# Impulsive Control for the Synchronization of Chaotic Systems with Time Delay

Authors: Ming Han; Yang Liu; Jianquan Lu
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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 Section2, 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 thatR is the set of real numbers, R+ is the set of nonnegative real numbers, Rn is the n-dimensional real space, and N is the set of positive integers. Let Ψ(t+)=lims→t+Ψ(s) and Ψ(t-)=lims→t-Ψ(s). For a,b∈R with a<b and for S⊂Rn, 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],Rn), the norm of Ψ is, respectively, defined by ∥Ψ∥=sup-τ≤s≤0∥Ψ(s)∥.et,et-∈PC([-τ,0],Rn) are defined by et(s)=e(t+s) and et-(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(2.1)x˙(t)=Ax(t)+Φ1(x(t)),
and, at the receiver end, we have
(2.2)y˙(t)=Ay(t)+Φ2(y(t),x(t-τ),y(t-τ)),t≠tk,Δy(t)=-Bke(t),t=tk,y(t0-s)=φ(s),s∈[0,τ],
where the matrices A,Bk∈Rn×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 {tk}k=1+∞ satisfies 0=t0<t1<t2<⋯<tk<⋯,  limk→∞tk=+∞, the time delay τ>0, and Δy(tk)=y(tk+)-y(tk-), in which y(tk+)=lims→0+y(tk+s),y(tk-)=lims→0-y(tk+s) for k=1,2,…. A typical form of the function Φ2 is given as follows:
(2.3)Φ2(y(t),x(t-τ),y(t-τ))=Φ1(y(t))+FK(x(t-τ)-y(t-τ)),
where F∈Rn×m,K∈Rm×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:(2.4)e˙(t)=Ae(t)+Ψ(x(t),y(t),x(t-τ),y(t-τ)),t≠tk,Δe(t)=Bke(t),t=tk,k=1,2,…,
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 givent>t0 and φ∈PC([-τ,0],Rn), the initial value problem of (2.4) is
(2.5)e˙(t)=Ae(t)+Ψ(x(t),y(t),x(t-τ),y(t-τ)),t≠tk,Δe(t)=Bke(t),t=tk,t∈Ne(t0-s)=φ(s),s∈(0,τ).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 Di∈Rn×n,  i=1,2, such that
(2.6)(x-y)TPΨ(x,y,u,v)≤(x-y)TPD1(x-y)+(x-y)TPD2(u-v).
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 et0=φ∈PC([-τ,0],Rn) and ∥Ψ∥τ=sup-τ≤s≤0∥Ψ(s)∥, there exist some constants λ>0 and M≥1 such that
(2.7)∥e(t,t0,φ)∥≤M∥φ∥τe-λ(t-t0),t≥t0.Definition 2.2.
Given a functionV:R+×Rn→R+, the upper right-hand derivative of V with respect to system (2.4) is defined by
(2.8)D+V(t,e(t))=limδ→0+sup1δ[V(t+δ,e(t+δ))-V(t,e(t))].


## 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.
LetP∈Rn×n be a positive definite matrix and λk(0<λk≤1,k∈N) be largest eigenvalue of P-1(I+Bk)TP(I+Bk). Assume that there exist constants αi,  i=1,2 with α2∈R+ and λ>0,  σ>0, such that for all k∈N, one has(i)
(3.1)[ATP+PA+PD1-α1PPD2D2TP-α2P]≤0,(ii)
(3.2)σ-λ-[α1+(α2λk-1)eλτ]≥0,
where 0<λ0≤1,(iii)
(3.3)lnλk-1<-(σ+λ)(tk-tk-1).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.
Lete(t)=e(t,t0,φ) be any solution of system (2.5), and consider a Lyapunov function as follows:
(3.4)V(t,e(t))=eT(t)Pe(t).
Let μ1>0 and μ2>0 be the smallest and largest eigenvalues of positive definite matrix P, respectively. Then, we can obtain
(3.5)μ1∥e(t)∥2≤V(t,e(t))≤μ2∥e(t)∥2.
Now, we are in a position to prove that, for any t∈[tk-1,tk),
(3.6)V(t,e(t))≤μ2M∥φ∥τ2e-λ(t-t0),k∈N.
