Abstract

We investigate a class of fuzzy neural networks with Hebbian-type unsupervised learning on time scales. By using Lyapunov functional method, some new sufficient conditions are derived to ensure learning dynamics and exponential stability of fuzzy networks on time scales. Our results are general and can include continuous-time learning-based fuzzy networks and corresponding discrete-time analogues. Moreover, our results reveal some new learning behavior of fuzzy synapses on time scales which are seldom discussed in the literature.

1. Introduction

It is well known that many applications of neural networks exist in diverse areas such as optimization, signal and image processing, pattern recognition, and control system. These applications are based on stability of equilibrium points of the network models. Hence, stability criteria of equilibrium points of networks have been greatly investigated in the literature [1–4]. Meanwhile, more recently, there have been several publications on the theme of neural networks where fuzzy logic is used. Yang and Yang [2, 5] and Yang et al. [6] have proposed a fuzzy cellular neural network to include and analyze the ambiguity or vagueness inherent in the inputs and outputs of neural networks. Further analysis of this type of networks can be found in the works of Yuan et al. [4], Liu and Tang [7], Huang and Zhang [8], Huang [9, 10], Chen and Liao [11], and the references therein.

It is welknown that the theory of time scales has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, population dynamics, biotechnology, economics, and so on. Meanwhile, it is unsuitable to study the stability for continuous and discrete system, respectively. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations. Many authors incorporate time scales into stability analysis of neural network models; we can refer to [12–15].

Stimulated by [16], we consider a class of networks of somatically crisp neurons with fuzzy learnable synapses on time scales described by where , denotes the state of neuron at time , denotes the passive negative stabilizing feedback of neuron , , ,, , and denote the synaptic weights of the various fuzzy and nonfuzzy synapses of neuron , are disposable constants, and denotes a learnable synaptic weight of neuron when it is presented with a constant input signal vector ; the external bias to the network is denoted by the constant vector . The operators and denote, respectively, the “max” and “min” operators used in fuzzy logic. The learning equation is based on the Hebbian-type [16, 17] unsupervised algorithm modified by the introduction of a forgetting term as proposed by Amari [18]. By using auxiliary variables , one gets where , , . Equation (2) is quite general and it includes several well known neural networks [16] and its difference analogue is where and are the forward difference operators and , .

For convenience, we let , , , , and . Then, (2) reduces to Correspondingly, synaptic dynamic equation is as follows: where . In this paper, we will study learning-based fuzzy networks (4) on time scales. Without the learning component and , (4) will include fuzzy networks discussed by several authors recently (see [2, 4, 7, 9–11]). In the absence of fuzzy synapses, our model reduces to the most commonly studied Hopfield-type neural network. Moreover, by using the calculus theory on time scale to unify and generalize discrete-time and continuous-time learning-based fuzzy networks, we can establish new sufficient conditions to ensure existence and global exponential stability of equilibrium of (4).

The paper is organized as follows. In Section 2, we present some basic definitions concerning the calculus on time scales. In Section 3, we develop Lyapunov functions technique on time scale to give some sufficient conditions of global exponential stability for (4). In Section 4, an example is given to illustrate the effectiveness of our main results. Conclusions remarks are given in Section 5.

2. Preliminaries on Time Scales

The basic calculus theory on time scales was initiated by Hilger [19, 20], and Agarwal et al. summarize and organize much of relative results in monograph [21–23]. In this section, we will introduce some basic definitions and lemmas.

Definition 1. A time scale is arbitrary nonempty closed subset of the real set with the topology and ordering inherited from .

Definition 2. On any time scale , one defines the forward and backward jump operators by and ; one puts and , where denotes the empty set. A point is said to be left-dense if and , right-dense if and , left-scattered if , and right-scattered if . The graininess function for a time scale is defined by . If has a left-scattered maximum , then one defined to be . Otherwise, .

Definition 3. For a function (the range of may be actually replaced by Banach space), the (delta) derivative is defined by if is continuous at and is right-scattered. If is not right-scattered then the derivative is defined by provided this limit exists.

Definition 4. A function is called a delta-antiderivative of provided holds for all . In this case, one defines the integral of by and one has the following formula:

Definition 5. A function is called right-dense continuous provided it is continuous at right-dense points of and the left sided limit exists (finite) at left-dense point of . The set of all right-dense continuous functions on is defined by .

Definition 6. One says that a function is regressive provided for all . The set of all regressive functions on a time scale forms an Abelian group under the addition defined by . The additive inverse in this group is denoted by . One then defines subtraction on the set of regressive functions by . It can be shown that . The set of all regressive and right-dense continuous functions will be denoted by .

Definition 7. One defines the set of all positively regressive elements of by for all .

Next, we give the definition of the exponential function and list some of its properties.

Definition 8. If , one defines the generalized exponential function as where , , and .

Remark 9. The exponential function is the unique solution of the IVP , for . As .

Lemma 10. If , then(i) and ;(ii);(iii);(iv);(v), for ;(vi);(vii).

Lemma 11 (see [3]). If and , then If , then

Lemma 12 (see [22]). Let and . Then, , , implies

Lemma 13 (see [2, 5]). Suppose , are any two vectors in :

Throughout this paper, we make the following basic assumptions:)The functions are Lipschitz continuous on with the Lipschitz constants and , respectively; that is, , .

3. Main Results

In this section, we study the global exponential stability of the unique equilibrium for (4) on time scale by using Lyapunov method.

Theorem 14. Suppose that (4) satisfies ; if there exist positive constants , , and such that()where and , , then there exists a unique equilibrium of (4) which is globally exponentially stable; that is, every solution of (4) satisfies where , , and

Proof. Similar to the proof of [16], we can prove (4) possesses a unique equilibrium . Let and let ; then, we can rewrite (4) into Now, we construct the Lyapunov function , where By using , we can conclude that which implies that for .
Consider where , , , , and Observe that where . Due to Lemma 11, we get where is a constant. This completes the proof.