Abstract
The fixed point technique has been employed in the stability analysis of time-delays bidirectional associative memory (BAM) neural networks with impulse. By formulating a contraction mapping in a product space, a new LMI-based exponential stability criterion was derived. Lately, fixed point methods have educed various good results inspiring this work, but those criteria cannot be programmed by a computer. In this paper, LMI conditions of the obtained result can be applicable to computer Matlab LMI toolbox which meets the need of the large-scale calculation in real engineering. Moreover, a numerical example and a comparable table are presented to illustrate the effectiveness of the proposed methods.
1. Introduction
Bidirectional associative memory (BAM) neural networks model was originally introduced by Kosko [1, 2]: Thanks to its generalization of the single-layer autoassociative Hebbian correlation to two-layer pattern-matched heteroassociative circuits, widespread applications have been found in various areas, such as automatic control, signal and image processing, pattern recognition, artificial intelligence, parallel computation, and optimization problems. Often, a stable equilibrium of BAM neural networks is the important precondition of the successful applications. There are many factors influencing the stability of neural networks, in which pulse and time delay are usually the main reasons [3–8]. So, the stability analysis of impulse or delays system has become a hot topic. All the time, the Lyapunov functional method is employed to deduce stability criteria [9–20]. But every method may have its limitations. During the recent decades, other techniques have been developed to investigate the stability, in which the fixed point method is always one of those alternatives [21–27]. For example, in 2015, Zhou utilized Brouwer’s fixed point theorem to prove the existence and uniqueness of equilibrium of the hybrid BAM neural networks with proportional delays and finally constructed appropriate delay differential inequalities to derive the stability of equilibrium [28]. In [29], Banach fixed point theorem was applied to show the existence of the unique equilibrium of BAM neural networks with time-varying delays in the leakage terms, and then the Lyapunov functional method was for demonstrating the global exponential stability. Different from [28, 29], we shall use Banach fixed point theorem deriving straightway the stability criterion of impulsive time-delay BAM neural networks, in which LMI conditions facilitate computer programming. Finally, a numerical example is presented to illustrate the effectiveness of the proposed methods.
For the sake of convenience, we introduce the following standard notations [18]:(i): a positive (negative) definite matrix; that is, for any .(ii): a semipositive (seminegative) definite matrix; that is, for any .(iii): this means matrix is a semipositive (seminegative) definite matrix.(iv): this means matrix is a positive (negative) definite matrix.(v) and denote the largest and the smallest eigenvalue of matrix , respectively.(vi)Denote for any matrix .(vii) for any vector .(viii) implies that , , and implies that , , for any vectors and .(ix) is identity matrix with compatible dimension.
Remark 1. Different from the methods of [28, 29], it will be the first time to utilize contraction mapping principle to infer directly the LMI-based stability criterion of BAM neural networks, convenient for computer programming. Recently, there have been a lot of good results and methods [21–29] enlightening our current work. In this paper, we shall propose the LMI-based criterion, novel against the existing results, published from 2013 to 2016 (see Remark 10 and Table 1).
2. Preliminaries
Consider the following delayed differential equations:where , , and . Here, represents the space of continuous functions from to . Active functions , , impulsive function , and time delays . The fixed impulsive moments satisfy with . and stand for the right-hand and left-hand limit of at time , respectively. We always assume , for all . Similar to [23], we assume in this paper that . Constant matrices , are positive definite diagonal matrices, and both and are matrices with dimension.
Throughout this paper, we assume that , , and are diagonal matrices, satisfying(A1), .(A2), .(A3), .
Definition 2. Dynamic equations (2) are said to be globally exponentially stable if, for any initial condition , there exist a pair of positive constants and such that where the norm , and , .
Definition 3. For a diagonal constants matrix , one denotes the matrix exponential function for all .
Lemma 4 (see [31] contraction mapping theorem). Let be a contraction operator on a complete metric space ; then, there exists a unique point for which .
Lemma 5. Let be a diagonal constants matrix and be the matrix exponential function of . Then, one has(1), ;(2), , where and each is a constant.
3. Main Result
Theorem 6. Assume that there exists a positive constant such that . In addition, there exists a constant such that and then the impulsive fuzzy dynamic equations (2) are globally exponentially stable, where .
Proof. To apply the fixed point theory, we firstly define a complete metric space as follows.
