Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 934592, 10 pages

http://dx.doi.org/10.1155/2014/934592

Research Article

Global Exponential Stability for DCNNs with Impulses on Time Scales

Yongkun Li and Yuanhong Zhi

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received 25 September 2013; Accepted 17 December 2013; Published 2 January 2014

Academic Editor: Lu Zhen

Copyright © 2014 Yongkun Li and Yuanhong Zhi. 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.

A class of delayed cellular neural networks (DCNNs) with impulses on time scales is considered. By using the topological degree theory, and the time scale calculus theory some sufficient conditions are derived to ensure the existence, uniqueness, and global exponential stability of equilibria for this class of neural networks. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and the discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new and complementary to the previously known results.

1. Introduction

Chua and Yang [1] proposed a novel class of information-processing systems called cellular neural networks (CNNs) in 1988. The CNNs can be applied in signal processing and can also be used to solve some image processing and pattern recognition problems [2]. Since time delays are unavoidable due to finite switching speeds of the amplifiers, delayed cellular neural networks (DCNNs) have been widely studied and successfully applied to pattern recognition, associative memories, and signal processing and optimization, especially in image processing. The dynamic behavior of the networks plays an important role in such applications [3–8]. Therefore, there are many works on the stability of equilibrium point of delayed cellular neural networks (DCNNs) [5–13].

Most neural networks can be classified into two types: continuous or discrete. However, many real-world systems and natural processes cannot be categorized into one of them. They display characteristics of both continuous and discrete styles. For instance, some biological neural networks in biology, bursting rhythm models in pathology, and optimal control models in economics are characterized by abrupt changes of state. These are the familiar impulsive phenomena. Other examples can also be found in information science, electronics, automatic control systems, computer networking, artificial intelligence, robotics, telecommunications, and so forth. Such a kind of phenomena, in which sudden and sharp changes often occur in a continuous process, cannot be well described by pure continuous or pure discrete models. Therefore, it is important and, in effect, necessary to study a new type of neural networks—impulsive neural networks—as an appropriate description of these phenomena of abrupt qualitative dynamical changes of essentially continuous systems. The fundamental theory of impulsive differential equations has been developed in [14]. Since delays and impulses can affect the dynamical behaviors of the system, it is necessary to investigate both delay and impulsive effects on the stability of neural networks. For more details, one can refer to [10, 13, 15–23].

The theory of time scale was initiated by Hilger in 1988, which has recently received a lot of attention [24–26]. The field of dynamic equations on time scale contains links and extends the classical theory of differential and difference equations. It is well known that both continuous and discrete systems are very important in implementation and applications (see [27–30]). But it is troublesome to study the stability for continuous and discrete systems, respectively. Therefore, it is significant to study that on time scales which can unify the continuous and discrete situations [21, 31–40].

Motivated by above, in this paper, we are concerned with the following impulsive DCNN on time scales:

where corresponds to the numbers of units in a neural network; corresponds to the state of the th unit at time ; denotes the output of the th unit at time . is the -interval , and denotes a time scale, which is an arbitrary nonempty closed subset of the real number and with bounded graininess . For the simplicity, we assume that and is unbounded above; that is, . Further, , , , and are constants. , denote the strength of the th unit at time and , respectively. denotes the external bias on the th unit and represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs. , are the moments of impulsive perturbations and satisfy and , (see Definition 3). represents the abrupt change of the state at the impulsive moment . To the best of our knowledge, this is first paper to study DCNNs with impulses on time scales.

Throughout this paper, we assume that and

(H1)functions satisfy for all , ;(H2) and there exists a positive number such that for all , .

Remark 1. The neural network (1) is a system of differential equations with state-dependent deviating arguments and from (H1), one can see that deviating arguments in (1) may be delayed type, advanced type, or mixed type.

Our main purpose of this paper is to study the existence and global exponential stability of the equilibria of (1) by using the topological degree theory and the time scale calculus theory. The results of this paper are completely new and complementary to the previously known results.

