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

Authors: Wenjiao Sun; Guojian Ren; Yongguang Yu; Xudong Hai
Journal: Complexity
 (2020)

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## Abstract

This 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.


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## Body

## 1. 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]. That caused its wide range of applications [4–6].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. This kind of model called fractional-order 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 [8–10]. 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]. Thus, global synchronization of FNNs has been extensively investigated to date [13–15].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. Therefore, the analysis of FMNNs has become increasingly attractive to researchers. At present, there are many excellent results on synchronization of FMNNs [18–21]. Velmurugan investigated finite-time synchronization of FMNNs with the method of Laplace transform and generalized Gronwalls inequality [18]. The 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 [24–27]. Thus, 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. That means, not only the time but also their spatial positions affect the dynamic behaviors of networks. The 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 reaction-diffusion. 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 [34–36]. Therefore, 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. The 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 time-spatial 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.This paper is structured as follows. In Section2, 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.
letℝn be the space of n-dimensional Euclidean space. The norm of vector x=x1,x2,…,xnT∈ℝn is defined as x=∑q=1nxq21/2. A bounded open-set Ω⊂ℝn containing the origin equips with the smooth boundary ∂Ω and mesΩ>0.


## 2. 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.


### 2.1. 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 α0<α<1 for a function ft,x∈C10,b×Ω,ℝ is defined by(1)∂αft,x∂tα=1Γ1−α∫0t∂fs,x∂sdst−sα,where Γz=∫0∞e−ttz−1dt. Particularly, when ft,⋅=ft, then(2)∂αft,⋅∂tα=dαftdtα=0CDtαft.
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 notationDα is used to replace Dtα0C.Definition 2.
(see [37]). Riemann–Liuville fractional integral of order α for a function ft:0,+∞⟶R is defined by(3)IR0tαft=1Γα∫0tfτt−τ1−αdτ,where 0<α<1.Proposition 1.
(see [37]). For arbitrary constants v1 and v2, there is(4)Dαv1ft+v2gt=v1Dαft+v2Dαgt.Proposition 2.
(see [37]). If 0<α<1 and ft∈C10,t, then IR0tαDαft=ft−f0.
Several important conclusions about the fractional-order system are listed below.Lemma 1.
(see [33]). Let ft,x:0,b×Ω⟶ℝ, be a continuous and differentiable function on t; then, for any t≥0, x∈Ω,(5)12∂αf2t,x∂tα≤ft,x∂αft,x∂tα,where 0<α<1. Particularly, when α=1, there is(6)12∂f2t,x∂t≤ft,x∂ft,x∂t.Lemma 2.
(see [38]). Consider the system as follows:(7)DαVxt=−ρVxt+kVxt−τ,xt=ϕt,t∈−τ,0,where ρ,k≥0 and Vx≥0. The solution of the system is asymptotically stable, when k<ρsinαπ/2.Lemma 3 (fractional comparison theorem; see [38]).
Consider the following fractional order inequality:(8)DαVxt≤−ρVxt+kVxt−τ,Vxt=ht,t∈−τ,0,and fractional order system:(9)DαVyt=−ρVyt+kVyt−τ,Vyt=ht,t∈−τ,0,where α∈0,1, τ>0, xt,yt∈Rn, and VxandVy are continuous and nonnegative functions in 0,+∞, ht≥0 on −τ,0. If ρ>0 and k>0, then the inequality holds:(10)Vxt≤Vyt,for all t∈0,+∞.Lemma 4.
(see [37, 39]). Suppose that ex,t:Ω×R+⟶R is integrable on Ω and is derivable respect to t. Denote Vt=∫Ωex,tdx, then(11)DαVt=∫ΩDαex,tdx.Lemma 5.
(see [40]). If xt,yt∈C1t0,b with xt≤yt, then ∀t∈t0,b there is Itαt0Rxt≤t0RItαyt.Lemma 6.
(see [41]). Let Ω be a cube xq<lq,q=1,2,…,n, and let fx be a real-valued function belonging to C1Ω which vanishes on the boundary of the Ω, i.e., fx∂Ω=0, then(12)∫Ωf2xdx≤lq2∫Ω∂fx∂xq2dx.Lemma 7.
(Gronwall–Bellman inequality; see [42]). If zt satisfies zt≤∫0taτzτdτ+bt, where atandbt are the known real functions, then(13)zt≤b0exp∫0taτdτ+∫0tb˙τexp∫τtardrdτ.Ifbt is a constant,(14)zt≤b0exp∫0taτdτ.


### 2.2. System Description

Reaction-diffusion FMNNs with time delay are described as(15)∂αuit,x∂tα=∑q=1ndiqΔuit,x−ciuit,x+∑j=1maijfjt,x+∑j=1mbijgjt−τ,x+Ii,where i=1,2,…,m, m≥2 is the quantity of units; 0<α<1 refers to the order of the equation; uit,x represents the state of the ith unit at time t and in space x; t∈J=0,T represents the time variable; Δ is a Laplace operator, which means Δuit,x=∂2uit,x/∂xq2; the smooth constants diq≥0 work as the transmission diffusion operator along the ith unit at qth position; aijfjt,x=aijuit,xfjujt,x, bijgjt−τ,x=bijuit,xgjujt−τ,x, where aijuit,x and bijuit,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; ci>0 is the self-feedback connection weight; fj⋅ and gj⋅ denote the discontinuous activation functions; Ii signifies the external input; and τ≥0 stands for time delay. In addition, the memristor-based connection weights aijuit,xandbijuit,x are given by(16)aijuit,x=a^ij,uit,⋅>Tja,unsureness,uit,⋅=Tja,aˇij,uit,⋅<Tja,bijuit,x=b^ij,uit,⋅>Tjb,unsureness,uit,⋅=Tjb,bˇij,uit,⋅<Tjb,where i,j=1,2,…,m, t≥0, switching jumps Tja,Tjb>0 and a^ij,aˇij,b^ij,andbˇij are all constants.Consider the following Dirichlet-type boundary conditions of (15)(17)uit,x=0,t∈−τ,∞,x∈∂Ω,and initial conditions(18)uis,x=φ0is,x,s∈−τ,0,x∈Ω,where φ0=φ01,φ02,…,φ0mT are real-value continuous functions defined on −τ,0×Ω.Take (15) as the drive system, then the corresponding response system is defined by(19)∂αvit,x∂tα=∑q=1ndiqΔvit,x−civit,x+∑j=1maijfjt,x+∑j=1mbijgjt−τ,x+Ii+κit,x,where i=1,2,…,m and κit,x is the controller which will be designed later.Similarly, there are(20)vit,x=0,t∈−τ,∞,x∈∂Ω,vis,x=Φ0is,x,s∈−τ,0,x∈Ω,and Φ0=Φ01,Φ02,…,Φ0mT, which have the same property to φ0.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. The solution’s global existence for fractional differential inclusions with time delay is given by [44].Assumption 1.
For eachi∈ℕ, fi⋅:ℝ⟶ℝ is piecewise continuous function. Moreover, fi⋅ has at most limited the number of discontinuous points on any compact interval of ℝ. The abovementioned properties are the same to gi⋅. Based on the definition of Filippov solution, (15) can be rewritten as(21)∂uit,x∂tα∈∑q=1ndiqΔuit,x−ciuit,x+∑j=1mKaijtco¯fjujt,x+∑j=1mKbijtco¯gjujt−τ,x+Ii,where i=1,2,…,m, uit,x are absolutely continuous on any compact subinterval of t0,∞×Ω:(22)Kaijt=a^ij,ujt,⋅>Tja,co¯a^ij,aˇij,ujt,⋅=Tja,aˇij,ujt,⋅<Tja,Kbijt=b^ij,ujt,⋅>Tjb,co¯b^ij,bˇij,ujt,⋅=Tjb,bˇij,ujt,⋅<Tjb.
Or equivalently, there are measurable functionsa˜ij⋅∈Kaij⋅, b˜ij⋅∈Kbij⋅, f˜1j⋅∈co¯fjuj⋅, and g˜1j⋅∈co¯gjuj⋅ satisfy(23)∂αuit,x∂tα=∑q=1ndiqΔuit,x−ciuit,x+∑j=1ma˜ijtf˜1jt,x+∑j=1mb˜ijtg˜1jt−τ,x+Ii,for a.e.t∈t0,∞, x∈Ω, i=1,2,…,m.
For writing convenience, we denotemaxa^ij,aˇij as aijmax and maxb^ij,bˇij as bijmax.
The IVP of the driven system (15) can be written as(24)∂αuit,x∂tα=∑q=1ndiqΔuit,x−ciuit,x+∑j=1ma˜ij,f˜1jt,x+∑j=1mb˜ij,g˜1jt−τ,x+Ii,uit,x=0,t∈−τ,∞,x∈∂Ω,uis,x=φ0is,x,t∈−τ,0,x∈Ω,f˜1js,x=γjs,x,a.e.t∈−τ,0,x∈Ω,g˜1js,x=ηjs,x,a.e.t∈−τ,0,x∈Ω.
The IVP of the response system (19) under the controller could be given by(25)∂αvit,x∂tα=∑q=1ndiqΔvit,x−civit,x+∑j=1ma˜ij,f˜2jt,x+∑j=1mb˜ij,g˜2jt−τ,x+Ii+κit,x,vit,x=0,t∈−τ,∞,x∈∂Ω,vis,x=Φ0is,x,t∈−τ,0,x∈Ω,f˜2js,x=ξjs,x,a.e.t∈−τ,0,x∈Ω,g˜2js,x=ωjs,x,a.e.t∈−τ,0,x∈Ω,where i,j=1,2,…,m. a˜ij,f˜1jt,x=a˜ijtf˜1jt,x, b˜ij,g˜1jt−τ,x=b˜ijtg˜1jt−τ,x, a˜ij,f˜2jt,x=a˜ijtf˜2jt,x, and b˜ij,g˜2jt−τ,x=b˜ijtg˜2jt−τ,x, f˜2j⋅∈co¯fjvj⋅andg˜2j⋅∈co¯gjvj⋅.
Now, leteit,x=uit,x−vit,x be the synchronization error between (15) and (19). The IVP about the error system is of the form(26)∂αeit,x∂tα=∑q=1ndiqΔeit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+∑j=1mbij,g¯jt−τ,x+κit,x,eit,x=0,t∈−τ,∞,x∈∂Ω,eis,x=φ¯0is,x,s∈−τ,0,x∈Ω,where i,j=1,2,…,m, the symbol a˜ij,f¯jt,x is written as a˜ijtf˜2jt,x−f˜1jt,x and bij,g¯jt−τ,x which stands for b˜ijtg˜2jt−τ,x−g˜1jt−τ,x.
Denoteeit,x=∫Ω∑i=1mei2t,xdx1/2 as the norm of vector et,x=e1t,x,e2t,x,…,emt,xT∈Rm.Definition 3.
The drive system (15) and response system (19) is said to be globally synchronized, if limt⟶∞eit,x=0.Remark 1.
Owing to the zero Dirichlet boundary conditions of (24) and (25), a zero equilibrium state of error system (26) exists. The global synchronization of systems (24) and (25) corresponds to the global asymptotic stability of the zero equilibrium of system (26).


## 2.1. 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 α0<α<1 for a function ft,x∈C10,b×Ω,ℝ is defined by(1)∂αft,x∂tα=1Γ1−α∫0t∂fs,x∂sdst−sα,where Γz=∫0∞e−ttz−1dt. Particularly, when ft,⋅=ft, then(2)∂αft,⋅∂tα=dαftdtα=0CDtαft.
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 notationDα is used to replace Dtα0C.Definition 2.
(see [37]). Riemann–Liuville fractional integral of order α for a function ft:0,+∞⟶R is defined by(3)IR0tαft=1Γα∫0tfτt−τ1−αdτ,where 0<α<1.Proposition 1.
(see [37]). For arbitrary constants v1 and v2, there is(4)Dαv1ft+v2gt=v1Dαft+v2Dαgt.Proposition 2.
(see [37]). If 0<α<1 and ft∈C10,t, then IR0tαDαft=ft−f0.
Several important conclusions about the fractional-order system are listed below.Lemma 1.
(see [33]). Let ft,x:0,b×Ω⟶ℝ, be a continuous and differentiable function on t; then, for any t≥0, x∈Ω,(5)12∂αf2t,x∂tα≤ft,x∂αft,x∂tα,where 0<α<1. Particularly, when α=1, there is(6)12∂f2t,x∂t≤ft,x∂ft,x∂t.Lemma 2.
(see [38]). Consider the system as follows:(7)DαVxt=−ρVxt+kVxt−τ,xt=ϕt,t∈−τ,0,where ρ,k≥0 and Vx≥0. The solution of the system is asymptotically stable, when k<ρsinαπ/2.Lemma 3 (fractional comparison theorem; see [38]).
Consider the following fractional order inequality:(8)DαVxt≤−ρVxt+kVxt−τ,Vxt=ht,t∈−τ,0,and fractional order system:(9)DαVyt=−ρVyt+kVyt−τ,Vyt=ht,t∈−τ,0,where α∈0,1, τ>0, xt,yt∈Rn, and VxandVy are continuous and nonnegative functions in 0,+∞, ht≥0 on −τ,0. If ρ>0 and k>0, then the inequality holds:(10)Vxt≤Vyt,for all t∈0,+∞.Lemma 4.
(see [37, 39]). Suppose that ex,t:Ω×R+⟶R is integrable on Ω and is derivable respect to t. Denote Vt=∫Ωex,tdx, then(11)DαVt=∫ΩDαex,tdx.Lemma 5.
(see [40]). If xt,yt∈C1t0,b with xt≤yt, then ∀t∈t0,b there is Itαt0Rxt≤t0RItαyt.Lemma 6.
(see [41]). Let Ω be a cube xq<lq,q=1,2,…,n, and let fx be a real-valued function belonging to C1Ω which vanishes on the boundary of the Ω, i.e., fx∂Ω=0, then(12)∫Ωf2xdx≤lq2∫Ω∂fx∂xq2dx.Lemma 7.
(Gronwall–Bellman inequality; see [42]). If zt satisfies zt≤∫0taτzτdτ+bt, where atandbt are the known real functions, then(13)zt≤b0exp∫0taτdτ+∫0tb˙τexp∫τtardrdτ.Ifbt is a constant,(14)zt≤b0exp∫0taτdτ.


## 2.2. System Description

Reaction-diffusion FMNNs with time delay are described as(15)∂αuit,x∂tα=∑q=1ndiqΔuit,x−ciuit,x+∑j=1maijfjt,x+∑j=1mbijgjt−τ,x+Ii,where i=1,2,…,m, m≥2 is the quantity of units; 0<α<1 refers to the order of the equation; uit,x represents the state of the ith unit at time t and in space x; t∈J=0,T represents the time variable; Δ is a Laplace operator, which means Δuit,x=∂2uit,x/∂xq2; the smooth constants diq≥0 work as the transmission diffusion operator along the ith unit at qth position; aijfjt,x=aijuit,xfjujt,x, bijgjt−τ,x=bijuit,xgjujt−τ,x, where aijuit,x and bijuit,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; ci>0 is the self-feedback connection weight; fj⋅ and gj⋅ denote the discontinuous activation functions; Ii signifies the external input; and τ≥0 stands for time delay. In addition, the memristor-based connection weights aijuit,xandbijuit,x are given by(16)aijuit,x=a^ij,uit,⋅>Tja,unsureness,uit,⋅=Tja,aˇij,uit,⋅<Tja,bijuit,x=b^ij,uit,⋅>Tjb,unsureness,uit,⋅=Tjb,bˇij,uit,⋅<Tjb,where i,j=1,2,…,m, t≥0, switching jumps Tja,Tjb>0 and a^ij,aˇij,b^ij,andbˇij are all constants.Consider the following Dirichlet-type boundary conditions of (15)(17)uit,x=0,t∈−τ,∞,x∈∂Ω,and initial conditions(18)uis,x=φ0is,x,s∈−τ,0,x∈Ω,where φ0=φ01,φ02,…,φ0mT are real-value continuous functions defined on −τ,0×Ω.Take (15) as the drive system, then the corresponding response system is defined by(19)∂αvit,x∂tα=∑q=1ndiqΔvit,x−civit,x+∑j=1maijfjt,x+∑j=1mbijgjt−τ,x+Ii+κit,x,where i=1,2,…,m and κit,x is the controller which will be designed later.Similarly, there are(20)vit,x=0,t∈−τ,∞,x∈∂Ω,vis,x=Φ0is,x,s∈−τ,0,x∈Ω,and Φ0=Φ01,Φ02,…,Φ0mT, which have the same property to φ0.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. The solution’s global existence for fractional differential inclusions with time delay is given by [44].Assumption 1.
For eachi∈ℕ, fi⋅:ℝ⟶ℝ is piecewise continuous function. Moreover, fi⋅ has at most limited the number of discontinuous points on any compact interval of ℝ. The abovementioned properties are the same to gi⋅. Based on the definition of Filippov solution, (15) can be rewritten as(21)∂uit,x∂tα∈∑q=1ndiqΔuit,x−ciuit,x+∑j=1mKaijtco¯fjujt,x+∑j=1mKbijtco¯gjujt−τ,x+Ii,where i=1,2,…,m, uit,x are absolutely continuous on any compact subinterval of t0,∞×Ω:(22)Kaijt=a^ij,ujt,⋅>Tja,co¯a^ij,aˇij,ujt,⋅=Tja,aˇij,ujt,⋅<Tja,Kbijt=b^ij,ujt,⋅>Tjb,co¯b^ij,bˇij,ujt,⋅=Tjb,bˇij,ujt,⋅<Tjb.
Or equivalently, there are measurable functionsa˜ij⋅∈Kaij⋅, b˜ij⋅∈Kbij⋅, f˜1j⋅∈co¯fjuj⋅, and g˜1j⋅∈co¯gjuj⋅ satisfy(23)∂αuit,x∂tα=∑q=1ndiqΔuit,x−ciuit,x+∑j=1ma˜ijtf˜1jt,x+∑j=1mb˜ijtg˜1jt−τ,x+Ii,for a.e.t∈t0,∞, x∈Ω, i=1,2,…,m.
For writing convenience, we denotemaxa^ij,aˇij as aijmax and maxb^ij,bˇij as bijmax.
The IVP of the driven system (15) can be written as(24)∂αuit,x∂tα=∑q=1ndiqΔuit,x−ciuit,x+∑j=1ma˜ij,f˜1jt,x+∑j=1mb˜ij,g˜1jt−τ,x+Ii,uit,x=0,t∈−τ,∞,x∈∂Ω,uis,x=φ0is,x,t∈−τ,0,x∈Ω,f˜1js,x=γjs,x,a.e.t∈−τ,0,x∈Ω,g˜1js,x=ηjs,x,a.e.t∈−τ,0,x∈Ω.
The IVP of the response system (19) under the controller could be given by(25)∂αvit,x∂tα=∑q=1ndiqΔvit,x−civit,x+∑j=1ma˜ij,f˜2jt,x+∑j=1mb˜ij,g˜2jt−τ,x+Ii+κit,x,vit,x=0,t∈−τ,∞,x∈∂Ω,vis,x=Φ0is,x,t∈−τ,0,x∈Ω,f˜2js,x=ξjs,x,a.e.t∈−τ,0,x∈Ω,g˜2js,x=ωjs,x,a.e.t∈−τ,0,x∈Ω,where i,j=1,2,…,m. a˜ij,f˜1jt,x=a˜ijtf˜1jt,x, b˜ij,g˜1jt−τ,x=b˜ijtg˜1jt−τ,x, a˜ij,f˜2jt,x=a˜ijtf˜2jt,x, and b˜ij,g˜2jt−τ,x=b˜ijtg˜2jt−τ,x, f˜2j⋅∈co¯fjvj⋅andg˜2j⋅∈co¯gjvj⋅.
Now, leteit,x=uit,x−vit,x be the synchronization error between (15) and (19). The IVP about the error system is of the form(26)∂αeit,x∂tα=∑q=1ndiqΔeit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+∑j=1mbij,g¯jt−τ,x+κit,x,eit,x=0,t∈−τ,∞,x∈∂Ω,eis,x=φ¯0is,x,s∈−τ,0,x∈Ω,where i,j=1,2,…,m, the symbol a˜ij,f¯jt,x is written as a˜ijtf˜2jt,x−f˜1jt,x and bij,g¯jt−τ,x which stands for b˜ijtg˜2jt−τ,x−g˜1jt−τ,x.
Denoteeit,x=∫Ω∑i=1mei2t,xdx1/2 as the norm of vector et,x=e1t,x,e2t,x,…,emt,xT∈Rm.Definition 3.
The drive system (15) and response system (19) is said to be globally synchronized, if limt⟶∞eit,x=0.Remark 1.
Owing to the zero Dirichlet boundary conditions of (24) and (25), a zero equilibrium state of error system (26) exists. The global synchronization of systems (24) and (25) corresponds to the global asymptotic stability of the zero equilibrium of system (26).


## 3. Main Results

The abovementioned synchronization problem in Section2 are considered in two cases, namely, the drive system (15) has known or unknown parameters.


### 3.1. Global Synchronization for Systems with Known Parameters

First of all, we assume that the parameters in (15) are known. Then, the following adaptive controllers are proposed to implement global synchronization.Assumption 2.
Fori=1,2,…,m, there are positive constants pi and qi such that(27)supαi∈co¯fiμ,βi∈co¯fiϑαi−βi≤piμi−ϑi+qi,∀μ,ϑ∈ℝ,supαi∈co¯giμ,βi∈co¯giϑαi−βi≤riμi−ϑi+si,∀μ,ϑ∈ℝ,where for any θ∈ℝ,(28)co¯fiθ=minfj−θ,fj+θ,maxfj−θ,fj+θ,co¯giθ=mingj−θ,gj+θ,maxgj−θ,gj+θ.
For the convenience of description, the following notations are introduced:(29)σi=2−ci+∑q=1n−diqlq2+∑j=1maijmaxpj+ajimaxpi+∑j=1mbijrj,σi∗=2∑j=1maijmaxqj+∑j=1mbijmaxsj,σi∗∗=∑j=1mbijmaxri,σ¯i=−2ci+∑j=1maijmaxpj+ajimaxpi+bijrj,σ˜i=−2ci+∑q=1ndiqlq2+∑j=1maijmaxpj+ajimaxpi,σ¯i∗=2∑j=1maijmaxqj.Theorem 1.
Under Assumption1 and Assumption 2, the global synchronization of (24) and (25) can be achieved based on the following adaptive controllers:(30)κit,x=−ℓit,x+ℓ¯ieit,x−ωisgneit,x,Dαℓit,x=ϖiei2t,x,i=1,2,…,m,where ℓi0,x≥0, ϖi>0, and ℓ¯i and ωi are constants which satisfy the following inequalities:(31)ℓ¯i>12σi+σi∗∗cscαπ2>0,ωi>12σi∗.Proof.
Employ the following auxiliary function:(32)Vt,et,⋅=∫Ω12∑i=1mei2t,xdx.
By the means of Lemma1, the inequality can be established:(33)dαdtα∫Ωei2t,xdx≤2∫Ωeit,x∂αeit,x∂tαdx.
According to Lemma4, taking the derivative of (32) yields(34)DαVt,et,⋅≤∑i=1m∫Ωeit,x∑q=1ndiqΔeit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+∑j=1mbij,g¯jt−τ,x+κit,xdx.
Using Green’s formula and Lemma6, it can be obtained that(35)∑q=1n∫Ωeit,xdiqΔeit,xdx≤−∑q=1n∫Ωdiqlqei2t,xdx.
Recalling Assumption1 and Assumption 2, the inequalities hold:(36)∑j=1m∫Ωeit,xa˜ij,f¯jt,xdx≤12∑j=1maijmaxpj∫Ωei2t,x+ej2t,xdx+∑j=1maijmaxqj∫Ωeit,xdx,∑j=1m∫Ωeit,xbij,g¯jt−τ,xdx≤12∑j=1mbijmaxrj∫Ωei2t,x+ej2t−τ,xdx+∑j=1mbijmaxsj∫Ωeit,xdx.
Combining controller (30) designed in Theorem 1, it follows that(37)DαVt,et,⋅≤12∑i=1mσi∫Ωei2t,xdx+12∑i=1mσi∗∗∫Ωei2t−τ,xdx+12∑i=1mσi∗∫Ωeit,xdx−∑i=1m∫Ωlit,x+ℓ¯iei2t,xdx−∑i=1mωi∫Ωeit,xdx.
From Definition2 and Property 2, one obtains(38)ℓit,x=ℓi0,x+1Γα∫0tt−ταϖiei2t,xdx.
Substituting (38) into (37) and noting that ℓi and ωi meet inequality (31), it can be deduced that(39)DαVt,et,⋅≤−λ1Vt,et,⋅+ρ1Vt−τ,et,⋅,where λ1=∑i=1m2ℓ¯i−σi and ρ1=∑i=1mσi∗∗.
According to Lemma2 and Lemma 3, it can be assert that the global synchronization between (24) and (25) can be realized.
The proof of Theorem1 completes.Remark 2.
Whenα=1, the error system (26) be changed into the integer form. The conclusion is the same as the result in Theorem 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. Thus, the error system (26) can be converted to(40)Dαeit=−cieit+∑j=1ma˜ij,f¯jt+∑j=1mbij,g¯jt−τ+κit,eis=φ¯0is,s∈−τ,0,i=1,2,…,m.
For convenience, the inequalities are given by(41)ℓ¯i>12σ¯i+σi∗∗cscαπ2>0,ωi>12σi∗.Corollary 1.
Under Assumption1 and Assumption 2, the global synchronization between (24) and (25) can be achieved based on the following adaptive controllers:(42)κit=−ℓit+ℓ¯ieit−ωisgneit,Dαℓit=ϖiei2t,i=1,2,…,m,where ℓi0>0, ϖi>0, and ℓ¯i and ωi are constants which satisfy inequality (41).Remark 3.
In fact, synchronization of FMNNs with time delay has been investigated in [20]. The results about synchronization of the systems with single delay in Theorem 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(43)∂αeit,x∂tα=∑q=1ndiqΔeit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+κit,x,eit,x=0,t∈0,∞,x∈∂Ω,eis,x=φ¯0is,x,s=0,x∈Ω.
For convenience, the inequalities are given by(44)ℓ¯i>12σ˜i>0,ωi>12σ¯i∗.Corollary 2.
Under Assumption1 and Assumption 2, the global synchronization between (24) and (25) can be realized based on the following adaptive controllers:(45)κit,x=−ℓit,x+ℓ¯ieit,x−ωisgneit,x,Dαℓit,x=ϖiei2t,x,i=1,2,…,m,where ℓi0,x≥0, ϖi>0, and ℓ¯i and ωi 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.


### 3.2. 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 weightci and input Ii in the drive system (15) are unknown. Then, the IVP of the drive system (15) can be written as(46)∂αuit,x∂tα=∑q=1ndiqΔuit,x−ciuit,x+∑j=1ma˜ij,f˜1jt,x+∑j=1mb˜ij,g˜1jt−τ,x+Ii,uit,x=0,t∈−τ,∞,x∈∂Ω,uis,x=φ0is,x,t∈−τ,0,x∈Ω.The IVP of the response system (19) can be shown as(47)∂αvit,x∂tα=∑q=1ndiqΔvit,x−c^it,xvit,x+∑j=1ma˜ij,f˜2jt,x+∑j=1mb˜ij,g˜2jt−τ,x+I^it,x+κit,x,vit,x=0,t∈−τ,∞,x∈∂Ω,vis,x=Φ0is,x,t∈−τ,0,x∈Ω,where i,j=1,2,..,m and c^it,x and I^it,x are treated as the compensation of ci and Ii.Then, fori,j=1,2,…,m, the IVP of the synchronization error can be described as(48)∂αeit,x∂tα=−c^it,x−civit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+∑j=1mbij,g¯jt−τ,x+I^it,x−Ii+κit,x,eis,x=0,s∈−τ,∞,x∈∂Ω,eis,x=φ¯0is,x,s∈−τ,0,x∈Ω.Assumption 3.
There are positive constantspi, qi, and g¯i such that(49)supαi∈co¯fiμ,βi∈co¯fiϑαi−βi≤piμi−ϑi+qi,∀μ,ϑ∈ℝ,giθ≤g¯i,where i=1,2,…,m, θ∈ℝ, and the notation co¯fiθ represents minfj−θ,fj+θ,maxfj−θ,fj+θ.Theorem 2.
Under Assumption1 and Assumption 3, the global asymptotic synchronization between (46) and (47) can realized based on the following adaptive controllers:(50)κit,x=−ℓit,x+ℓ¯ieit,x−ωisgneit,x,Dαℓit,x=ϖiei2t,x,Dαc^it,x=ϱieit,xvit,x,DαI^it,x=−νieit,x,where i=1,2,…,m, ℓi0,x, ϖi, ϱi, and νi are positive constants, and ℓ¯i and ωi are constants which satisfy the following inequalities:(51)ℓ¯i>12σ˜i>0,ωi>12σ˜i∗,with σ˜i=−2ci+∑q=1ndiq/lq2+∑j=1maijmaxpj+ajimaxpi and σ˜i∗=2∑j=1maijmaxqj+bijmaxg¯j.Proof.
Employ the following auxiliary function:(52)Vt=V1t+V2t+V3t,where(53)V1t=∫Ω12∑i=1mei2t,xdx,V2t=∫Ω12ϱi∑i=1mc^it,x−ci2dx,V3t=∫Ω12νi∑i=1mI^it,x−Ii2dx.
Similarly to the proof of Theorem1, one obtains(54)DαV1t≤∑i=1m12σ˜i∫Ωei2t,xdx+∑i=1m12σ˜i∗∫Ωeit,xdx+∑i=1m∫ΩI^it,x−Iieit,xdx−∑i=1m∫Ωc^it,x−civit,xeit,xdx+∑i=1m∫Ωeit,xκit,xdx,DαV3t≤−∑i=1m∫Ωeit,xI^it,x−Iidx,DαV2t≤∑i=1m∫Ωeit,xc^it,x−civit,xdx.
Sinceℓi and ωi fulfill condition (51), it follows(55)DtαVt≤−λ2V1t,where λ2=∑i=1m2ℓ¯i−σ˜i.
By Lemma5, it can be concluded that(56)Vt−V0≤−λΓα∫0tt−τα−1V1τdτ.
According to the definition ofV1, thus(57)V1t≤Vt≤V0−λΓα∫0tt−τα−1V1τdτ.
By Lemma7, it reveals that(58)V1t≤V0exp−λΓα∫0tt−τα−1dτ=V0exp−λΓα+1tα.
Therefore,(59)limt⟶∞V1t=limt⟶∞∫Ω12∑i=1mei2t,xdx=0.
From (59), it is easy to see that the global synchronization between (46) and (47) can be realized.
The proof of Theorem2 completes.Remark 5.
The result obtained in Theorem2 is more practical for considering the situation where the parameters of the drive system (24) are unknown. Through the proof of Theorem 2, it can be asserted that the dynamical compensator is necessary for the unknown system. If the reaction-diffusion in space Ω disappears, the error system (48) can be converted to(60)Dαeit=−c^it−civit−cieit+∑j=1ma˜ij,f¯jt+∑j=1mbij,g¯jt−τ+I^it−Ii+κit,eis=φ¯0is,s∈−τ,0.Corollary 3.
Under Assumption1 and Assumption 3, the global synchronization between (46) and (47) can be achieved based on the following adaptive controllers:(61)κit=−ℓit+ℓ¯ieit−ωisgneit,Dαℓit=ϖiei2t,Dαc^it=νieit,DαI^it=−ϱieit,i=1,2,…,m,where ℓi0≥0, ϖi>0, ℓ¯i>1/2σ⌣i>0, ωi>1/2σ˜i∗, and σ⌣i=−2ci+∑j=1maijmaxpj+∑j=1majimaxpi.Suppose that there is no time delayτ in the course of communication; then, the error system (48) can be changed into(62)∂αeit,x∂tα=−c^it,x−civit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+I^it,x−Ii+κit,x,eis,x=0,s∈0,∞,x∈∂Ω,eis,x=φ¯0is,x,s=0,x∈Ω..Corollary 4.
Under Assumption1 and Assumption 3, (46) and (47) can achieve global synchronization based on the following adaptive controllers:(63)κit,x=−ℓ¯it,x+ℓieit,x−ωisgneit,x,Dαℓit,x=ϖiei2t,x,Dαc^it,x=νieit,x,DαI^it,x=−ϱieit,x,i=1,2,…,m,where ℓi0,x≥0, ϖi>0, ℓ¯i>1/2σ⌣i>0, ωi>1/2σ⌣i∗, and σ⌣i∗=2∑j=1maijmaxqj.


## 3.1. Global Synchronization for Systems with Known Parameters

First of all, we assume that the parameters in (15) are known. Then, the following adaptive controllers are proposed to implement global synchronization.Assumption 2.
Fori=1,2,…,m, there are positive constants pi and qi such that(27)supαi∈co¯fiμ,βi∈co¯fiϑαi−βi≤piμi−ϑi+qi,∀μ,ϑ∈ℝ,supαi∈co¯giμ,βi∈co¯giϑαi−βi≤riμi−ϑi+si,∀μ,ϑ∈ℝ,where for any θ∈ℝ,(28)co¯fiθ=minfj−θ,fj+θ,maxfj−θ,fj+θ,co¯giθ=mingj−θ,gj+θ,maxgj−θ,gj+θ.
For the convenience of description, the following notations are introduced:(29)σi=2−ci+∑q=1n−diqlq2+∑j=1maijmaxpj+ajimaxpi+∑j=1mbijrj,σi∗=2∑j=1maijmaxqj+∑j=1mbijmaxsj,σi∗∗=∑j=1mbijmaxri,σ¯i=−2ci+∑j=1maijmaxpj+ajimaxpi+bijrj,σ˜i=−2ci+∑q=1ndiqlq2+∑j=1maijmaxpj+ajimaxpi,σ¯i∗=2∑j=1maijmaxqj.Theorem 1.
Under Assumption1 and Assumption 2, the global synchronization of (24) and (25) can be achieved based on the following adaptive controllers:(30)κit,x=−ℓit,x+ℓ¯ieit,x−ωisgneit,x,Dαℓit,x=ϖiei2t,x,i=1,2,…,m,where ℓi0,x≥0, ϖi>0, and ℓ¯i and ωi are constants which satisfy the following inequalities:(31)ℓ¯i>12σi+σi∗∗cscαπ2>0,ωi>12σi∗.Proof.
Employ the following auxiliary function:(32)Vt,et,⋅=∫Ω12∑i=1mei2t,xdx.
By the means of Lemma1, the inequality can be established:(33)dαdtα∫Ωei2t,xdx≤2∫Ωeit,x∂αeit,x∂tαdx.
According to Lemma4, taking the derivative of (32) yields(34)DαVt,et,⋅≤∑i=1m∫Ωeit,x∑q=1ndiqΔeit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+∑j=1mbij,g¯jt−τ,x+κit,xdx.
Using Green’s formula and Lemma6, it can be obtained that(35)∑q=1n∫Ωeit,xdiqΔeit,xdx≤−∑q=1n∫Ωdiqlqei2t,xdx.
Recalling Assumption1 and Assumption 2, the inequalities hold:(36)∑j=1m∫Ωeit,xa˜ij,f¯jt,xdx≤12∑j=1maijmaxpj∫Ωei2t,x+ej2t,xdx+∑j=1maijmaxqj∫Ωeit,xdx,∑j=1m∫Ωeit,xbij,g¯jt−τ,xdx≤12∑j=1mbijmaxrj∫Ωei2t,x+ej2t−τ,xdx+∑j=1mbijmaxsj∫Ωeit,xdx.
Combining controller (30) designed in Theorem 1, it follows that(37)DαVt,et,⋅≤12∑i=1mσi∫Ωei2t,xdx+12∑i=1mσi∗∗∫Ωei2t−τ,xdx+12∑i=1mσi∗∫Ωeit,xdx−∑i=1m∫Ωlit,x+ℓ¯iei2t,xdx−∑i=1mωi∫Ωeit,xdx.
From Definition2 and Property 2, one obtains(38)ℓit,x=ℓi0,x+1Γα∫0tt−ταϖiei2t,xdx.
Substituting (38) into (37) and noting that ℓi and ωi meet inequality (31), it can be deduced that(39)DαVt,et,⋅≤−λ1Vt,et,⋅+ρ1Vt−τ,et,⋅,where λ1=∑i=1m2ℓ¯i−σi and ρ1=∑i=1mσi∗∗.
According to Lemma2 and Lemma 3, it can be assert that the global synchronization between (24) and (25) can be realized.
The proof of Theorem1 completes.Remark 2.
Whenα=1, the error system (26) be changed into the integer form. The conclusion is the same as the result in Theorem 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. Thus, the error system (26) can be converted to(40)Dαeit=−cieit+∑j=1ma˜ij,f¯jt+∑j=1mbij,g¯jt−τ+κit,eis=φ¯0is,s∈−τ,0,i=1,2,…,m.
For convenience, the inequalities are given by(41)ℓ¯i>12σ¯i+σi∗∗cscαπ2>0,ωi>12σi∗.Corollary 1.
Under Assumption1 and Assumption 2, the global synchronization between (24) and (25) can be achieved based on the following adaptive controllers:(42)κit=−ℓit+ℓ¯ieit−ωisgneit,Dαℓit=ϖiei2t,i=1,2,…,m,where ℓi0>0, ϖi>0, and ℓ¯i and ωi are constants which satisfy inequality (41).Remark 3.
In fact, synchronization of FMNNs with time delay has been investigated in [20]. The results about synchronization of the systems with single delay in Theorem 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(43)∂αeit,x∂tα=∑q=1ndiqΔeit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+κit,x,eit,x=0,t∈0,∞,x∈∂Ω,eis,x=φ¯0is,x,s=0,x∈Ω.
For convenience, the inequalities are given by(44)ℓ¯i>12σ˜i>0,ωi>12σ¯i∗.Corollary 2.
Under Assumption1 and Assumption 2, the global synchronization between (24) and (25) can be realized based on the following adaptive controllers:(45)κit,x=−ℓit,x+ℓ¯ieit,x−ωisgneit,x,Dαℓit,x=ϖiei2t,x,i=1,2,…,m,where ℓi0,x≥0, ϖi>0, and ℓ¯i and ωi 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.


## 3.2. 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 weightci and input Ii in the drive system (15) are unknown. Then, the IVP of the drive system (15) can be written as(46)∂αuit,x∂tα=∑q=1ndiqΔuit,x−ciuit,x+∑j=1ma˜ij,f˜1jt,x+∑j=1mb˜ij,g˜1jt−τ,x+Ii,uit,x=0,t∈−τ,∞,x∈∂Ω,uis,x=φ0is,x,t∈−τ,0,x∈Ω.The IVP of the response system (19) can be shown as(47)∂αvit,x∂tα=∑q=1ndiqΔvit,x−c^it,xvit,x+∑j=1ma˜ij,f˜2jt,x+∑j=1mb˜ij,g˜2jt−τ,x+I^it,x+κit,x,vit,x=0,t∈−τ,∞,x∈∂Ω,vis,x=Φ0is,x,t∈−τ,0,x∈Ω,where i,j=1,2,..,m and c^it,x and I^it,x are treated as the compensation of ci and Ii.Then, fori,j=1,2,…,m, the IVP of the synchronization error can be described as(48)∂αeit,x∂tα=−c^it,x−civit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+∑j=1mbij,g¯jt−τ,x+I^it,x−Ii+κit,x,eis,x=0,s∈−τ,∞,x∈∂Ω,eis,x=φ¯0is,x,s∈−τ,0,x∈Ω.Assumption 3.
There are positive constantspi, qi, and g¯i such that(49)supαi∈co¯fiμ,βi∈co¯fiϑαi−βi≤piμi−ϑi+qi,∀μ,ϑ∈ℝ,giθ≤g¯i,where i=1,2,…,m, θ∈ℝ, and the notation co¯fiθ represents minfj−θ,fj+θ,maxfj−θ,fj+θ.Theorem 2.
Under Assumption1 and Assumption 3, the global asymptotic synchronization between (46) and (47) can realized based on the following adaptive controllers:(50)κit,x=−ℓit,x+ℓ¯ieit,x−ωisgneit,x,Dαℓit,x=ϖiei2t,x,Dαc^it,x=ϱieit,xvit,x,DαI^it,x=−νieit,x,where i=1,2,…,m, ℓi0,x, ϖi, ϱi, and νi are positive constants, and ℓ¯i and ωi are constants which satisfy the following inequalities:(51)ℓ¯i>12σ˜i>0,ωi>12σ˜i∗,with σ˜i=−2ci+∑q=1ndiq/lq2+∑j=1maijmaxpj+ajimaxpi and σ˜i∗=2∑j=1maijmaxqj+bijmaxg¯j.Proof.
Employ the following auxiliary function:(52)Vt=V1t+V2t+V3t,where(53)V1t=∫Ω12∑i=1mei2t,xdx,V2t=∫Ω12ϱi∑i=1mc^it,x−ci2dx,V3t=∫Ω12νi∑i=1mI^it,x−Ii2dx.
Similarly to the proof of Theorem1, one obtains(54)DαV1t≤∑i=1m12σ˜i∫Ωei2t,xdx+∑i=1m12σ˜i∗∫Ωeit,xdx+∑i=1m∫ΩI^it,x−Iieit,xdx−∑i=1m∫Ωc^it,x−civit,xeit,xdx+∑i=1m∫Ωeit,xκit,xdx,DαV3t≤−∑i=1m∫Ωeit,xI^it,x−Iidx,DαV2t≤∑i=1m∫Ωeit,xc^it,x−civit,xdx.
Sinceℓi and ωi fulfill condition (51), it follows(55)DtαVt≤−λ2V1t,where λ2=∑i=1m2ℓ¯i−σ˜i.
By Lemma5, it can be concluded that(56)Vt−V0≤−λΓα∫0tt−τα−1V1τdτ.
According to the definition ofV1, thus(57)V1t≤Vt≤V0−λΓα∫0tt−τα−1V1τdτ.
By Lemma7, it reveals that(58)V1t≤V0exp−λΓα∫0tt−τα−1dτ=V0exp−λΓα+1tα.
Therefore,(59)limt⟶∞V1t=limt⟶∞∫Ω12∑i=1mei2t,xdx=0.
From (59), it is easy to see that the global synchronization between (46) and (47) can be realized.
The proof of Theorem2 completes.Remark 5.
The result obtained in Theorem2 is more practical for considering the situation where the parameters of the drive system (24) are unknown. Through the proof of Theorem 2, it can be asserted that the dynamical compensator is necessary for the unknown system. If the reaction-diffusion in space Ω disappears, the error system (48) can be converted to(60)Dαeit=−c^it−civit−cieit+∑j=1ma˜ij,f¯jt+∑j=1mbij,g¯jt−τ+I^it−Ii+κit,eis=φ¯0is,s∈−τ,0.Corollary 3.
Under Assumption1 and Assumption 3, the global synchronization between (46) and (47) can be achieved based on the following adaptive controllers:(61)κit=−ℓit+ℓ¯ieit−ωisgneit,Dαℓit=ϖiei2t,Dαc^it=νieit,DαI^it=−ϱieit,i=1,2,…,m,where ℓi0≥0, ϖi>0, ℓ¯i>1/2σ⌣i>0, ωi>1/2σ˜i∗, and σ⌣i=−2ci+∑j=1maijmaxpj+∑j=1majimaxpi.Suppose that there is no time delayτ in the course of communication; then, the error system (48) can be changed into(62)∂αeit,x∂tα=−c^it,x−civit,x−cieit,x+∑j=1ma˜ij,f¯jt,x+I^it,x−Ii+κit,x,eis,x=0,s∈0,∞,x∈∂Ω,eis,x=φ¯0is,x,s=0,x∈Ω..Corollary 4.
Under Assumption1 and Assumption 3, (46) and (47) can achieve global synchronization based on the following adaptive controllers:(63)κit,x=−ℓ¯it,x+ℓieit,x−ωisgneit,x,Dαℓit,x=ϖiei2t,x,Dαc^it,x=νieit,x,DαI^it,x=−ϱieit,x,i=1,2,…,m,where ℓi0,x≥0, ϖi>0, ℓ¯i>1/2σ⌣i>0, ωi>1/2σ⌣i∗, and σ⌣i∗=2∑j=1maijmaxqj.


## 4. Numerical Simulations

Several numerical simulations are presented to illustrate the main results. The first example is used to validate Theorem1. The second one is used to text Theorem 2. Now, let us consider the 2D (q=1) reaction-diffusion FMNNs with time delay and zero Direclet boundary values. Here, m=2, n=1, and Ω=−2,2.Example 1.
Consider error system (26) with such parameters:(64)A=a11−0.020.13−0.01,B=0.35b120.360.37,C=0.35000.23,I=0.60000.40,where the memristive coefficients are defined by(65)a11=0.19,δ<1,0.26,δ=1,0.55,δ>1,b12=−0.13,δ<1,−0.17,δ=1,−0.24,δ>1.Suppose that time delay τ=1.1 and discontinuous activation functions are fδ=gδ=0.98δ+0.05sgnδ. Let initial values of (24) and (25) be(66)u1x,t=8sin−2+x+t2,u2x,t=6sin−2+x+t,v1x,t=−5cos−2+x+t,v2x,t=−6cos−2+x+t,where t∈−1.1,0 and x∈−2,2. Given the fractional order α=0.98 and d11=d21=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 fδ and gδ have only one discontinuous point 0, Assumption 1 is valid. According to Assumption 2, we can choose p1=p2=r1=r2=0.98 and q1=q2=s1=s2=0.05. By calculation, the parameters of controllers are selected as ℓ¯1=0.70>0.39, w1=0.06>0.05, ℓ¯2=0.60>0.36, w2=0.05>0.04, ℓ10,x=ℓ20,x=0.01>0, and ϖ1=ϖ2=0.01>0. It is easy to verify that condition (9) in Theorem 1 is satisfied. Figure 2 suggests that the synchronization errors asymptotically converge to zero.
Theorem1 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 reaction-diffusion 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.Figure 1
Synchronization errors between systems (24) and (25), with no controller.
(a)(b)Figure 2
Synchronization errors between systems (24) and (25), under controller (30).
(a)(b)Figure 3
Trajectories of error system (26) at spatial position x=55 under different orders.
(a)(b)Figure 4
Trajectories of error system (26) at spatial position x=55 under different diffusion coefficients.
(a)(b)Example 2.
Consider the error system (48) with the following parameters:(67)A=a11−0.950.580.76,B=0.22b120.150.03,C=0.58000.49,I=0.23000.25,where(68)a11=0.19,δ<1,0.80,δ=1,0.26,δ>1,b12=0.05,δ<1,0.11,δ=1,0.06,δ>1.Assume α=0.98, d11=d21=0.85, and (47) with unknown parameters ci and Ii, and initial values of compensators in (47) are given by(69)C^0=1001,I^0=2002.Activation functions are fδ=0.98δ+0.05sgnδ and gδ=0.5sgnδ and time delay τ=3.5. Initial values of (46) and (47) are chosen as(70)u1x,t=1.7sin−2+x+t2,u2x,t=−2.1sin−2+x+t,v1x,t=1.5cos−2+x+8t,v2x,t=−1.9cos−2+x+8t2,where t∈−3.4,0 and x∈−2,2.
As seen in Figure5, the error state is instable. Choosing p1=p2=0.98, q1=q2=0.05, and g¯1=g¯2=0.5, we obtain ℓ¯1=1.10>1.06, w1=0.11>0.10, ℓ¯2=1.00>0.97, and w2=0.08>0.07 and ℓi0,x=0.01>0, ϖi=0.01>0, ϱi=0.001>0, and νi=0.02>0, i=1,2. Condition (51) in Theorem 2 is content clear. Figure 6 shows that the synchronization errors achieve stability rapidly under the adaptive controller (50).
For the error system (48), under different orders α=0.6,0.7,0.8,0.9 with the same reaction-diffusion coefficients d=0.6 and different reaction-diffusion coefficients d=0.8,1.1,1.5,2.0 with the same order α=0.7, simulations are carried out. Corresponding error state trajectories are depicted in Figures 7 and 8. It reveals that the asymptotic stability of the system varies with orders and reaction-diffusion coefficients. Compared with the former case, the figures show little differences with different parameters.Figure 5
Synchronization errors between systems (46) and (47), with no controller.
(a)(b)Figure 6
Synchronization errors between systems (46) and (47), with controller (50).
(a)(b)Figure 7
Trajectories of error system (26) at spatial position x=55 under different orders.
(a)(b)Figure 8
Trajectories of error system (26) at spatial position x=55 under different diffusion coefficients.
(a)(b)


## 5. Conclusion and Discussion

The 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.



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