Synchronization of Chaotic Delayed Neural Networks via Impulsive Control

This paper is concerned with the impulsive synchronization problem of chaotic delayed neural networks. By employing Lyapunov stability theorem, impulsive control theory and linearmatrix inequality (LMI) technique, several new sufficient conditions ensuring the asymptotically synchronization for coupled chaotic delayed neural networks are derived. Based on these new sufficient conditions, an impulsive controller is designed. Moreover, the stable impulsive interval of synchronized neural networks is objectively estimated by combining the MATLAB LMI toolbox and one of the two given equations. Two examples with numerical simulations are given to illustrate the effectiveness of the proposed method.


Introduction
During implementation of artificial neural networks, time delay is a very familiar phenomenon due to the finite switching speed of neurons and amplifiers.The existence of time delay possesses an important source in what cause instability and even chaotic behavior in some type of delayed neural networks (DNNs) if the parameters and time delays are appropriately chosen (see [1][2][3][4][5][6]).These kinds of chaotic neural networks have been successfully applied in chemical biology [7], combinatorial optimization [8], associative memory [9], biological simulation [10], and so on.Especially, the synchronization problem of chaotic delayed neural networks has been extensively studied over the past few decades due to its potential applications in many areas, such as secure communications [11][12][13], image encryption [14], image processing [15], and harmonic oscillation generation [16].Hence, it is of great theoretical and practical significance to investigate synchronization problem of chaotic delayed neural networks.
A wide variety of schemes have been proposed for the synchronization of chaotic systems, for example, adaptive control [17], slide mode control [18], coupling control [19], feedback control [20], and impulsive control [21][22][23][24][25].It is worth noting that impulsive control is characterized by the abrupt changes in the system dynamics at certain instants, which is an advantage in reducing the amount of information transmission and improving the security and robustness against disturbances especially in telecommunication network and power grid, orbital transfer of satellite.In addition, impulsive control allows the stabilization and synchronization of chaotic systems using only small control impulses.Thus, it has been widely used to synchronize chaotic neural networks (see [21,23,24]).In [21], Zhao and Zhang obtained some new criteria for the impulsive exponential antisynchronization of two chaotic delayed neural networks by establishing an integral delay inequality via the inequality method.Li et al. [23] investigated the synchronization scheme of coupled neural networks with time delays by utilizing the stability theory for impulsive functional differential equations.Zhang and Sun [24] studied the robust synchronization model of coupled delayed neural networks under general impulsive control.However, all above results do not efficiently utilize the so-called sector nonlinearity property of the activation functions of the neural networks, which leads to some conservatism of the results.
Motivated by the above discussions, the impulsive synchronization problem for chaotic delayed neural networks has not been fully investigated yet, which is still open and remains challenging.The aim of this paper is to study the synchronization of chaotic neural networks with timevarying delay.Some novel sufficient conditions which guarantee the coupled chaotic delayed neural networks can be asymptotically synchronized are derived based on Lyapunov stability theorem, impulsive control theory, and linear matrix inequality (LMI) technique.Moreover, the stable impulsive interval of synchronized neural networks is objectively estimated by combining the MATLAB LMI toolbox and one of the two given equations.
The organization of this paper is as follows.In Section 2, the impulsive synchronization problem is described and some necessary definitions and lemmas are given.Some new synchronization criteria are obtained in Section 3. In Section 4, two illustrative examples are given to show the effectiveness of the proposed method.Finally, conclusions are given in Section 5.
Notations.Let R denote the set of real numbers, let R + denote the set of positive real numbers, and let R  and R × denote the  dimensional Euclidean space and the set of all  ×  real matrices, respectively.N denotes the set of positive integers.‖‖ is the Euclidean norm of the vector .For any matrix  ∈ R × ,  > 0 denotes that  is a symmetric and positive definite matrix.If  1 ,  2 are symmetric matrices, then  1 ≤  2 means that  1 −  2 is a negative semidefinite matrix.  (),   () denote the minimum and maximum eigenvalue of matrix , respectively.  and  −1 mean the transpose of  and the inverse of a square matrix . denotes the identity matrix with appropriate dimensions.