For t≠tk(k∈N), it follows from assumption (2.6) and condition (i) that
(3.7)D+V(t,e(t))=e˙TPe(t)+eT(t)Pe˙(t)=[eT(t)AT+ΨT]Pe(t)+eT(t)P[Ae(t)+Ψ]=eT[ATP+PA]e(t)+eT(t)PΨ+ΨTPe(t)≤eT(t)[ATP+PA+2PD1]e(t)+eT(t)PD2×e(t-τ)+eT(t-τ)D2TPe(t)=[eT(t)eT(t-τ)][ATP+PA+2PD1PD2D2TP0][e(t)e(t-τ)].
It further follows from condition (i) that
(3.8)D+V(t,e(t))≤[eT(t)eT(t-τ)][α1P00α2P][e(t)e(t-τ)]=α1V(t,e(t))+α2V(t-τ,e(t-τ)).
Defining γ=supk∈N(1/λk-1)≥1 in condition (iii), we have
(3.9)lnγ+λτ-(σ+λ)(tk-tk-1)>0.
Then, in view of the inequality in (3.9), we can find constant M≥1 such that
(3.10)1<e(σ+λ)(t1-t0)≤M≤γeλτ-(σ+λ)(t1-t0)e(σ+λ)(t1-t0),
which shows that
(3.11)∥φ∥τ2<∥φ∥τ2eσ(t1-t0)≤M∥φ∥τ2e-λ(t1-t0).
Firstly, we prove that
(3.12)V(t,e(t))≤μ2M∥φ∥τ2e-λ(t-t0),t∈[t0,t1).
To do this, we only need to prove that
(3.13)V(t,e(t))≤μ2M∥φ∥τ2e-λ(t1-t0),t∈[t0,t1).
If the inequality (3.13) is not true, it follows from (3.11) that there must exist some t-∈[t0,t1) such that
(3.14)V(t-,e(t-))>μ2M∥φ∥τ2e-λ(t1-t0)≥μ2∥φ∥τ2eσ(t1-t0)>μ2∥φ∥τ2≥V(t0+s,e(t0+s)),s∈[-τ,0],
which implies that there exists t*∈[t0,t-) such that
(3.15)V(t*,e(t*))=μ2M∥φ∥τ2e-λ(t1-t0),V(t,e(t))≤V(t*,e(t*)),t∈[t0-τ,t*],
and there exists t**∈[t0,t*) such that
(3.16)V(t**,e(t**))=λ2∥φ∥τ2,V(t**,e(t**))≤V(t,e(t)),t∈[t**,t*].
Hence, for any s∈[-τ,0], we can obtain that
(3.17)V(t+s,e(t+s))≤μ2M∥φ∥τ2e-λ(t1-t0)≤μ2γeλτ-(σ+λ)(t1-t0)e(σ+λ)(t1-t0)∥φ∥τ2=γeλτμ2∥φ∥τ2=γeλτV(t**,e(t**))≤γeλτV(t,e(t)),t∈[t**,t*].
By condition (ii), (3.8), and (3.17), we can obtain
(3.18)D+V(t,e(t))≤(α1+γα2eλτ)V(t,e(t))≤(α1+α2λ0eλτ)V(t,e(t))≤(σ-λ)V(t,e(t)),t∈[t**,t*],
which shows that
(3.19)V(t*,e(t*))≤V(t**,e(t**))e(σ-λ)(t*-t**)=μ2∥φ∥τ2e(σ-λ)(t*-t**)<μ2∥φ∥τ2eσ(t1-t0)≤μ2M∥φ∥τ2e-λ(t1-t0)=V(t*,e(t*)).
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,
(3.20)V(t,e(t))≤μ2M∥φ∥τ2e-λ(t-t0),t∈[tk-1,tk).
Next, we will prove that (3.6) holds for k=m+1, that is,
(3.21)V(t,e(t))≤μ2M∥φ∥τ2e-λ(t-t0),t∈[tm,tm+1).
For the sake of contradiction, we suppose that (3.21) is not true. Then, we define
(3.22)t-=inf{t∈[tm,tm+1)∣V(t,e(t))>μ2M∥φ∥τ2e-λ(t-t0)}.
From condition (iii) and (3.20), we can get
(3.23)V(tm,e(tm))=eT(tm-)(I+Bm)TP(I+Bm)e(tm-)≤λmV(tm-,e(tm-))≤λmμ2M∥φ∥τ2e-λ(tm-t0)=λmμ2M∥φ∥τ2eλ(t--tm)e-λ(t--t0)<λmμ2eλ(tm+1-tm)M∥φ∥τ2e-λ(t--t0)<μ2M∥φ∥τ2e-λ(t--t0),
and sot-≠tm. By the continuity of V(t,e(t)) in the interval [tm,tm+1), we obtain
(3.24)V(t*,e(t*))=μ2M∥φ∥τ2e-λ(t--t0),V(t,e(t))≤V(t*,e(t*)),t∈[tm,t*].
