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Based on the theory of fractional calculus, the generalized Gronwall inequality and estimates of mittag-Leffer functions, the finite-time stability of Caputo fractional-order BAM neural networks with distributed delay is investigated in this paper. An illustrative example is also given to demonstrate the effectiveness of the obtained result.

Fractional calculus (integral and differential operations of noninteger order) was firstly introduced 300 years ago. Due to lack of application background and the complexity, it did not attract much attention for a long time. In recent decades fractional calculus is applied to physics, applied mathematics, and engineering [

We know that the next state of a system not only depends upon its current state but also upon its history information. Since a model described by fractional-order equations possesses memory, it is precise to describe the states of neurons. Moreover, the superiority of the Caputo’s fractional derivative is that the initial conditions for fractional differential equations with Caputo derivatives take on the similar form as those for integer-order differentiation. Therefore, it is necessary and interesting to study fractional-order neural networks both in theory and in applications.

Recently, fractional-order neural networks have been presented and designed to distinguish the classical integer-order models [

The integer-order bidirectional associative memory (BAM) model known as an extension of the unidirectional autoassociator of Hopfield [

Motivated by the above-mentioned works, we were devoted to establishing the finite-time stability of Caputo fractional-order BAM neural networks with distributed delay. In this paper, we will apply Laplace transform, generalized Gronwall inequality, and estimates of Mittag-Leffler functions to establish the finite-time stability criterion of fractional-order distributed delayed BAM neural networks.

This paper is organized as follows. In Section

For the convenience of the reader, we first briefly recall some definitions of fractional calculus; for more details, see [

The Riemann-Liouville fractional integral of order

The Caputo fractional derivative of order

The Mittag-Leffler function in two parameters is defined as

The Laplace transform of Mittag-Leffler function is

In this paper, we are interested in the finite-time stability of fractional-order BAM neural networks with distributed delay by the following state equations:

Let

The initial conditions associated with (

In order to obtain main result, we make the following assumptions.

For

The neurons activation functions

The neurons activation functions

Since the Caputo’s fractional derivative of a constant is equal to zero, the equilibrium point of system (

Define the functions as follows:

System (

A technical result about norm upper-bounding function of the matrix function

If

Let

We first give a key lemma in the proof of our main result as follows.

Let

Substituting (

For convenience, let

In the following, sufficient conditions for finite-time stability of fractional-order BAM neural networks with distributed delay are derived.

Let

By Laplace transform and inverse Laplace transform, system (

In this section, we give an example to illustrate the effectiveness of our main result.

Consider the following two-state Caputo fractional BAM type neural networks model with distributed delay

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Natural Science Foundation of Jiangsu Province (BK2011407) and the Natural Science Foundation of China (11271364 and 10771212).