The organization of this paper is as follows. In the next section, some notations, definitions, and lemmas are presented. Section 3 addresses the existence and uniqueness of equilibria of system (1) by using the method of topological degree theory. In Section 4, we give the criteria of global exponential stability of the equilibrium point of system (1). In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections 3 and 4.

2. Notations and Preliminaries

In this section, we will first recall some basic definitions and lemmas which will be useful for the proof of our main results.

Definition 2 (see [33, 34]). A time scale is arbitrary nonempty closed subset of the real set with the topology and ordering inherited from .

Definition 3 (see [33, 34]). On any time scale , we define the forward and backward jump operators by

A point is said to be left-dense if and , right-dense if and , left-scattered if , and right-scattered if . The graininess function for a time scale is defined by . If has a left-scattered maximum , then we defined to be . Otherwise, .

Definition 4 (see [33, 34]). For a function (the range of may be actually replaced by Banach space), the (delta) derivative is defined by

if is continuous at and is right-scattered. If is not right-scattered, then the derivative is defined by

provided this limit exists.

Lemma 5 (see [33, 34]). If , are differential at , one has

(1);(2).

Definition 6 (see [33, 34]). A function is called a delta-antiderivative of provided holds for all . In this case, we define the integral of by

and we have the following formula:

Definition 7 (see [33, 34]). A function is called right-dense continuous (rd-continuous) provided it is continuous at right-dense points of and the left-sided limit exists (finite) at left-dense point of . The set of all right-dense continuous functions on is defined by . If is continuous at each right-dense point and each left-dense point, then is said to be continuous function on . We define .

Lemma 8 (see [33, 34]). If , and , then one has

(1);(2)if for all , then ;(3)if on , then .

Definition 9 (see [33, 34]). A function is called regressive if for all . If is regressive function, then the generalized exponential function is defined by

with the cylinder transformation

Let be two regressive functions; we define

Then, the generalized exponential function has the following properties.

Lemma 10 (see [33, 34]). Assume that are two regressive functions; then

(1) and ;(2);(3);(4);(5);(6);(7);(8).

Definition 11. A point is called an equilibrium point of model (1) if is a solution of (1).

Throughout this paper, we always assume that the impulsive jump vector satisfies

That is, if is an equilibrium point of the following non-impulsive system:

then it is also the equilibrium point of impulsive system (1).

Definition 12 (see [41]). A real matrix is said to be a nonsingular -matrix if , and all successive principal minors of are positive.

Lemma 13 (see [41]). Let with ; then is a nonsingular -matrix if and only if the diagonal elements of are all positive and there exists a positive vector such that or .

3. Existence and Uniqueness of Equilibrium Point

In this section, we will discuss the existence and uniqueness of equilibria of the DCNN with impulses on time scales and give their proofs.

Theorem 14. Under assumptions (H1) and (H2), if the following condition is satisfied

(H), ,

then there is exactly one equilibrium point of model (1).

Remark 15. From Lemma 13, we can easily prove that (H) holds implying that the following condition is true:

(H0)there exists a vector such that

For convenience, we set , , , and . Let , . From assumption (H), we have

which implies that is a nonsingular -matrix. So we know that is a nonsingular -matrix. Hence, there exists a vector such that

It follows that (H0) holds.

Now, we prove our theorem.

Proof. Let be an equilibrium point of system (1); then, we have

We denote , where

Obviously, the equilibrium points of model (1) are solutions of system

Define the following homotopic mapping:

where . Let denote the th component of ; then, we can get

It follows that

Let

From the assumption of the theorem, we can easily see that . Let

Then, for any , we have

As , we have

Hence, all the above conclusions mean that

From the homotopy invariance theorem, we obtain

where is the identity operator. By topological degree theory, we can easily know that system (11) has at least one solution in . That means model (1) has at least an equilibrium point.

In order to prove the uniqueness of the equilibrium point, let and be two equilibrium points of system (1). So, we have

Then,

By using assumption (H2), we get

It follows that

Hence,

So, we get

From the assumption (H0), we get , . Therefore, system (1) has one unique equilibrium point. The proof is complete.

