Discrete Dynamics in Nature and Society

Volume 2016, Article ID 2762960, 11 pages

http://dx.doi.org/10.1155/2016/2762960

## Exponential Stability of Cohen-Grossberg Neural Networks with Impulse Time Window

College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China

Received 29 March 2016; Accepted 19 May 2016

Academic Editor: Zhengqiu Zhang

Copyright © 2016 Mei Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper concerns the problem of exponential stability for a class of Cohen-Grossberg neural networks with impulse time window and time-varying delays. In our letter, the impulsive effects we considered can stochastically occur at a definitive time window and the impulsive controllers we considered can be nonlinear and even rely on the states of all the neurons. Hence, the impulses here can be more applicable and more general. By utilizing Lyapunov functional theory, inequality technique, and the analysis method, we obtain some novel and effective exponential stability criteria for the Cohen-Grossberg neural networks. These results generalize a few previous known results and numerical simulations are given to show the effectiveness of the derived results.

#### 1. Introduction

Cohen-Grossberg neural network, which was first proposed by Cohen and Grossberg in 1983 [1], is one of the most typical and popular neural network models because it contains some well-known neural networks such as recurrent neural network, cellular neural network, and Hopfield neural network as a special case. Recently, the studies of Cohen-Grossberg neural networks are many since Cohen-Grossberg neural networks have been widely applied to various problems arising in scientific and engineering areas, such as optimization problem, system control, signal processing, associative memory, pattern recognition, and new class of artificial neural network.

Time delays, in implementation of neural networks, are inevitably encountered in the signal transmission attribute to the finite switching speed of amplifiers. Moreover, besides delay influence, it has been observed that many a physical system may suffer instantaneous perturbations which may exhibit impulsive effects. So, the control of impulsive neural networks with delays is of both theoretical significance and practical significance. In recent years, neural networks with delays and impulse have been extensively investigated by a large quantity of researchers [2–4]. Global exponential stability of Cohen-Grossberg neural systems with time-varying delays via impulsive control was investigated in [2]. Existence and exponential stability of periodic solution for shunting inhibitory cellular neural networks with impulses and delay were considered in [3]. Exponential stability of fuzzy Cohen-Grossberg networks with impulsive effects and time delays were introduced in [4].

Under the current situation, in order to stabilize or synchronize nonlinear dynamical systems, impulsive control strategy, as an important control means, has been widely concerned. From the control point of view, impulsive control theory has extensive applications; for example, it can be applied in HIV prevention model [5], pest control model [6], and nanoelectronics [7]. Thus, it is necessary to investigate the stability of nonlinear system based on the impulsive control.

In the past decades, the analysis for impulsive systems has attracted attention widely [8–10]. But in the existing literatures of impulsive control [11–13], the impulsive instants were fixed or the impulsive occurrence can be calculated. In fact, it has been known that any computer/machine cannot put impulses in exact time, and there will be errors between the expected time and the actual one. For example, in pest control, we should carry out regular spraying of crops. However, due to man-made reasons or natural causes, in the specified day, we cannot carry out on schedule. But as long as we spray the crops within a few days before or after the expected day, this does not affect the growth of crops. Therefore, it is significant to investigate a more practical impulsive scheme concerning the above case.

It is well known that, in the practical application of impulsive control, the impulsive moments at certain instants almost cannot be determined, but an effective impulse interval can be selected; namely, impulse occurs in a time window. For instance, we intend to add an input of impulse at time ; our computer/machine may add the impulse in a short time window , where is a small positive number. This time error interval is called impulsive time window which widely exists in our society. Thus, it is essential and urgent to study this class of impulsive system with impulse time window.

Motivated by the above discussion, this paper investigates the exponential stability of Cohen-Grossberg neural networks model with impulsive time window and time-varying discrete delays. The main contributions in this paper can be summarized as follows. First, the impulsive effects we considered can stochastically occur at a definitive time window which is more applicable and more general. Then, the impulsive controllers we considered can be nonlinear and even rely on the states of all the neurons, which remove the restriction that the impulsive functions are linear. Moreover, our theorems do not require the activation functions to be differentiable, bounded, or monotonically nondecreasing. Finally, by utilizing Lyapunov functional theory, inequality technique, and the analysis method, we obtain some novel and effective exponential stability criteria for the Cohen-Grossberg neural networks, and the conditions utilized in this paper are easy to be verified and improve the conditions derived in [14–16].

The rest of this paper is organized as follows. In Section 2, preliminaries and model of Cohen-Grossberg neural networks with time-varying delays are given. Some stability criteria are obtained in Section 3 under the impulsive intermittent controller we assumed. In Section 4, the feasibility and effectiveness of the developed methods are shown by a numerical example. Conclusions are finally reached in Section 5.

#### 2. Preliminaries

In the paper, we consider a type of Cohen-Grossberg neural network model with time-varying delays which is described by where denotes the state of the th neuron at time , and represent the amplification function and behaved function at time , respectively, the time-varying delay corresponds to the time-varying transmission delay and satisfies , and represent the activation functions of the th neuron, corresponds the external input to the th neuron, and and are constant connection weights and constant delayed connection weights of the th neuron on the th neuron, respectively.

The initial condition of system (1) is given by where , , and denotes the Banach space of all continuous functions mapping into with norms defined by the following forms:

In order to achieve main results, the following assumptions and definition are needed.

For each , is continuous and there exist positive constants and such that

For each , function is continuous and monotonically increasing and there exists real number such that

For each , functions and are Lipschitz-continuous on . That is, there exist positive constants and such that

For each , is differentiable and there exists a constant such that

*Definition 1. *A constant vector is said to be an equilibrium point of system (1) if satisfies the following equality:

For stabilizing the equilibrium point of system (1) by means of periodically impulsive control systems with time windows, we mean that in every period we input an impulse in the time of , , where is unknown but is within impulse time window . Figure 1 shows distribution diagram of pulsed occurrences. This method is called single impulse control of impulse time window.