Finite-Time Synchronization for Fractional Order Fuzzy Inertial Cellular Neural Networks with Piecewise Activations and Mixed Delays

: This paper investigates a class of ﬁnite-time synchronization problems of fractional order fuzzy inertial cellular neural networks (FFICNNs) with piecewise activations and mixed delays. First, the Caputo FFICNNs are established. A suitable transformation variable is constructed to rewrite FFICNNs with mixed delays into a ﬁrst-order differential system. Secondly, some new effective criteria are constructed on the basis of the ﬁnite-time stability theory and Lyapunov functionals to realize the synchronization of the drive-response system. Finally, two numerical simulation examples show that the proposed method is effective.


Introduction
Yang et al. [1,2] put forward cellular neural networks (CNNs) in 1988, which have since been extensively investigated and applied in such fields as secondary optimization, pattern classification, pattern recognition, associative memory, and image processing.Yang et al. [3] put forward the fuzzy cellular neural networks (FCNNs) in 1996.Compared with the CNNs, the FCNNs contains fuzzy logics, and there is a local connection between cells [4][5][6][7][8][9][10].FCNNs have superior performance applications in image encryption, psychophysics, perception, robots, secure communication, and medical diagnosis, etc.
Babcock et al. [11] supplemented an inertial term to neural networks to make up inertial neural networks (INNs) in 1986, which caused more complex bifurcation and chaotic dynamic behaviors of neuron coupling as instability and concussion behaviors.Adding an inertia term to electronic neural networks may bring about complex behaviors such as spontaneous concussion, instability, and chaotic behavior.Moreover, there exist significant biological backgrounds for bringing the inertial term into neural systems.Many researchers have paid extensive attention to inertial neural networks, and have made important progress (see [12][13][14][15][16]).
Fractional order calculus is an extension of integer calculus to arbitrary order, has become a mathematical tool to solve practical problems in pattern recognition, information processing, robot control, physics, statistics, and other fields for its superiority, which is characterized by infinite memory considering the current states and all its previous ones.Scholars [17][18][19][20][21][22] used fractional operators to build fractional order neural network models.
In practical application, the signals of processing and transmission between neurons are limited by the switching speed of the amplifier, the delay time is inevitable, and this will influence the stability of the neural networks and give rise to divergence, instability, and oscillation in network systems.In fact, mixed delays include constant ones, time-varying ones, and distributed ones, which are considered to be more effective than single ones at modeling network systems in that these simple delays are often impractical, while network systems become more complex.
Synchronization is the dynamic behavior of the drive system and reponse system to attain the same state after a certain time.Nowadays, much attention has been paid to synchronization, owing to its potential applications in medicine, information science, optimized calculation, automatic control, and other fields.At present, there are many achievements in synchronous research.According to the convergence time, it is mainly divided into three categories: global asymptotical synchronization, exponential synchronization, and finite-time synchronization.In comparison with the global asymptotical synchronization of neural networks, finite-time synchronization has better convergence performance.However, the final convergence time is closely related to the initial state of the drive-response system.To achieve synchronization quickly, we can rely on control methods-for example, feedback control, adaptive control, intermittent control, etc.Of course, we sincerely hope that systems can achieve synchronization with practical engineering applications in a finite time.Therefore, the research for finite-time synchronization of neural networks is of great significance.At present, the research result of the inertial neural networks' synchronization mainly includes the following aspects.Yang et al. [23] designed three different types of controllers by using the maximum analysis method to ensure the global asymptotic synchronization between the drive-response system.Feng et al. [24] investigated the exponential synchronization control of inertial neural networks with time-varying delays.By designing Lyapunov-Krasovskii functionals and utilizing new weighted integral inequalities, the delay-related criteria of linear matrix inequality (LMI) are obtained.Zhang et al. [25] obtained two new strategies to solve the synchronization for fractional neural networks with two inertia terms and a time delay by constructing Lyapunov function and using the LMI method.Tang et al. [26] researched the exponential synchronization of INNs with finite and discrete distributed mixed delays by using intermittent control.Liang et al. [27] introduced the exponential synchronization control of inertial Cohen-Grossberg neural networks with time-varying delays.Shi et al. [28] studied the lag synchronization and global exponential stabilization of INNs via adaptive control by constructing nonnegative function and employing inequality techniques, and obtained several new results.Zhang et al. [29] introduced the finite-time synchronization control of FINNs with the maximum method of functions.Hua et al. [30] introduced a new control method to solve the finite-time synchronization problem of IMNNs with time-varying delay.It can effectively dispose of the problems caused by mixed delay and memristor connection weight.Chen et al. [31] design four different kinds of feedback controllers, under which the considered inertial memristive neural networks can realize fixed-time synchronization perfectly.Alimi et al. [32] introduced the finite/fixed time synchronization of the INNs with multiple proportional delays.Yang et al. [33] studies the synchronization control of a new FICNNs with piecewise activation and mixed delay.However, few reports are found in the existing works for finite-time synchronization of FFICNNs with piecewise activation and mixed delay.
Motivated by these, this paper will investigate the problems of finite-time synchronization analysis for a class of FFICNNs with piecewise activation and mixed delay.There are the following primary innovations in this paper.

