Pinning adaptive synchronization of neutral-type coupled neural networks with stochastic perturbation
© Sun et al.; licensee Springer. 2014
Received: 3 January 2014
Accepted: 25 February 2014
Published: 7 March 2014
In this paper, by using a pinning adaptive control scheme, we investigate the almost surely synchronization of neutral-type coupled neural networks with stochastic perturbation. Based on Lyapunov stability theory, stochastic analysis, and matrix theory, some sufficient conditions for almost surely synchronization are derived. Furthermore, a numerical example is exhibited to illustrate the validity of the theoretical results.
In recent years, the neural networks (NNs) have played a significant role in the fields of science and engineering due to their practical applications including signal and image processing, associative memories, combinatorial optimization, automatic control, and so on (see [1–4]). Among dynamical behaviors of the neural networks, synchronization is one of the most important topics that have received considerable research attention [5–9]. At the same time, many different kinds of synchronization have been proposed, such as generalized synchronization , cluster synchronization , phase synchronization , lag synchronization , and so on.
Due to the finite speed of transmitting signals between neurons and the transmission process randomly perturbed by the environmental elements, time delays and stochastic noises exist in various neural networks, which have become one of the main sources for causing instability and poor performance of neural networks. So far, most of the existing results related to the synchronization analysis for neural networks have concerned time delays and stochastic noises; see [2, 5–7, 12–14] and the references therein.
In the case where the network cannot synchronize by itself, many control techniques have been developed to drive the network to achieve synchronize, such as linear state feedback control [15, 16], state observer-based control , impulsive control , and adaptive control . All of them have the feature that the controller needs to be added to each node. But in practice, it is too difficult to add controllers to all nodes in a large-scale network. To reduce the number of controlled nodes, pinning control is introduced, in which controllers are only applied to partial nodes. In , it has been shown that a single controller can ensure that the whole network synchronizes asymptotically with large enough coupling strength and without any prior knowledge of the structure of the network topology. In , the cluster synchronization issue is considered for a class of delayed coupled complex dynamical networks by using the pinning control strategy. In addition, the pinning adaptive control method has received considerable research attention, which is utilized to get the appropriate control gains effectively. By using the adaptive pinning approach, the robust synchronization of a class of nonlinearly coupled complex networks is investigated in . An adaptive pinning control method is proposed in  to synchronize a delayed complex dynamical network with free coupling matrix. In , the adaptive pinning synchronization is investigated for complex networks with non-delayed and delayed couplings and vector-form stochastic perturbations.
Recently, the stability and synchronization of neutral-type systems have been studied widely [23–27], in which the time delays occur not only in the system states but also in the derivatives of the system states. In , the problem of stochastic stability of neutral-type neural networks with Markovian jumping parameters is considered. By using the adaptive control approach, the exponential synchronization in the p th moment of neutral-type delayed neural networks is investigated in . In , an adaptive control method is proposed to synchronize for a class of coupled neutral-type complex dynamical networks by adding adaptive controller to all nodes. To the best of our knowledge, the problem of adaptive pinning synchronization for neutral-type neural networks with stochastic perturbation has received very little research attention.
A new class of neutral-type neural networks with pinning adaptive controller is considered.
A new pinning adaptive law is designed.
The notations are quite standard. Throughout this paper, , , and denote the set of non-negative real numbers, n dimensional Euclidean space and the set of all real matrices, respectively. The superscript T denotes matrix transposition, denotes the trace of the corresponding matrix and I denotes the identity matrix. stands for the Euclidean norm in . stands for the block diagonal matrix. Let be a complete probability space with a filtration satisfying the usual conditions (i.e. the filtration contains all P-null sets and is increasing and right continuous). Denote by the family of all -measurable, bounded, and -value random variables.
2 Model and preliminaries
where is the state vector associated with n neurons, and denotes the neuron activation functions, represents the time-varying delay with , , and are the connection weight and the delay connection weight matrices, respectively, is a positive diagonal matrix, is called the neutral-type parameter matrix, and is the constant external input vector.
where and .
To prove our main results, the following assumptions are necessary.
hold, for any .
hold, for any .
