Periodicity of Nonautonomous Fuzzy Neural Networks with Reaction-Diffusion terms and Distributed Time Delays

Abstract

In this paper, the periodicity of a class of nonautonomous fuzzy neural networks with impulses, reaction-diffusion terms, and distributed time delays are investigated. By establishing an integro-differential inequality with impulsive initial conditions and time-varying coefficients, employing the -matrix theory, Poincar mappings, and fixed point theory, several new sufficient conditions to ensure the periodicity and global exponential stability of the formulated system are obtained. It is worthwhile to mention that our technical methods are practical, in the sense that all new stability conditions are stated in simple algebraic forms, and an optimization method is provided to estimate the exponential convergence rate, so their verification and applications are straightforward and convenient. The validity and generality of our methods are illustrated by two numerical examples.

1. Introduction

In 1996, Yang et al. studied fuzzy cellular neural networks (FCNNs) [1–3] by combining fuzzy logic with traditional cellular neural networks based on the previous cellular neural networks [4]. It was shown that FCNNs play an important role in image processing problems and pattern recognition. These applications rely heavily on the dynamic behavior of FCNNs. Therefore, it is particularly important to analyze the dynamics of FCNNs. As we all know, neural networks often have delays in the process of information processing. The existence of time delays may cause the systems to oscillate, diverge or become unstable. Neural dynamics considering the delay problems are very important for the stability and balance of the neural networks. Some scholars have studied the stability of FCNNs with constant and time-varying delays [5, 6], and some have studied the stability of FCNNs with distributed time delays [7] and leaky time delays [8]. Furthermore, diffusion effects in neural networks are unavoidable when electrons move in asymmetric electromagnetic fields. Therefore, we must consider that the activation is different in time and space. A number of neural network models with reaction-diffusion terms and various delays have been developed and studied [9–11].

On the other hand, in neural network systems, in addition to time delays and diffusion effects, there are impulse effects, which are because of the fact that many neural networks undergo abrupt changes at a given moment due to transient disturbances. These changes occur in the fields of physics, chemistry, population dynamics, etc. At the same time, impulse controls also have important theoretical significance and application values in the fields of optimal controls [12–14]. These changes occur in the fields of physics, chemistry, population dynamics, optimal control, etc. Some results about impulse effects have been obtained in time delays neural networks [12–20]. In particular, when we consider the long-term dynamic behaviours of systems, the parameters of the systems usually change over time and this nonautonomous phenomenons often occur in many practical systems. In [21], the authors studied the stability of a nonautonomous fuzzy neural network with reaction-diffusion terms without impulses. Long [22] studied the dynamic behaviors of nonautonomous cellular neural networks with time-varying delays. In [23], the authors studied the existence, uniqueness, and global stability of periodic solutions of general nonautonomous impulsive cellular neural networks and obtained some criteria.

Based on what we know, there are no results on the exponential stability of FCNNs with impulses, distributed time delays, and reaction-diffusion terms at the same time, which is very important in theories and applications. In terms of mathematical models, FCNNs have not only fuzzy logic but also impulse effects between its template input and/or output, except the sum of product operations. The models include reaction-diffusion terms, fuzzy logic, and impulse characteristics, which have complex dynamic behaviors. It is therefore necessary to further investigate the dynamic behaviours of FCNNs. We have used the properties of M-matrices and inequality tricks to establish a new differential inequality that yields a sufficient condition for global exponentially stable periodic solutions of the systems. Finally, the validity of the results are verified by means of arithmetic examples and numerical simulations using [24].

Consider the non-autonomous FCNNs, which contain the reaction-diffusion terms, distributed time delays, and impulses.where , , , is a bounded compact set with smooth boundary and mes in space ; represent the delay kernel function which is real-valued piecewise continuous. denotes the th neuron in space and at time ; and represent the signal activation function of the th neuron; denotes the rate of potential recovery to the isolated state of the th neuron at moment . and denote the elements of the feedback template and the feed-forward template at moment ; and represent the fuzzy AND and fuzzy OR operations, respectively; and , represent elements of fuzzy feedback MIN template and fuzzy feedback MAX template at time , respectively; and represent elements of fuzzy feed-forward MIN template and fuzzy feed-forward MAX template at time , respectively; and represent the input and bias of the th neuron at moment ; represents the transmission diffusion coefficient. In (1b), satisfies and represent the left and right limits at , respectively; shows impulsive perturbation of the th neuron at time . Let . Equation (1c) denote the Dirichlet boundary conditions and (1d) denote the initial conditions.

If the impulsive operator , we obtain the following systems (2a)–(2c):

Systems (2a)–(2c) are continuous forms of systems (1a)–(1d).

The main contributions of this manuscript are as follows:(a)A new neural network model is formulated, this model assembles nonautonomous fuzzy neural networks, and reaction-diffusion neural networks with distributed time delays, impulses, and Dirichlet boundary conditions. The systems formulated in this paper are more complex than previously considered models, but there is no doubt that some of the existing systems are also special cases of this system.(b)A new integro-differential inequality with impulsive initial conditions and time-varying coefficients is established, and it may be a powerful tool for analyzing impulsive nonautonomous systems.(c)Several new criteria to ensure the global exponential periodicity and stability of the studied system are obtained, these results are expressed in the form of simple algebraic inequalities which depend only on the formulated systems parameters, and their verification and applications are straightforward and convenient.

