒^𝑇(𝑡−𝜏)𝐷^𝑇₂(𝑡)=𝑒𝑃𝑒_𝑇(𝑡)𝑒^𝑇()𝐴𝑡−𝜏^𝑇𝑃+𝑃𝐴+2𝑃𝐷₁𝑃𝐷₂𝐷^𝑇₂𝑒(𝑡)𝑒().𝑃0𝑡−𝜏 (3.7) It further follows from condition (i) that 𝐷⁺𝑉(𝑡,𝑒(𝑡))≤𝑒^𝑇(𝑡)𝑒^𝑇()𝛼𝑡−𝜏¹𝑃00𝛼₂𝑃𝑒(𝑡)𝑒(𝑡−𝜏)=𝛼₁𝑉(𝑡,𝑒(𝑡))+𝛼₂𝑉(𝑡−𝜏,𝑒(𝑡−𝜏)). (3.8) Defining 𝛾=sup_𝑘∈𝑁(1/𝜆_𝑘−1)≥1 in condition (iii), we have ln𝛾+𝜆𝜏−(𝜎+𝜆)𝑡_𝑘−𝑡_𝑘−1>0. (3.9) Then, in view of the inequality in (3.9), we can find constant 𝑀≥1 such that 1<𝑒^{(𝜎+𝜆)(𝑡1−𝑡0)}≤𝑀≤𝛾𝑒^{𝜆𝜏−(𝜎+𝜆)(𝑡1−𝑡0)}𝑒^{(𝜎+𝜆)(𝑡1−𝑡0)}, (3.10) which shows that 𝜑²_𝜏<𝜑²_𝜏𝑒^{𝜎(𝑡1−𝑡0)}≤𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡1−𝑡0)}. (3.11) Firstly, we prove that 𝑉(𝑡,𝑒(𝑡))≤𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)},𝑡∈𝑡₀,𝑡₁. (3.12) To do this, we only need to prove that 𝑉(𝑡,𝑒(𝑡))≤𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡1−𝑡0)},𝑡∈𝑡₀,𝑡₁. (3.13) If the inequality (3.13) is not true, it follows from (3.11) that there must exist some 𝑡∈[𝑡₀,𝑡₁) such that 𝑉𝑡,𝑒𝑡>𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡1−𝑡0)}≥𝜇₂𝜑²_𝜏𝑒^{𝜎(𝑡1−𝑡0)}>𝜇₂𝜑²_𝜏≥𝑉𝑡₀+𝑠,𝑒𝑡₀+𝑠,𝑠∈[−𝜏,0], (3.14) which implies that there exists 𝑡^∗∈[𝑡₀,𝑡) such that 𝑉𝑡^∗,𝑒𝑡^∗=𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡1−𝑡0)},𝑉(𝑡,𝑒(𝑡))≤𝑉𝑡^∗,𝑒𝑡^∗,𝑡∈𝑡₀−𝜏,𝑡^∗, (3.15) and there exists 𝑡^∗∗∈[𝑡₀,𝑡^∗) such that 𝑉𝑡^∗∗,𝑒𝑡^∗∗=𝜆₂𝜑²_𝜏,𝑉𝑡^∗∗,𝑒𝑡^∗∗≤𝑉(𝑡,𝑒(𝑡)),𝑡∈𝑡^∗∗,𝑡^∗. (3.16) Hence, for any 𝑠∈[−𝜏,0], we can obtain that 𝑉(𝑡+𝑠,𝑒(𝑡+𝑠))≤𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡1−𝑡0)}≤𝜇₂𝛾𝑒^{𝜆𝜏−(𝜎+𝜆)(𝑡1−𝑡0)}𝑒^{(𝜎+𝜆)(𝑡1−𝑡0)}𝜑²_𝜏=𝛾𝑒^𝜆𝜏𝜇₂𝜑²_𝜏=𝛾𝑒^𝜆𝜏𝑉𝑡^∗∗,𝑒𝑡^∗∗≤𝛾𝑒^𝜆𝜏𝑉(𝑡,𝑒(𝑡)),𝑡∈𝑡^∗∗,𝑡^∗. (3.17) By condition (ii), (3.8), and (3.17), we can obtain 𝐷⁺𝑉(𝑡,𝑒(𝑡))≤𝛼₁+𝛾𝛼₂𝑒^𝜆𝜏𝑉((𝑡≤𝛼𝑡,𝑒))₁+𝛼₂𝜆₀𝑒^𝜆𝜏𝑉((𝑡≤()𝑉((𝑡𝑡𝑡,𝑒))𝜎−𝜆𝑡,𝑒)),𝑡∈^∗∗,𝑡^∗, (3.18) which shows that 𝑉𝑡^∗,𝑒𝑡^∗≤𝑉𝑡^∗∗,𝑒𝑡^∗∗𝑒^{(𝜎−𝜆)(𝑡∗−𝑡∗∗)}=𝜇₂𝜑²_𝜏𝑒^{(𝜎−𝜆)(𝑡∗−𝑡∗∗)}<𝜇₂𝜑²_𝜏𝑒^{𝜎(𝑡1−𝑡0)}≤𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡1−𝑡0)}=𝑉𝑡^∗,𝑒𝑡^∗. (3.19) It is obviously a contradiction. Hence, (3.12) holds, and then (3.6) is true for 𝑘=1. Now, we suppose that (3.6) holds for 𝑘=1,2,…,𝑚  (𝑚∈𝑁,𝑚≥1), that is, 𝑉(𝑡,𝑒(𝑡))≤𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)},𝑡∈𝑡_𝑘−1,𝑡_𝑘. (3.20) Next, we will prove that (3.6) holds for 𝑘=𝑚+1, that is, 𝑉(𝑡,𝑒(𝑡))≤𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)},𝑡∈𝑡_𝑚,𝑡_𝑚+1. (3.21) For the sake of contradiction, we suppose that (3.21) is not true. Then, we define 𝑡=inf𝑡∈𝑡_𝑚,𝑡_𝑚+1∣𝑉(𝑡,𝑒(𝑡))>𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}. (3.22) From condition (iii) and (3.20), we can get 𝑉𝑡_𝑚,𝑒𝑡_𝑚=𝑒^𝑇𝑡⁻_𝑚𝐼+𝐵_𝑚^𝑇𝑃𝐼+𝐵_𝑚𝑒𝑡⁻_𝑚≤𝜆_𝑚𝑉𝑡⁻_𝑚,𝑒𝑡⁻_𝑚≤𝜆_𝑚𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡𝑚−𝑡0)}=𝜆_𝑚𝜇₂𝑀𝜑²_𝜏𝑒^{𝜆(𝑡−𝑡𝑚)}𝑒^{−𝜆(𝑡−𝑡0)}<𝜆_𝑚𝜇₂𝑒^{𝜆(𝑡𝑚+1−𝑡𝑚)}𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}<𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}, (3.23)
