New Methods of Finite-Time Synchronization for a Class of Fractional-Order Delayed Neural Networks

1School of Mathematics and Computational Science, Anqing Normal University, Anqing 246011, China 2School of Mathematics, Southeast University, Nanjing 210096, China 3School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China 4Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia 5Department of Electrical and Computer Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia


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][4][5][6][7][8][9][10][11].
Fractional-order neural networks have attracted great attention due to their potential properties of memory and hereditary.Particularly, the dynamical analysis of fractionalorder 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][16][17][18][19][20][21][22][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 finitetime synchronization were demonstrated on integer-order systems [24][25][26][27][28].Note that time delay [29][30][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 2 Mathematical Problems in Engineering achieve the finite-time synchronization of fractional delayed neural networks with orders 0 <  ≤ 1/2 and 1/2 <  < 1.

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.

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.
Remark 13.Generally, time-delayed differential models unavoidably exist in neural networks [37].However, quite a few researches investigated the finite-time synchronization of neural networks with delay.Hence, it is foremost important and necessary to consider finite-time synchronization of fractional-order delayed neural networks.
Remark 14.In this paper, Assumptions 4 and 5 are general, not too strict; many functions can satisfy the assumptions; for example, [15,17,29] considered these assumptions too.
Remark 15.In [33], the authors discussed the finite-time synchronization of fractional-order memristor-based neural networks with time delays by using Laplace transform, generalized Gronwall's inequality and Mittag-Leffler functions, and the results showed Mittag-Leffler functions.In this paper, employing Lemma 4 proposed in [35] and Bernoulli inequality can obtain results that only include exponential functions, given that the form is simpler and the calculation is easier.
Remark 16.In Theorem 10, from ( 12) and ( 14) and inequality (16), it is obvious that the convergence time  is proportional to the inverse of the control gain   .Therefore, a greater   results in the shorter convergence time , and therefore   should be selected in accordance with the convergence time  to be short.
Remark 17.In [32], the authors discussed finite-time stability of fractional-order neural networks with delay by utilizing Gronwall inequality.Unlike the previous work [32], in this paper, we discussed the finite-time synchronization for a class of fractional-order delayed neural networks by employing Lemma 4 proposed in [35], Bernoulli inequality, and inequality skills; two new delay-dependent sufficient conditions have been established.
Remark 18. Different from integer-order delayed systems, it is difficult to construct Lyapunov functions for fractional-order nonlinear systems with time delay; results have been obtained using different techniques from approaches used in the area of finite synchronization.By employing inequality skills, new sufficient conditions ensuring finite-time synchronization are derived.
When  = 0.4, take  = 0.01,  = 1,  = 0.1 and select the control gain  1 = 4,  2 = 5; apparently, the condition of Theorem 11 is satisfied.It could be verified that  = 8.0815,  = 1.0674, and the estimated time of finitetime synchronization is  = 0.7099.Synchronization errors between master and slave systems are shown in Figures 1 and  2. It is clearly seen that the synchronization errors converge to zero, indicating that master system and slave system are synchronized in a finite-time.For comparison purposes, the curves of the state variable of the master system and the slave system are shown in Figures 3 and 4.
When  = 0.7, take  = 0.01,  = 1,  = 0.1 and select the control gain  1 = 2,  2 = 4; apparently, the condition of Theorem 10 is satisfied.It is easy to obtain  = 4.2950,  = 0.6992, and the estimated time of finite-time synchronization is  = 0.5937.Synchronization errors between master and slave systems are shown in Figures 5 and 6.It can be seen that the synchronization errors converge to zero, confirming that master system and slave system are synchronized in a finitetime.Also, for comparison purposes, the curves of the state  variable of the master system and the slave system are shown in Figures 7 and 8.

Conclusions
In this paper, finite-time synchronization for a class fractional-order delayed neural networks with order , 0 <  ≤ 1/2 and 1/2 <  < 1, was discussed.Some new sufficient conditions are derived to ensure the finite-time synchronization for this class of fractional-order systems.Numerical example is presented to verify the effectiveness of the theoretical results.The proposed methods are novel and solve well the finite-time synchronization of fractionalorder delayed neural networks.We would like to point out that it is possible to extend the methods to other fractional-order models, such as fractional-order delayed neutral-type neural networks and fractional neural networks with incommensurate.These issues will be further worth discussing.