Research Article | Open Access
Stability of the Stochastic Reaction-Diffusion Neural Network with Time-Varying Delays and p-Laplacian
The main aim of this paper is to discuss moment exponential stability for a stochastic reaction-diffusion neural network with time-varying delays and p-Laplacian. Using the Itô formula, a delay differential inequality and the characteristics of the neural network, the algebraic conditions for the moment exponential stability of the nonconstant equilibrium solution are derived. An example is also given for illustration.
In many neural networks, time delays cannot be avoided. For example, in electronic neural networks, time delays will be present due to the finite switching speed of amplifies. In fact, time delays are often encountered in various engineering, biological, and economical systems. On the other hand, when designing a neural network to solve a problem such as optimization or pattern recognition, we need foremost to guarantee that the neural networks model is globally asymptotically stable. However, the existence of time delay frequently causes oscillation, divergence, or instability in neural networks. In recent years, the stability of neural networks with delays or without delays has become a topic of great theoretical and practical importance (see [1–16]).
The stability of neural networks which depicted by partial differential equations was studied in [6, 7]. Stochastic differential equations were employed to research the stability of neural networks in [8–11], while [12, 13] used stochastic partial differential equations to analysis this question. In , the authors studied almost exponential stability for a stochastic recurrent neural network with time-varying delays. In addition, moment exponential stability for a stochastic reaction-diffusion neural network with time-varying delays is discussed in .
In this paper, we consider the stochastic reaction-diffusion neural network with time-varying delays and -Laplacian as follows: In (1.1), , is a common number. is a bounded convex domain with smooth boundary and measure . denotes the numbers of neurons in the neural network, corresponds to the state of the th neurons at time and in space , the is an amplification function. is output. denotes the output of the th neuron at time and in space , namely, activation function which shows how neurons respond to each other. is an m-dimensional Brownian motion which is defined on a complete probability space with a natural filtration (i.e., ). , . denotes the intensity of the stochastic perturbation. Functions and are subject to certain conditions to be specified later. is a real constant matrix and represents weight of the neuron interconnections, namely, denotes the strength of th neuron on the th neuron at time and in space , and corresponds to axonal signal transmission delay.
2. Definitions and Lemmas
Throughout this paper, unless otherwise specified, let denote Euclidean norm. Define that and where . Denote by the family of continuous functions from to . For every and , denote by × the family of all -measurable × valued random variables such that , where , denotes the expectation of random variable .
Definition 2.3. The nonconstant equilibrium solution of (1.1) about the given norm is called exponential stability in th moment, if there are constants , for every stochastic field solution of problem (1.1)–(1.3) such that
The constant on the right hand side in (2.3) is called Lyapunov exponent of every solution of problem (1.1)–(1.3) converging on equilibrium about norm .
In order to obtain th moment exponential stability for a nonconstant equilibrium solution of problem (1.1)–(1.3), we need the following lemmas.
Lemma 2.4 (see ). Let and be two real matrices. The continuous function satisfies the delay differential inequalities If for and and is an M-matrix, then there are constants , such that where are initial functions. is right-hand upper derivate. represents a norm.
Lemma 2.5 (see ). Let , then there are positive constants and for any such that
Remark 2.6. If , Lemma 2.5 also holds with .
Suppose that , , and are Lipschitz continuous such that the following conditions hold: , ,, ,, for all , , where , , and are positive constants.
3. Main Result
Theorem 3.1. Let and – hold. Assume that there are positive constants such that the matrix is an M-matrix, where then the nonconstant equilibrium solution of problem (1.1)–(1.3) about norm is exponential stability in th moment, that is, there are constants and , for any and any such that
Proof. Set . For every and , by means of Itô formula and , one has that
Both sides of Inequality (3.3) are integrated about over . Set . One has that
Set . By (1.2), one has that
where is unit outer cotangent vector on . By (3.4), (3.5), (H1), and Young’s inequality, one has that
where and are defined by (3.1).
For , both sides of (3.6) are integrated about from to , then both sides of (3.6) are calculated expectation. By the properties of Brownian motion, one has that Since the integrals and are finite, by Fubini theorem  and (3.7), one obtain that
Set . Both sides of Inequality (3.8) are divided by , let , one has the following inequality: By Lemma 2.4, there are positive constants , such that where is initial value. Set , then By (3.11) and Lemma 2.5, one obtains that
In order to prove Theorem 3.1, we need the following lemma.
Proof. Set . Similar to (3.8) in proof of Theorem 3.1, one has that
By (3.13) and the assumption that is an M-matrix, one obtains that
Because of and , one has that .
We continue the proof of Theorem 3.1 as the following.
By Lemma 3.2, one has that where is a common number. We derive every solution of problem (1.1)–(1.3) such that then a nonconstant equilibrium solution of problem (1.1)–(1.3) about norm is exponential stability in th moment. The proof of Theorem 3.1 is complete.
In order to illustrate the application of the theorem, we give an example.
Example 3.3. Discuss the stochastic reaction-diffusion neural network with time-varying delays and -Laplacian as the following: where Set as a nonconstant equilibrium solution of (3.17) and (3.18). One can derive that Taking and , one has that and is an -matrix. The nonconstant equilibrium solution of (3.17) and (3.18) about norm is exponential stability in the 3rd moment.
The authors thank the reviewers for their constructive comments.This work is supported by the National Science Foundation of China (no. 10971240).
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