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The Caputo fractional order fuzzy inertial cellular neural networks model is established.It can be used to describe many systems of internal coherence in the real environment.In addition, it is easy to implement in engineering applications and has important application prospects.

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A novel nonlinear controller is designed to realize the finite-time synchronization of FFICNNs with piecewise activation and mixed delay.It is of high reliability and great accuracy, and can better synchronize the position and motion of the system.

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The Lyapunov direct method is applied in the analysis of the inertial system to avoid the loss of the inertia.The numerical simulation results show that the designed method is effective.Therefore, it is of more important practical significance.
The other parts of this paper are structured as follows.Section 2 introduces the network system model of FICNNs with piecewise activations and mixed delays, assumptions, lemmas, and definitions of finite-time synchronization.In Section 3, the synchronization of the neural network system is analyzed.In Section 4, two examples are given to demonstrate the effectiveness of the obtained results.The conclusion is revealed in Section 5.

Preliminaries
In this article, the system model is defined by Caputo fractional order.First, some basic definitions, lemmas, and assumptions about fractional calculus are introduced.Definition 1 ([34]).By using Equation (1) to define the fractional integral of the function z(t), we have where α > 0, t > t 0 , Γ(•) is gamma function.

Definition 2 ([34]
).By using Equation (2) to define the the fractional derivative of the function z(t), we have For simplicity, we use D α t z(t) to represent C 0 D α t z(t) in this article.Some useful lemmas are given as follows.

Problem Formulation
In this paper, the system of the m-dimensional FFCNNs with piecewise activation and mixed delays is defined as where 0 < α ≤ We assume that the following assumptions about activation functions are satisfied.Assumption 1.There exist nonnegative constants L k and S k such that where ; co represents the convex closure of the set.
Assumption 2. x i : R → R is continuous except on a countable set of isolation points { i k } for each i = 1, 2, . . ., n, where there exists a finite left limit x i ( i− k ) and right limit x i ( i+ k ).Moreover, g i has at most a finite number of jump discontinuities in every compact interval of R.
Remark 2. Assumptions 1 and 2 are not restrictive.Generally, the neural network model can satisfy Assumptions 1 and 2 in [41][42][43].In addition, to obtain the main results, the concept of Filippov solution, which comes from Filippov for system (14), is given.Definition 3. If (i) and (ii) are satisfied, then the function z = (z 1 , z 2 , . . ., z m ) T is said to be a solution of system (14) , where ξ k is constant.According to Lemmas 2 and 3, the system ( 16) can be transformed into the Equation (17), and it is chosen as the drive system.We have where , and Equation ( 18) is the initial value of system.We have Now, we choose the system (19) as the response system.We have where u k (t), u k (t) represent the designed controller.Equation ( 20) is the error system: According to Equation (17) and Equation ( 19), we obtain The structure of this FICNN is shown in Figure 1.Definition 4 ( [32]).The drive system (17) and response systems (19) are synchronized within a finite time, if T > t 0 exists under properly designed controllers u k (t), u k (t), such that

Finite-Time Synchronization
In this section, we will design controller u k (t), u k (t) to get the finite-time synchroniza- tion between the drive-response systems.The controller is designed, where 0 < µ < 1, ρ > 0, k k , ω k , η k1 , and η k2 are the control parameters.
Theorem 1. Assumptions 1 and 2 are true if the following conditions (23) hold; then, the system ( 17) and ( 19) can achieve finite-time synchronization under controller (22).We have for k = 1, 2, . . ., m.Furthermore, the T can be calculated by Equation (24), where Proof.Select the following Lyapunov function: According to Lemmas 3 and 4, the derivative of V(t) is calculated, and we obtain Substituting the controller (22) into Equation ( 26), and we have .
According to Assumption 1, we have .
By using Assumption 1 and Lemma 6, we have Similarly, . By using the above inequality, we have By using Lemma 7, it follows that . Therefore, the drive (17) and the response ( 19) system can achieve to synchronization in the finite-time by Lemma 8.

