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

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 controlmethod 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.


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 matrixbased 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 [5][6][7].
In addition, much attention has been paid to the potential applications of the synchronization of coupled neural networks, for example, secure communication [8][9][10].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 [13][14][15].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 [16][17][18].It is obvious that the definition of FPS includes comprehensive synchronization and projective synchronization.In order for scaling function 2 Complexity 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 timevarying 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 [25][26][27][28][29][30][31][32][33].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 [27][28][29], 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 timevarying 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,[27][28][29].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.

Model Description and Preliminaries
Notations.The following notation will be used in this paper: R  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  ∈ R  × R  , the th row and the th column of  are called the th row-column pair of .  ∈ R (−)×(−) is the minor matrix of  ∈ R × by removing arbitrary  (1 ≤  ≤ ) row-column pairs of .The symbol ⊗ denotes the Kronecker product.
The isolated dynamic network is where () = ( where ‖⋅‖ stands for the Euclidean vector norm and () ∈ R  can be an equilibrium point, or a (quasi-)periodic orbit, or an orbit of a chaotic attractor.

Complexity
Hence, our network model ( 1) includes previous network model, which can be regarded as a special case of neural network (1).

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.

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 U  () such as where the updating laws are where  1 ,  2 , and  3 are positive constants and () is a solution of an isolated node.The controller U  () is different type of controller, and  1 () is the nonlinear control and  2 () is the adaptive linear pinning control.Then, substituting ( 14) into ( 8), it can be derived that where and, for convenience, we denote Complexity Then 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 that where Applying Lemmas 6 and 7, we have () ) .
Remark 14.The nodes pinned for directed networks are chosen as follows.
Step 2. The  pinned nodes are sorted according to the pinned-node selection scheme studied [26], for the pinning controlled error neural network (17); so, the nodes to be pinned are chosen in the particular order.Let  = 1, if the first inequalities of Theorem 12 are satisfied, and then the least number is 1; otherwise, go to the next step.
Step 3. If condition (20) is not satisfied, increase  ( =  + 1) gradually with more network nodes to the pinned node based on the order in Step 2 particularly until condition (20) holds.
For undirected networks, for example, the small-world network [31], the scale-free network [32], and the Watts-Strogatz network [39], we can randomly choose a set of pinned nodes to satisfy condition (20) by increasing the number of pinned nodes .

FPS under Hybrid Adaptive Control.
The nonlinear and adaptive controls are designed to realize FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling.Then we have the following controlled form: where () ,  = 1, 2, . . ., .
Then, substituting ( 34) into ( 8), we have where By using adaptive controlling method, we get the following theorem.
Proof.Construct the following Lyapunov-Krasovskii functional candidate: where By taking the derivative of () along the trajectories of system (36) that is similar to the proof of Theorem 12, we obtain It is obvious that there exist sufficiently large positive constants  * 1 ,  * 2 , and  * 3 such that We can choose  * 1 ,  * 2 , and  * 3 satisfying (42), (43), and (44), respectively.The remaining proof is similar to Theorem 12 and omitted.

Numerical Simulations
In this section, we provide several numerical examples to demonstrate the feasibility of the proposed method.
Afterwards, the FPS problems for the nonlinear and adaptive pinning controlled network consisting of  twodimensional neural networks are described as follows: where   () = [ 1 (),  2 ()]  ∈ R 2 is the state variable of the th node.The inner-coupling matrices with uncertainties are and the other parameters are the same as those in (45).
Directed Neural Network.We consider the directed neural networks as shown in Figure 2.

Conclusions
In this paper, the hybrid adaptive pinning control for FPS of neural networks with mixed time-varying delays and uncertainties asymmetric coupling were investigated.We have applied the use of nonlinear and adaptive pinning controls and the nonlinear and adaptive controls.Some sufficient conditions are derived to guarantee the FPS by   use of the Lyapunov-Krasovskii function method.Moreover, the drive and response systems could be synchronized up to the desired scaling functions based on the adaptive control technique.Furthermore, numerical examples are given to illustrate the effectiveness of the proposed theoretical results in this paper as well.