Finite-Time Mittag–Leffler Synchronization of Neutral-Type Fractional-Order Neural Networks with Leakage Delay and Time-Varying Delays

Eva Kaslik,Călin-Adrian Popa

doi:10.3390/math8071146

Abstract

This paper studies fractional-order neural networks with neutral-type delay, leakage delay, and time-varying delays. A sufficient condition which ensures the finite-time synchronization of these networks based on a state feedback control scheme is deduced using the generalized Gronwall–Bellman inequality. Then, a different state feedback control scheme is employed to realize the finite-time Mittag–Leffler synchronization of these networks by using the fractional-order extension of the Lyapunov direct method for Mittag–Leffler stability. Two numerical examples illustrate the feasibility and the effectiveness of the deduced sufficient criteria.

Highlights

Fractional calculus studies the different possibilities of defining real or complex orders for the differentiation and integration operators. It has a long history, only recently it has been successfully applied to physics and engineering problems
We will use a different state feedback control scheme to realize finite-time Mittag–Leffler synchronization between master system (1) and slave system (2), for which the controller is given by α ui (t) = k i1 ei (t) + k i2 sign(ei (t))|ei (t − μ)| + k i3 sign(ei (t))|ei (t − τ (t))| + k i4 sign(ei (t))| D−
We give a sufficient condition that ensures the finite-time synchronization of master system (1) and slave system (2), based on the controller (4)

Summary

Introduction

Fractional calculus studies the different possibilities of defining real or complex orders for the differentiation and integration operators. Finite-time stability criteria for delayed memristor-based fractional-order neural networks were deduced in [24]. Finite-time projective synchronization sufficient criteria were deduced in [33], for fractional-order complex-valued memristor-based neural networks with delay. The property of finite-time Mittag–Leffler synchronization was introduced in [36], where sufficient criteria to attain this type of synchronization were given for fractional-order memristive BAM neural networks. “fractional-order complex-valued memristor-based neural networks with both leakage and time-varying delays” were the focus of [22], where sufficient criteria for finite-time stability were deduced for these networks. Taking all the above into account, we consider neutral-type fractional-order neural networks with leakage delay and time-varying delays in this paper, and study their finite-time synchronization and finite-time. The smallest eigenvalue of positive definite matrix P is λmin ( P). || · || represents the vector Euclidean norm or the matrix Frobenius norm, and | · | is the element-wise vector norm or the element-wise matrix norm

Preliminaries

Main Results

Numerical Examples

Conclusions