Global Synchronization of Reaction-Diffusion Fractional-Order Memristive Neural Networks with Time Delay and Unknown Parameters

+is paper investigated the global synchronization of fractional-order memristive neural networks (FMNNs). To deal with the effect of reaction-diffusion and time delay, fractional partial and comparison theorem are introduced. Based on the set value mapping theory and Filippov solution, the activation function is extended to discontinuous case. Adaptive controllers with a compensator are designed owing to the existence of unknown parameters, with the help of Gronwall–Bellman inequality. Numerical simulation examples demonstrate the availability of the theoretical results.


Introduction
As a generalization of differential and integral calculus from an integer order to arbitrary order, fractional calculus has more than 300 years history [1]. Since Mandelbort proposed the fact that lots of fractals exist in the field of technology, a whole new array of possibilities for natural science research has been opened [2]. Compared with the conventional integer order calculus, it shows some superiority in the aspect of memory and heredity [3]. at caused its wide range of applications [4][5][6].
Neural networks model, as we know, has great potential for controlling highly nonlinear and severe uncertain systems. Due to the ability of studying arbitrary continuous nonlinear functions, it has been fruitfully applied to optimization, secure communication, cryptography, and so on. Taking into account these facts, the combination of fractional calculus and neural networks model is a remarkably great improvement. is kind of model called fractionalorder neural networks (FNNs) was proposed by Arena in 1998 [7]. Afterwards, a myriad of studies have focused on the dynamical behaviors of FNNs. One of the most interesting issues is to explore the synchronization problem of FNNs. Various definition types of synchronization have been built [8][9][10]. Among them, global synchronization is the most basic and widely applied one, since it guarantees the convergence of the system. From the perspective of engineering applications, it has achieved the required realistic standards [11,12].
Another hot topic of research is memristor-based neural networks, which were first put forward by Chua [16]. In 2008, a nanoscale memory was made by Williams group [17], which was named memristor. Memory characteristics and nanometer dimensions make the memristor show properties the same as the neurons in the human brain. From this standpoint, fractional-order memristive neural networks (FMNNs) have been suggested replacing the FNNs to emulate the human brain. erefore, the analysis of FMNNs has become increasingly attractive to researchers. At present, there are many excellent results on synchronization of FMNNs [18][19][20][21]. Velmurugan investigated finitetime synchronization of FMNNs with the method of Laplace transform and generalized Gronwalls inequality [18]. e lag complete synchronization criteria of FMNNs are proposed by period intermittent control [19]. Delay-independent synchronization criteria for FMNNs are established by using the maximum modulus principle, the spectral radii of matrix, and Kakutani fixed point theorem [20]. Chen et al. designed a stabilizing state-feedback controller by employing the FO Razumikhin theorem and linear matrix inequalities [21]. Besides the abovementioned methods, adaptive mechanism has been proposed as a more effective strategy [22]. Compared with the conventional feedback control, the parameters of adaptive controllers will be updated automatically, which makes the energy be saved [23]. It also has significant effect on the analysis of fractional order nonlinear system, especially when the system has unknown parameters [24][25][26][27]. us, it is a natural idea to investigate the synchronization of FMNNs using adaptive control scheme.
It is a remarkable fact that time delays emerge in the communication of neurons [28,29]. Furthermore, the occurrence of diffusion phenomena has a significant impact in the electrons' transportation through a nonuniform electromagnetic field. at means, not only the time but also their spatial positions affect the dynamic behaviors of networks. e models covering time delay and reaction-diffusion are good mimicry for the real neural networks in terms of application, but the existence of time delays and reaction-diffusion could bring about some undesirable behaviors [30,31]. Drawing on the abovementioned concerns, special attention should be paid on time delay and reactiondiffusion. However, the research on FMNNs with both reaction-diffusion and time delay is still in its infancy [32,33].
For realistic neural networks systems, one must include uncertainties. Uncertainty could occur in parameters in the mathematical model, for they are hardly to be exactly known in advance [34][35][36]. erefore, the study on synchronization of the drive-response system considering unknown parameters is one of the most challenging tasks. Expanding on the existing literature, the global synchronization problem for reaction-diffusion FMNNs with time delay and unknown parameters is proposed firstly. e main contents of this paper are the following. (1) Based on fractional comparison theorem, the problem of synchronization for the integral delayed neural networks is generalized to the fractional order case. (2) By utilizing Caputo partial fractional derivative, reaction-diffusion is emphasized for the FMNNs, which extends the problem from the time domain to timespatial dimension. (3) With the help of functional differential inclusions, the solution of the discontinuous systems is guaranteed in the sense of Filippov. (4) Employing Gronwall-Bellman inequality, the adaptive controllers with compensator are designed to achieve the synchronization, despite the presence of parametric uncertainties. is paper is structured as follows. In Section 2, we briefly introduce preliminaries and model description. Two types of adaptive controllers for global synchronization of the drive-response system are designed in Section 3. In Section 4, two simulations are shown to verify the correctness of the proposed results. A short conclusion and some discussions are given in Section 5.
Notations. let R n be the space of n-dimensional Euclidean space. e norm of vector containing the origin equips with the smooth boundary zΩ and mes Ω > 0.

