Research ArticleGlobal Synchronization of Reaction-Diffusion Fractional-OrderMemristive Neural Networks with Time Delay andUnknown Parameters

Wenjiao Sun Guojian Ren Yongguang Yu and Xudong Hai

Department of Mathematics Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Yongguang Yu ygyubjtueducn

Received 18 October 2019 Revised 3 January 2020 Accepted 1 April 2020 Published 31 May 2020

Academic Editor Hiroki Sayama

Copyright copy 2020Wenjiao Sun et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

+is paper investigated the global synchronization of fractional-order memristive neural networks (FMNNs) To deal with theeffect of reaction-diffusion and time delay fractional partial and comparison theorem are introduced Based on the set valuemapping theory and Filippov solution the activation function is extended to discontinuous case Adaptive controllers with acompensator are designed owing to the existence of unknown parameters with the help of GronwallndashBellman inequalityNumerical simulation examples demonstrate the availability of the theoretical results

1 Introduction

As a generalization of differential and integral calculus froman integer order to arbitrary order fractional calculus hasmore than 300 years history [1] Since Mandelbort proposedthe fact that lots of fractals exist in the field of technology awhole new array of possibilities for natural science researchhas been opened [2] Compared with the conventional in-teger order calculus it shows some superiority in the aspectof memory and heredity [3] +at caused its wide range ofapplications [4ndash6]

Neural networks model as we know has great potentialfor controlling highly nonlinear and severe uncertain sys-tems Due to the ability of studying arbitrary continuousnonlinear functions it has been fruitfully applied to opti-mization secure communication cryptography and so onTaking into account these facts the combination of frac-tional calculus and neural networks model is a remarkablygreat improvement +is kind of model called fractional-order neural networks (FNNs) was proposed by Arena in1998 [7] Afterwards a myriad of studies have focused on thedynamical behaviors of FNNs One of the most interestingissues is to explore the synchronization problem of FNNsVarious definition types of synchronization have been built

[8ndash10] Among them global synchronization is the mostbasic and widely applied one since it guarantees the con-vergence of the system From the perspective of engineeringapplications it has achieved the required realistic standards[11 12] +us global synchronization of FNNs has beenextensively investigated to date [13ndash15]

Another hot topic of research is memristor-based neuralnetworks which were first put forward by Chua [16] In2008 a nanoscale memory was made by Williams group[17] which was named memristor Memory characteristicsand nanometer dimensions make the memristor showproperties the same as the neurons in the human brainFrom this standpoint fractional-order memristive neuralnetworks (FMNNs) have been suggested replacing the FNNsto emulate the human brain +erefore the analysis ofFMNNs has become increasingly attractive to researchersAt present there are many excellent results on synchroni-zation of FMNNs [18ndash21] Velmurugan investigated finite-time synchronization of FMNNs with the method of Laplacetransform and generalized Gronwalls inequality [18]+e lagcomplete synchronization criteria of FMNNs are proposedby period intermittent control [19] Delay-independentsynchronization criteria for FMNNs are established by usingthe maximum modulus principle the spectral radii of

HindawiComplexityVolume 2020 Article ID 4145826 14 pageshttpsdoiorg10115520204145826

matrix and Kakutani fixed point theorem [20] Chen et aldesigned a stabilizing state-feedback controller byemploying the FO Razumikhin theorem and linear matrixinequalities [21] Besides the abovementioned methodsadaptive mechanism has been proposed as a more effectivestrategy [22] Compared with the conventional feedbackcontrol the parameters of adaptive controllers will beupdated automatically which makes the energy be saved[23] It also has significant effect on the analysis of fractionalorder nonlinear system especially when the system hasunknown parameters [24ndash27] +us it is a natural idea toinvestigate the synchronization of FMNNs using adaptivecontrol scheme

It is a remarkable fact that time delays emerge in thecommunication of neurons [28 29] Furthermore the oc-currence of diffusion phenomena has a significant impact inthe electronsrsquo transportation through a nonuniform elec-tromagnetic field +at means not only the time but alsotheir spatial positions affect the dynamic behaviors of net-works +e models covering time delay and reaction-dif-fusion are good mimicry for the real neural networks interms of application but the existence of time delays andreaction-diffusion could bring about some undesirable be-haviors [30 31] Drawing on the abovementioned concernsspecial attention should be paid on time delay and reaction-diffusion However the research on FMNNs with both re-action-diffusion and time delay is still in its infancy [32 33]

For realistic neural networks systems one must includeuncertainties Uncertainty could occur in parameters in themathematical model for they are hardly to be exactly knownin advance [34ndash36] +erefore the study on synchronizationof the drive-response system considering unknown pa-rameters is one of the most challenging tasks Expanding onthe existing literature the global synchronization problemfor reaction-diffusion FMNNs with time delay and unknownparameters is proposed firstly +e main contents of thispaper are the following (1) Based on fractional comparisontheorem the problem of synchronization for the integraldelayed neural networks is generalized to the fractionalorder case (2) By utilizing Caputo partial fractional de-rivative reaction-diffusion is emphasized for the FMNNswhich extends the problem from the time domain to time-spatial dimension (3) With the help of functional differ-ential inclusions the solution of the discontinuous systems isguaranteed in the sense of Filippov (4) Employing Gron-wallndashBellman inequality the adaptive controllers withcompensator are designed to achieve the synchronizationdespite the presence of parametric uncertainties

+is paper is structured as follows In Section 2 webriefly introduce preliminaries and model description Twotypes of adaptive controllers for global synchronization ofthe drive-response system are designed in Section 3 InSection 4 two simulations are shown to verify the cor-rectness of the proposed results A short conclusion andsome discussions are given in Section 5

Notations let Rn be the space of n-dimensional Euclideanspace +e norm of vector x (x1 x2 xn)T isin Rn isdefined as x (1113936

nq1 x2

q)12 A bounded open-set Ω sub Rn

containing the origin equips with the smooth boundary zΩand mesΩgt 0

2 Preliminaries and Model Description

Preliminaries about fractional calculus and the model ofreaction-diffusion FMNNs with time delay are introduced inthis section Concept of Filippov solution fractional com-parison theorem and GronwallndashBellman inequality need tobe emphasized

21 Preliminaries Definition and the properties aboutCaputo RiemannndashLiuville fractional calculus are given in thefollowing

Definition 1 (see [37]) For any tgt 0 the time Caputofractional derivative of order α(0lt αlt 1) for a functionf(t x) isin C1[[0 b] timesΩR] is defined by

zαf(t x)

ztα

1Γ(1 minus α)

1113946t

0

zf(s x)

zs

ds

(t minus s)α (1)

where Γ(z) 1113938infin0 eminus ttzminus1dt Particularly when

f(t middot) f(t) thenzαf(t middot)

ztαdαf(t)

dtα

C0 D

αt f(t) (2)

It should be noted that the initial conditions of Caputofractional derivative are same as integer-order derivativewhich makes Caputo fractional derivative equip specificphysical sense in the real world In this paper simple no-tation Dα is used to replace C

0 Dαt

Definition 2 (see [37]) RiemannndashLiuville fractional integralof order α for a function f(t) ([0 +infin)⟶ R) is definedby

0RI

αt f(t)

1Γ(α)

1113946t

0

f(τ)

(t minus τ)1minusα dτ (3)

where 0lt αlt 1

Proposition 1 (see [37]) For arbitrary constants v1 and v2there is

Dα

v1f(t) + v2g(t)( 1113857 v1Dαf(t) + v2D

αg(t) (4)

Proposition 2 (see [37]) If 0lt αlt 1 and f(t) isin C1[0 t]then 0

RIαt Dαf(t) f(t) minus f(0)

Several important conclusions about the fractional-ordersystem are listed below

Lemma 1 (see [33]) Let f(t x) [0 b] timesΩ⟶ R be acontinuous and differentiable function on t then for anytge 0 x isin Ω

12

zαf2(t x)

ztαlef(t x)

zαf(t x)

ztα (5)

where 0lt αlt 1 Particularly when α 1 there is

2 Complexity

12

zf2(t x)

ztlef(t x)

zf(t x)

zt (6)

Lemma 2 (see [38]) Consider the system as followsDαV(x(t)) minusρV(x(t)) + kV(x(t minus τ))

x(t) ϕ(t)

t isin [minusτ 0]

⎧⎪⎪⎨

⎪⎪⎩(7)

where ρ kge 0 and V(x)ge 0 4e solution of the system isasymptotically stable when klt ρ sin(απ2)

Lemma 3 (fractional comparison theorem see [38])Consider the following fractional order inequality

DαV(x(t)) le minus ρV(x(t)) + kV(x(t minus τ))

V(x(t)) h(t) t isin [minusτ 0]1113896 (8)

and fractional order system

DαV(y(t)) minusρV(y(t)) + kV(y(t minus τ))

V(y(t)) h(t) t isin [minusτ 0]1113896 (9)

where α isin (0 1) τ gt 0 x(t) y(t) isin Rn and V(x) andV(y)

are continuous and nonnegative functions in (0 +infin)h(t)ge 0 on [minusτ 0] If ρgt 0 and kgt 0 then the inequalityholds

V(x(t)) leV(y(t)) (10)

for all t isin [0 +infin)

Lemma 4 (see [37 39]) Suppose that e(x t) Ω times R+⟶ R

is integrable on Ω and is derivable respect to t DenoteV(t) 1113938Ωe(x t)dx then

DαV(t) 1113946

ΩD

αe(x t)dx (11)

Lemma 5 (see [40]) If x(t) y(t) isin C1[t0 b] withx(t)ley(t) then forallt isin [t0 b] there is R

t0Iαt x(t)le R

t0Iαt y(t)

Lemma 6 (see [41]) LetΩ be a cube |xq|lt lq q 1 2 nand let f(x) be a real-valued function belonging to C1(Ω)

which vanishes on the boundary of the Ω ie f(x)|zΩ 0then

1113946Ω

f2(x)dxle l

2q1113946Ω

zf(x)

zxq

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

dx (12)

Lemma 7 (GronwallndashBellman inequality see [42]) If z(t)

satisfies z(t)le 1113938t

0 a(τ)z(τ)dτ + b(t) where a(t) and b(t)

are the known real functions then

z(t)le b(0)exp 1113946t

0a(τ)dτ1113888 1113889 + 1113946

t

0_b(τ)exp 1113946

t

τa(r)dr1113888 1113889dτ

(13)

If b(t) is a constant

z(t)le b(0)exp 1113946t

0a(τ)dτ1113888 1113889 (14)

22 System Description Reaction-diffusion FMNNs withtime delay are described as

zαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x) + 1113944

m

j1langaijfjrang(t x)

+ 1113944

m

j1langbijgjrang(t minus τ x) + Ii

(15)

where i 1 2 m mge 2 is the quantity of units 0lt αlt 1refers to the order of the equation ui(t x) represents thestate of the ith unit at time t and in space x t isin J [0 T]

represents the time variable Δ is a Laplace operator whichmeans Δui(t x) z2ui(t x)zx2

q the smooth constantsdiq ge 0 work as the transmission diffusion operator along theith unit at qth position langaijfjrang(t x) aij(ui(t x))

fj(uj(t x)) langbijgjrang(t minus τ x) bij(ui(t x))gj(uj(t minus τ

x)) where aij(ui(t x)) and bij(ui(t x)) stand for thestrength of the jth unit on the ith unit at time t in space x

and at time t minus τ in space x respectively ci gt 0 is the self-feedback connection weight fj(middot) and gj(middot) denote thediscontinuous activation functions Ii signifies the externalinput and τ ge 0 stands for time delay In addition thememristor-based connection weights aij(ui(t

x)) and bij(ui(t x)) are given by

aij ui(t x)( 1113857

1113954aij ui(t middot)1113868111386811138681113868

1113868111386811138681113868gtTaj

unsureness ui(t middot)1113868111386811138681113868

1113868111386811138681113868 Taj

aij ui(t middot)1113868111386811138681113868

1113868111386811138681113868ltTaj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

bij ui(t x)( 1113857

1113954bij ui(t middot)1113868111386811138681113868

1113868111386811138681113868gtTbj

unsureness ui(t middot)1113868111386811138681113868

1113868111386811138681113868 Tbj

bij ui(t middot)1113868111386811138681113868

1113868111386811138681113868ltTbj

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(16)

where i j 1 2 m tge 0 switching jumps Taj Tb

j gt 0 and1113954aij aij

1113954bij and bij are all constantsConsider the following Dirichlet-type boundary condi-

tions of (15)

ui(t x) 0 t isin [minusτinfin) x isin zΩ (17)

and initial conditions

ui(s x) φ0i(s x) s isin [minusτ 0] x isin Ω (18)

where φ0 (φ01φ02 φ0m)T are real-value continuousfunctions defined on [minusτ 0] timesΩ

Take (15) as the drive system then the correspondingresponse system is defined by

Complexity 3

zαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus civi(t x) + 1113944

m

j1langaijfjrang(t x)

+ 1113944m

j1langbijgjrang(t minus τ x) + Ii + κi(t x)

(19)

where i 1 2 m and κi(t x) is the controller which willbe designed later

Similarly there are

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) s isin [minusτ 0] x isin Ω(20)

and Φ0 (Φ01Φ02 Φ0m)T which have the sameproperty to φ0

Since the right-hand sides of equations are discon-tinuous the solutions of the equations in the conven-tional sense have been shown to be invalid In this caseFilippov solutions [43] are introduced to deal with thediscontinuous equations in functional differential in-clusions framework +e solutionrsquos global existence forfractional differential inclusions with time delay is givenby [44]

Assumption 1 For each i isin N fi(middot) R⟶ R is piecewisecontinuous function Moreover fi(middot) has at most limited thenumber of discontinuous points on any compact interval ofR +e abovementioned properties are the same to gi(middot)Based on the definition of Filippov solution (15) can berewritten as

zui(t x)

ztαisin 1113944

n

q1diqΔui(t x) minus ciui(t x)

+ 1113944

m

j1K aij1113960 1113961(t)co fj uj(t x)1113872 11138731113960 1113961

+ 1113944

m

j1K bij1113960 1113961(t)co gj uj(t minus τ x)1113872 11138731113960 1113961 + Ii

(21)

where i 1 2 m ui(t x) are absolutely continuous onany compact subinterval of [t0infin) timesΩ

K aij1113960 1113961(t)

1113954aij uj(t middot)gt Taj

11138681113868111386811138681113868

11138681113868111386811138681113868

co 1113954aij aij1113966 1113967 uj(t middot) Taj

11138681113868111386811138681113868

11138681113868111386811138681113868

aij uj(t middot)lt Taj

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

K bij1113960 1113961(t)

1113954bij uj(t middot)gt Tbj

11138681113868111386811138681113868

11138681113868111386811138681113868

co 1113954bijbij1113966 1113967 uj(t middot) Tb

j

11138681113868111386811138681113868

11138681113868111386811138681113868

bij uj(t middot)lt Tbj

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)

Or equivalently there are measurable functions1113957aij(middot) isin K[aij](middot) 1113957bij(middot) isin K[bij](middot) 1113957f1j(middot) isinco[fj(uj(middot))]and 1113957g1j(middot) isinco[gj(uj(middot))] satisfyzαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x) + 1113944

m

j11113957aij(t)1113957f1j(t x)

+ 1113944m

j1

1113957bij(t)1113957g1j(t minus τ x) + Ii

(23)

for ae t isin [t0infin) x isin Ω i 1 2 mFor writing convenience we denote max |1113954aij| |aij|1113966 1113967 as

amaxij and max |1113954bij| |bij|1113966 1113967 as bmax

ij +e IVP of the driven system (15) can be written aszαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x) + 1113944

m

j1lang1113957aij

1113957f1jrang(t x)

+ 1113944m

j1lang1113957bij 1113957g1jrang(t minus τ x) + Ii

ui(t x) 0 t isin [minusτinfin) x isin zΩ

ui(s x) φ0i(s x) t isin [minusτ 0] x isin Ω

1113957f1j(s x) cj(s x) ae t isin [minusτ 0] x isin Ω

1113957g1j(s x) ηj(s x) ae t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

+e IVP of the response system (19) under the controllercould be given by

zαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus civi(t x) + 1113944

m

j1lang1113957aij

1113957f2jrang(t x)

+ 1113944m

j1lang1113957bij 1113957g2jrang(t minus τ x) + Ii + κi(t x)

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) t isin [minusτ 0] x isin Ω

1113957f2j(s x) ξj(s x) ae t isin [minusτ 0] x isin Ω

1113957g2j(s x) ωj(s x) ae t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)

where i j 1 2 m lang1113957aij1113957f1jrang(t x) 1113957aij(t)1113957f1j(t x)

lang1113957bij 1113957g1jrang(t minus τ x) 1113957bij(t)1113957g1j(t minus τ x) lang1113957aij1113957f2jrang(t x)

1113957aij(t)1113957f2j(t x) and lang1113957bij 1113957g2jrang(t minus τ x) 1113957bij(t)1113957g2j(t minus τ x)1113957f2j(middot) isinco[fj(vj(middot))] and 1113957g2j(middot) isin co[gj(vj(middot))]

4 Complexity

Now let ei(t x) ui(t x) minus vi(t x) be the synchroni-zation error between (15) and (19) +e IVP about the errorsystem is of the form

zαei(t x)

ztα 1113944

n

q1diqΔei(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang(t x)

+ 1113944m

j1langbij gjrang(t minus τ x) + κi(t x)

ei(t x) 0 t isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

where i j 1 2 m the symbol lang1113957aij fjrang(t x) is writtenas 1113957aij(t)[1113957f2j(t x) minus 1113957f1j(t x)] and langbij gjrang(t minus τ x) whichstands for 1113957bij(t)[1113957g2j(t minus τ x) minus 1113957g1j(t minus τ x)]

Denote ei(t x) [1113938Ω1113936mi1 e2i (t x)dx]12 as the norm

of vector e(t x) (e1(t x) e2(t x) em(t x))T isin Rm

Definition 3 +e drive system (15) and response system (19)is said to be globally synchronized if limt⟶infinei(t x) 0

Remark 1 Owing to the zero Dirichlet boundary conditionsof (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 zeroequilibrium of system (26)

3 Main Results

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

31 Global Synchronization for Systems with KnownParameters First of all we assume that the parameters in(15) are known +en the following adaptive controllers areproposed to implement global synchronization

Assumption 2 For i 1 2 m there are positive con-stants pi and qi such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

supαiisinco gi(μ)[ ]βiisinco gi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868le ri μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + si forallμ ϑ isin R

(27)

where for any θ isin R

co fi(θ)1113858 1113859 min fminusj (θ) f

+j (θ)1113966 1113967 max f

minusj (θ) f

+j (θ)1113966 11139671113960 1113961

co gi(θ)1113858 1113859 min gminusj (θ) g

+j (θ)1113966 1113967 max g

minusj (θ) g

+j (θ)1113966 11139671113960 1113961

(28)

For the convenience of description the following no-tations are introduced

σi 2 minusci + 1113944n

q1minus

diq

l2q

⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ + 1113944m

j1amaxij pj + a

maxji pi1113872 1113873 + 1113944

m

j1bijrj

σ lowasti 2 1113944m

j1amaxij qj + 1113944

m

j1bmaxij sj

⎛⎝ ⎞⎠

σlowastlowasti 1113944m

j1bmaxij ri

σi minus2ci + 1113944m

j1amaxij pj + a

maxji pi + bijrj1113872 1113873

1113957σi minus2 ci + 1113944n

q1

diq

l2q

⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ + 1113944m

j1amaxij pj + a

maxji pi1113872 1113873

σ lowasti 21113944m

j1amaxij qj

(29)

Theorem 1 Under Assumption 1 and Assumption 2 theglobal synchronization of (24) and (25) can be achieved basedon the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x) i 1 2 m

⎧⎪⎨

⎪⎩

(30)

where ℓi(0 x)ge 0 ϖi gt 0 and ℓi and ωi are constants whichsatisfy the following inequalities

ℓi gt12

σi + σlowastlowasti cscαπ2

1113874 11138751113874 1113875gt 0

ωi gt12σ lowasti

(31)

Proof Employ the following auxiliary function

V(t e(t middot)) 1113946Ω

12

1113944

m

i1e2i (t x)dx (32)

By the means of Lemma 1 the inequality can beestablished

dα

dtα1113946Ω

e2i (t x)dx1113874 1113875le 21113946

Ωei(t x)

zαei(t x)

ztαdx (33)

According to Lemma 4 taking the derivative of (32)yields

Complexity 5

DαV(t e(t middot))le 1113944

m

i11113946Ω

ei(t x) 1113944n

q1diqΔei(t x)⎡⎢⎢⎣ ⎤⎥⎥⎦ minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113944

m

j1langbij gjrang(t minus τ x)

+ κi(t x)dx

(34)

Using Greenrsquos formula and Lemma 6 it can be obtainedthat

1113944

n

q11113946Ω

ei(t x)diqΔei(t x)dxle minus 1113944n

q11113946Ω

diq

lqe2i (t x)dx

(35)

Recalling Assumption 1 and Assumption 2 the in-equalities hold

1113944

m

j11113946Ω

ei(t x)lang1113957aij fjrang(t x)dxle12

1113944

m

j1amaxij pj1113946

Ωe2i (t x)

+ e2j(t x)dx + 1113944

m

j1amaxij qj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

1113944

m

j11113946Ω

ei(t x)langbij gjrang(t minus τ x)dxle12

1113944

m

j1bmaxij rj1113946

Ωe2i (t x)

+ e2j(t minus τ x)dx + 1113944

m

j1bmaxij sj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

(36)

Combining controller (30) designed in +eorem 1 itfollows that

DαV(t e(t middot))le

12

1113944

m

i1σi1113946Ω

e2i (t x)dx +

12

1113944

m

i1σlowastlowasti

1113946Ω

e2i (t minus τ x)dx +

12

1113944

m

i1σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

minus 1113944m

i11113946Ω

li(t x) + ℓi( 1113857e2i (t x)dx

minus 1113944m

i1ωi1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

(37)

From Definition 2 and Property 2 one obtains

ℓi(t x) ℓi(0 x) +1Γ(α)

1113946t

0(t minus τ)

αϖie2i (t x)dx (38)

Substituting (38) into (37) and noting that ℓi and ωi meetinequality (31) it can be deduced that

DαV(t e(t middot))le minusλ1V(t e(t middot)) + ρ1V(t minus τ e(t middot))

(39)

where λ1 1113936mi1(2ℓi minus σi) and ρ1 1113936

mi1 σlowastlowasti

According to Lemma 2 and Lemma 3 it can be assertthat the global synchronization between (24) and (25) can berealized

+e proof of +eorem 1 completes

Remark 2 When α 1 the error system (26) be changedinto the integer form+e conclusion is the same as the resultin +eorem 31 in article [45] If the reaction-diffusion inspace Ω disappears then the states of the systemrsquos variableare only relevant to time but independent of space +us theerror system (26) can be converted to

Dαei(t) minusciei(t) + 1113944m

j1lang1113957aij fjrang(t) + 1113944

m

j1langbij gjrang(t minus τ) + κi(t)

ei(s) φ0i(s) s isin [minusτ 0] i 1 2 m

⎧⎪⎪⎨

⎪⎪⎩

(40)

For convenience the inequalities are given by

ℓi gt12

σi + σlowastlowasti cscαπ2

1113874 11138751113874 1113875gt 0

ωi gt12σ lowasti

(41)

Corollary 1 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t) i 1 2 m

⎧⎨

⎩ (42)

where ℓi(0)gt 0 ϖi gt 0 and ℓi and ωi are constants whichsatisfy inequality (41)

Remark 3 In fact synchronization of FMNNs with timedelay has been investigated in [20] +e results about syn-chronization of the systems with single delay in +eorem 3in [20] are the same as Corollary 1 obtained in this paperSuppose that there is no time delay τ in the course ofcommunication between the different neurons then errorsystem (26) can be reduced to

zαei(t x)

ztα 1113944

n

q1diqΔei(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + κi(t x)

ei(t x) 0 t isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

6 Complexity

For convenience the inequalities are given by

ℓi gt12

1113957σi gt 0

ωi gt12σ lowasti

(44)

Corollary 2 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be realizedbased on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

i 1 2 m

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(45)

where ℓi(0 x)ge 0ϖi gt 0 and ℓi and ωi satisfy inequality (44)

Remark 4 Comparing the coefficients of the adaptivecontrollers (31) and (44) it can be concluded that thecontrollers for global synchronization of the system withtime delay are more consumptive

32 Global Synchronization for Systems with UnknownParameters In real neural networks the parameters of thesystem may be unknown so it is difficult to achieve syn-chronization by the conventional adaptive control schemesIn the following text adaptive controllers with the dynamiccompensators are introduced

Here suppose that self connection weight ci and input Ii

in the drive system (15) are unknown +en the IVP of thedrive system (15) can be written as

zαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x)

+ 1113944m

j1lang1113957aij

1113957f1jrang(t x) + 1113944m

j1lang1113957bij 1113957g1jrang(t minus τ x) + Ii

ui(t x) 0 t isin [minusτinfin) x isin zΩ

ui(s x) φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

+e IVP of the response system (19) can be shown aszαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus 1113954ci(t x)vi(t x) + 1113944

m

j1lang1113957aij

1113957f2jrang

(t x) + 1113944m

j1lang1113957bij 1113957g2jrang(t minus τ x) + 1113954Ii(t x) + κi(t x)

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where i j 1 2 m and 1113954ci(t x) and 1113954Ii(t x) are treated asthe compensation of ci and Ii

+en for i j 1 2 m the IVP of the synchroni-zation error can be described as

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + 1113944m

j1langbij gjrang(t minus τ x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(48)

Assumption 3 +ere are positive constants pi qi and gi

such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

gi(θ)legi

(49)

where i 1 2 m θ isin R and the notation co[fi(θ)]

represents [min fminusj (θ) f+

j (θ)1113966 1113967 max fminusj (θ) f+

j (θ)1113966 1113967]

Theorem 2 Under Assumption 1 and Assumption 3 theglobal asymptotic synchronization between (46) and (47) canrealized based on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ϱiei(t x)vi(t x)

Dα1113954Ii(t x) minus]iei(t x)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(50)

where i 1 2 m ℓi(0 x) ϖi ϱi and ]i are positiveconstants and ℓi and ωi are constants which satisfy thefollowing inequalities

ℓi gt12

1113957σi gt 0

ωi gt12

1113957σ lowasti

(51)

with 1113957σi minus2[ci + 1113936nq1(diql2q)] + 1113936

mj1(amax

ij pj + amaxji pi) and

1113957σ lowasti 21113936mj1(amax

ij qj + bmaxij gj)

Proof Employ the following auxiliary function

V(t) V1(t) + V2(t) + V3(t) (52)

where

Complexity 7

V1(t) 1113946Ω

12

1113944

m

i1e2i (t x)dx

V2(t) 1113946Ω

12ϱi

1113944

m

i11113954ci(t x) minus ci( 1113857

2dx

V3(t) 1113946Ω

12]i

1113944

m

i1

1113954Ii(t x) minus Ii1113872 11138732dx

(53)

Similarly to the proof of +eorem 1 one obtains

DαV1(t)le 1113944

m

i1

12

1113957σi1113946Ω

e2i (t x)dx

+ 1113944m

i1

12

1113957σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

+ 1113944m

i11113946Ω

1113954Ii(t x) minus Ii1113872 1113873ei(t x)dx

minus 1113944m

i11113946Ω

1113954ci(t x) minus ci( 1113857vi(t x)ei(t x)dx

+ 1113944m

i11113946Ω

ei(t x)κi(t x)dx

DαV3(t)le minus 1113944

m

i11113946Ω

ei(t x) 1113954Ii(t x) minus Ii1113872 1113873dx

DαV2(t)le 1113944

m

i11113946Ω

ei(t x) 1113954ci(t x) minus ci( 1113857vi(t x)dx

(54)

Since ℓi and ωi fulfill condition (51) it follows

Dαt V(t)le minus λ2V1(t) (55)

where λ2 1113936mi1(2ℓi minus 1113957σi)

By Lemma 5 it can be concluded that

V(t) minus V(0)le minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (56)

According to the definition of V1 thus

V1(t)leV(t)leV(0) minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (57)

By Lemma 7 it reveals that

V1(t)leV(0)exp minusλΓ(α)

1113946t

0(t minus τ)

αminus 1dτ1113888 1113889

V(0)exp minusλΓ(α + 1)

tα

1113888 1113889

(58)

+erefore

limt⟶infin

V1(t) limt⟶infin

1113946Ω

12

1113944

m

i1e2i (t x)dx 0 (59)

From (59) it is easy to see that the global synchroni-zation between (46) and (47) can be realized

+e proof of +eorem 2 completes

Remark 5 +e result obtained in +eorem 2 is morepractical for considering the situation where the parametersof the drive system (24) are unknown +rough the proof of+eorem 2 it can be asserted that the dynamical compen-sator is necessary for the unknown system If the reaction-diffusion in space Ω disappears the error system (48) can beconverted to

Dαei(t) minus 1113954ci(t) minus ci( 1113857vi(t) minus ciei(t) + 1113944m

j1lang1113957aij fjrang(t)

+ 1113944m

j1langbij gjrang(t minus τ) + 1113954Ii(t) minus Ii1113872 1113873 + κi(t)

ei(s) φ0i(s) s isin [minusτ 0]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(60)

Corollary 3 Under Assumption 1 and Assumption 3 theglobal synchronization between (46) and (47) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t)

Dα1113954ci(t) ]iei(t)

Dα1113954Ii(t) minusϱiei(t)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where ℓi(0)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)1113957σ lowasti andσi minus2ci + (1113936

mj1 amax

ij pj + 1113936mj1 amax

ji pi)

Suppose that there is no time delay τ in the course ofcommunication then the error system (48) can be changedinto

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(62)

Corollary 4 Under Assumption 1 and Assumption 3 (46)and (47) can achieve global synchronization based on thefollowing adaptive controllers

8 Complexity

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ]iei(t x)

Dα1113954Ii(t x) minusϱiei(t x)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(63)

where ℓi(0 x)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)σlowasti and

σlowasti 21113936

mj1 amax

ij qj

4 Numerical Simulations

Several numerical simulations are presented to illustrate themain results+e first example is used to validate+eorem 1+e second one is used to text +eorem 2 Now let usconsider the 2D (q 1) reaction-diffusion FMNNs withtime delay and zero Direclet boundary values Here m 2n 1 and Ω [minus2 2]

Example 1 Consider error system (26) with suchparameters

A a11 minus002

013 minus0011113888 1113889

B 035 b12

036 0371113888 1113889

C 035 0

0 0231113888 1113889

I 060 0

0 0401113888 1113889

(64)

where the memristive coefficients are defined by

a11

019 δ lt 1

026 δ 1

055 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

minus013 δ lt 1

minus017 δ 1

minus024 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(65)

Suppose that time delay τ 11 and discontinuous activa-tion functions are f(δ) g(δ) 098δ + 005sgn(δ) Letinitial values of (24) and (25) be

u1(x t) 8 sin minus2 + x + t2( 1113857

u2(x t) 6 sin(minus2 + x + t)1113896

v1(x t) minus5 cos(minus2 + x + t)

v2(x t) minus6 cos(minus2 + x + t)1113896

(66)

where t isin [minus11 0] and x isin (minus2 2) Given the fractionalorder α 098 and d11 d21 085 Under these parame-ters the dynamical behaviors of the synchronization errors

between (24) and (25) are shown in Figure 1 From thefigure it can be observed that the synchronization errorsgradually increase as time goes on Notice that f(δ) andg(δ) have only one discontinuous point 0 Assumption 1 isvalid According to Assumption 2 we can choose p1 p2

r1 r2 098 and q1 q2 s1 s2 005 By calculationthe parameters of controllers are selected as ℓ1 070gt 039w1 006gt 005 ℓ2 060gt 036 w2 005gt 004ℓ1(0 x) ℓ2(0 x) 001gt 0 and ϖ1 ϖ2 001gt 0 It iseasy to verify that condition (9) in +eorem 1 is satisfiedFigure 2 suggests that the synchronization errors asymp-totically converge to zero

+eorem 1 is not only dependent on order but alsorelated with reaction-diffusion coefficients Figure 3 showsthe trajectories of the error system at spatial position x 55with different orders α 06 07 08 09 when the reaction-diffusion coefficients d 18 While if the order is fixedα 07 the trajectories of the error system at spatial positionx 55 with different reaction-diffusion coefficientsd 06 14 22 30 are shown in Figure 4 From the nu-merical results we could note that the error system wouldachieve the asymptotic stability faster with the higher orderIt also holds true with increasing reaction-diffusioncoefficient

Example 2 Consider the error system (48) with the fol-lowing parameters

A a11 minus095

058 0761113888 1113889

B 022 b12

015 0031113888 1113889

C 058 0

0 0491113888 1113889

I 023 0

0 0251113888 1113889

(67)

where

a11

019 δ lt 1

080 δ 1

026 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

005 δ lt 1

011 δ 1

006 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(68)

Assume α 098 d11 d21 085 and (47) with unknownparameters ci and Ii and initial values of compensators in(47) are given by

1113954C0 1 0

0 11113888 1113889

1113954I0 2 0

0 21113888 1113889

(69)

Complexity 9

100100

Space x50

Time t50

0 0

ndash20

ndash10

0

10

20

30

Erro

r 1(t

x)

ndash10

ndash5

0

5

10

15

20

(a)

100100

5050

0 0

ndash10

0

10

20

30

40

Erro

r 2(t

x)

ndash5

0

5

10

15

20

25

30

Space xTime t

(b)

Figure 1 Synchronization errors between systems (24) and (25) with no controller

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 1(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Space xTime t

(a)

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 2(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Space xTime t

(b)

Figure 2 Synchronization errors between systems (24) and (25) under controller (30)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 1 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(a)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(b)

Figure 3 Trajectories of error system (26) at spatial position x 55 under different orders

10 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

12

zf2(t x)

ztlef(t x)

zf(t x)

zt (6)

Lemma 2 (see [38]) Consider the system as followsDαV(x(t)) minusρV(x(t)) + kV(x(t minus τ))

x(t) ϕ(t)

t isin [minusτ 0]

⎧⎪⎪⎨

⎪⎪⎩(7)

where ρ kge 0 and V(x)ge 0 4e solution of the system isasymptotically stable when klt ρ sin(απ2)

Lemma 3 (fractional comparison theorem see [38])Consider the following fractional order inequality

DαV(x(t)) le minus ρV(x(t)) + kV(x(t minus τ))