In order to derive the main results, the following definitions and lemmas are necessary in this paper.
on with the initial data given by .
Definition 1 
for all initial conditions .
Definition 2 The network (2) is said to have almost surely asymptotical synchronization if network (4) is almost surely asymptotically stable.
Lemma 1 
for every .
This lemma is called the LaShall-type invariance principle.
Lemma 2 
for any .
3 Main result
Our object is to design an adaptive controller such that the neutral-type coupled neural network (2) can realize synchronization. The main results are stated as follows.
where , , and .
where and .
It can be seen that for any . Therefore, applying a LaSalle-type invariance principle for the stochastic differential delay equation, we can conclude that the controlled network (2) can be synchronized with the trajectory for almost every initial data. The proof is completed. □
Remark 1 In this paper, we investigate the almost surely synchronization for neutral-type coupled neural networks by adding adaptive controllers to the partial nodes. It is different from that in , where the authors consider the exponential synchronization in mean square of the coupled complex dynamical networks by adding adaptive controllers to all nodes.
When , from Theorem 1 we obtain the following corollary.
When the time-varying delay is constant (i.e. ), from Theorem 1 we have the following corollary.
4 Numerical simulation
In this section, we present a numerical simulation to illustrate the feasibility and effectiveness of our results.
In this paper, we have investigated the almost surely synchronization problem for an array of linearly coupled neutral-type neural networks by using adaptive pinning control. By utilizing Lyapunov stability theory and the adaptive pinning control method, some novel conditions for synchronization are derived. Furthermore, a numerical example has verified the effectiveness of the presented method.
This work is supported by the Scientific Research Foundation Program of Shaanxi Railway Institute (2011-27).
- Gopalsamy K, He X: Delay-independent stability in bidirectional associative memory networks. Automatica 1987, 23: 311–326. 10.1016/0005-1098(87)90005-7View ArticleGoogle Scholar
- Arik S: Stability analysis of delayed neural networks. IEEE Trans. Circuits Syst. I 2000, 47: 1089–1092. 10.1109/81.855465MathSciNetView ArticleMATHGoogle Scholar
- Perez-Munuzuri V, Perez-Villar V, Chua L: Autowaves for image processing on a two-dimensional CNN array of excitable nonlinear circuits: flat and wrinkled labyrinths. IEEE Trans. Circuits Syst. I 1993, 40: 174–181.View ArticleMATHGoogle Scholar
- Principle J, Kuo J, Celebi S: An analysis of the gamma memory in dynamic neural networks. IEEE Trans. Neural Netw. 1994, 5: 337–361.Google Scholar
- Wang Z, Shu H, Liu Y, Ho D, Liu X: Robust stability analysis of generalized neural networks with discrete and distributed time delays. Chaos Solitons Fractals 2006, 30: 886–896. 10.1016/j.chaos.2005.08.166MathSciNetView ArticleMATHGoogle Scholar
- Wu W, Chen T: Global synchronization criteria of linearly coupled neural network systems with time-varying coupling. IEEE Trans. Neural Netw. 2008, 19: 319–332.View ArticleGoogle Scholar
- Lu W, Chen T: Synchronization of coupling connected neural networks with delays. IEEE Trans. Circuits Syst. I 2004, 51: 2491–2503. 10.1109/TCSI.2004.838308View ArticleGoogle Scholar
- He W, Cao J: Exponential synchronization of hybrid coupled networks with delayed coupling. IEEE Trans. Neural Netw. 2010, 21: 571–583.View ArticleGoogle Scholar
- Ding X, Gao Y, Zhou W, Tong D, Su H: Adaptive almost surely asymptotically synchronization for stochastic delayed neural networks with Markovian switching. Adv. Differ. Equ. 2013., 2013: Article ID 211Google Scholar