Preliminaries and Model Description
Preliminaries about fractional calculus and the model of reaction-diffusion FMNNs with time delay are introduced in this section. Concept of Filippov solution, fractional comparison theorem, and Gronwall-Bellman inequality need to be emphasized.

Preliminaries. Definition and the properties about
Caputo Riemann-Liuville fractional calculus are given in the following.
Definition 1 (see [37]). For any t > 0, the time Caputo fractional derivative of order α(0 < α < 1) for a function where It should be noted that the initial conditions of Caputo fractional derivative are same as integer-order derivative, which makes Caputo fractional derivative equip specific physical sense in the real world. In this paper, simple notation D α is used to replace C 0 D α t .
where a(t) and b(t) are the known real functions, then 2.2. System Description. Reaction-diffusion FMNNs with time delay are described as where i � 1, 2, . . . , m, m ≥ 2 is the quantity of units; 0 < α < 1 refers to the order of the equation; u i (t, x) represents the state of the ith unit at time t and in space x)) stand for the strength of the jth unit on the ith unit at time t in space x and at time t − τ in space x, respectively; c i > 0 is the selffeedback connection weight; f j (·) and g j (·) denote the discontinuous activation functions; I i signifies the external input; and τ ≥ 0 stands for time delay. In addition, the memristor-based connection weights a ij (u i (t, x)) and b ij (u i (t, x)) are given by Consider the following Dirichlet-type boundary conditions of (15) and initial conditions Take (15) as the drive system, then the corresponding response system is defined by where i � 1, 2, . . . , m and κ i (t, x) is the controller which will be designed later.
and Φ 0 � (Φ 01 , Φ 02 , . . . , Φ 0m ) T , which have the same property to φ 0 . Since the right-hand sides of equations are discontinuous, the solutions of the equations in the conventional sense have been shown to be invalid. In this case, Filippov solutions [43] are introduced to deal with the discontinuous equations in functional differential inclusions framework. e solution's global existence for fractional differential inclusions with time delay is given by [44].
has at most limited the number of discontinuous points on any compact interval of R. e abovementioned properties are the same to g i (·). Based on the definition of Filippov solution, (15) can be rewritten as Or equivalently, there are measurable functions ij . e IVP of the driven system (15) can be written as e IVP of the response system (19) under the controller could be given by x) be the synchronization error between (15) and (19). e IVP about the error system is of the form Definition 3.
e drive system (15) and response system (19) is said to be globally synchronized, if lim t⟶∞ ‖e i (t, x)‖ � 0.

