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Complexity
Volume 2017 (2017), Article ID 4654020, 18 pages
https://doi.org/10.1155/2017/4654020
Research Article

Hybrid Adaptive Pinning Control for Function Projective Synchronization of Delayed Neural Networks with Mixed Uncertain Couplings

1Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
2Department of Applied Mathematics and Statistics, Rajamangala University of Technology Isan, Nakhon Ratchasima 30000, Thailand
3Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
4Department of Mathematics, University of Phayao, Phayao 56000, Thailand

Correspondence should be addressed to N. Yotha

Received 25 March 2017; Revised 14 June 2017; Accepted 2 July 2017; Published 8 August 2017

Academic Editor: Fathalla A. Rihan

Copyright © 2017 T. Botmart et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents the function projective synchronization problem of neural networks with mixed time-varying delays and uncertainties asymmetric coupling. The function projective synchronization of this model via hybrid adaptive pinning controls and hybrid adaptive controls, composed of nonlinear and adaptive linear feedback control, is further investigated in this study. Based on Lyapunov stability theory combined with the method of the adaptive control and pinning control, some novel and simple sufficient conditions are derived for the function projective synchronization problem of neural networks with mixed time-varying delays and uncertainties asymmetric coupling, and the derived results are less conservative. Particularly, the control method focuses on how to determine a set of pinned nodes with fixed coupling matrices and strength values and randomly select pinning nodes. Based on adaptive control technique, the parameter update law, and the technique of dealing with some integral terms, the control may be used to manipulate the scaling functions such that the drive system and response systems could be synchronized up to the desired scaling function. Finally, numerical examples are given to illustrate the effectiveness of the proposed theoretical results.

1. Introduction

Presently, neural networks are under extensive consideration because of their significant application in various fields such as image processing, pattern recognition, and associative memories because the switching speed of information processing and the inherent neuron communication is limited [1, 2]. Global asymptotic stability of a general class of recurrent neural networks with time-varying delays is discussed in [3]. Zhang and Han [4] investigated the global asymptotical stability analysis for delayed neural networks using a matrix-based quadratic convex approach. It is well know that the time delay is continually necessarily existent in neural networks because it might lead to inconstancy or considerably inferior performances. So, the neural networks with time delays have attracted considerable attention of many researchers [57].

In addition, much attention has been paid to the potential applications of the synchronization of coupled neural networks, for example, secure communication [810]. The synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay were discussed in [11]. Chen and Cao [12] suggested projective synchronization of neural networks with mixed time-varying delays and parameter mismatch. Moreover, there is synchronization problem that is called function projective synchronization that has received increasing attention in [1315]. FPS is the driver and response system that can be synchronized up to a scaling function. Many researches mentioned that function projective synchronization (FPS) is the greater general definition of chaotic synchronization [1618]. It is obvious that the definition of FPS includes comprehensive synchronization and projective synchronization. In order for scaling function to achieve unity or be constant, only one complete synchronization or projective synchronization could be obtained since the unpredictability of the scaling function in FPS can also raise the security of communication. Thus, FPS has drawn the attention of many researchers in various fields. In Gao et al. [19] the generalized function projective synchronization of weighted cellular neural networks with multiple time-varying coupling delays was studied. Adaptive projective synchronization in complex networks with time-varying coupling delay discussed in [20]. Tang and Wong [21] studied on the distributed synchronization problem of coupled neural networks by randomly occurring control method. Hu et al. [22] suggested a pinning synchronization control scheme for a class of linearly coupled neural networks. Huang et al. [23] investigated the stabilization of delayed chaotic neural networks by periodically intermittent control. Further, Cai et al. [24] considered the outer synchronization between two hybrid-coupled delayed dynamical networks via aperiodically adaptive intermittent pinning control. However, not all of the neural networks could synchronize by themselves. So, they need to bring the suitable controllers in order to make them synchronize. One of the most important existing control methods is the pinning control.

Pinning control is the strategy that employs the local feedback injection to a small fraction of nodes to carry out the global performances of the total networks. It is a competent and useful strategy especially for the large size networks. The pinning synchronization of neural networks has been generally examined at the present [2533]. Meanwhile, various selection rules of pinned nodes have been introduced in the existing literatures. The pinned nodes selection rules according to the out-degree and in-degree of the nodes and the synchronization problem were studied for undirected and directed networks which was presented in [25, 26, 34]. The pinning control problem of neural networks was considered and then some unexceptional pinning conditions were found out [2729], while Chen et al. [35] expressed that, even applying one single pinned node, the whole networks could be controlled as long as the coupling strength was large enough. Furthermore, Wang and Chen [36] summarized that the most highly connected nodes are pinned in order to get the better performance for the undirected networks.

As discussions mentioned above, hybrid adaptive pinning control for FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling is an interesting topic for investigating. Therefore, this paper will be focused on this topic in order to facilitate clear comprehension and the purposes of this paper are given as follows:(i)The mixed time-varying delays with discrete and distributed time-varying delays are considered in the dynamical nodes and in uncertainties asymmetric coupling, simultaneously, which are different from time-delay case in [23, 2729]. So, our systems are general ones.(ii)For the control method, FPS is studied by using the nonlinear and adaptive pinning controls and using the nonlinear and adaptive controls which contain error linear term, time-varying delay error linear term, and distributed time-varying delay error linear term.(iii)The FPS of this paper focuses on how to determine a set of pinned nodes for a linearly coupled delayed neural network with fixed coupling matrices and strength values. Moreover, this paper used random selection of pinning nodes which is different from the pinning control method in [13, 37]. Based on constructing a novel Lyapunov-Krasovskii functional, adaptive control technique, the parameter update law, and the technique of dealing with Jensen’s and Cauchy inequalities, some novel sufficient conditions for guaranteeing the existence of the FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling are derived. Finally, numerical examples are included to show the effectiveness of using the nonlinear and adaptive pinning controls and the nonlinear and adaptive controls.