4. Global Exponential Stability of the Equilibrium Point

In this section, we consider the following DCNN system with impulses of the type

where , , , , and are defined as those in (1) and are positive constants which satisfy for all , . Let . Then, the initial conditions associated with (33) are of the form

where , are rd-continuous.

Definition 16. Let be an equilibrium point of (33) with initial value . If there exists a positive constant with such that for , there exists such that for an arbitrary solution of (33) with initial value satisfies

where , Then the equilibrium point is said to be exponentially stable.

Now, we study the global exponential stability of the unique equilibrium to (33) on time scales by using Lyapunov method. We have the following.

Theorem 17. Let (H2) and (H) hold. Suppose further that

(H₃), , .

Then, the equilibrium of system (33) is globally exponentially stable.

Remark 18. We denote the -interval as .

Now, we prove Theorem 17.

Proof. Let . Then, we can rewrite (33) as

Multiplying both sides of the first equation of (4.2) by and integrating on , where , we get

For positive constant with , we have , where . Take

In view of (H), we have . Hence, it is obvious that

We claim that

To prove this claim, we show that for any , the following inequality holds

By way of contradiction, assume that (41) does not hold. Then, there exist and such that

Therefore, there must be a constant such that

Note that, in view of (37), we have

Thus, we get a contradiction. Hence, (41) holds. Let ; then (40) holds. From (40), we have that , . Since , , it follows that

Thus, for , we may repeat the above procedure and obtain

Similarly, we have

Take ; then and . Hence, we have that

which means that the equilibrium point of (33) is exponentially stable. This completes the proof.

Remark 19. If the time scale , then and system (33) becomes the following model:

From Theorem 17, we can immediately derive the following result.

Corollary 20. Suppose that system (49) satisfies condition (H2) and (H), and the following assumptions hold:

(1);(2).

Then, the equilibrium of system (49) is globally exponentially stable.

Remark 21. In [42], by utilizing the time scale calculus theory, topological degree theory, and Hölder’s inequality on time scales, authors studied the existence and the global exponential stability of equilibrium point to a class of impulsive BAM neural networks with distributed delays on time scales. But, results obtained in [42] cannot be applied to (1). Also, for establishing the global exponential stability of equilibrium point to (1), our method used in this paper is totally different from that used in [42].

5. An Example

In this section, an example is given to show the effectiveness of the result obtained in the previous section. Because the condition (4.2) is not dependent on the impulses, we just need to check it with the nonimpulsive system.

Consider the following simple DCNN on time scale :

where , , ,

Taking , we can easily see that .

Let , then for , , we have , and for , we have .

We have that

which imply that the assumption (H) of Theorem 14 holds. Thus, it follows from Theorems 14 and 17 that system (50) has a unique equilibrium point which is globally exponentially stable (see Figure 1).

Figure 1: Transient responses of states , in Example when .

Since for and for , from the discussion above one can easily see that for or , (50) always has a unique equilibrium point which is globally exponentially stable. That is, the following continuous-time system

and its discrete-time analogue

have the same dynamical properties, where , , , and are the same as those in (50) (see Figures 2 and 3).

Figure 2: Transient responses of states , in Example when .

Figure 3: Transient responses of states , in Example when .

6. Conclusion

Using the topological degree theory and the time scale calculus theory, some sufficient conditions are obtained to ensure the existence and the global exponential stability of equilibria for DCNNs neural networks with impulses on time scales. This is the first time to apply the time scale calculus theory to unify the study of the stability of the equilibrium for DCNNs with impulses on time scales under the same framework. The results obtained in this paper possess highly important significance and are easily checked in practice. In addition, the method in this paper may be applied to some other systems such as the BAM and Cohen-Grossberg systems with impulses and so on.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant no. 11361072 and the YNU Postdoctoral Science Foundation Project under Grant no. W4030002.

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