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.
Remark 2. It is well known that when someone designs a suitable controller for a dynamical system, impulsive controller is inevitable. However, an accurately impulsive moment chosen is more time-consuming and even is impossible. But it is possible to choose an impulse time window which not only can guarantee the control results but also costs less. In [17, 18], the authors considered aperiodically impulse time window, namely, impulse instants . Yet, in many practical applications, we need periodically impulse time window. For instance, we usually carry out regular spraying of crops in pest control, fish, or prevent HIV virus in a fixed period of time. Therefore, the research problem of periodically impulse time window needs to be solved urgently.
Now, to study the stability behaviors of system (1), the following control system is introduced: where is Dirac delta function, functions denote the external control inputs which satisfy the following condition.
There exist nonnegative matrices such that hold for any , ; and .
If impulsive functions for all and , assumption is reduced to the following form.
There exist nonnegative diagonal matrices such that where . Such function and assumption have been required in [19, 20].
Integrating from to both sides of system (9), we have where is sufficiently small and . As , by applying the properties of the Dirac delta function, we have
In the paper, we assume that is left continuous at ; that is, . In this case, we can get
Then, from the above equation, system (9) can be rewritten as
In the sequel, for further study, the following definition is given.
Assume that be the space of -dimensional real column vectors. For any , denotes vector norms defined by
Definition 3. The equilibrium point of system (1) is said to be globally exponentially stable, if there exist and such that where is any solution of system (1) with initial condition .
In order to stabilize , the appropriate control functions can be chosen satisfying the following condition.
Remark 4. Obviously, the impulsive control functions satisfying always exist in applications. For instance, the following controllers can be chosen: where ; that is, , which has been given in [21, 22] and satisfies . In addition, under assumption , if a constant vector is an equilibrium point of system (1), then is an equilibrium of system (15).
Transform to the origin by using transformation for . Then system (15) can be rewritten as the following form: where , , , and .
3. Periodically Impulsive Control Systems with Impulse Time Windows
In the section, we will consider the globally exponential stability of the equilibrium point of neural networks (1) under impulsive intermittent control role. System (20) can be rewritten as follows: where denotes the control period; is unknown but is determined by a random function within impulse time window .
3.1. Stability Based on 1-Norm
For convenience, we denote
Firstly, we introduce the following condition.
for any
Remark 5. From , the inequality is satisfied by choosing suitable . The detailed proof is given as follows. Consider the following function: where for each . It is easy to see that Moreover, is continuous on , as , and there exists positive number such that and for any . Denoting , we have
Theorem 6. Suppose that assumptions and hold; then the equilibrium point of system (1) is globally exponentially stable under the impulsive intermittent control (14) if the following condition is also satisfied.
There exists a constant such that where , , and is the unique positive root of equation .
Proof. Now we define From , it is easy to obtain that for and , ; calculating the upper right derivative of along the solution of system (21), we derived Now let us define for ; calculating the upper right derivative of along the solution of system (21), we derived So, if , we have if , we have On the other hand, from (21), , and , we have for ; from the definition of , we have From (33) and (34), we obtain the following cases.
If , then we have that where . If , we have that If , then we have that If , we have that Below we will use induction method to prove that the following statements are true.
For , and, for ,As for , from the above analysis, we know that inequalities (41) and (42) hold.
Assume that inequalities (41) and (42) are true for all ; then, for any integer satisfying , and, for , When , if , thenFor , Then, by the induction, we see that (41) and (42) hold.
In view of , we obtain So, for , we can getand, for , we can getTherefore, for any , we always have It follows that where which means that Therefore, the proof of Theorem 6 is completed.
3.2. Stability Based on -Norm
For convenience, we denote
Firstly, we introduce the below condition.
for any
Remark 7. From , the inequality is satisfied by choosing appropriate . The detailed proof is presented as follows. Consider the following function: where for each . Similar to Remark 7, we can easily obtain that there exists positive number such that and for any . Denoting , we have which implies that
Theorem 8. Suppose that assumptions , , , and hold; then the equilibrium point of system (1) is globally exponentially stable under the impulsive intermittent control (14) if the following condition is also satisfied.
There exists a constant such that where , , and is the unique positive root of equation .
Proof. From and (21), for , we haveDenote where and . It is easy to see that where is a constant.
In the following, we will prove that Otherwise, there exist and such that and, for any , From (62) and (65), for any , If for any , then, using (55) and (60), we obtain If there exist some such that , then, from (66), we have From this together with (65), we have From what has been discussed above, we obtain Then there exists such that Integrating above inequality from to , we have which leads to a contradiction with (65); then inequality (63) holds.
Moreover, from and the definition of , we have Now, we prove that, for and , Otherwise, there exist and such that and, for any , For , if , it follows from (76) that and if , from (62) and (63), we have Hence, for any , we always have Then From what has been discussed above, we obtain Then there exists such that Integrating above inequality from to , we have which leads to a contradiction with (76); then inequality (74) holds.
Below we will use induction method to prove that the following statements are true.
For , and, for ,where .
As for , from (63) and (74), we know that inequalities (84) and (85) hold.
Assume that inequalities (84) and (85) are true for all ; then, for any integer satisfying , and, for , For , it follows that Similar to the proof of (63), we can prove that holds for . And, similar to (74), we can verify that holds for . Here, for simplicity, the process of those proofs is omitted. Then, by the induction, we see that (84) and (85) hold.
Let ; we have In view of , we obtain So, for , we derive and, for , we obtain Therefore, for any , we always have which implies that for . The proof of Theorem 8 is completed.
Remark 9. If impulsive controllers (14) reduce to (19), it is obvious that conditions and are satisfied and for and , where denotes the identity matrix. Moreover, it is easy to see that for in Theorems 6 and 8.
Corollary 10. Suppose that assumptions and hold; then the equilibrium point of system (1) is exponentially stable under the impulsive intermittent control (14) if the following condition is also satisfied.
Corollary 11. Suppose that assumptions , , , and hold; then the equilibrium point of system (1) is exponentially stable under the impulsive intermittent control (14) if the following condition is also satisfied.
Remark 12. In the paper, Theorems 6 and 8 are proven based on 1-norm and -norm to ensure the global exponential stability of the equilibrium. Compared with Theorem 6, the condition for time-varying delays, , is removed in Theorem 8.
Remark 13. When , the Cohen-Grossberg neural networks in the paper turn out to the models considered in [12, 14–16, 19, 21–23]. Therefore, from this point, we can get the conclusion that the model in this paper is general. In [14–16, 24, 25], the authors considered delayed Cohen-Grossberg neural networks by using fixed point impulsive theorem. In fact, we know that there will be errors between the expected time and the actual one. In many applications, it may be impossible to determine the impulsive sequences. After introducing the time window, impulse interval is described by parameters of the time window. Moreover, in [14–16], the authors require the following condition: Obviously, this condition is very strong. However, in our paper, we only need the following condition: So, our results are less weak than results of [14–16].
Remark 14. From Theorems 6 and 8, we have respectively. One can easily observe that the left hand of the above inequality is formed by the parameters of time window, and the right hand expresses requirement of the minimum time interval of impulse. Hence, the above inequality describes the relationship between the parameter of impulse time window of the system and the exponential rate.
4. Numerical Simulations
In the section, we present an example to show the effectiveness of the results obtained in this paper.
Example. Consider the following Cohen-Grossberg neural networks model with time-varying variable delays: where , , , , , , , , , , , , , and take the activation function as , , and .
In the following, we introduce the following impulsive controllers: We can verify that is an equilibrium of system (103). Let ; about the jumps, we suppose that So that, in interval , we have only one jump.
By simple computing, we get , , , , , , , , , , , , and then the conditions of Theorem 8 are satisfied. It follows from Theorem 8 that system (103) is exponentially stable. The face can be seen by the numerical simulation in Figures 2, 3, and 4.
5. Conclusion
In this paper, the global exponential stability of Cohen-Grossberg neural networks model with time-varying delays has been studied. Particularly, several new sufficient conditions ensuring the exponential stability of Cohen-Grossberg neural networks are obtained via periodically impulsive intermittent control. Evidently, compared with correspondingly previous works [14, 15], our results are less conservative and more general. In addition, in our paper, the activation functions are not differentiable, bounded, or monotonically nondecreasing, and the impulsive controllers, which remove the condition that the impulsive functions are linear, can be nonlinear and even rely on the states of all the neurons. Finally, some numerical simulations are given to illustrate the effectiveness of the proposed theories.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work was supported by National Natural Science Foundation of China (Grants nos. 61473244, 61563048, and 11402223), Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20136501120001), Natural Science Foundation of Xinjiang (Grant no. 2014211B002), Natural Science Foundation of Xinjiang University (Grant no. BS130108), and Project Funded by China Postdoctoral Science Foundation (Grant no. 2013M540782 and no. 2014T70953).