Let be the space consisting of functions , satisfying the following:(a) is continuous on .(b), , for .(c), and exists, for all .(d) as , where is a positive constant, satisfying .It is not difficult to verify that the product space is a complete metric space if it is equipped with the following metric: where and , , .
Next, we are to construct a contraction mapping , which may be divided into three steps.
Step 1 (formulating the mapping). Let be a solution of system (2).
Then, for , , we have Further, we get by the integral property where is the vector to be determined.
Next, we claim that , and Indeed, on the one hand, we can conclude from (9) that For , , taking the time derivative of both sides leads to or which is the first equation of system (2).
Moreover, as , we can gain by (9) On the other hand, multiplying both sides of the first equation of system (2) by yields Moreover, integrating from to gives which yields, after letting ,Throughout this paper, we assume that is a sufficient small positive number. Now, taking in the above equation yields which yields by letting So, we have actually got for all , . Furthermore, (19) generates Making a synthesis of the above equations results in (9).
Similarly, we can obtain So, we may define the mapping on the space as follows: Step 2. We claim that for any . In other words, we need to prove that must satisfy conditions (a)–(d) of .
Indeed, satisfies condition (b) due to (23).
Besides, let be a real number; we have In (24), letting brings about which implies that is continuous on , . Here, the convergences are under the metric defined on the space . Below, all the convergences of vector functions are in this sense. Thus, condition (a) is satisfied.
Furthermore, letting , , in (24), we have which implies that condition (c) is also satisfied.
Next, condition (d) is satisfied if only Indeed, obviously, and as .
Below, we may firstly prove In fact, it follows from that, for any given , there exists a corresponding constant such that Next, we get by (A1) On the one hand, On the other hand, obviously, there exists a positive number such that . So, we haveCombining (30)–(32) yields (28). Similarly, we can prove Next, we need to prove that if , Indeed, it is obvious that Below, we shall prove Firstly, we may assume that and for any given . Hence, which together with the arbitrariness of the positive number implies that Similarly, we can also get Hence, we have proved (36) and (34). And so we can conclude (27) from (28), (33), and (34). This means that condition (d) is satisfied, too.
Therefore, for any .
Step 3. Below, we only need to prove that is a contraction mapping.
Indeed, for any , we have and hence where and are the inverse matrices of and , respectively.
Therefore, is a contraction mapping such that there exists the fixed point of in , which implies that is a solution of the impulsive fuzzy dynamic equations (2), satisfying as . And the proof is completed.
Remark 7. Impulsive BAM neural networks model brings some mathematical difficulties to contraction mapping technique. However, in this paper, we set up the contraction mapping on the complete product space to overcome obstacles.
4. Numerical Example
Example 8. We equip the impulsive system with the following parameters: Let . Then, we can use Matlab LMI toolbox to solve LMI conditions (4), obtaining the datum feasible as follows: Obviously, . Thereby, we can conclude from Theorem 6 that the impulsive equations (2) are globally exponentially stable.
Remark 9. Example 8 illustrates the effectiveness of LMI-based criterion (Theorem 6). Table 1 presents a comparable result among the related literature, mainly published from 2013 to 2016.
Remark 10. From Table 1, we know that there are many existing literatures involving fixed point technique and stability analysis, and a lot of interesting conclusions are derived [21–23, 28–30]. Motivated by some methods of those literatures, we utilized Banach contraction mapping theorem to obtain the LMI-based stability criterion applicable to computer Matlab LMI toolbox. It is well known that computer software can solve large-scale computations in actual engineering, which demonstrates the superiority of the proposed method in this paper to a certain extent.
5. Conclusion
Recently, fixed point technique and methods are employed to the stability analysis to BAM neural networks, and some stability criteria are derived (see, e.g., [28, 29]). Some good methods and results of related literature [21–23, 28–30] inspire our current work. Different from existing papers, we utilized Banach fixed point theorem deriving immediately the exponential stability criterion applicable to computer Matlab LMI toolbox. Computer programming is suitable for large-scale computation in practical engineering.
Competing Interests
The authors declare that they have no competing interests.
Authors’ Contributions
Ruofeng Rao wrote the original manuscript, Zhilin Pu and Shouming Zhong checked it, and Xinggui Li and Ruofeng Rao were in charge of correspondence. All authors typed, read, and approved the final manuscript.
Acknowledgments
This work was supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2010JY0057, 2012JYZ010) and the Scientific Research Fund of Sichuan Provincial Education Department (12ZB349).