Global Mittag-Leffler synchronization of delayed fractional-order memristive neural networks

Yingjie Fan,Yuxia Li,Xia Huang,Jianwei Xia,Zhen Wang

doi:10.1186/s13662-018-1800-y

Abstract

This paper deals with the global Mittag-Leffler synchronization of fractional-order memristive neural networks (FMNNs) with time delay. Since the FMNNs are essentially a class of switched systems with irregular switching laws, it is more difficult to achieve synchronization than with the traditional neural networks. First, under the framework of fractional-order differential inclusions and set-valued maps, the FMNNs are transformed into a continuous system with uncertainties. Then a linear state feedback combined with switching control law is designed in order to achieve the Mittag-Leffler synchronization. In addition, several synchronization criteria are obtained by constructing appropriate Lyapunov functionals, together with the help of some inequality techniques. Finally, an example is given to demonstrate the effectiveness of the obtained results.

Highlights

The memristor was first postulated by Chua in 1971, as the fourth fundamental circuit element together with the resistor, inductor and capacitor [1]
Since considerable efforts have been made devoted to the investigation of memristive neural networks (MNNs), such as stability [7], stabilization [8,9,10], nonlinear dynamics analysis [11], and synchronization [12,13,14,15,16,17,18]
J=1 j=1 where i = 1, 2, . . . , n, n is the number of neurons; ci > 0 denotes the self-feedback connection weight; 0 < α < 1 is the fractional order; xi(t) corresponds to the state variable connected with the ith neuron; τ is the time delay; fj(·) denotes the activation function satisfying fj(0) = 0; aij(xj(t)) and bij(xj(t)) are memristive synaptic connection weights, which are defined by aij xj(t)

Summary

Introduction

The memristor was first postulated by Chua in 1971, as the fourth fundamental circuit element together with the resistor, inductor and capacitor [1]. N, n is the number of neurons; ci > 0 denotes the self-feedback connection weight; 0 < α < 1 is the fractional order; xi(t) corresponds to the state variable connected with the ith neuron; τ is the time delay; fj(·) denotes the activation function satisfying fj(0) = 0; aij(xj(t)) and bij(xj(t)) are memristive synaptic connection weights, which are defined by aij xj(t). In order to achieve the global Mittag-Leffler synchronization, the feedback gains ρi should be well designed such that ρi – nj=1(|a∗ij – a∗ij∗| + |b∗ij – b∗ij∗|)Mj ≥ 0 are ensured. Remark 2 In this paper, by constructing appropriate Lyapunov functionals and designing suitable hybrid controllers, some sufficient criteria are established to achieve the globally Mittag-Leffler synchronization of delayed FMNNs. Notice that the memristive synaptic connection weights aij(·), bij(·) are both state-dependent, which makes it impossible to achieve the complete synchronization only via linear state feedback control.

Numerical simulations

Conclusions