Abstract

Finite-time synchronization for a class of fractional-order delayed neural networks with fractional order , and , is investigated in this paper. Through the use of Hölder inequality, generalized Bernoulli inequality, and inequality skills, two sufficient conditions are considered to ensure synchronization of fractional-order delayed neural networks in a finite-time interval. Numerical example is given to verify the feasibility of the theoretical results.

1. Introduction

Since the 17th century, the theory of fractional calculus was mainly focused on the pure theoretical field of mathematics [1, 2]. In the past two decades, it has been found that the dynamical behaviors of many systems can be described by the fractional calculus. Furthermore, fractional-order models can help exhibit the dynamical behaviors of systems. In fact, many physical systems show fractional dynamical behaviors because of special properties [3–11].

Fractional-order neural networks have attracted great attention due to their potential properties of memory and hereditary. Particularly, the dynamical analysis of fractional-order neural networks can be used to describe the dynamical characteristics of neural networks. For instance, synchronization is considered as an important topic, as reported in [12, 13], where chaotic synchronization of fractional-order neural networks was proposed. Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks were discussed in [14]. In addition, the results for stability analysis and synchronization of fractional-order networks were presented in [15–23].

However, the majority of the results were demonstrated to ensure the asymptotic stability of error systems. Asymptotic synchronization indicates that the trajectories of the slave system reach to the trajectories of the master system over the infinite horizon. In fact, it is more desirable that the networks can reach synchronization in a finite-time in physical and engineering systems, achieving an optimality in convergence time. Thus, it is necessary to study the finite-time synchronization of neural networks. In order to achieve faster synchronization in control systems, an effective finite-time control method is utilized. Some important results on finite-time synchronization were demonstrated on integer-order systems [24–28]. Note that time delay [29–31] occurs in many physical and engineering systems. So it is natural to study fractional-order systems with delays. Till now few results are obtained with the consideration of the finite-time stability and synchronization of fractional-order neural networks with time delays [32, 33]. For instance, finite-time synchronization of fractional-order memristor-based neural networks with time delays was considered by using Laplace transform, such as the generalized Gronwall inequality and Mittag-Leffler functions, in [33].

Motivated by the above discussion, the main goal of this paper is to adopt new methods and obtain some new sufficient conditions that can assist master-slave systems to achieve the finite-time synchronization of fractional delayed neural networks with orders and .

Throughout the paper, denote and which are the Euclidean vector norm and the matrix norm, respectively; and are the element of the vector and the matrix .

2. Preliminaries and Model Description

There are some definitions of the fractional-order integrals and derivatives. Due to the advantages of the Caputo fractional derivative, the definition of Caputo derivative is used in this paper.

Definition 1 (see [1]). The fractional integral with noninteger order of a function is defined bywhere , is the Gamma function, and .

Definition 2 (see [1]). The Caputo derivative of fractional order of a function is defined bywhere ; .
In this paper, we consider a class of fractional-order neural networks with time delay as master system, which is described byor equivalentlyfor , where is the number of units in a neural network, corresponds to the state of the th unit at time and denotes , and is the self-regulating parameter of the neurons. represents the external input; and are the connective weights matrix in the presence and absence of delay, respectively. Functions and denote the output of the th unit at time and , respectively, where is the transmission delay and denotes , .
The initial conditions associated with system (1) are of the form , where denotes the real-valued continuous function defined on , with the norm given by .

The slave system is given:or equivalentlywhere is the state vector of the system response and is a suitable controller. The initial conditions associated with system (3) are of the form , where denotes the real-valued continuous function defined on , with the norm given by .

For generalities, the following definition, assumptions, and lemmas are presented.

Definition 3. System (1) is said to be synchronized with system (3) in a finite-time with respect to , for a suitable designed controller , if and only if , implying , , where are real positive numbers and .

Assumption 4. The neuron activation functions are Lipschitz continuous, with the existence of positive constants , such thatfor all .

Assumption 5. and are continuous and bounded functions defined on , and let .

Lemma 6 (Hölder inequality [34]). Assume that and , and if , then andwhere is the Banach space of all Lebesgue measurable functions with .
Let ; it reduces to the Cauchy-Schwartz inequality as follows:

Lemma 7 (generalized Bernoulli inequality [34]). If , and , then, for , , or .

Lemma 8 (see [35]). If and , then(1),(2), ,(3), .

Lemma 9 (see [36]). Let , and be nonnegative continuous functions on and let be a real number. Ifthenwhere .

3. Finite-Time Synchronization

In this section, master-slave finite-time synchronization of delayed fractional-order neural networks is discussed. The aim here is to design a suitable controller that can achieve the synchronization between the slave system and the master system.

Let be the synchronization errors.

Select the linear control input functions as the following form:where each denotes the control gain.

Then the error systems are obtained:The vector form is as follows:where and .

The initial conditions of system (14) are of the following form:

Theorem 10. When , suppose that Assumptions 4 and 5 hold, ifwhere , ; master system (3) is synchronized with slave system (5) in a finite-time with respect to under the control (12).

Proof. Let be the initial condition of system (14), based on Lemma 8, the solution of system (7) in the form of the equivalent Volterra fractional integral equation is as follows:Taking the norm on both sides of the above system, according to the Assumptions 4 and 5, gives the following:By using the Cauchy-Schwartz inequality, (18) gets the following:Note thatNoting that and , one obtainsThenLet and .
ThenAccording to Lemmas 7 and 9, one hasThereforeHence, if (16) is satisfied and , then ; master system (3) is synchronized with slave system (5) in a finite-time.