2. Preliminaries

In this section, we explain some of the necessary assumptions, associated notations, and definitions. (H1) There exist diagonal matrices  = diag and  = diag such that, For all . (H2) There exists a non-negative diagonal matrix such that, For all . (H3) and are continuous bounded function defined on . (H4) there exists a positive number such that, is continuous for .

Remark 1. Assumptions (H1)–(H4) are widely used in the work on exponential stability of cellular neural networks (see, [1–11, 16, 17, 19–26]).
Let is bounded on and for . exists for and for all but a finite number of points . For , is defined as.where , .
Let is bounded on and for . exists for and for all but a finite number of points .
Let , and , then the Schur product of and is defined by .
and denotes a identity matrix.

Definition 1. If satisfies,(i) is piecewise continuous and right-continuous at every discontinuity point which are the first kind of discontinuity points;(ii) satisfies systems (1a)–(1d) for all .Hence, represents the special solution of systems (1a)–(1d) under initial condition .

Definition 2. In the initial condition of , is any solution to the systems (1a)–(1d). If there exist two positive numbers and such that,Then, systems (1a)–(1d) are globally exponentially stable.

Definition 3 (see [28]). If is a real matrix, suppose that,(i)(ii)All successive principal minors of are positiveThen, is a nonsingular -matrix.

Lemma 1 (see [28]). Setting with for all . Then, the necessary and sufficient conditions for to be a nonsingular M-matrix are that there exists a vector such that or .

Lemma 2 (see [29]). Let is a real-valued function and is a cube. If meets , that is is equal to zero at the boundary of . Then,

Lemma 3 (see [28]). Let and be two states of neural networks (1a)–(1d), and be a real-valued function. Then, following inequalities hold:

Lemma 4. Let , If makes the following differential inequality hold:where with , , with . If the initial condition meets,where and satisfy the following inequality:Then, .

Proof 1. For , , let . Then,If the above-given is false, that is there exist a number and several integer such that,According to Lemma 4 and (13), we obtain the following equation:From , and , it follows that,From (15), we obtain the following equation:That is,This is contradictory to (14). Therefore, the inequality (13) holds for all .
Now, letting in (13), we have that,That is,

3. Main Results

In this section, we introduce the main results of systems (1a)–(1d) and their proof process.

Theorem 1. .If assumptions (H1)–(H4) are satisfied, suppose that, (C1) There exist and such that, where with , , , , , diag , diag ; (C2) There exists a constant such that,where ;
Then systems (1a)–(1d) are globally exponentially stable.

Proof 2. For , let and be solutions of systems (1a)–(1d) through and , respectively. Define , that is , for all .
Let , then,for .
Multiply both sides of (23) by and integrate it, one can obtain,Due to Green formula and Dirichlet boundary conditions, one has.From Lemma 2, we can obtain the following equation:According to (H1) and Hoder inequality, we have,By assumption (H1), Lemma 2 and Hoder inequality, one obtains,By the same way, we can obtain the following equation:Applying (24)–(29) to (23), we can obtain the following equation:i.e.,Let , , . Then, (31) can be simplified into the following form:From condition (C1), There exist and , then,Here, taking , we have the following equation:From Lemma 4, one can obtain,If the following inequality is true for When , we can obtain the following equation:By (36) and (37) and , we have the following equation:Combining (33), (34) and (38) and Lemma 3, one has,According to mathematical induction, then we have,applying the condition and (40), one can obtain,for all .
This implies that,That is,where .

Remark 2. Condition (C1) is equivalent to that is a nonsingular M-matrix for all . As a matter of fact, if is a nonsingular M-matrix for any , by using Lemma 1, there exist such that,By the uniform continuity, there exists satisfies:It tells us that (C1) is true. Reversely, setting in (C1), we can easily get that is a non-singular M-matrix for all .

Corollary 1. If assumptions , and are satisfied, suppose that condition holds. Then, systems (2a)–(2c) are globally exponentially stable.

Corollary 2. When the coefficient of system (1a)–(1d) are constants, they degenerate into the following autonomous FCNNs with reaction-diffusion terms and distribution delays:For assumption and , Theorem 1 can be expressed in the following form:  There exist and such that where with , , , , , , diag , diag ;  There exists a constant such that,where ; then systems (46a)–(46d) are globally exponentially stable.

Remark 3. Some existing neural network models (see, [24, 30]) are special cases such as systems (2a)–(2c) and systems (46a)–(46d). Compared with the methods of constructing Lyapunov functional in [24], our results are more concise, and it is not difficult to find that some of the standards have been improved. Moreover, in [30], the authors gave sufficient conditions for the existence of uniqueness and global exponential stability of the equilibrium point of impulsive FCNNs with distributed time delays and reaction-diffusion terms, but we have to say that the method we using is similar.
Next, in order to consider the periodic solution of the systems (1a)–(1d), we add the following two assumptions. (H5) and are periodic continuous functions with a common period for all . (H6) For and the impulsive time , there exists a positive integer such that,

Remark 4. Assumptions (H5) and (H6) indicate that both the system parameters and the impulse function are periodic, which is a necessary prerequisite for solving the periodic solution of the systems. These assumptions can be seen in many studies on the periodicity of cellular neural networks (see, [15, 18, 26]).
Combined with assumptions (H5) and (H6), we have the following results for periodic systems (1a)–(1d), based on the discussion of the global exponential stability of the systems in Theorem 1.

Theorem 2. If assumptions (H1)–(H6) are satisfied, suppose that, (C1) There exist and such that, where with , , , , , diag , diag