V(x(t)) h(t) t isin [minusτ 0]1113896 (8)

and fractional order system

DαV(y(t)) minusρV(y(t)) + kV(y(t minus τ))

V(y(t)) h(t) t isin [minusτ 0]1113896 (9)

where α isin (0 1) τ gt 0 x(t) y(t) isin Rn and V(x) andV(y)

are continuous and nonnegative functions in (0 +infin)h(t)ge 0 on [minusτ 0] If ρgt 0 and kgt 0 then the inequalityholds

V(x(t)) leV(y(t)) (10)

for all t isin [0 +infin)

Lemma 4 (see [37 39]) Suppose that e(x t) Ω times R+⟶ R

is integrable on Ω and is derivable respect to t DenoteV(t) 1113938Ωe(x t)dx then

DαV(t) 1113946

ΩD

αe(x t)dx (11)

Lemma 5 (see [40]) If x(t) y(t) isin C1[t0 b] withx(t)ley(t) then forallt isin [t0 b] there is R

t0Iαt x(t)le R

t0Iαt y(t)

Lemma 6 (see [41]) LetΩ be a cube |xq|lt lq q 1 2 nand let f(x) be a real-valued function belonging to C1(Ω)

which vanishes on the boundary of the Ω ie f(x)|zΩ 0then

1113946Ω

f2(x)dxle l

2q1113946Ω

zf(x)

zxq

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

dx (12)

Lemma 7 (GronwallndashBellman inequality see [42]) If z(t)

satisfies z(t)le 1113938t

0 a(τ)z(τ)dτ + b(t) where a(t) and b(t)

are the known real functions then

z(t)le b(0)exp 1113946t

0a(τ)dτ1113888 1113889 + 1113946

t

0_b(τ)exp 1113946

t

τa(r)dr1113888 1113889dτ

(13)

If b(t) is a constant

z(t)le b(0)exp 1113946t

0a(τ)dτ1113888 1113889 (14)

22 System Description Reaction-diffusion FMNNs withtime delay are described as

zαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x) + 1113944

m

j1langaijfjrang(t x)

+ 1113944

m

j1langbijgjrang(t minus τ x) + Ii

(15)

where i 1 2 m mge 2 is the quantity of units 0lt αlt 1refers to the order of the equation ui(t x) represents thestate of the ith unit at time t and in space x t isin J [0 T]

represents the time variable Δ is a Laplace operator whichmeans Δui(t x) z2ui(t x)zx2

q the smooth constantsdiq ge 0 work as the transmission diffusion operator along theith unit at qth position langaijfjrang(t x) aij(ui(t x))

fj(uj(t x)) langbijgjrang(t minus τ x) bij(ui(t x))gj(uj(t minus τ

x)) where aij(ui(t x)) and bij(ui(t x)) stand for thestrength of the jth unit on the ith unit at time t in space x

and at time t minus τ in space x respectively ci gt 0 is the self-feedback connection weight fj(middot) and gj(middot) denote thediscontinuous activation functions Ii signifies the externalinput and τ ge 0 stands for time delay In addition thememristor-based connection weights aij(ui(t

x)) and bij(ui(t x)) are given by

aij ui(t x)( 1113857

1113954aij ui(t middot)1113868111386811138681113868

1113868111386811138681113868gtTaj

unsureness ui(t middot)1113868111386811138681113868

1113868111386811138681113868 Taj

aij ui(t middot)1113868111386811138681113868

1113868111386811138681113868ltTaj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

bij ui(t x)( 1113857

1113954bij ui(t middot)1113868111386811138681113868

1113868111386811138681113868gtTbj

unsureness ui(t middot)1113868111386811138681113868

1113868111386811138681113868 Tbj

bij ui(t middot)1113868111386811138681113868

1113868111386811138681113868ltTbj

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(16)

where i j 1 2 m tge 0 switching jumps Taj Tb

j gt 0 and1113954aij aij

1113954bij and bij are all constantsConsider the following Dirichlet-type boundary condi-

tions of (15)

ui(t x) 0 t isin [minusτinfin) x isin zΩ (17)

and initial conditions

ui(s x) φ0i(s x) s isin [minusτ 0] x isin Ω (18)

where φ0 (φ01φ02 φ0m)T are real-value continuousfunctions defined on [minusτ 0] timesΩ

Take (15) as the drive system then the correspondingresponse system is defined by

Complexity 3

zαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus civi(t x) + 1113944

m

j1langaijfjrang(t x)

+ 1113944m

j1langbijgjrang(t minus τ x) + Ii + κi(t x)

(19)

where i 1 2 m and κi(t x) is the controller which willbe designed later

Similarly there are

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) s isin [minusτ 0] x isin Ω(20)

and Φ0 (Φ01Φ02 Φ0m)T which have the sameproperty to φ0

Since the right-hand sides of equations are discon-tinuous the solutions of the equations in the conven-tional sense have been shown to be invalid In this caseFilippov solutions [43] are introduced to deal with thediscontinuous equations in functional differential in-clusions framework +e solutionrsquos global existence forfractional differential inclusions with time delay is givenby [44]

Assumption 1 For each i isin N fi(middot) R⟶ R is piecewisecontinuous function Moreover fi(middot) has at most limited thenumber of discontinuous points on any compact interval ofR +e abovementioned properties are the same to gi(middot)Based on the definition of Filippov solution (15) can berewritten as

zui(t x)

ztαisin 1113944

n

q1diqΔui(t x) minus ciui(t x)

+ 1113944

m

j1K aij1113960 1113961(t)co fj uj(t x)1113872 11138731113960 1113961

+ 1113944

m

j1K bij1113960 1113961(t)co gj uj(t minus τ x)1113872 11138731113960 1113961 + Ii

(21)

where i 1 2 m ui(t x) are absolutely continuous onany compact subinterval of [t0infin) timesΩ

K aij1113960 1113961(t)

1113954aij uj(t middot)gt Taj

11138681113868111386811138681113868

11138681113868111386811138681113868

co 1113954aij aij1113966 1113967 uj(t middot) Taj

11138681113868111386811138681113868

11138681113868111386811138681113868

aij uj(t middot)lt Taj

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

K bij1113960 1113961(t)

1113954bij uj(t middot)gt Tbj

11138681113868111386811138681113868

11138681113868111386811138681113868

co 1113954bijbij1113966 1113967 uj(t middot) Tb

j

11138681113868111386811138681113868

11138681113868111386811138681113868

bij uj(t middot)lt Tbj

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)

Or equivalently there are measurable functions1113957aij(middot) isin K[aij](middot) 1113957bij(middot) isin K[bij](middot) 1113957f1j(middot) isinco[fj(uj(middot))]and 1113957g1j(middot) isinco[gj(uj(middot))] satisfyzαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x) + 1113944

m

j11113957aij(t)1113957f1j(t x)

+ 1113944m

j1

1113957bij(t)1113957g1j(t minus τ x) + Ii

(23)

for ae t isin [t0infin) x isin Ω i 1 2 mFor writing convenience we denote max |1113954aij| |aij|1113966 1113967 as

amaxij and max |1113954bij| |bij|1113966 1113967 as bmax

ij +e IVP of the driven system (15) can be written aszαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x) + 1113944

m

j1lang1113957aij

1113957f1jrang(t x)