Remark 1.
Owing to the zero Dirichlet boundary conditions of (24) and (25), a zero equilibrium state of error system (26) exists. e global synchronization of systems (24) and (25) corresponds to the global asymptotic stability of the zero equilibrium of system (26).

Main Results
e abovementioned synchronization problem in Section 2 are considered in two cases, namely, the drive system (15) has known or unknown parameters.

Global Synchronization for Systems with Known
Parameters. First of all, we assume that the parameters in (15) are known. en, the following adaptive controllers are proposed to implement global synchronization.

Assumption 2.
For i � 1, 2, . . . , m, there are positive constants p i and q i such that where for any θ ∈ R, For the convenience of description, the following notations are introduced: Theorem 1. Under Assumption 1 and Assumption 2, the global synchronization of (24) and (25) can be achieved based on the following adaptive controllers: where ℓ i (0, x) ≥ 0, ϖ i > 0, and ℓ i and ω i are constants which satisfy the following inequalities: Proof. Employ the following auxiliary function: By the means of Lemma 1, the inequality can be established: According to Lemma 4, taking the derivative of (32) yields Using Green's formula and Lemma 6, it can be obtained that Recalling Assumption 1 and Assumption 2, the inequalities hold: Combining controller (30) designed in eorem 1, it follows that From Definition 2 and Property 2, one obtains Substituting (38) into (37) and noting that ℓ i and ω i meet inequality (31), it can be deduced that D α V(t, e(t, ·)) ≤ −λ 1 V(t, e(t, ·)) + ρ 1 V(t − τ, e(t, ·)), (39) where λ 1 � m i�1 (2ℓ i − σ i ) and ρ 1 � m i�1 σ * * i . According to Lemma 2 and Lemma 3, it can be assert that the global synchronization between (24) and (25) can be realized. e proof of eorem 1 completes.
□ Remark 2. When α � 1, the error system (26) be changed into the integer form. e conclusion is the same as the result in eorem 3.1 in article [45]. If the reaction-diffusion in space Ω disappears, then the states of the system's variable are only relevant to time but independent of space. us, the error system (26) can be converted to For convenience, the inequalities are given by (24) and (25) can be achieved based on the following adaptive controllers:
Remark 3. In fact, synchronization of FMNNs with time delay has been investigated in [20]. e results about synchronization of the systems with single delay in eorem 3 in [20] are the same as Corollary 1 obtained in this paper. Suppose that there is no time delay τ in the course of communication between the different neurons; then, error system (26) can be reduced to

Complexity
For convenience, the inequalities are given by Corollary 2. Under Assumption 1 and Assumption 2, the global synchronization between (24) and (25) can be realized based on the following adaptive controllers: where ℓ i (0, x) ≥ 0, ϖ i > 0, and ℓ i and ω i satisfy inequality (44).

Remark 4.
Comparing the coefficients of the adaptive controllers (31) and (44), it can be concluded that the controllers for global synchronization of the system with time delay are more consumptive.

Global Synchronization for Systems with Unknown
Parameters. In real neural networks, the parameters of the system may be unknown, so it is difficult to achieve synchronization by the conventional adaptive control schemes. In the following text, adaptive controllers with the dynamic compensators are introduced.
Here, suppose that self connection weight c i and input I i in the drive system (15) are unknown. en, the IVP of the drive system (15) can be written as e IVP of the response system (19) can be shown as where i, j � 1, 2, .., m and c i (t, x) and I i (t, x) are treated as the compensation of c i and I i . en, for i, j � 1, 2, . . . , m, the IVP of the synchronization error can be described as Assumption 3. ere are positive constants p i , q i , and g i such that where i � 1, 2, . . . , m, θ ∈ R, and the notation co (47) can realized based on the following adaptive controllers: 1, 2, . . . , m, ℓ i (0, x), ϖ i , ϱ i , and ] i are positive constants, and ℓ i and ω i are constants which satisfy the following inequalities:

Theorem 2. Under Assumption 1 and Assumption 3, the global asymptotic synchronization between (46) and
Proof. Employ the following auxiliary function: where Complexity 7 Similarly to the proof of eorem 1, one obtains Since ℓ i and ω i fulfill condition (51), it follows where λ 2 � m i�1 (2ℓ i − σ i ). By Lemma 5, it can be concluded that According to the definition of V 1 , thus erefore, From (59), it is easy to see that the global synchronization between (46) and (47) can be realized. e proof of eorem 2 completes. □ Remark 5. e result obtained in eorem 2 is more practical for considering the situation where the parameters of the drive system (24) are unknown. rough the proof of eorem 2, it can be asserted that the dynamical compensator is necessary for the unknown system. If the reactiondiffusion in space Ω disappears, the error system (48) can be converted to

Corollary 3. Under Assumption 1 and Assumption 3, the global synchronization between (46) and (47) can be achieved based on the following adaptive controllers:
Suppose that there is no time delay τ in the course of communication; then, the error system (48) can be changed into

Numerical Simulations
Several numerical simulations are presented to illustrate the main results. e first example is used to validate eorem 1. e second one is used to text eorem 2. Now, let us consider the 2D (q � 1) reaction-diffusion where the memristive coefficients are defined by Suppose that time delay τ � 1.1 and discontinuous activation functions are f(δ) � g(δ) � 0.98δ + 0.05sgn(δ). Let initial values of (24) and (25) be where t ∈ [−1.1, 0] and x ∈ (−2, 2). Given the fractional order α � 0.98 and d 11 � d 21 � 0.85. Under these parameters, the dynamical behaviors of the synchronization errors between (24) and (25) are shown in Figure 1. From the figure, it can be observed that the synchronization errors gradually increase as time goes on. Notice that f(δ) and g(δ) have only one discontinuous point 0, Assumption 1 is valid. According to Assumption 2, we can choose p 1 � p 2 � r 1 � r 2 � 0.98 and q 1 � q 2 � s 1 � s 2 � 0.05. By calculation, the parameters of controllers are selected as ℓ 1 � 0.70 > 0.39, w 1 � 0.06 > 0.05, ℓ 2 � 0.60 > 0.36, w 2 � 0.05 > 0.04, ℓ 1 (0, x) � ℓ 2 (0, x) � 0.01 > 0, and ϖ 1 � ϖ 2 � 0.01 > 0. It is easy to verify that condition (9) in eorem 1 is satisfied. Figure 2 suggests that the synchronization errors asymptotically converge to zero. eorem 1 is not only dependent on order but also related with reaction-diffusion coefficients. Figure 3 shows the trajectories of the error system at spatial position x � 55 with different orders α � 0.6, 0.7, 0.8, 0.9 when the reactiondiffusion coefficients d � 1.8. While if the order is fixed α � 0.7, the trajectories of the error system at spatial position x � 55 with different reaction-diffusion coefficients d � 0.6, 1.4, 2.2, 3.0 are shown in Figure 4. From the numerical results, we could note that the error system would achieve the asymptotic stability faster with the higher order. It also holds true with increasing reaction-diffusion coefficient.    (24) and (25), under controller (30).

Conclusion and Discussion
e global synchronization problem for reaction-diffusion FMNNs with time delay was emphasized firstly in this paper. Both the known and unknown parameters cases were investigated, separately. Based on fractional comparison theorem and Gronwall-Bellman inequality, two kinds of adaptive control schemes were proposed. Numerical simulations were implemented to examine the effectiveness of the obtained results.
Conventional adaptive controllers have mostly been used for controlling neural networks system, while they cannot guarantee the synchronization of unknown systems. An adaptive controller with a dynamic compensator such as the one described in this paper can eliminate this problem. However, the true values of the unknown parameters could not be obtained. Further research is still in progress to design an adaptive controller with dynamic compensator to achieve the identity of parameters.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.