Abstract
This paper is concerned with the pth moment exponential stability of fuzzy cellular neural networks with timevarying delays under impulsive perturbations and stochastic noises. Based on the Lyapunov function, stochastic analysis and differential inequality technique, a set of novel sufficient conditions on pth moment exponential stability of the system are derived. These results generalize and improve some of the existing ones. Moreover, an illustrative example is given to demonstrate the effectiveness of the results obtained.
1 Introduction
In the last decades, cellular neural networks [1,2] have been extensively studied and applied in many different fields such as associative memory, signal processing and some optimization problems. In such applications, it is of prime importance to ensure that the designed neural networks are stable. In practice, due to the finite speeds of the switching and transmission of signals, time delays do exist in a working network and thus should be incorporated into the model equation. In recent years, the dynamical behaviors of cellular neural networks with constant delays or timevarying delays or distributed delays have been studied; see, for example, [311] and the references therein.
In addition to the delay effects, recently, studies have been intensively focused on stochastic models. It has been realized that the synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes, and it is of great significance to consider stochastic effects on the stability of neural networks or dynamical system described by stochastic functional differential equations (see [1223]). On the other hand, most neural networks can be classified as either continuous or discrete. Therefore most of the investigations focused on the continuous or discrete systems, respectively. However, there are many realworld systems and neural processes that behave in piecewise continuous style interlaced with instantaneous and abrupt change (impulses). Motivated by this fact, several new neural networks with impulses have been recently proposed and studied (see [2433]).
In this paper, we would like to integrate fuzzy operations into cellular neural networks. Speaking of fuzzy operations, Yang and Yang [34] first introduced fuzzy cellular neural networks (FCNNs) combining those operations with cellular neural networks. So far researchers have found that FCNNs are useful in image processing, and some results have been reported on stability and periodicity of FCNNs [3440]. However, to the best of our knowledge, few author investigated the stability of stochastic fuzzy cellular neural networks with timevarying delays and impulses.
Motivated by the above discussions, in this paper, we consider the following stochastic fuzzy cellular neural networks with timevarying delays and impulses
where n corresponds to the number of units in the neural networks. For , corresponds to the state of the ith neuron. , are signal transmission functions. denotes the rate at which a cell i resets its potential to the resting state when isolated from other cells and inputs; corresponds to the transmission delay. represents the elements of the feedback template. , , , and are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feedforward MIN template and fuzzy feedforward MAX template, respectively; ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operation, respectively; denotes the external input of the ith neurons. is the external bias of the ith unit. is a transmission delay satisfying ; is the diffusion coefficient matrix and is the ith row vector of : is an ndimensional Brownian motion defined on a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all Pnull sets). is the impulses at moment , the fixed moments of time satisfy , ; , .
System (1) is supplemented with the initial condition given by
where is measurable and continuous everywhere except at a finite number of points , at which and exist and .
Let denote the family of all nonnegative functions on which are continuous once differentiable in t and twice differentiable in x. If , define an operator LV associated with (1) as
where
For convenience, we introduce several notations. denotes a column vector. denotes a vector norm defined by ; denotes the space of continuous mappings from topological space X to topological space Y. Denoted by is the family of all bounded measurable, valued random variables ϕ, satisfying , where denotes the expectation of a stochastic process.
Throughout the paper, we give the following assumptions.
(A1) The signal transmission functions , () are Lipschitz continuous on R with Lipschitz constants and , namely, for any ,
(A2) There exist nonnegative numbers , such that for all , ,
(A3) , where is the equilibrium point of (1) with the initial condition (2), satisfies .
Definition 1.1 The equilibrium point of system (1) is said to be pth moment exponentially stable if there exist positive constants , such that
where is any solution of system (1) with initial value , .
When , it is usually said to be exponentially stable in mean square.
Lemma 1.1[34]
Supposexandyare two states of system (1), then we have
and
Lemma 1.2If () denotepnonnegative real numbers, then
wheredenotes an integer. A particular form of (4), namely
2 Main results
In this section, we consider the existence and global pth moment exponential stability of system (1).
Lemma 2.1For two positive real numbersaandb, assume that there exists a constant numbersuch that. Then the equation
Proof Let . It is easy to see , , . Thus, is strictly increasing on . Therefore, Eq. (5) has a unique positive solution . □
Lemma 2.2[18]
For two positive real numbersaandb, assume that there exists a constant numbersuch that. Assume thatis a nonnegative continuous function onand satisfies the following inequality:
then, whereλis a solution of (5) and the upper right Dini derivative ofis defined as
Theorem 2.1Under conditions (A1)(A3), if there exist constants (), such that
where
Thenis a unique equilibrium which is globallypth moment exponential stable.
Proof The proof of existence and uniqueness of equilibrium for the system is similar to that of [39]. So we omit it. Suppose that is the unique equilibrium of system (1).
Let
then system (1) can be transformed into the following equation, for :
We define a Lyapunov function . Let , , then we can get the operator associated with system (8) of the following form:
where
Firstly, for , applying the Ito formula, we obtain that
Since , taking expectations on both sides of equality (10) and applying the inequality (9) yields
denote , the preceding result (11) leads directly to
Hence, from Lemma 2.2, we have
Namely,
where , λ is the unique positive solution of the following equation:
Next, suppose that for , the inequality
holds. From (A3), we get
This, together with (14), leads to
On the other hand, for , applying the Ito formula, we get
Then, we have
So, for small enough , we have
From (16) and (17), we have
Similarly, we obtain
Hence, by the mathematical induction, for any , we conclude that
which implies that the equilibrium point of the impulsive system (1) is pth moment exponentially stable. This completes the proof of the theorem. □
3 Comparisons and remarks
It can be easily seen that many neural networks are special cases of system (1). Thus, in this section, we give some comparisons and remarks.
Suppose that , system (1) becomes the stochastic fuzzy cellular neural networks with timevarying delays.
For (20), we have the following corollary by Theorem 2.1.
Corollary 3.1If (A1)(A2) hold, if there exist constants (), such that
where
then the unique equilibriumof system (20) is globallypth moment exponential stable.
If we do not consider fuzzy AND and fuzzy OR operations and when in system (1), then system (1) becomes impulsive stochastic cellular neural networks with timevarying delays
Remark 3.1 The stability of system (21) has been investigated in [31]. In [31], authors required the differentiability and monotonicity of a time delays function which satisfied . Hence, this assumption may impose a very strict constraint on the model because time delays sometimes vary dramatically with time in real circuits. Obviously, Theorem 2.1 does not require these conditions.
Remark 3.2 In Theorem 2.1, if we do not consider fuzzy AND and OR operation, it becomes traditional cellular neural networks. The results in [32] are the corollary of Theorem 2.1. Therefore the results of this paper extend the previous known publication.
4 An example
Example 4.1 Consider the following stochastic fuzzy neural networks with timevarying delays and timevarying delays and impulses
Let , obviously, , . By simple computation, we can easily get that , . Letting , we have
Thus, system (22) satisfies assumptions (A1)(A3). It follows from Theorem 2.1 that system (22) is exponentially stable in mean square.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors indicated in parentheses made substantial contributions to the following tasks of research: drafting the manuscript (PX); participating in the design of the study (LH); writing of paper (PX).
Acknowledgements
The authors would like to thank the editor and the referees for their helpful comments and valuable suggestions regarding this paper. This work is partially supported by 973 program (2009CB326202), the National Natural Science Foundation of China (11071060, 60835004, 11101133), Fundamental Research Fund for the Central Universities of China (531107040222) and Scientific Research Fund of Hunan Provincial Education Department (Grant No. 12A206).
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