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Research Article | Open Access
Exponential Stability of Impulsive Delayed Reaction-Diffusion Cellular Neural Networks via Poincaré Integral Inequality
This work is devoted to the stability study of impulsive cellular neural networks with time-varying delays and reaction-diffusion terms. By means of new Poincaré integral inequality and Gronwall-Bellman-type impulsive integral inequality, we summarize some novel and concise sufficient conditions ensuring the global exponential stability of equilibrium point. The provided stability criteria are applicable to Dirichlet boundary condition and show that not only the reaction-diffusion coefficients but also the regional features including the boundary and dimension of spatial variable can influence the stability. Two examples are finally illustrated to demonstrate the effectiveness of our obtained results.
Cellular neural networks (CNNs), proposed by Chua and Yang in 1988 [1, 2], have been the focus of a number of investigations due to their potential applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision [3–7]. As the switching speed of neurons and amplifiers is finite in the implementation of neural networks, time delays are inevitable and therefore a type of more effective models is afterwards introduced, called delayed cellular neural networks (DCNNs). Actually, DCNNs have been found to be helpful in solving some dynamic image processing and pattern recognition problems.
As we all know, all the applications of CNNs and DCNNs depend heavily on the dynamic behaviors such as stability, convergence, and oscillatory [8, 9], wherein stability analysis is a major concern in the designs and applications. Correspondingly, the stability of CNNs and DCNNs is a subject of current interest and considerable theoretical efforts have been put into this topic with many good results reported (see, e.g., [10–13]).
With reference to neural networks, however, it is noteworthy that the state of electronic networks is often subject to instantaneous perturbations which may be caused by a switching phenomenon, frequency change, or other sudden noise. On this account, neural networks will experience abrupt change at certain instants, exhibiting impulse effects [14, 15]. For instance, according to Arbib  and Haykin , when a stimulus from the body or the external environment is received by receptors, the electrical impulses will be conveyed to the neural net and impulse effects arise naturally in the net. In view of this discovery, many scientists have shown growing interests in the influence that the impulses may have on CNNs or DCNNs with a result that a large number of relevant results have been achieved (see, e.g., [18–24]).
Besides impulsive effects, diffusing effects are also nonignorable in reality since the diffusion is unavoidable when the electrons are moving in asymmetric electromagnetic fields. Therefore, the model of impulsive delayed reaction-diffusion neural networks appears as a natural description of the observed evolution phenomena of several real world problems. This one acknowledgement poses a new challenge to the stability research of neural networks.
So far, there have been some theoretical achievements [25–33] on the stability of impulsive delayed reaction-diffusion neural networks. Previously, authors of [27–32] studied the stability of impulsive delayed reaction-diffusion neural networks and put forward several stability criteria by impulsive differential inequality and Green formula, wherein the reaction-diffusion term is evaluated to be less than zero by means of Green formula and thereby the presented stability criteria are shown to be wholly independent of diffusion. According to this result, we fail to see the influence of diffusion on stability.
Recently, it is encouraging that, for impulsive delayed reaction-diffusion neural network, some new stability criteria involving diffusion are obtained in [25, 26, 33–36]. Meanwhile the estimation of reaction-diffusion term is not merely less than zero, instead a more accurate one is given; that is, the reaction-diffusion term is verified to be less than a negative definite term by using some inequalities together with Green formula. It is thereby testified that the diffusion does contribute to the stability of impulsive neural networks.
In , the authors quoted the following inequality to deal with the reaction-diffusion terms: where is a cube and is a real-valued function belonging to . We can easily derive from this inequality that For better exploring the influence of diffusion on stability, we wonder if we can get a more accurate estimate of reaction-diffusion term. Fortunately, we find the following new Poincaré integral inequality supporting this idea: One can refer to Lemma 3 in Section 2 for the details of this inequality.
On the other hand, it is well known that the theory of differential and integral inequalities plays an important role in the qualitative and quantitative study of solution to differential equations. Up till now, there have been many applications of impulsive differential inequalities to impulsive dynamic systems, followed by lots of stability criteria provided. However, these stability criteria appear a bit complicated and we wonder if we can deduce relatively concise stability criteria by using impulsive integral inequalities.
Motivated by these, we attempt to, for impulsive delayed neural networks, employ new Poincaré integral inequality to further investigate the influence of diffusion on the stability and combine Gronwall-Bellman-type impulsive integral inequality so as to provide some new and concise stability criteria. The rest of this paper is organized as follows. In Section 2, the model of impulsive cellular neural networks with time-varying delays and reaction-diffusion terms as well as Dirichlet boundary condition is outlined; in addition, some facts and lemmas are introduced for later reference. In Section 3, we provide a new estimate on the reaction-diffusion term by the agency of new Poincaré integral inequality and then discuss the global exponential stability of equilibrium point by utilizing Gronwall-Bellman-type impulsive integral inequality with a result of some novel and concise stability criteria presented. To conclude, two illustrative examples are given in Section 4 to verify the effectiveness of our obtained results.
Let and . Let denote the n-dimensional Euclidean space, and let be a fixed rectangular region in and . As usual, denote
Consider the following impulsive cellular neural network 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; activation function stands for the output of the th unit at time and in space ; , , and are constants: indicates the connection strength of the th unit on the th unit at time and in space , 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 th unit, satisfying and ), 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 at time and in space . The fixed moments () are called 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 the impulsive moment and in space .
The solution of problem (5)–(8) is, for the time variable , a piecewise continuous function with the first kind discontinuity at the points , where it is continuous from the left; that is, the following relations are true:
Throughout this paper, the norm of is defined by
Before proceeding, we introduce two hypotheses as follows:(H1) satisfies , and there exists a constant such that for all and .(H2) is continuous and , ,??.
Lemma 2 (see  Gronwall-Bellman-type Impulsive Integral Inequality). Assume that(A1) the sequence satisfies , with ,(A2) and is left-continuous at , ,(A3) and for , where and const. Then,
Lemma 3 (see  Poincaré integral inequality). Let be a fixed rectangular region in and . For any ,
Remark 4. According to Lemma??2.1 in , we know if is a cube () and is a real-valued function belonging to , then which yields Through the simple example as follows, we can find that in some cases the estimate shown in Lemma 3 can do better. Let , we derive from Lemma??2.1 in  that whereas the application of Lemma 3 of this paper will give which is obviously superior to .
3. Main Results
Theorem 5. Provided that one has the following:(1)let and denote ;(2), ;(3)there exists a constant satisfying as well as , where ,??;
then, the equilibrium point of problem (5)–(8) is globally exponentially stable with convergence rate .
Proof. Multiplying both sides of (5) by and integrating with respect to spatial variable on , we get
Regarding the right-hand part of (19), the first term becomes by using Green formula, Dirichlet boundary condition, Lemma 3, and condition of Theorem 5
Moreover, From (H1), we have
Consequently, substituting (20)–(22) into (19) produces for , , ,??.
Define a Lyapunov function as . It is easy to find that is a piecewise continuous function with the first kind discontinuous points (), where it is continuous from the left, that is, ??(). In addition, we also see as and the following estimate derived from condition of Theorem 5:
Put , . It then results from (23) that
Choose of the form . From (26), one reads where and .
Now construct again, where satisfies and . Evidently, is also a piecewise continuous function with the first kind discontinuous points ??(), where it is continuous from the left, that is, . Moreover, at ??(), we find by use of (24)
Set ,??. By virtue of (27), one has
Choose small enough . Integrating (29) from to gives which yields, after letting in (30),
Next we will estimate the value of at , . For small enough , we put . An application of (31) leads to, for ,
If we let in (32), there results
Note that is applicable for . Thus, holds for . By synthesizing (31) and (34), we then arrive at
This, together with (28), results in for ,??.
Recalling assumptions that and (), we obtain
By induction argument, we reach
Since we claim
According to Lemma 2, we know which reduces to This completes the proof.
Remark 6. According to Theorem 5, we see that the diffusion can really influence the stability of equilibrium point of problem (5)–(8), wherein the factors embrace not only the reaction-diffusion coefficients but also the regional features including the dimension and boundary of spatial variable. Owing to the employ of new Poincaré integral inequality, in this paper, the estimation of reaction-diffusion terms is superior to that in  in some cases, and this will be helpful to further know the influence of diffusion on stability. What is more, from condition of Theorem 5, we also see that the dimension of spatial variable has an impact on the stability while this is not mentioned in .
Remark 7. Among the three conditions of Theorem 5, condition is critical and therefore we must ensure the existence of constant . Fortunately, it is not difficult to find that there must exist a constant satisfying condition if which is easily checked.
Theorem 8. Providing that one has the following:(1)let and denote ;(2), , ;(3);(4)there exists a constant which satisfies and ,??where and ;then, the equilibrium point of problem (5)–(8) 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, when ??(), it follows from condition of Theorem 8 that
Construct another Lyapunov function , where satisfies and . Then, is also a piecewise continuous function with the first kind discontinuous points , , where it is continuous from the left, and for ?(), it results from (46) that
Set ,??. Following the same procedure as in Theorem 5, we get
The relations (47) and (48) yield
By induction argument, we reach Hence,
Introducing as shown in the proof of Theorem 5 into (51), (51) becomes It then results from Lemma 2 that, for ,
On the other hand, since , one has . Thereby, and (53) can be rewritten as which implies The proof is completed.
Theorem 9. Provided that one has the following:(1)let and denote ;(2), ;(3)there exist constants and such that and , where and ;then, the equilibrium point of problem (5)–(8) is globally exponentially stable with convergence rate .
Theorem 11. Assume that one has the following:(1)let and denote ;(2), , ;(3);(4)there exist constants and such that and , where and ;then, the equilibrium point of problem (5)–(8) is globally exponentially stable with convergence rate .
Further, on the condition that , where and , we obtain, for ),
Identical with the proof of Theorem 8, we reach the following.
Theorem 12. Assume that one has the following:(1)let and denote ;(2), where and ;(3);(4)there exist constants and such that and , where and ;then, the equilibrium point of problem (5)–(8) is globally exponentially stable with convergence rate .
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.
For and , we compute . This, together with , yields
Let . Since , we conclude from Theorem 5 that the equilibrium point of this system is globally exponentially stable.
For and , we compute . This, together with and , yields
Letting , , , and , we can find satisfying