From (3.23), we can derive that there exists t**∈[tm,t*) such that
(3.25)V(t**,e(t**))=λmμ2eλ(tm+1-tm)M∥φ∥τ2e-λ(t--t0),V(t**,e(t**))≤V(t,e(t))≤V(t*,e(t*)),t∈[t**,t*].
As for anyt∈[t**,t*],  s∈[-τ,0], then either t+s∈[t0-τ,tm) or t+s∈[tm,t*]. These two cases will be discussed in the following.
Ift+s∈[t0-τ,tm), from (3.20), we have
(3.26)V(t+s,e(t+s))≤λ2M∥φ∥τ2e-λ(t-t0)e-λs≤μ2M∥φ∥τ2e-λ(t--t0)eλ(t--t)eλτ≤μ2eλτeλ(tm+1-tm)M∥φ∥τ2e-λ(t--t0),
whereas if t+s∈[tm,t*], from (3.24), then
(3.27)V(t+s,e(t+s))≤μ2M∥φ∥τ2e-λ(t--t0)≤μ2eλτeλ(tm+1-tm)M∥φ∥τ2e-λ(t--t0).
Above all, from (3.25)–(3.27), we can obtain, for any s∈[-τ,0],
(3.28)V(t+s,e(t+s))=eλτλmV(t**,e(t**))≤eλτλmV(t,e(t)),t∈[t**,t*].
Hence, by condition (ii), (3.8), and (3.28), we can conclude that
(3.29)D+V(t,e(t))≤(α1+α2λmeλτ)V(t,e(t))≤(σ-λ)V(t,e(t)).
Thus, in view of condition (iii), we have
(3.30)V(t*,e*)≤V(t**,e**)e(σ-λ)(t*-t**)=λmμ2eλ(tm+1-tm)M∥φ∥τ2e-λ(t--t0)e(σ-λ)(t*-t**)<μ2e-(σ+λ)(tm+1-tm)eλ(tm+1-tm)×M∥φ∥τ2e-λ(t--t0)e(σ-λ)(t*-t**)=μ2M∥φ∥τ2e-σ(tm+1-tm)e(σ-λ)(t*-t**)e-λ(t--t0)<μ2M∥φ∥τ2e-λ(t--t0)=V(t*,e*),
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
(3.31)∥e(t)∥≤μ2μ1M1/2∥φ∥τe-(λ/2)(t-t0),t≥t0,
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) tk-tk-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 Theorem3.1, we can obtain the following result.Corollary 3.4.
LetP∈Rn×n be a symmetric and positive definite matrix, 0<β=supk∈Nλk, where λk is given by Theorem 3.1 for each k∈N. Assume that there exist constants αi,  i=1,2 with α2∈R+, such that for all k∈N the following are satisfied:(i)
(3.32)[ATP+PA+2PD1-α1PPD2D2TP-α2P]≤0,(ii)
(3.33)(α1+α2β)(tk-tk-1)<-lnβ.
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 Theorem3.1 can be reduced to the following form:
(3.34)σ-λ+[α1+(α2β)eλτ]≥0.
Then, it is easy to get the following inequality, which is equivalent to (3.34) and condition (iii) in Theorem 3.1:
(3.35)[2λ+α1+(α2β)eλτ](tk-tk-1)<-lnβ.
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 Corollary3.4 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 tk-tk-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:
(4.1)x˙1=-σx1+σx2,x˙2=rx1-x2-x1x3,x˙3=x1x2-bx3,
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:
(4.2)x˙(t)=Ax(t)+Φ1(x(t)),
where
(4.3)A=[-1010028-1000-83],Φ1=[0-x1x3x1x2].
We also choose Φ2 defined by (2.3) with FK=I. Set the delay τ=1 and the impulsive matrices Bk=-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 Bk=-0.5I, we can get β=0.25. According to condition (ii) in Corollary 3.4, when
(4.4)tk-tk-1<-lnβα1+α2/β=0.0396,
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 tk-tk-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).
Corollary3.4
[13]
[14]
[15, 16]
0.0396
0.0072
0.0108
0.0178Figure 1
The trajectories of the error Lorenz system under impulsive controller withtk-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.



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