+ 1113944m

j1lang1113957bij 1113957g1jrang(t minus τ x) + Ii

ui(t x) 0 t isin [minusτinfin) x isin zΩ

ui(s x) φ0i(s x) t isin [minusτ 0] x isin Ω

1113957f1j(s x) cj(s x) ae t isin [minusτ 0] x isin Ω

1113957g1j(s x) ηj(s x) ae t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

+e IVP of the response system (19) under the controllercould be given by

zαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus civi(t x) + 1113944

m

j1lang1113957aij

1113957f2jrang(t x)

+ 1113944m

j1lang1113957bij 1113957g2jrang(t minus τ x) + Ii + κi(t x)

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) t isin [minusτ 0] x isin Ω

1113957f2j(s x) ξj(s x) ae t isin [minusτ 0] x isin Ω

1113957g2j(s x) ωj(s x) ae t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)

where i j 1 2 m lang1113957aij1113957f1jrang(t x) 1113957aij(t)1113957f1j(t x)

lang1113957bij 1113957g1jrang(t minus τ x) 1113957bij(t)1113957g1j(t minus τ x) lang1113957aij1113957f2jrang(t x)

1113957aij(t)1113957f2j(t x) and lang1113957bij 1113957g2jrang(t minus τ x) 1113957bij(t)1113957g2j(t minus τ x)1113957f2j(middot) isinco[fj(vj(middot))] and 1113957g2j(middot) isin co[gj(vj(middot))]

4 Complexity

Now let ei(t x) ui(t x) minus vi(t x) be the synchroni-zation error between (15) and (19) +e IVP about the errorsystem is of the form

zαei(t x)

ztα 1113944

n

q1diqΔei(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang(t x)

+ 1113944m

j1langbij gjrang(t minus τ x) + κi(t x)

ei(t x) 0 t isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

where i j 1 2 m the symbol lang1113957aij fjrang(t x) is writtenas 1113957aij(t)[1113957f2j(t x) minus 1113957f1j(t x)] and langbij gjrang(t minus τ x) whichstands for 1113957bij(t)[1113957g2j(t minus τ x) minus 1113957g1j(t minus τ x)]

Denote ei(t x) [1113938Ω1113936mi1 e2i (t x)dx]12 as the norm

of vector e(t x) (e1(t x) e2(t x) em(t x))T isin Rm

Definition 3 +e drive system (15) and response system (19)is said to be globally synchronized if limt⟶infinei(t x) 0

Remark 1 Owing to the zero Dirichlet boundary conditionsof (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 zeroequilibrium of system (26)

3 Main Results

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

31 Global Synchronization for Systems with KnownParameters First of all we assume that the parameters in(15) are known +en the following adaptive controllers areproposed to implement global synchronization

Assumption 2 For i 1 2 m there are positive con-stants pi and qi such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

supαiisinco gi(μ)[ ]βiisinco gi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868le ri μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + si forallμ ϑ isin R

(27)

where for any θ isin R

co fi(θ)1113858 1113859 min fminusj (θ) f

+j (θ)1113966 1113967 max f

minusj (θ) f

+j (θ)1113966 11139671113960 1113961

co gi(θ)1113858 1113859 min gminusj (θ) g

+j (θ)1113966 1113967 max g

minusj (θ) g

+j (θ)1113966 11139671113960 1113961

(28)

For the convenience of description the following no-tations are introduced

σi 2 minusci + 1113944n

q1minus

diq

l2q

⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ + 1113944m

j1amaxij pj + a

maxji pi1113872 1113873 + 1113944

m

j1bijrj

σ lowasti 2 1113944m

j1amaxij qj + 1113944

m

j1bmaxij sj

⎛⎝ ⎞⎠

σlowastlowasti 1113944m

j1bmaxij ri

σi minus2ci + 1113944m

j1amaxij pj + a

maxji pi + bijrj1113872 1113873

1113957σi minus2 ci + 1113944n

q1

diq

l2q

⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ + 1113944m

j1amaxij pj + a

maxji pi1113872 1113873

σ lowasti 21113944m

j1amaxij qj

(29)

Theorem 1 Under Assumption 1 and Assumption 2 theglobal synchronization of (24) and (25) can be achieved basedon the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x) i 1 2 m

⎧⎪⎨

⎪⎩

(30)

where ℓi(0 x)ge 0 ϖi gt 0 and ℓi and ωi are constants whichsatisfy the following inequalities

ℓi gt12

σi + σlowastlowasti cscαπ2

1113874 11138751113874 1113875gt 0

ωi gt12σ lowasti

(31)

Proof Employ the following auxiliary function

V(t e(t middot)) 1113946Ω

12

1113944

m

i1e2i (t x)dx (32)

By the means of Lemma 1 the inequality can beestablished

dα

dtα1113946Ω

e2i (t x)dx1113874 1113875le 21113946

Ωei(t x)

zαei(t x)

ztαdx (33)

According to Lemma 4 taking the derivative of (32)yields

Complexity 5

DαV(t e(t middot))le 1113944

m

i11113946Ω

ei(t x) 1113944n

q1diqΔei(t x)⎡⎢⎢⎣ ⎤⎥⎥⎦ minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113944

m

j1langbij gjrang(t minus τ x)

+ κi(t x)dx

(34)

Using Greenrsquos formula and Lemma 6 it can be obtainedthat

1113944

n

q11113946Ω

ei(t x)diqΔei(t x)dxle minus 1113944n

q11113946Ω

diq

lqe2i (t x)dx

(35)

Recalling Assumption 1 and Assumption 2 the in-equalities hold

1113944

m

j11113946Ω

ei(t x)lang1113957aij fjrang(t x)dxle12

1113944

m

j1amaxij pj1113946

Ωe2i (t x)

+ e2j(t x)dx + 1113944

m

j1amaxij qj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

1113944

m

j11113946Ω

ei(t x)langbij gjrang(t minus τ x)dxle12

1113944

m

j1bmaxij rj1113946

Ωe2i (t x)

+ e2j(t minus τ x)dx + 1113944

m

j1bmaxij sj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

(36)

Combining controller (30) designed in +eorem 1 itfollows that

DαV(t e(t middot))le

12

1113944

m

i1σi1113946Ω

e2i (t x)dx +

12

1113944

m

i1σlowastlowasti

1113946Ω

e2i (t minus τ x)dx +

12

1113944

m

i1σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

minus 1113944m

i11113946Ω

li(t x) + ℓi( 1113857e2i (t x)dx

minus 1113944m

i1ωi1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

(37)

From Definition 2 and Property 2 one obtains

ℓi(t x) ℓi(0 x) +1Γ(α)

1113946t

0(t minus τ)

αϖie2i (t x)dx (38)

Substituting (38) into (37) and noting that ℓi and ωi meetinequality (31) it can be deduced that

DαV(t e(t middot))le minusλ1V(t e(t middot)) + ρ1V(t minus τ e(t middot))