- Li T, Wang T, Yang X, Fei S: Pinning cluster synchronization for delayed dynamical networks via Kronecker product. Circuits Syst. Signal Process. 2013, 32: 1907–1929. 10.1007/s00034-012-9523-xMathSciNetView ArticleGoogle Scholar
- Wang R, Zhang Z, Qu J, Cao J: Phase synchronization motion and neural coding in dynamic transmission of neural information. IEEE Trans. Neural Netw. 2011, 22: 1097–11106.View ArticleGoogle Scholar
- Yang X, Cao J, Long Y, Rui W: Adaptive lag synchronization for competitive neural networks with mixed delays and uncertain hybrid perturbations. IEEE Trans. Neural Netw. 2010, 21: 1656–1667.View ArticleGoogle Scholar
- Chen G, Zhou J, Liu Z: Global synchronization of coupled delayed neural networks and applications to chaotic CNN models. Int. J. Bifurc. Chaos 2004, 14: 2229–2240. 10.1142/S0218127404010655MathSciNetView ArticleMATHGoogle Scholar
- Li C, Yang S: Synchronization in linearly coupled dynamical networks with distributed time-delay coupling. Chaos 2008, 18: 2039–2047.MathSciNetMATHGoogle Scholar
- Li Z, Fang G, Hill D: Controlling complex dynamical networks with coupling delays to a desired orbit. Phys. Lett. A 2006, 359: 42–46. 10.1016/j.physleta.2006.05.085View ArticleMATHGoogle Scholar
- Huang T, Li C, Liu X: Synchronization of chaotic systems with delay using intermittent linear state feedback. Chaos 2008., 18: Article ID 033122Google Scholar
- Jiang G, Tang W, Chen G: A state-oberver-based approach for synchronization in complex dynamical networks. IEEE Trans. Circuits Syst. I 2006, 53: 2739–2745.MathSciNetView ArticleGoogle Scholar
- Dai A, Zhou W, Feng J, Fang JA, Xu S: Exponential synchronization of the coupling delayed switching complex dynamical networks via impulsive control. Adv. Differ. Equ. 2013., 2013: Article ID 195Google Scholar
- Chen T, Liu X, Lu W: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I 2007, 54: 1317–1326.MathSciNetView ArticleGoogle Scholar
- Jin X, Yang G: Adaptive pinning synchronization of a class of nonlinearly coupled complex networks. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 316–326. 10.1016/j.cnsns.2012.07.011MathSciNetView ArticleMATHGoogle Scholar
- Lee T, Park J, Jung H, Lee S, Kwon O: Synchronization of a delayed complex dynamical network with free coupling matrix. Nonlinear Dyn. 2012, 69: 1081–1090. 10.1007/s11071-012-0328-zMathSciNetView ArticleMATHGoogle Scholar
- Yang X, Cao J: Adaptive pinning synchronization of complex networks with stochastic perturbations. Discrete Dyn. Nat. Soc. 2010., 2010: Article ID 416182Google Scholar
- Zhu Q, Zhou W, Tong D, Fang J: Adaptive synchronization for stochastic neural networks of neutral-type with mixed time-delays. Neurocomputing 2013, 99: 477–485.View ArticleGoogle Scholar
- Lee S, Kwon O, Park J: A novel delay-dependent criterion for delayed neural networks of neutral type. Phys. Lett. A 2010, 374: 1843–1848. 10.1016/j.physleta.2010.02.043MathSciNetView ArticleMATHGoogle Scholar
- Chen W, Wang L: Delay-dependent stability for neutral-type neural networks with time-varying delays and Markovian jumping parameters. Neurocomputing 2013, 120: 569–576.View ArticleGoogle Scholar
- Zhou W, Gao Y, Tong D, Ji C, Fang J: Adaptive exponential synchronization in p th moment of neutral-type neural networks with time delays and Markovian switching. Int. J. Control. Autom. Syst. 2013, 11: 845–851. 10.1007/s12555-012-9308-9View ArticleGoogle Scholar
- Zhang Y, Gu D, Xu S: Global exponential adaptive synchronization of complex dynamical networks with neutral-type neural network nodes and stochastic disturbances. IEEE Trans. Circuits Syst. I 2013, 60: 2709–2718.MathSciNetView ArticleGoogle Scholar
- Mao XR, Yuan CG: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London; 2006.View ArticleMATHGoogle Scholar
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