and so 𝑡≠𝑡_𝑚. By the continuity of 𝑉(𝑡,𝑒(𝑡)) in the interval [𝑡_𝑚,𝑡_𝑚+1), we obtain 𝑉𝑡^∗,𝑒𝑡^∗=𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)},𝑉(𝑡,𝑒(𝑡))≤𝑉𝑡^∗,𝑒𝑡^∗,𝑡∈𝑡_𝑚,𝑡^∗. (3.24) From (3.23), we can derive that there exists 𝑡^∗∗∈[𝑡_𝑚,𝑡^∗) such that 𝑉𝑡^∗∗,𝑒𝑡^∗∗=𝜆_𝑚𝜇₂𝑒^{𝜆(𝑡𝑚+1−𝑡𝑚)}𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)},𝑉𝑡^∗∗,𝑒𝑡^∗∗≤𝑉(𝑡,𝑒(𝑡))≤𝑉𝑡^∗,𝑒𝑡^∗,𝑡∈𝑡^∗∗,𝑡^∗. (3.25)
As for any 𝑡∈[𝑡^∗∗,𝑡^∗],  𝑠∈[−𝜏,0], then either 𝑡+𝑠∈[𝑡₀−𝜏,𝑡_𝑚) or 𝑡+𝑠∈[𝑡_𝑚,𝑡^∗]. These two cases will be discussed in the following.
If 𝑡+𝑠∈[𝑡₀−𝜏,𝑡_𝑚), from (3.20), we have 𝑉(𝑡+𝑠,𝑒(𝑡+𝑠))≤𝜆₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}𝑒^−𝜆𝑠≤𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}𝑒^{𝜆(𝑡−𝑡)}𝑒^𝜆𝜏≤𝜇₂𝑒^𝜆𝜏𝑒^{𝜆(𝑡𝑚+1−𝑡𝑚)}𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}, (3.26) whereas if 𝑡+𝑠∈[𝑡_𝑚,𝑡^∗], from (3.24), then 𝑉(𝑡+𝑠,𝑒(𝑡+𝑠))≤𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}≤𝜇₂𝑒^𝜆𝜏𝑒^{𝜆(𝑡𝑚+1−𝑡𝑚)}𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}. (3.27) Above all, from (3.25)–(3.27), we can obtain, for any 𝑠∈[−𝜏,0], 𝑉((=𝑒𝑡+𝑠,𝑒𝑡+𝑠))^𝜆𝜏𝜆_𝑚𝑉𝑡^∗∗,𝑒𝑡^∗∗≤𝑒^𝜆𝜏𝜆_𝑚𝑉(𝑡,𝑒(𝑡)),𝑡∈𝑡^∗∗,𝑡^∗. (3.28) Hence, by condition (ii), (3.8), and (3.28), we can conclude that 𝐷⁺𝑉(𝑡,𝑒(𝑡))≤𝛼₁+𝛼₂𝜆_𝑚𝑒^𝜆𝜏𝑉((𝑡≤()𝑉((𝑡.𝑡,𝑒))𝜎−𝜆𝑡,𝑒)) (3.29) Thus, in view of condition (iii), we have 𝑉𝑡^∗,𝑒^∗≤𝑉𝑡^∗∗,𝑒^∗∗𝑒^{(𝜎−𝜆)(𝑡∗−𝑡∗∗)}=𝜆_𝑚𝜇₂𝑒^{𝜆(𝑡𝑚+1−𝑡𝑚)}𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}𝑒^{(𝜎−𝜆)(𝑡∗−𝑡∗∗)}<𝜇₂𝑒^{−(𝜎+𝜆)(𝑡𝑚+1−𝑡𝑚)}𝑒^{𝜆(𝑡𝑚+1−𝑡𝑚)}×𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}𝑒^{(𝜎−𝜆)(𝑡∗−𝑡∗∗)}=𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜎(𝑡𝑚+1−𝑡𝑚)}𝑒^{(𝜎−𝜆)(𝑡∗−𝑡∗∗)}𝑒^{−𝜆(𝑡−𝑡0)}<𝜇₂𝑀𝜑²_𝜏𝑒^{−𝜆(𝑡−𝑡0)}=𝑉𝑡^∗,𝑒^∗, (3.30) which is a contradiction. It implies that the supposition is not true. Hence, (3.6) holds for 𝑘=𝑚+1, then, we can conclude by some induction that (3.6) holds for any 𝑘∈𝑁. It immediately follows from (3.6) that ‖𝑒(𝑡)‖≤𝜇₂𝜇₁𝑀^1/2𝜑_𝜏𝑒^{−(𝜆/2)(𝑡−𝑡0)},𝑡≥𝑡₀, (3.31) that is, the trivial solution of the impulsive delayed system (2.5) is globally and exponentially stable with a convergence rate of 𝜆/2 for any fixed delays 𝜏∈(0,∞). Then, it implies that system (2.1) and system (2.2) are globally synchronized. The proof is thus complete.

