Hybrid Control Scheme for Projective Lag Synchronization of Riemann–Liouville Sense Fractional Order Memristive BAM NeuralNetworks with Mixed Delays

Grienggrai Rajchakit,Chuangxia Huang,Jinde Cao,Anbalagan Pratap,Ramachandran Raja,Jehad Alzabut

doi:10.3390/math7080759

Abstract

This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop control and adaptive state feedback control is designed to guarantee the global projective lag synchronization of the addressed FOMBNNs model. Secondly, by using a Lyapunov–Krasovskii functional and Barbalet’s lemma, a new brand of sufficient criterion is proposed to ensure the projective lag synchronization of the FOMBNNs model considered. Moreover, as special cases by using a hybrid control scheme, some sufficient conditions are derived to ensure the global projective synchronization, global complete synchronization and global anti-synchronization for the FOMBNNs model considered. Finally, numerical simulations are provided to check the accuracy and validity of our obtained synchronization results.

Highlights

Differential equation and dynamic system modeling have become important research topics in natural science and engineering technology [1,2,3,4,5,6,7,8,9,10,11,12]
The attention of distributed delays is significant in fractional order neural networks (FONNs) dynamical systems, and there is a huge amount of research works on FONNs with distributed delay—see, for instance, [35,36]
A novel hybrid controller, which is the combination of open loop control and adaptive state feedback control are designed to ensure the projective lag synchronization criteria for fractional order memristive bidirectional associative memory (BAM) neural networks (FOMBNNs) with mixed time delays

Summary

Introduction

Differential equation and dynamic system modeling have become important research topics in natural science and engineering technology [1,2,3,4,5,6,7,8,9,10,11,12]. During the last two decades, the study of fractional differential equations has been widely applicable to many real world problems They have already been successfully applied in many fields of engineering including but not restricted to market dynamics [17], neural networks [18] and polarization [19]. Otherwise, during a particular period, the signal propagation is distributed because the variety of axon sizes and lengths are too large In this manner, the attention of distributed delays is significant in fractional order neural networks (FONNs) dynamical systems, and there is a huge amount of research works on FONNs with distributed delay—see, for instance, [35,36]. In [49], the sufficient conditions are derived to ensure the finite-time Mittag–Leffler synchronization of delayed FOMNNs with different orders by applying the Lyapunov-stability theory and a state feedback control. In contrast to the existing results in the literature, the hybrid control BAM type neural networks and mixed time delays have not taken into consideration; our proposed results make it up

Preliminaries and Problem Statement

Main Results

Numerical Example

Conclusions