Global Asymptotic Stability for a New Class of Neutral Neural Networks

Ricai Luo

doi:10.11648/j.ijamtp.20180403.13

Abstract

In the present world, due to the complicated dynamic properties of neural cells, many dynamic neural networks are described by neutral functional differential equations including neutral delay differential equations. These neural networks are called neutral neural networks or neural networks of neural-type. The differential expression not only defines the derivative term of the current state but also explains the derivative term of the past state. In this paper, global asymptotic stability of a neutral-type neural networks, with time-varying delays, are presented and analyzed. The neural network is made up of parts that include: linear, non-linear, non-linear delayed, time delays in time derivative states, as well as a part of activation function with the derivative. Different from prior references, as part of the considered networks, the last part involves an activation function with the derivative rather than multiple delays; that is a new class of neutral neural networks. This paper assumes that the activation functions satisfy the Lipschitz conditions so that the considered system has a unique equilibrium point. By constructing a Lyapunov-Krasovskii-type function and by using a linear matrix inequality analysis technique, a sufficient condition for global asymptotic stability of this neural network has been obtained. Finally, we present a numerical example to show the effectiveness and applicability of the proposed approach.

Highlights

Since Hopfield proposed a neural network model which was named after him in 1984, the Hopfield neural network has been applied to various fields, such as combinatorial optimization [1,2,3,4], image processing [5, 6], pattern recognition [7], signal processing [8], and communication [9]
Neutral-type neural networks are usually described by the following ordinary differential equations: International Journal of Applied Mathematics and Theoretical Physics 2018; 4(3): 78-83 xɺ(t) = −Cx(t) + Af (x(t)) + Bf (x(t −τ (t)))
Because the activation functions satisfies (H), by using the Brouwer fixed-point theorem, it would be inferred that the neural network model (3) has an unique equilibrium point y* for each I; it is similar to the proof provided by the literature [25]

Summary

Introduction

Since Hopfield proposed a neural network model which was named after him in 1984, the Hopfield neural network has been applied to various fields, such as combinatorial optimization [1,2,3,4], image processing [5, 6], pattern recognition [7], signal processing [8], and communication [9]. Neutral-type neural networks are a special type of time-delayed neural networks, in which the information relating to derivatives of the past states is introduced to describe the system dynamics [23,24,25,26,27]. Lakshmanan et al have studied the following neutral delay Hopfield neural network model: yɺ(t) = − Ay(t) + Bg( y(t)) + Cg( y(t −τ )). Because the activation functions satisfies (H), by using the Brouwer fixed-point theorem, it would be inferred that the neural network model (3) has an unique equilibrium point y* for each I; it is similar to the proof provided by the literature [25].

Main Results

A numerical Example

Conclusion