Remark 3.2. In LMI (3.1) of Theorem 3.1, the constant 𝛼₁(if𝛼₁>0) is used to measure the level of instability for delay-free system and is determined by the matrix 𝐴+𝑃𝐷₁, while 𝛼₂ is decided by the matrix 𝐷₂.

Remark 3.3. Compared with Theorem 3.1 in [28], a distinct feature of Theorem 3.1 in this paper is to eliminate the restriction that the impulsive interval can not be too large, that is, the additional assumption (iv) 𝑡_𝑘−𝑡_𝑘−1≥𝜏 in Theorem 3.1 of [28] is indeed deleted here. Moreover, a controlled parameter 𝜎 is introduced to adjust the degree of convergence rate 𝜆/2 on globally exponential stability of the error system (2.4). It should be mentioned that our results allow us to develop an effective impulse control strategy to exponentially synchronize chaotic systems, and it is particularly meaningful for some practical applications.

Similarly to Theorem 3.1, we can obtain the following result.

Corollary 3.4. Let 𝑃∈𝑅^𝑛×𝑛 be a symmetric and positive definite matrix, 0<𝛽=sup_𝑘∈𝑁𝜆_𝑘, where 𝜆_𝑘 is given by Theorem 3.1 for each 𝑘∈𝑁. Assume that there exist constants 𝛼_𝑖,  𝑖=1,2 with 𝛼₂∈𝑅⁺, such that for all 𝑘∈𝑁 the following are satisfied:(i)𝐴^𝑇𝑃+𝑃𝐴+2𝑃𝐷₁−𝛼₁𝑃𝑃𝐷₂𝐷^𝑇₂𝑃−𝛼₂𝑃≤0, (3.32)(ii)𝛼₁+𝛼₂𝛽𝑡_𝑘−𝑡_𝑘−1<−ln𝛽. (3.33)
Then, the trivial solution of system (2.4) is globally and exponentially stable for any fixed delay 𝜏∈(0,+∞), that is, system (2.1) and system (2.2) are globally and exponentially synchronized.