The rest of the paper is organized as follows. Section 2 provides some mathematical preliminaries and network model. Section 3 presents FPS of neural network with mixed time-varying delays and hybrid uncertainties asymmetric coupling by hybrid adaptive control and hybrid adaptive pinning control, respectively. In Section 4, some numerical examples illustrate given theoretical results. The paper ends with conclusions in Section 5 and cited references.

2. Model Description and Preliminaries

Notations. The following notation will be used in this paper: denotes the -dimensional space and denotes the Euclidean vector norm; denotes the transpose of matrix ; is symmetric if ; denotes an -dimensional identity matrix; for the matrix , the th row and the th column of are called the th row-column pair of . is the minor matrix of by removing arbitrary row-column pairs of . The symbol denotes the Kronecker product.

Consider an array of delayed neural networks consisting of identical nodes with uncertainties asymmetric couplingwhere is the number of coupled nodes, is the neuron state vector of the th node, denotes the number of neurons in a neural network, denote the rate with which the cell resets its potential to the resting state when being isolated from other cells and inputs, , and are connection weight matrices, and are the time-varying delays, , denote the neuron activation function vector, the positive constants , and are the strength values for the constant coupling and delayed couplings, respectively, are the control input of the node , , and are constant inner-coupling matrices and it is assumed that , and are positive definite matrix, , and are the uncertainties of inner-coupling matrices, and are the outer-coupling matrices and satisfy the following conditions:

Assumption 1. The time-varying delay function is differential function and satisfies the conditions , and .

Assumption 2. The activation functions , satisfy the Lipschitz constants : where are positive constant matrices and we denote

Assumption 3. The parameter uncertainties are assumed to satisfy the following conditions: where , and are known real constant matrices and is and unknown time-varying matrix function satisfying

The isolated dynamic network iswhere with and the parameters , and and the nonlinear functions have the same definitions as in (1).

Definition 4 (FPS). Network (1) with time delay is said to achieve function projective synchronization if there exists a continuously differentiable scaling function such that where stands for the Euclidean vector norm and can be an equilibrium point, or a (quasi-)periodic orbit, or an orbit of a chaotic attractor.

To investigate the stability of the synchronized states (1), we set the synchronization error in the form . Then, substituting it into (1), it is easy to get the following:The initial condition of (8) is defined bywhere and

Remark 5. If neural networks (1) are without parameter uncertainties and , the networks model turns into the neural networks proposed in [2729]: Hence, our network model (1) includes previous network model, which can be regarded as a special case of neural network (1).

Lemma 6 (see [5] (Cauchy inequality)). For any symmetric positive definite matrixes and we have

Lemma 7 (see [5]). For any constant symmetric matrixes , and , vector function such that the integrations concerned are well defined, and then

Lemma 8 (see [38]). Let and , , , and be matrices with appropriate dimensions. Then (i),(ii),(iii),(iv),(v)

Lemma 9 (see [25]). Assume that and are the Hermitian matrices. Suppose that , , and are eigenvalues of matrices , and , respectively. Then one has

Lemma 10 (see [35]). If is irreducible and satisfies , Then, for any constant , all eigenvalues of the matrix are negative definite, where .

Lemma 11 (see [25]). For a symmetric matrix and a diagonal matrix with , let where is the minor matrix of by removing its first row-column pairs, and are matrices with appropriate dimensions, and . If is equivalent to .

3. Main Results

In this section, we present hybrid control scheme to synchronize neural networks (1) to the homogenous trajectory (6). Then, we will give some sufficient conditions in FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling.

3.1. FPS under Hybrid Adaptive Pinning Control

We design nonlinear and adaptive pinning controls to realize FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling. In order to stabilize the origin of neural networks (1) by means of nonlinear and adaptive pinning controls such aswhere the updating laws arewhere , and are positive constants and is a solution of an isolated node. The controller is different type of controller, and is the nonlinear control and is the adaptive linear pinning control. Then, substituting (14) into (8), it can be derived thatwhere and, for convenience, we denote

Theorem 12. For some given synchronization scaling function , neural networks (1) satisfying Assumptions 1, 2, and 3, and target system can realize function projective synchronization by the nonlinear and adaptive pinning control law as shown in (14) if there exist positive constants , and by taking appropriate , and such thatThen controlled neural network (1) is function projective synchronization.

Proof. Construct the following Lyapunov-Krasovskii functional candidate:where By taking the derivative of along the trajectories of system (17), we have the following: After were calculated, we will get thatwhere , , , , , and .
From Assumption 2, we obtain the following three inequalities: Applying Lemmas 6 and 7, we have Therefore, we havewhere . Note that the matrix is symmetric. Let where is the minor matrix of by removing its first row-column pairs, and are matrices with appropriate dimensions, and diag. If and with Lemma 9, we have . Therefore, one can choose suitable positive constants , such that . It follows from Lemma 11 and that . Then, by and (29), we can conclude thatWe only need to choose the suitable positive constants