Remark 3.
The finite-time synchronization of FFICNNs with piecewise activations and mixed delays is achieved by constructing a suitable controller in Theorem 1.However, the control laws u k (t) and u k (t) are not easily adaptive, and they must meet some special conditions.Therefore, we will optimize the applicable laws u k (t) and u k (t) to improve the feasibility.
Theorem 2. Assumptions 1 and 2 are true if the following conditions ( 29) hold, and then the system ( 17) and ( 19) can achieve finite-time synchronization under controller (28).We have for k = 1, 2, . . ., m.Furthermore, the T can be calculated by Equation (30), where Proof.Select the following Lyapunov function: Calculating the derivative of V(t), according to Lemma 3, 5, we have . ( 32) Substituting the controller (28) into Equation ( 32), we have According to Assumption 1 and Lemma 6, we have .

Numerical Simulations
In this section, we use two numerical examples to verify the validity of the results of Theorems 1 and 2.
Example 1.Consider a two-dimensional FFINNs with piecewise activations and mixed delay.
The drive system is given by and the response system where h k (x) = tanh(x) + 0.8, x ≥ 0 tanh(x) − 0.8, x < 0 .
Obviously, h k (x) satisfies Assumptions 1 and 2 with L k = 1, S k = 1, k = 1, 2. The parameters of drive (33) and response (34) system are chosen as follows: The controller is designed as follows: .
The controller parameters are as follows: The initial conditions are given as The time T is calculated by Theorem 1, which means that (34) and (35) can achieve synchronization in a finite time T. Furthermore, here T = 1.7220.
Figures 2-4 show the simulation results of the drive (33) and response(34) system.Figures 2 and 3 show the trajectories of states p(t) and z(t), and q(t) and g(t), respectively.Equation (34) indeed converges to (33) with the controller (35) in T, and the convergence error continues to remain zero.The time evolution of synchronization errors θ(t) and θ(t) between systems (33) and (34) are presented in Figure 4. Simulation results show that the main results of the finite-time synchronization entrenched are correct in this article.
The time T is calculated by Theorem 2, which means that (37) and (38) can achieve synchronization in a finite time T. Furthermore, here T = 1.9082.
Figures 5-7 show the simulation results of the drive(36)-response(37) system.Figures 5 and 6 show the trajectories of states p(t) and z(t) , q(t) and g(t), respectively.Equation (37) indeed converges to (36) with the controller (38) in T, and the convergence error continues to remain zero.The time evolution of synchronization errors θ(t) and θ(t) between systems (36) and (37) are presented in Figure 7. Simulation results show that the main results of the finite-time synchronization entrenched are correct in this article.

Conclusions
In this paper, a class of fractional order fuzzy inertial cellular neural network system with piecewise activation and mixed delay is proposed, and the finite-time synchronization of the system is discussed.By using the finite-time stability theory, Lyapunov functionals, and analytical techniques, some novel methods for finite-time synchronization of driveresponse systems are obtained.Finally, two numerical simulation examples show that the proposed method is effective.In the future, we propose to expand our results to better apply FFINNs and implement it by designing hardware.The more efficient control laws will be considered, which is a challenging topic.

Figure 1 .
Figure 1.Framework of feedback control.
1. z k (t) represent the state of the kth neuron.a k , b k represent the passive decay rate to the state of kth neuron.c kr , d kr represent elements of feedback template and g kr denotes the feedforward template.v k and I k are input and bias of the kth neuron, respectively.αkr , β kr are fuzzy feedback MIN and MAX template, P kr , Q kr are fuzzy feedforward MIN and MAX template.hk (•) denotes activation functions, σ 0 (t) and σ k (t) represent the delay of the kth neuron.Remark 1.The mixed delay σ meets the following conditions: σ ) − ρsign e k (t) e k (t) µ + ω k e k (t − σ k (t)) u k (t) = −η k1 e k (t) − sign e k (t) η k2 + ρ e k (t)