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This paper studies the global exponential stability for a class of impulsive disturbance complex-valued Cohen-Grossberg neural networks with both time-varying delays and continuously distributed delays. Firstly, the existence and uniqueness of the equilibrium point of the system are analyzed by using the corresponding property of

Recently, several kinds of complex-valued neural networks have been proposed and attracted researchers’ attention due to the broader range of their applications in electromagnetics, quantum waves, optoelectronics, filtering, speech synthesis, remote sensing, signal processing, and so on [

As a very important issue, dynamical behavior analysis including existence, uniqueness, and stability of the equilibrium point of neural networks has attracted growing interest in the past decades (see [

Due to the finite switching speed of amplifiers, time delay inevitably exists in neural networks. It can cause oscillation and instability behavior of systems. As pointed out in [

The impulsive disturbances are also likely to exist in the system of neural networks and can affect the dynamical behaviors of the system states, the same as the time delays effect. For instance, in the implementation of electronic networks, the states of the neural networks are subject to instantaneous perturbations and experience abrupt changes at certain instants, which may be caused by the switching phenomenon, frequency change, or other sudden noises. This phenomenon of instantaneous perturbations to the system exhibits an impulsive effect [

The model of Cohen-Grossberg neural networks was proposed by Cohen and Grossberg in 1983 [

Based on the above analysis, in this paper, we will investigate the dynamical behavior for a class of impulsive complex-valued Cohen-Grossberg neural networks with time-varying delays and continuously distributed delays. In this paper, advantages and contributions can be listed as follows.

To make reading easier, the following notations will be used throughout this paper. Let

In this paper, we consider a class of impulsive disturbance complex-valued Cohen-Grossberg neural networks with time-varying delays and continuously distributed delays, which can be described by

It is assumed that initial conditions of (

Denote by

The equilibrium point

It is assumed that there exist positive numbers

The authors in [

Each function

The choice of the activation function is the main challenge in the dynamical behavior analysis of complex-valued neural networks compared to the study of real-valued neural networks. The complex-valued activation functions were supposed to need explicit separation into a real part and an imaginary part in [

It is supposed that the amplification function

The amplification functions of complex-valued Cohen-Grossberg neural networks were supposed to be with upper bounds and lower bounds in [

Let

Let

The impulsive effect is introduced into (

To proceed with our results, we quote the following lemmas in the proof of the theorems in this paper.

Let

The real parts of all eigenvalues of

There exists a positive vector

If

then

It is supposed that Assumptions

The proof of the theorem is separated into two steps.

Define a map

It is well known that if

① We prove that the map

From inequalities (

According to Lemma

Moreover, because inequalities (

It is assumed that there exist

Taking absolute value on both sides of (

Considering Assumptions

Furthermore, inequalities (

Obviously,

② In what follows, we will prove that

Let

Multiplying by the conjugate complex number of

Taking the conjugate operation on both sides of (

Taking the summation operation on both sides of (

Multiplying by

Considering inequalities (

Namely,

Combining ① and ② above, we know that

For analysis convenience, we translate the coordinate of (

Let the initial conditions of (

Obviously, if the zero solution of (

Choose the vector Lyapunov function as follows:

Let

When

Let

Define the curve

Furthermore, we can claim that

This is a contradiction with the above assumption

Next, the mathematical induction method will be applied to prove that the following inequalities hold:

When

It is assumed that the following inequalities hold:

When

Due to

Furthermore, we can conclude that the following inequalities hold:

If inequalities (

Substituting (

This is a contradiction with the assumption

Based on the idea of the mathematical induction method, we have

It follows from the condition of the theorem

Furthermore, we have

According to Definition

To sum up, it can be concluded from Steps

Although there have been various methods for studying the diverse complex-valued neural networks, the scalar Lyapunov function method combined with the LMI method is nearly the most popular method to research the stability problem and synchronization problem (see [

From Theorem

If there are no continuously distributed delays in (

Suppose that Assumptions

Similarly, if there are no time-varying delays in (

Suppose that Assumptions

When

It is easy to obtain sufficient conditions for ensuring the existence, uniqueness, and global exponential stability of the equilibrium point of system (

Suppose that Assumptions

The same as the preceding analysis, we can easily obtain the corresponding criteria for guaranteeing the stability of the equilibrium point of impulsive complex-valued Hopfield neural networks with only time-varying delays or continuously distributed delays. Therefore, we omit the similar works here.

When there is no impulsive disturbance in model (

All variables and functions in model (

It is supposed that Assumptions

According to Step

We construct functions as follows:

From inequalities (

Because

We choose the vector Lyapunov function as follows:

According to the proof of Step

When there are only time-varying delays or continuously distributed delays in system (

Separating the model of complex-valued neural networks into its real and imaginary parts is a routine method (e.g., [

In this section, we will give three examples with numerical simulations to demonstrate the correctness of the above results.

Consider a class of two-order system described by (

The interconnected matrices are given as

By calculation, we obtain

From the above computing analysis, the assumption conditions in Theorem

The numerical simulations of the above system are shown in Figures

State curves of

State curves of

State curves of

State curves of

In [

Consider a class of two-order system described by (

Let

By calculation, we have

According to Theorem

The numerical simulation of the system is shown in Figure

State curves of

According to results of calculation and simulation, we find that the convergence rate in Example

Consider a class of two-order system described by (

It is assumed that activation functions are

By calculation, we have

Obviously, it is known from Lemma

The numerical simulation of the system is shown in Figure

State curves of

This paper has studied the dynamical behavior for a class of impulsive disturbance complex-valued Cohen-Grossberg neural networks with both time-varying delays and continuously distributed delays. Based on the idea of the vector Lyapunov function method, some sufficient conditions have been established for ensuring the existence, uniqueness, and global exponential stability of the equilibrium point of the system by using the corresponding properties of

It is well known that the synchronization problem of chaotic neural networks can be translated into the stability problem of the corresponding error system of driving system and driven system. There have been some literatures concerning the analysis of synchronization control for some complex-valued neural networks with time delays by using the idea of adaptive control [

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grants nos. 11402214, 51375402, and 11572264); the Science & Technology Department of Sichuan Province (Grants nos. 2017TD0035, 2017TD0026, 2015TD0021, and 2016HH0010); the Scientific Research Foundation of the Education Department of Sichuan Province (Grants nos. 17ZA0364 and 16ZB0163); the Open Research Subject of Key Laboratory of Fluid and Power Machinery (Xihua University), Ministry of Education (Grant no. szjj2016-007); the Open Research Fund of Key Laboratory of Automobile Measurement and Control & Safety (Xihua University), Sichuan Province (Grant no. szjj2017-074); and the Open Research Fund of Key Laboratory of Automobile Engineering (Xihua University), Sichuan Province (Grant no. szjj2016-017). The authors deeply appreciate Ling Zhao from Xihua University for her helpful constructive suggestions in the revision of the language of this article.