Proof. Condition (ii) in Theorem 3.1 can be reduced to the following form: 𝜎−𝜆+𝛼₁+𝛼₂𝛽𝑒^𝜆𝜏≥0. (3.34) Then, it is easy to get the following inequality, which is equivalent to (3.34) and condition (iii) in Theorem 3.1: 2𝜆+𝛼₁+𝛼₂𝛽𝑒^𝜆𝜏𝑡_𝑘−𝑡_𝑘−1<−ln𝛽. (3.35) Hence, there must exist a small-enough real number 𝜆>0 such that (3.35) holds. And if 𝜆 is extremely tiny, we can obtain that condition (ii) in Corollary 3.4 holds. From Theorem 3.1, the proof is completed.

Remark 3.5. As Corollary 3.4 can be applied to deal with globally exponential stability of impulsive differential equations with any time delays 𝜏∈(0,+∞), it is obviously more applicable than those existing in the works in the recent literature [23–25, 28] where the time delays need to be assumed not bigger than the length of impulsive interval. In view of Corollary 3.4, it is clear that we have removed the restriction of time delay indeed.

Remark 3.6. Since the amount of transmitted information decreases leading to reduced control cost, the cost of impulsive synchronization of chaotic systems is closely related to the impulse distances and having a minimum level of synchronization error with the largest impulsive intervals as generally a desired [26]. From the proposed result in Corollary 3.4, the upper bound of impulsive intervals for the globally and exponentially synchronization can be given by 𝑡_𝑘−𝑡_𝑘−1<−ln𝛽/(𝛼₁+𝛼₂/𝛽) for each 𝑘∈𝑁. From the following numerical example, we will show that our upper bound of impulsive intervals is bigger than some existing results.

4. Numerical Simulation

In this section, an example is presented here to illustrate our main results.

Example 4.1. Consider the Lorenz system in [13] as follows: 𝑥̇₁=−𝜎𝑥₁+𝜎𝑥₂,𝑥̇₂=𝑟𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥̇₃=𝑥₁𝑥₂−𝑏𝑥₃, (4.1) where 𝜎,𝑟,𝑏∈𝑅⁺,  𝜎=10,  𝑟=28, and 𝑏=8/3. Let 𝑥=[𝑥₁,𝑥₂,𝑥₃]^𝑇, then we can rewrite the Lorenz system in the form of (2.1) as follows: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+Φ₁(𝑥(𝑡)), (4.2) where 8𝐴=−1010028−1000−3,Φ₁=0−𝑥₁𝑥₃𝑥₁𝑥₂. (4.3) We also choose Φ₂ defined by (2.3) with 𝐹𝐾=𝐼. Set the delay 𝜏=1 and the impulsive matrices 𝐵_𝑘=−0.5𝐼 for all 𝑘=1,2,…. Then, it can be observed that inequality (2.6) holds with 𝐷₁=𝐼and𝐷₂=−𝐹𝐾=−𝐼.
Solving (3.1) in Theorem 3.1, we get that 𝛼₁=31,  𝛼₂=1, and 𝑃=𝐼. Since 𝐵_𝑘=−0.5𝐼, we can get 𝛽=0.25. According to condition (ii) in Corollary 3.4, when 𝑡_𝑘−𝑡_𝑘−1<−ln𝛽𝛼₁+𝛼₂/𝛽=0.0396, (4.4) the trivial solution of error system based on impulsive delayed differential equation (4.1) is globally exponentially stable for any fixed delay 𝜏∈(0,+∞). With the same parameter, a comparison of the upper bound of impulsive intervals with [13–16] is presented in Table 1, which shows that our upper bound is much bigger. Choose 𝑡_𝑘−𝑡_𝑘−1=0.035, and the simulation result of error system 𝑒(𝑡) is shown in Figure 1.

Table 1 

A comparison of the upper bound of impulsive intervals of (4.1).

Figure 1 

The trajectories of the error Lorenz system under impulsive controller with 𝑡𝑘−𝑡𝑘−1=0.035.

5. Conclusion

In this paper, we have investigated the synchronization of coupled chaotic systems with time delay by using impulsive control. The time delays in systems need not to be smaller than the length of impulsive interval as many works existing in the literature assumed. A numerical example has been given to demonstrate the effectiveness of the theoretical results, and the estimation of the stable region of the impulsive intervals has also been presented. By comparison, the upper bound of impulsive intervals is greater than it was in some existing results. The obtained results can be easily used to control many systems, especially to stabilize and synchronize chaotic systems.

Acknowledgments

The authors would like to take this opportunity to thank Professor. Jinde Cao and the reviewers for their constructive comments and useful suggestions. This work was partially supported by the NNSF of China (Grants nos. 61175119, 11101373, 11271333, and 61074011), the NSF of Jiangsu Province of China (Grant no. BK2010408), Huo Ying-Dong Education Foundation (Grant no. 132037), and Zhejiang Innovation Project, China (Grant no. T200905).