- Open Access
The exponential stability of BAM neural networks with leakage time-varying delays and sampled-data state feedback input
© Li et al.; licensee Springer. 2014
Received: 31 October 2013
Accepted: 6 January 2014
Published: 27 January 2014
In this paper, the exponential stability of bidirectional associative memory neural networks with leakage time-varying delays and sampled-data state feedback input is considered. By applying the time-delay approach, some conditions for ensuring the stability of a system are obtained. In addition, a numerical example is given to demonstrate the effectiveness of the obtained results.
In the past few decades, neural networks have been widely investigated by researchers. In 1987, bidirectional associative memory (BAM) neural networks were firstly introduced by Kosko [1, 2]. Due to its better abilities as information associative memory, the BAM neural network has attracted considerable attention in different fields such as signal processing, pattern recognition, optimization, and so on.
It is well known that time delay is unavoidable in the hardware implementation of neural networks due to the finite switching speed of neurons and amplifiers. The delay can cause instability, oscillation, or poor dynamical behavior. In practical applications, there exist many types of time delays such as discrete delays , time-varying delays , distributed delays [5, 6], random delays  and leakage delays (or forgetting delays) [8, 9]. Up to now, a large number of results about delay BAM neural networks have been reported [10–13]. All of the conclusions could be roughly summarized into two types: in terms of the stability analysis of equilibrium points, and of the existence and stability of periodic or almost periodic solutions.
The leakage delay, which exists in the negative feedback term of a neural network system, emerges as a research topic of primary importance recently. Gopalsamy  investigated the stability of the BAM neural networks with constant leakage delays. Further, Liu  discussed the global exponential stability for BAM neural networks with time-varying leakage delays, which extend and improve the main results of Gopalsamy. Peng et al. [15–17] derived the stability criteria for the BAM neural networks with leakage delays, unbounded distributed delays and probabilistic time-varying delays.
Sampled-data state feedback is a practical and useful control scheme and has been studied extensively over the past decades. There are some results dealing with synchronization [18, 19], state estimate [20–22] and stability [23–29]. Recently, the work in  has studied the problem of the stability of sampled-data piecewise affine systems via the input delay approach. Although the importance of the stability of neural networks has been widely recognized, no related results have been established for the sampled-data stability of BAM neural networks with leakage time-varying delays. Motivated by the works above, we consider the sampled-data stability of BAM neural networks with leakage time-varying delays under variable sampling with a known upper bound on the sampling intervals.
The organization of this paper is as follows. In Section 2, the problem is formulated and some basic preliminaries and assumptions are given. The main results are presented in Section 3. In Section 4, a numerical example is given to demonstrate the effectiveness of the obtained results. Some conclusions are proposed in Section 5.
where , the and are neuron state variables, the positive constants and denote the time scales of the respective layers of the networks, , , , are connection weights of the network. and denote the leakage delays, and are time-varying delays, , are neuron activation functions, , are sampled-data state feedback inputs, denotes the sample time point, , , ℕ denotes the set of all natural numbers.
Assume that there exists a positive constant L such that the sample interval , . Let , for , then with .
Before ending this section, we introduce two assumptions, which will be used in next section.
for all , and .
3 Main results
The initial conditions of (3) are: , , , , where and are continuous functions on .
The main results are stated as follows.
Theorem 1 Let Assumptions 1 and 2 hold; then the BAM neural network (3) is exponentially stable, i.e., there exists a positive constant λ such that , , .
where , .
which contradicts (a).
This is a contradiction with (b).
Using mathematical induction, the inequalities (8) hold. By a similar proof to (11), we have , , for , which implies , , . This completes the proof. □
Remark 2 If the leakage delays in (3) are constant, that is, , . Assumption 2 is changed into the following form.
Similar to the proof of Theorem 1, we get the following result.
Corollary 1 If Assumptions 1 and 2′ hold, the BAM neural networks with constant leakage delays and the sampled-data state feedback inputs are exponentially stable.
4 Simulation example
In this section, we give an illustrative example to show the efficiency of our theoretical results.
The activation functions are taken as . Time-varying delays are chosen as , and the leakage delays are chosen as , , respectively.
In this paper, we investigate the stability of BAM neural networks with leakage delays and a sampled-data input. By using the time-delay approach, the conditions for ensuring the exponential stability of the system are derived. It should be pointed out that there are many papers focusing on the stability problem of sampled-data systems, leakage delay, and sampled-data state feedback that have never been taken into consideration in the BAM neural networks. To the best of our knowledge, this is the first time to consider the stability of BAM neural networks with both leakage delays and sampled-data state feedback at the same time. The results of this paper are worthy as complementary to the existing results. Finally, a numerical example and its computer simulations have been presented to show the effectiveness of our theoretical results.
This work was jointly supported by the National Natural Science Foundation of China under Grant 60875036, the Foundation of Key Laboratory of Advanced Process Control for Light Industry (Jiangnan university), Ministry of Education, P.R. China the Fundamental Research Funds for the Central Universities (JUSRP51317B, JUDCF13042).
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