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).
- Kosko B: Adaptive bi-directional associative memories. Appl. Opt. 1987, 26: 4947-4960. 10.1364/AO.26.004947View ArticleGoogle Scholar
- Kosko B: Bi-directional associative memories. IEEE Trans. Syst. Man Cybern. 1988, 18: 49-60. 10.1109/21.87054MathSciNetView ArticleGoogle Scholar
- Gao M, Cui B: Global robust exponential stability of discrete-time interval BAM neural networks with time-varying delays. Appl. Math. Model. 2009, 33(3):1270-1284. 10.1016/j.apm.2008.01.019MathSciNetView ArticleMATHGoogle Scholar
- Ailong W, Zhigang Z: Dynamic behaviors of memristor-based recurrent neural networks with time-varying delays. Neural Netw. 2012, 36: 1-10.View ArticleMATHGoogle Scholar
- Wang Z, Zhang H: Global asymptotic stability of reaction-diffusion Cohen-Grossberg neural networks with continuously distributed delays. IEEE Trans. Neural Netw. 2010, 21(1):39-48.View ArticleGoogle Scholar
- Li Y, Yang C: Global exponential stability analysis on impulsive BAM neural networks with distributed delays. J. Math. Anal. Appl. 2006, 324(2):1125-1139. 10.1016/j.jmaa.2006.01.016MathSciNetView ArticleMATHGoogle Scholar
- Lou X, Ye Q, Cui B: Exponential stability of genetic regulatory networks with random delays. Neurocomputing 2010, 73: 759-769. 10.1016/j.neucom.2009.10.006View ArticleGoogle Scholar
- Gopalsamy K: Leakage delays in BAM. J. Math. Anal. Appl. 2007, 325: 1117-1132. 10.1016/j.jmaa.2006.02.039MathSciNetView ArticleMATHGoogle Scholar
- Zhang H, Shao J: Existence and exponential stability of almost periodic solutions for CNNs with time-varying leakage delays. Neurocomputing 2013, 74: 226-233.View ArticleGoogle Scholar
- Arik S, Tavsanoglu V: Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays. Neurocomputing 2005, 68: 161-176.View ArticleGoogle Scholar
- Cao J, Wang L: Periodic oscillatory solution of bidirectional associative memory networks with delays. Phys. Rev. E 2000, 61: 1825-1828.MathSciNetView ArticleGoogle Scholar
- Liu Z, Chen A, Cao J: Existence and global exponential stability of almost periodic solutions of BAM neural networks with distributed delays. Phys. Lett. A 2003, 319: 305-316. 10.1016/j.physleta.2003.10.020MathSciNetView ArticleMATHGoogle Scholar
- Chen A, Huang L, Cao J: Existence and stability of almost periodic solution for BAM neural networks with delays. Appl. Math. Comput. 2003, 137: 177-193. 10.1016/S0096-3003(02)00095-4MathSciNetView ArticleMATHGoogle Scholar
- Liu B: Global exponential stability for BAM neural networks with time-varying delays in the leakage terms. Nonlinear Anal., Real World Appl. 2013, 14: 559-566. 10.1016/j.nonrwa.2012.07.016MathSciNetView ArticleMATHGoogle Scholar
- Peng S: Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms. Nonlinear Anal., Real World Appl. 2010, 11: 2141-2151. 10.1016/j.nonrwa.2009.06.004MathSciNetView ArticleMATHGoogle Scholar
- Balasubramaniam P, Kalpana M, Rakkiyappan R: Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays. Math. Comput. Model. 2011, 53: 839-853. 10.1016/j.mcm.2010.10.021MathSciNetView ArticleMATHGoogle Scholar
- Lakshmanan S, Park J, Lee T, Jung H, Rakkiyappan R: Stability criteria for BAM neural networks with leakage delays and probabilistic time-varying delays. Appl. Math. Comput. 2013, 219: 9408-9423. 10.1016/j.amc.2013.03.070MathSciNetView ArticleMATHGoogle Scholar
- Wu Z, Shi P, Su H, Chu J: Sampled-data synchronization of chaotic Lur’e systems with time delays. IEEE Trans. Neural Netw. Learn. Syst. 2013, 24(3):410-421.View ArticleGoogle Scholar
- Gan Q: Synchronisation of chaotic neural networks with unknown parameters and random time-varying delays based on adaptive sampled-data control and parameter identification. IET Control Theory Appl. 2012, 6(10):1508-1515. 10.1049/iet-cta.2011.0426MathSciNetView ArticleGoogle Scholar
- Lee T, Park J, Kwon O, Lee S: Stochastic sampled-data control for state estimation of time-varying delayed neural networks. Neural Netw. 2013, 46: 99-108.View ArticleMATHGoogle Scholar
- Hu J, Li N, Liu X, Zhang G: Sampled-data state estimation for delayed neural networks with Markovian jumping parameters. Nonlinear Dyn. 2013, 73: 275-284. 10.1007/s11071-013-0783-1MathSciNetView ArticleMATHGoogle Scholar
- Rakkiyappan R, Sakthivel N, Park J, Kwon O: Sampled-data state estimation for Markovian jumping fuzzy cellular neural networks with mode-dependent probabilistic time-varying delays. Appl. Math. Comput. 2013, 221: 741-769.MathSciNetView ArticleMATHGoogle Scholar
- Fridman E, Blighovsky A: Robust sampled-data control of a class of semilinear parabolic systems. Automatica 2012, 48: 826-836. 10.1016/j.automatica.2012.02.006MathSciNetView ArticleMATHGoogle Scholar
- Samadi B, Rodrigues L: Stability of sampled-data piecewise affine systems: a time-delay approach. Automatica 2009, 45: 1995-2001. 10.1016/j.automatica.2009.04.025MathSciNetView ArticleMATHGoogle Scholar
- Rodrigues L: Stability of sampled-data piecewise-affine systems under state feedback. Automatica 2007, 43: 1249-1256. 10.1016/j.automatica.2006.12.016View ArticleMathSciNetMATHGoogle Scholar
- Zhang C, Jiang L, He Y, Wu H, Wu M: Stability analysis for control systems with aperiodically sampled data using an augmented Lyapunov functional method. IET Control Theory Appl. 2013, 7(9):1219-1226. 10.1049/iet-cta.2012.0814MathSciNetView ArticleGoogle Scholar
- Seuret A, Peet M: Stability analysis of sampled-data systems using sum of squares. IEEE Trans. Autom. Control 2013, 58(6):1620-1625.MathSciNetView ArticleGoogle Scholar
- Oishi Y, Fujioka H: Stability and stabilization of aperiodic sampled-data control systems using robust linear matrix inequalities. Automatica 2010, 46: 1327-1333. 10.1016/j.automatica.2010.05.006MathSciNetView ArticleMATHGoogle Scholar
- Feng L, Song Y: Stability condition for sampled data based control of linear continuous switched systems. Syst. Control Lett. 2011, 60: 787-797. 10.1016/j.sysconle.2011.07.006MathSciNetView ArticleMATHGoogle Scholar
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