Asymptotic Stability of Impulsive Reaction-Diffusion Cellular Neural Networks with Time-Varying Delays
This work addresses the asymptotic stability for a class of impulsive cellular neural networks with time-varying delays and reaction-diffusion. By using the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality as well as piecewise continuous Lyapunov functions, we summarize some new and concise sufficient conditions ensuring the global exponential asymptotic stability of the equilibrium point. The provided stability criteria are applicable to Dirichlet boundary condition and showed to be dependent on all of the reaction-diffusion coefficients, the dimension of the space, the delay, and the boundary of the spatial variables. 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]. Moreover, on the ground that time delays are unavoidably encountered for the finite switching speed of neurons and amplifiers in implementation of neural networks, it was followed by the introduction of the delayed cellular neural networks (DCNNs) so as to solve some dynamic image processing and pattern recognition problems. Such applications concerning CNNs and DCNNs depend heavily on the dynamical behaviors such as stability, convergence, and oscillatory [8, 9]. Particularly, stability analysis has been a major concern in the designs and applications of the CNNs and DCNNs. 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 actually subject to instantaneous perturbations more often than not. On this account, the networks experience abrupt change at certain instants which may be caused by a switching phenomenon, frequency change, or other sudden noise; that is, the networks often exhibit impulsive 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 impulsive effects arise naturally in the net. As a consequence, in the past few years, scientists have become increasingly interested in the influence that impulses may have on the CNNs and DCNNs and a large number of stability criteria have been derived (see, e.g. [18–22]).
In reality, besides impulsive effects, diffusion effects are also nonignorable since diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields. As such, the model of neural networks with both impulses and reaction-diffusion should be more accurate to describe the evolutionary process of the systems in question, and it is necessary to consider the effects of both diffusion and impulses on the stability of CNNs and DCNNs.
In the past years, there have been a few theoretical contributions to the stability of CNNs and DCNNs with impulses and diffusion. For instance, Qiu  formulated a mathematical model of impulsive neural networks with time-varying delays and reaction-diffusion terms described by impulsive partial differential equations and studied, via delay impulsive differential inequality, the problem of global exponential stability with some stability criteria presented. Remarkably, all of the obtained stability criteria in  are independent of the diffusion. In 2008, Li and song  investigated a class of impulsive Cohen-Grossberg networks with time-varying delays and reaction-diffusion terms. By establishing a delay inequality with impulsive initial conditions and M-matrix theory, some sufficient conditions ensuring global exponential stability of the equilibrium points are given. Analogous to , the proposed stability criteria in  are also independent of the diffusion. More recently Pan et al.  investigated a class of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion in 2010. By the aid of the delay impulsive differential inequality quoted in , several sufficient conditions are exploited ensuring global exponential stability of the equilibrium points. Especially, different from [23, 24], the estimate of the exponential convergence rate depends on reaction-diffusion in .
In this paper, unlike the methods of impulsive differential inequalities and Poincaré inequality used in , we attempt to adopt the new techniques of the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality to investigate the problem of global exponential asymptotic stability for impulsive cellular neural networks with time-varying delays and reaction-diffusion terms. Different from the existing research, we find, besides the reaction-diffusion coefficients, the dimension of the space and the boundary of the spatial variables do influence the stability.
The rest of the paper is organized as follows. In Section 2, the model of impulsive delayed cellular neural networks with reaction-diffusion terms and Dirichlet boundary condition is outlined, and some facts and lemmas are introduced for later reference. By the new agency of the impulsive integral inequality of Gronwall-Bellman type as well as Hardy-Sobolev inequality, we discuss the global exponential asymptotic stability and develop some new criteria in Section 3. To conclude, two illustrative examples are given to verify the effectiveness of our results in Section 4.
Let denote the n-dimensional Euclidean space, and is a bounded open set containing the origin. The boundary of is smooth and . Let and .
We consider the following impulsive neural networks with time delays and reaction-diffusion terms: where corresponds to the numbers of units in a neural network; , , denotes the state of the ith neuron at time and in space ; smooth functions represent transmission diffusion operators of the ith unit; activation functions stand for the output of the jth unit at time and in space ; , , are constants: indicates the strength of the jth unit on the ith unit at time and in space , denotes the strength of the jth unit on the ith unit at time and in space , where corresponds to the transmission delay along the axon of the jth unit and satisfies () as well as (), and represents the rate with which the ith 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 satisfying 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 system (2.1)-(2.2), satisfying the initial condition and Dirichlet boundary condition where the vector-valued function is such that is bounded on and () is first-order continuous differentiable as to on .
The solution of problems ((2.5)–(2.8)) 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 governed by
Before moving on, we introduce two hypotheses as follows.(H1) Activation function satisfies , and there exists constant such that holds for all and .(H2) The functions are continuous on and , , .
Lemma 2.2 (Gronwall-Bellman-type impulsive integral inequality ). Assume that
(A1) the sequence satisfies , with ,
(A2) and is left-continuous at , ,
(A3) and for where and . Then,
Lemma 2.3 (Hardy-Sobolev inequality ). Let () be a bounded open set containing the origin and . Then there exists a positive constant such that
Lemma 2.4. If and , then holds for any .
3. Main Results
Theorem 3.1. Provided that(1)for (), there exists a constant such that . In addition, there exists a constant such that . Denote ,(2), ,(3)there exists a constant satisfying as well as , where , ,then, the equilibrium point of problems ((2.5)–(2.8)) is globally exponentially stable with convergence rate – .
Proof. Multiplying both sides of (2.1) by and integrating with respect to spatial variable on , we get
Regarding the right-hand part of (3.1), the first term becomes by using Green formula, Dirichlet boundary condition, Lemma 2.3, and condition 1 of Theorem 3.1
Moreover, we derive from (H1) that Consequently, substituting ((2.10)–(3.14)) into (3.1) produces .
We define a Lyapunov function as . It is easy to find 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, due to and the following estimate derived from condition 2 of Theorem 3.1 we have holds for (). Put , . Then for the derivative of with respect to problems ((2.5)–(2.8)), it results from (3.4) that Choose of the form . From (3.7), one then reads
Construct , where satisfies and . Evidently, is also a piecewise continuous function with points of discontinuity of the first kind (), in which it is continuous from the left, that is (). Moreover, at (), we find by use of (3.6)
Set , . By virtue of (3.8), one has
Choose small enough . Integrating (3.10) from to gives
which yields after letting in (3.11)
We now proceed to estimate the value of at , . For small enough , we put . Now an application of (3.12) leads to, for ,
If we let in (3.13), there results
Note that is applicable for . Thus,
holds for . By synthesizing (3.12) and (3.15), we then arrive at
This, together with (3.9), results in .
Recalling assumptions that and (), we have Hence,
By induction argument, we reach
Therefore, Since we claim
According to Lemma 2.2, we claim which reduces to This completes the proof.
Remark 3.2. According to the conditions of Theorem 3.1, we see that the reaction-diffusion term do influence the stability of problem ((2.5)–(2.8)). Moreover, besides the reaction-diffusion coefficients, the dimension of the space and the boundary of spatial variables have also an effect on the stability of the equilibrium point .
Theorem 3.3 .. Providing that for (), there exist constants such that , in addition, there exists constant such that ; denote ,, , ,, there exists constant which satisfies and , where and , then, the equilibrium point of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate –.
Proof. Define a Lyapunov function of the form , where . Obviously, is a piecewise continuous function with points of discontinuity of the first kind , , where it is continuous from the left, that is, (). Furthermore, for (), it follows from condition 2 of Theorem 3.3 that
Construct another Lyapunov function defined by , where satisfies and . Then, is also a piecewise continuous function with points of discontinuity of the first kind , , where it is continuous from the left, that is (). And for (), it results from (3.27) that
Set , . Following the same procedure as in Theorem 3.1, we get
The relations (3.28) and (3.29) yield
By induction argument, we arrive at Hence,
Introducing as shown in the proof of Theorem 3.1 into (3.32), the expression becomes
It then results from Lemma 2.2 that
On the other hand, since , one has . Thereby,
And (3.34) can be rewritten as which implies The proof is completed.
Remark 3.4. Theorem 3.1 is in fact the special case of Theorem 3.3 by choosing . Due to Lemma 2.4, we know
hold for any .
In the sequel, we follow the same procedures as in Theorems 3.1 and 3.3 to find the following theorems.
Theorem 3.5. Provided thatfor (), there exists a constant such that . in addition, there exists a constant such that ; denote ,, ,there exist constants and such that and , where and , then, the equilibrium point of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate –.
Theorem 3.6. Assume thatfor (), there exists a constant such that ; In addition, there exists a constant such that ; denote ,, , ,,there exist constants and such that and , where and .
Then, the equilibrium point of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate .
Further, on the condition that , where and , we obtain, for (),
Identical with the proof of Theorem 3.3, we present the theorem as follows.
Theorem 3.7. Assume thatfor (), there exists a constant such that . in addition, there exists a constant such that . Denote ,, where and ,,there exist constantsandsuch thatandwhere.
Then, the equilibrium point of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate .
Remark 3.8. Different from Theorems 3.1–3.6, the impulsive part in Theorem 3.7 could be nonlinear and this will be of more applicability. Actually, Theorems 3.1–3.6 can be regarded as the special cases of Theorem 3.7.
Example 4.1. Consider the following impulsive reaction-diffusion delayed neural network:
with the impulsive effects characterized by
and initial condition (2.3) and Dirichlet condition (2.4), where , , , , , , , , , and . For and , we compute . This, together with the chosen , yields
By selecting , and , we estimate
According to Theorem 3.1, we therefore conclude that the system in Example 4.1 is globally exponential stable.
Example 4.2. Consider the following impulsive reaction-diffusion delayed neural network:
with the impulsive effects featured by
and initial condition (2.3) and Dirichlet condition (2.4), where , , , , , , , , , , and . For and , we compute . This, together with and , yields
Select by setting . Hence, we compute by letting , , , and that
It is then concluded from Theorem 3.7 that the system in Example 4.2 is globally exponentially stable.
The work is supported by National Natural Science Foundation of China under Grant 60904028.
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