Existence and Stability of Almost Periodic Solution for a Stochastic Cellular Neural Network with External Perturbation

Abstract

A class of stochastic cellular neural networks with external perturbation is investigated. By employing fixed points principle and some stochastic analysis techniques, we establish some sufficient conditions for existence and exponential stability of a quadratic mean almost periodic solution of the model. The new criteria not only improve some classical results but also are applied in real problems due to the changes of external input.

1. Introduction

Cellular neural networks (CNNs) and the various generalizations have attracted many scientists' attention due to their important applications, such as associative memory, optimization problems, parallel computation, and so on [1–7]. Huang et al. [7] studied almost periodic solutions of a delayed cellular neural networks as follows: The authors obtained some good criteria ensuring exponential global attractivity of almost periodic solution to (1).

The concept of almost periodic stochastic process is of great importance in probability for investigating stochastic process [8]. Recently, the existence and stability of almost periodic solution to stochastic cellular neural networks were considered [9]. To the best of our knowledge, there are few works about the quadratic mean almost periodic solution for stochastic cellular neural networks. Motivated by [7–13], in this paper, we will consider the existence and exponential stability of quadratic mean almost periodic for stochastic cellular neural networks with distributed delay as follows: where and is the number of neurons in the network; denotes the state variable of the th neuron; and denote the activation function of the th neuron; the feedback function and indicate the strength of the neuron interconnections within the network; represents an amplification function; represents external input; can be viewed as a stochastic perturbation on the neuron states and is a Brownian motion; is a variable delay function of the neuron and the kernel function satisfies . Some sufficient conditions ensuring the existence and stability of square mean almost periodic solutions are shown. The results in this paper improve some previous results and are applied in real problems such as signal processing and the design of networks and secure communication.

The rest of the paper is organized as follows. In Section 2, we introduce some definitions, lemmas, and some notations which would be useful to get the main results. In Section 3, the main results of existence and stability to (2) are obtained.

2. Preliminaries

Now let us state the following definitions and lemmas, which will be used to prove our main results.

Let be a Banach space and let be a probability space. Define for to be the space of all -value random variable such that It is easy to find that is a Banach space when it is equipped with its natural norm defined by

Definition 1. A stochastic process is said to be continuous whenever

Definition 2. A stochastic process is said to be stochastically bounded whenever

Definition 3. A stochastic process is said to be -mean almost periodic if for each there exists such that any interval of length contains at least a number for which The number will be called an -translation of and the set of all -translation of is denoted by .

The collection of all stochastic processes which are -mean almost periodic is denoted by . Let denote the collection of all stochastic processes , which are continuous and uniformly bounded. Obviously, is a Banach space when it is equipped with the norm

Lemma 4 (see [8]). is a closed subspace.

Lemma 5 (see [8]). If belongs to , then (i)the mapping is uniformly continuous;(ii)there exists a constant such that , for each ;(iii) is stochastically bounded.

Let and be Banach spaces and let and be their corresponding -spaces, respectively.

Definition 6. A function , which is jointly continuous, is said to be -mean almost periodic in uniformly in where is compact if, for any , there exists such that any interval of length constants at least a number for which for each stochastic process . The number will be called an -translation of and the set of all -translation of is denoted by .

Lemma 7 (see [8, 9]). Let , be a -mean almost periodic process in uniformly in , where is compact. Suppose that is Lipschitzian in the following sense: for all and for each , where . Then for any -mean almost periodic process ; then stochastic process is -mean almost periodic.

We need to introduce the following notations. For every real sequence and a continuous stochastic process , if exists, we define . Like the proof of Fink [14], we have the following lemma.

Lemma 8. is -mean almost periodic if and only if is continuous and, for each , there exists a subsequence of such that uniformly on .

Lemma 9. If are square almost periodic stochastic process, then is square mean almost periodic.

Proof. It is obvious that is continuous for ; that is, . For any sequence , since are square almost periodic, we have uniformly for . On the other hand, since is almost periodic, it is uniformly continuous on . For any , there exists a positive number , such that implies that . From (11), there exists a positive integer , when , we have Since , we have when . Thus, is square mean almost periodic.

3. Main Results

In this section, we state and prove our main results concerning the existence and stability of square mean almost solutions of (2). Throughout the rest of the paper, the following assumptions are satisfied.(H1)The functions are square mean almost periodic functions, where .(H2)The activation functions are square mean almost periodic, and and are Lipschitz in the following sense: there exist and for which (H3)The functions are square mean almost periodic in uniformly in being a compact subspace). Moreover, is Lipschitz in the following sense: there exists for which Let denote the signs ,  and .

Theorem 10. Assume that conditions (H1)–(H3) are satisfied and ; then (2) has a unique square mean almost periodic solution.

Proof. By (2), we can obtain that for all , given by (16) is the solution to (2).
Define , where For , we show that is square mean almost periodic if is. Assuming that is square mean almost periodic and applying conditions (H1) and (H2) and Lemmas 7 and 9, one can easily obtain that , is square mean almost periodic. Therefore, for each , there exists such that any interval of length contains at least for which for each .
Now, by using Cauchy-Schwarz inequality, we can write According to the above, for , for each ; that is, is square mean almost periodic.
Next, for , we show that is square mean almost periodic if is. Since is square mean almost periodic, for each there exists such that any interval of length contains at least for which for each .
So, we have Similarly, from (20)–(22), by using Cauchy-Schwarz inequality, we obtain that So, for , for each ; that is, is square mean almost periodic. From (H1), it is easy to see that .
Now, we prove that provided . From (H3) and Lemma 7, one can see that is square mean almost periodic for ; for each there exists such that any interval of length contains at least for which for each . We make extensive use of the Ito's isometry identity and the properties of defined by for each . And is also a Brownian motion and has the same distribution as . Now, By making a change of variables , we get Thus, applying Ito's isometry identity, we have which implies that .
From the above, for , it is clear that maps into itself. In the next section, we show that is a contraction mapping. Actually, for , we have Since , we can get Firstly, we evaluate the first term of the right hand side as follows: Secondly, we evaluate the second term of the right hand side as follows: As to the third term, we apply isometry identity and have Thus, for , by combining (30), (31), and (32), we can obtain that and it follows that From , it suffices to show that has a unique fixed point, which is clearly the unique square mean almost periodic solution to (2). The proof is completed.

Remark 11. According to the conditions of Theorem 10, we can find that the activation function of the neuron, the feedback function, the neuron interconnections, amplification function, external input, and stochastic perturbation have key effect on the unique of square mean almost periodic solution.

Theorem 12. Assume that the conditions of Theorem 10 are held; then the square mean almost periodic solution of system (2) is global exponential stability.

Proof. Take a positive constant such that . For , it follows from (16) that Now, we evaluate the first term on the right hand side of (35); we obtain For any and any , there exists such that for . Therefore When , , that is, there exists such that for any , one can get Thus, from (37) and (38), we have for any , when , Like the discussion of other terms on the right hand side of (35), as , we have Therefore, from the above, we get that () as . Thus we know that (2) has a unique square mean almost periodic solution, which is an exponentially stable.

Remark 13. From the assumptions of Theorem 10, we can see that stochastic perturbations on the neuron states also have key effect on the exponentially stability of square mean almost periodic.

Conflict of Interests

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

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11371027, 11071001, and 11201109) and Anhui Province Natural Science Foundation (1208085MA13 and 1408085QF116).