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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.

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 [

In addition, much attention has been paid to the potential applications of the synchronization of coupled neural networks, for example, secure communication [

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 [

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:

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 [

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.

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 [

The rest of the paper is organized as follows. Section

Consider an array of delayed neural networks consisting of

The time-varying delay function

The activation functions

The parameter uncertainties are assumed to satisfy the following conditions:

The isolated dynamic network is

Network (

To investigate the stability of the synchronized states (

If neural networks (

For any symmetric positive definite matrixes

For any constant symmetric matrixes

Let

Assume that

If

For a symmetric matrix

In this section, we present hybrid control scheme to synchronize neural networks (

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 (

For some given synchronization scaling function

Construct the following Lyapunov-Krasovskii functional candidate:

From Assumption

If there are no uncertain parameters in coupled delayed neural networks (

The nodes pinned for directed networks are chosen as follows.

For undirected networks, for example, the small-world network [

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:

For some given synchronization scaling function

Construct the following Lyapunov-Krasovskii functional candidate:

In this section, we provide several numerical examples to demonstrate the feasibility of the proposed method.

Consider a two-dimensional neural network with time-varying delay presented in the following system:

(a) Chaotic trajectory of neural network (

Afterwards, the FPS problems for the nonlinear and adaptive pinning controlled network consisting of

Simple directed neural network with 6 nodes.

Choose the following parameters: the time-varying scaling function

Figure

Chaotic behavior of isolate node

The FPS errors between the states of isolate node

The FPS errors between the states of isolate node

The evolution of adaptive pinning feedback gains

The topology structure of Watts-Strogatz neural network with

The topology structure of Watts-Strogatz neural network with

The topology structure of Watts-Strogatz neural network with

Figure

The FPS errors between the states of isolate node

The FPS errors between the states of isolate node

The evolution of adaptive pinning feedback gains

We consider the FPS problems for the nonlinear and adaptive controlled network consisting of

The inner-coupling matrices with uncertainties are the same as in (

Simple directed neural network with 5 nodes.

Figure

The FPS errors between the states of isolate node

The FPS errors between the states of isolate node

The evolution of adaptive feedback gains

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.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The first author was supported by National Research Council of Thailand and Khon Kaen University 2017 (Grant no. 600061). The second author was supported by Rajamangala University of Technology Isan and the Thailand Research Fund (TRF), the Office of the Higher Education Commission (OHEC) (Grant no. MRG5980027). The third author is also supported by Chiang Mai University, Chiang Mai, Thailand. The fourth author was financially supported by University of Phayao, Phayao, Thailand.