Problem Description and Preliminaries
The chaotic neural networks with variable delay can be described by where () = ( Based on assumption (H1), we set Remark 1.In usual Lipschitz condition, it is assumed that  −  = − +  .Clearly, the condition (H1) is quite general and includes the usual Lipschitz conditions as a special case.
At discrete time   , the state variables of the drive system are transmitted to the response system as the control input such that the state vectors of the response system are suddenly changed at these instants.Therefore, the impulsive controlled response system can be written as where Δ(  ) denotes the state jumping at impulsive time instant  =   , ( +  ), ( −  ) and ( +  ), ( −  ) are the righthand and left-hand limits of the functions () and () at   , respectively.Moreover, (  ) = ( +  ), (  ) = ( +  ).Suppose that the discrete time sequence Let () = () − () be the synchronization error, and then we can obtain the error system between (1) and ( 5): where ℎ((⋅)) = ((⋅) + (⋅)) − ((⋅)).
The following definitions and lemmas which are useful in deriving synchronization criteria are used in the paper.
Lemma 5 (see [29]).For any symmetric and positive definite matrix  ∈ R × , the following inequality holds: Lemma 6 (see [30]).If ,  are real matrices with appropriate dimensions, then there exists a number  > 0 such that

Main Results
In this section, we use the Lyapunov-like function  (,  ()) =   ()  () (11) to derive the asymptotically stability conditions of the zero solution of the error system (6), which implies that the drive system (1) and the response system (5) can be asymptotically synchronized.
Remark 8.The stable impulsive interval is associated with the impulsive matrix   and the choice of parameter  6 .In order to reduce the man-made misleading during the prediction of stable impulsive interval, here only fix parameter  3 ,  4 and the impulsive matrix   , and then let  5  ≤  1 ,  6  ≥  2 .Firstly, ,  1 ,  2 can be obtained by Theorem 7 via MATLAB LMI toolbox.Finally,  5 ,  6 can be calculated by following algebraic equations: or linear matrix inequalities: In [22,25], the parameters  4 and  which are corresponding to parameter  6 had been selected subjectively, which may cause result to lack fidelity.In addition, parameters  3 ,  4 are selectable variables which can increase the flexibility of the possible outcomes.Therefore, our results is more objective in some situations.
In particular, when  −  = − +  < 0 in (H1), the following corollary holds.x(t) x(0) x(end) y(t) y(0) y(end) x 1 (t) and y 1 (t) x 2 (t) and y 2 (t) Figure 2: Time response of state variables, the synchronization errors, and the phase plots of the drive system and response system in Example 12 under the impulsive controller (0.06, −1.3).
Remark 11.The sufficient conditions in Theorem 7 and Corollaries 9 and 10 are all independent of the delay parameter but rely on the maximum impulsive interval sup{  −  −1 } and the impulsive matrix   , which plays a fundamental role when the size of the delay is unknown.

Numerical Results
In order to illustrate the feasibility of our above-established criteria in the preceding sections, we provide two concrete examples.Throughout the simulations, we use the IMEX implicit Euler method.

Conclusion
In this paper, the impulsive synchronization problem of chaotic delayed neural networks has been investigated.Some new criterions which ensure that the coupled chaotic delayed neural networks can be asymptotically synchronized have been derived in terms of linear matrix inequalities (LMIs) by using Lyapunov stability theorem, impulsive control theory, and linear matrix inequality (LMI) technique.The desired impulsive controller which is with respect to the stable impulsive interval and the impulsive matrix is established, its existence can be verified effectively by the simulations.It is worthwhile to mention that the positive constants set ( 3 ,  4 ) can increase the flexibility for the design of the impulsive controller.Moreover, the stable impulsive interval can be calculated combining MATLAB LMI toolbox and one of the two given equations objectively.Finally, two illustrative examples are given to show the applicability and usefulness of the proposed results.

Corollary 9 .Figure 1 :
Figure 1: Time response of state variables and the phase plots of the drive system and response system in Example 12 without impulsive control.

Figure 3 :
Figure 3: Chaotic attractor of Ikeda-type neural network in Example 14 and the synchronization error of the drive system and response system in Example 14.
)-1(d) show the time response of state variables and the phase plots of the drive system and response system in Example 12 without impulsive control.By choosing  3 =  4 = 0.7,  = −1.3 and then using MATLAB LMI toolbox and (31), we can obtain the following feasible solutions to LMIs and  5 ,  6 in Corollary 9: