Research Article | Open Access

# Stability of Impulsive Cohen-Grossberg Neural Networks with Time-Varying Delays and Reaction-Diffusion Terms

**Academic Editor:**Qi Luo

#### Abstract

This work concerns the stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms as well as Dirichlet boundary condition. By means of Poincaré inequality and Gronwall-Bellman-type impulsive integral inequality, we summarize some new and concise sufficient conditions ensuring the global exponential stability of equilibrium point. The proposed criteria are relevant to the diffusion coefficients and the smallest positive eigenvalue of corresponding Dirichlet Laplacian. In conclusion, two examples are illustrated to demonstrate the effectiveness of our obtained results.

#### 1. Introduction

Cohen-Grossberg neural networks (CGNNs) were introduced by Cohen and Grossberg in 1983 [1] and have been a hot topic due to their important applications in various fields such as parallel computation, associative memory, image processing, and optimization problems.

By reason that time delays are unavoidably encountered for the finite switching speed of neurons and amplifiers in the implementation of neural networks, a more powerful model of delayed Cohen-Grossberg neural networks (DCGNNs) is afterwards proposed. This kind of mathematical models is widely applied in dynamic image processing and pattern recognition problems. It is worth noting that all these applications depend heavily on the dynamical behaviors such as stability, convergence, and oscillatory [2–6]. Meanwhile, stability is an important consideration in the designs and applications of neural networks. The stability of delayed neural networks is a subject of current interest, and therefore considerable theoretical efforts have been put into this topic followed by a large number of stability criteria reported; for example, see [7–12] and the references therein.

In real world, however, many evolutionary processes are characterized by abrupt changes at certain instants which may be caused by switching phenomena, frequency changes, or other sudden noises. As such, in the past few years, scientists have become gradually interested in the influence that impulses may have on the CGNNs and DCGNNs, thus obtaining some related results; for example, see [13–18] and the references therein.

Actually, besides impulsive effects, we have to recognize that diffusion effects are also nonignorable in reality as diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields. On this account, the model of neural networks with both impulses and diffusion should be more effective for describing the evolutionary process of practical systems. Based on this consideration, we wonder what the influence of diffusion on the stability of CGNNs and DCGNNs is.

So far there have appeared a few theoretical achievements [19–29] on the stability of impulsive reaction-diffusion neural networks with or without delays. Particularly, in [21–26], the main research technique is the impulsive differential inequality whereby the authors discussed the stability of equilibrium point and provided a series of sufficient conditions independent of diffusion. From these results, we fail to see the influence of diffusion on the stability of CGNNs and DCGNNs.

Encouragingly, recently there were reported some new results on the stability of CGNNs and DCGNNs in [19, 20, 27]; thereinto, the presented stability criteria derived from the impulsive differential inequality are related to the diffusion terms, and thereby we know the diffusion do contribute to the stability of impulsive neural networks.

In this paper, different from [20, 27], we shall consider the case where the boundary condition is Dirichlet boundary condition rather than Neumann boundary condition. Moreover, unlike [19], we shall utilize the new method of Poincarè inequality to deal with the reaction-diffusion terms, and Gronwall-Bellman-type impulsive integral inequality is also introduced for stability analysis. The obtained results show that not only the reaction-diffusion coefficients but also the first eigenvalue of corresponding Dirichlet Laplacian can affect the stability.

The rest of this paper is structured as follows. In Section 2, the model of impulsive delayed Cohen-Grossberg neural networks with reaction-diffusion terms as well as Dirichlet boundary condition is outlined and some facts and lemmas are introduced for later reference. By the new agencies of Gronwall-Bellman-type impulsive integral inequality and Poincaré inequality, we discuss the global exponential stability of equilibrium point and develop some new and concise algebraic criteria in Section 3. To conclude, two illustrative examples are given in Section 4 to verify the effectiveness of our results.

#### 2. Preliminaries

Let denote the -dimensional Euclidean space, and let be an open bounded domain with smooth boundary and . Let and .

Consider the following impulsive CGNN with time-varying delays and reaction-diffusion terms: where corresponds to the numbers of units in a neural network, , denotes the state of the th neuron at time and in space represents transmission diffusion of the th unit, represents the amplification function, is the appropriate behavior function, activation function stands for the output of the th unit at time and in space and and are constants: indicates the connection strength of the th unit on the th unit at time and in space , while denotes the connection weight of the th unit on the th unit at time and in space , where corresponds to the transmission delay along the axon of the jth unit satisfying and . is the sequence of impulsive moments meeting and and represent the right-hand and left-hand limit of at time and in space , respectively. stands for the abrupt change of at impulsive moment and in space .

Denote by , the solution of systems (1)-(2), satisfying the initial condition and Dirichlet boundary condition where the vector-valued function is such that is bounded on .

The solution of problems (1)–(4) is, for the time variable , a piecewise continuous function with the first kind discontinuity at the points , where it is left-continuous; that is, the following relations are valid:

Throughout this paper, we define the norm of as and make the following assumptions for convenience. is continuous and bounded; that is, there exist constants and such that is continuous and ; moreover, there exists constant such that is continuous and ; furthermore, there exists constant such that is continuous and for and .

In the light of (H1)–(H4), it is easy to see that problems (1)-(2) admit an equilibrium point .

*Definition 1. *The equilibrium point of problems (1)-(2) is said to be globally exponentially stable if there exist constants and such that
where .

Lemma 2 (see [30] (Gronwall-Bellman-type impulsive integral inequality)). *Assume the following.*(A1)* The sequence satisfies , with .*(A2)* and is left-continuous at .*(A3)* and for ,
**where and . Then,
*

Lemma 3 (see [31] (Poincaré inequality)). * Let be a bounded region in , and on the boundary of ; then
**
where is the smallest positive eigenvalue of the following problem:
*

Lemma 4. *If and , then holds for any .*

#### 3. Main Results

Theorem 5. *Assume the following.*(1)* and denote .*(2)*.*(3)*There exists a constant satisfying and , where
**Then, the equilibrium point of systems (1)-(2) is globally exponentially stable with convergence rate .*

*Proof. *Multiplying both sides of (1) by , we get
which yields, after integrating with respect to spatial variable on ,
where ,

By combining Green formula, Dirichlet boundary condition, Lemma 3, and condition (1) of Theorem 5, we obtain

Moreover, it follows from assumptions (H1), (H2), and (H3) that

Consequently, substituting (19)–(22) into (17) produces
for , , .

Now define Lyapunov function as . It is not difficult to see that is a piecewise continuous function with points of discontinuity of the first kind , where it is continuous from the left; that is, (). In addition, for (), we know
as and , supported by condition 2 of Theorem 5.

Put , . It is derived from (23) that

Define function of the form again. From (25), one then reads
where and .

Construct , where satisfies and . Evidently, is also a piecewise continuous function with the first kind discontinuous points , in which it is continuous from the left; that is, . Moreover, at , we find by the use of (24)

Set . By virtue of (26), one has

Choose small enough . Integrating (28) from to gives
which yields, after letting in (29),

Next, we estimate the value of at . For small enough , we put . Now an application of (30) leads to, for ,

If we let in (31), there results

Note that is applicable for . Thus,
holds for . By synthesizing (30) and (33), we then arrive at

This, together with (27), results in
for .

Recalling the assumptions that and , we therefore obtain
Hence,

By induction argument, we reach
Thus,

Since
we claim

According to Lemma 2, we assert that
which reduces to
This completes the proof.

*Remark 6. *According to the conditions of Theorem 5, we see that the reaction-diffusion terms can influence the stability of equilibrium point . Specifically, the acting factors include the reaction-diffusion coefficients and the first eigenvalue of corresponding Dirichlet Laplacian.

*Remark 7. *It is not difficult to see that there must exist constant satisfying condition 3 of Theorem 5 if .

Theorem 8. *Assume the following.*(1)* and denote .*(2)*, , .*(3)*.*(4)*There exists a constant satisfying and , where and .*

Then, the equilibrium point of systems (1)-(2) is globally exponentially stable with convergence rate .

*Proof. *Define Lyapunov function of the form , where . Obviously, is a piecewise continuous function with the first kind discontinuous points , where it is continuous from the left; that is, . Furthermore, for , we derive from condition 2 of Theorem 8 that
Thereby,

Construct function again, where satisfies and . Then, is also a piecewise continuous function with the first kind discontinuous points , where it is continuous from the left; that is, . And for , it follows from (45) that

Set . Following the same procedure as shown in the proof of Theorem 5, we get

The relations (46) and (47) yield
By induction argument, we obtain
Hence,

Introducing as shown in the proof of Theorem 5 into (50), (50) becomes, for ,
It then results from Lemma 2 that, for ,

On the other hand, since , one has . Thereby,
and (52) can be rewritten as
which implies
The proof is completed.

Due to Lemma 4, we know that the following inequalities: hold for any . Thus, in a similar way to the proofs of Theorems 5–8, we can prove the following theorems.

Theorem 9. *Assume the following.*(1)* and denote .*(2)*, .*(3)*There exist constants and such that and , where , and .**
Then, the equilibrium point of systems (1)-(2) is globally exponentially stable with convergence rate .*

*Remark 10. *There must exist constant satisfying condition 3 of Theorem 9 if there are constants such that .

Theorem 11. *Assume the following.*(1)* and denote .*(2)*, , .*(3)*.*(4)*There exist constants and satisfying and , where
**Further, on the condition that , where and , we obtain
**
for . In an identical way with the proof of Theorem 8, we can present the following.*

Theorem 12. *Assume the following.*(1)*Let and denote .*(2)*, where and .*(3)*.*(4)*There exist constants and such that and , where
*

*Remark 13. *Different from Theorems 5–11, the impulsive part in Theorem 12 could be nonlinear, and this will be of more applicability. Actually, Theorems 5–11 can be regarded as the special cases of Theorem 12.

#### 4. Examples

*Example 14. *Consider problems (1)–(4) with ; moreover, , , , , , , , , , , and .

As and , we know . Further, for , , , and , we compute

Let . Since , we therefore conclude from Theorem 5 that the zero solution of this system is globally exponential stable.

*Example 15. *Consider problems (1)–(4) with ; moreover, , , , , , , , , , , , and .

As and , we know . Further, for , , and , we compute

Let , and ; we can find such that