(39)

where λ1 1113936mi1(2ℓi minus σi) and ρ1 1113936

mi1 σlowastlowasti

According to Lemma 2 and Lemma 3 it can be assertthat the global synchronization between (24) and (25) can berealized

+e proof of +eorem 1 completes

Remark 2 When α 1 the error system (26) be changedinto the integer form+e conclusion is the same as the resultin +eorem 31 in article [45] If the reaction-diffusion inspace Ω disappears then the states of the systemrsquos variableare only relevant to time but independent of space +us theerror system (26) can be converted to

Dαei(t) minusciei(t) + 1113944m

j1lang1113957aij fjrang(t) + 1113944

m

j1langbij gjrang(t minus τ) + κi(t)

ei(s) φ0i(s) s isin [minusτ 0] i 1 2 m

⎧⎪⎪⎨

⎪⎪⎩

(40)

For convenience the inequalities are given by

ℓi gt12

σi + σlowastlowasti cscαπ2

1113874 11138751113874 1113875gt 0

ωi gt12σ lowasti

(41)

Corollary 1 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t) i 1 2 m

⎧⎨

⎩ (42)

where ℓi(0)gt 0 ϖi gt 0 and ℓi and ωi are constants whichsatisfy inequality (41)

Remark 3 In fact synchronization of FMNNs with timedelay has been investigated in [20] +e results about syn-chronization of the systems with single delay in +eorem 3in [20] are the same as Corollary 1 obtained in this paperSuppose that there is no time delay τ in the course ofcommunication between the different neurons then errorsystem (26) can be reduced to

zαei(t x)

ztα 1113944

n

q1diqΔei(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + κi(t x)

ei(t x) 0 t isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

6 Complexity

For convenience the inequalities are given by

ℓi gt12

1113957σi gt 0

ωi gt12σ lowasti

(44)

Corollary 2 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be realizedbased on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

i 1 2 m

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(45)

where ℓi(0 x)ge 0ϖi gt 0 and ℓi and ωi satisfy inequality (44)

Remark 4 Comparing the coefficients of the adaptivecontrollers (31) and (44) it can be concluded that thecontrollers for global synchronization of the system withtime delay are more consumptive

32 Global Synchronization for Systems with UnknownParameters In real neural networks the parameters of thesystem may be unknown so it is difficult to achieve syn-chronization by the conventional adaptive control schemesIn the following text adaptive controllers with the dynamiccompensators are introduced

Here suppose that self connection weight ci and input Ii

in the drive system (15) are unknown +en the IVP of thedrive system (15) can be written as

zαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x)

+ 1113944m

j1lang1113957aij

1113957f1jrang(t x) + 1113944m

j1lang1113957bij 1113957g1jrang(t minus τ x) + Ii

ui(t x) 0 t isin [minusτinfin) x isin zΩ

ui(s x) φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

+e IVP of the response system (19) can be shown aszαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus 1113954ci(t x)vi(t x) + 1113944

m

j1lang1113957aij

1113957f2jrang

(t x) + 1113944m

j1lang1113957bij 1113957g2jrang(t minus τ x) + 1113954Ii(t x) + κi(t x)

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where i j 1 2 m and 1113954ci(t x) and 1113954Ii(t x) are treated asthe compensation of ci and Ii

+en for i j 1 2 m the IVP of the synchroni-zation error can be described as

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + 1113944m

j1langbij gjrang(t minus τ x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(48)

Assumption 3 +ere are positive constants pi qi and gi

such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

gi(θ)legi

(49)

where i 1 2 m θ isin R and the notation co[fi(θ)]

represents [min fminusj (θ) f+

j (θ)1113966 1113967 max fminusj (θ) f+

j (θ)1113966 1113967]

Theorem 2 Under Assumption 1 and Assumption 3 theglobal asymptotic synchronization between (46) and (47) canrealized based on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ϱiei(t x)vi(t x)

Dα1113954Ii(t x) minus]iei(t x)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(50)

where i 1 2 m ℓi(0 x) ϖi ϱi and ]i are positiveconstants and ℓi and ωi are constants which satisfy thefollowing inequalities

ℓi gt12

1113957σi gt 0

ωi gt12

1113957σ lowasti

(51)

with 1113957σi minus2[ci + 1113936nq1(diql2q)] + 1113936

mj1(amax

ij pj + amaxji pi) and

1113957σ lowasti 21113936mj1(amax

ij qj + bmaxij gj)

Proof Employ the following auxiliary function

V(t) V1(t) + V2(t) + V3(t) (52)

where

Complexity 7

V1(t) 1113946Ω

12

1113944

m

i1e2i (t x)dx

V2(t) 1113946Ω

12ϱi

1113944

m

i11113954ci(t x) minus ci( 1113857

2dx

V3(t) 1113946Ω

12]i

1113944

m

i1

1113954Ii(t x) minus Ii1113872 11138732dx

(53)

Similarly to the proof of +eorem 1 one obtains

DαV1(t)le 1113944

m

i1

12

1113957σi1113946Ω

e2i (t x)dx

+ 1113944m

i1

12

1113957σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

+ 1113944m

i11113946Ω

1113954Ii(t x) minus Ii1113872 1113873ei(t x)dx

minus 1113944m

i11113946Ω

1113954ci(t x) minus ci( 1113857vi(t x)ei(t x)dx

+ 1113944m

i11113946Ω

ei(t x)κi(t x)dx

DαV3(t)le minus 1113944

m

i11113946Ω

ei(t x) 1113954Ii(t x) minus Ii1113872 1113873dx

DαV2(t)le 1113944

m

i11113946Ω

ei(t x) 1113954ci(t x) minus ci( 1113857vi(t x)dx

(54)

Since ℓi and ωi fulfill condition (51) it follows

Dαt V(t)le minus λ2V1(t) (55)

where λ2 1113936mi1(2ℓi minus 1113957σi)

By Lemma 5 it can be concluded that

V(t) minus V(0)le minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (56)

According to the definition of V1 thus

V1(t)leV(t)leV(0) minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (57)

By Lemma 7 it reveals that

V1(t)leV(0)exp minusλΓ(α)

1113946t

0(t minus τ)

αminus 1dτ1113888 1113889

V(0)exp minusλΓ(α + 1)

tα

1113888 1113889

(58)

+erefore

limt⟶infin

V1(t) limt⟶infin

1113946Ω

12

1113944

m

i1e2i (t x)dx 0 (59)

From (59) it is easy to see that the global synchroni-zation between (46) and (47) can be realized

+e proof of +eorem 2 completes

Remark 5 +e result obtained in +eorem 2 is morepractical for considering the situation where the parametersof the drive system (24) are unknown +rough the proof of+eorem 2 it can be asserted that the dynamical compen-sator is necessary for the unknown system If the reaction-diffusion in space Ω disappears the error system (48) can beconverted to

Dαei(t) minus 1113954ci(t) minus ci( 1113857vi(t) minus ciei(t) + 1113944m

j1lang1113957aij fjrang(t)

+ 1113944m

j1langbij gjrang(t minus τ) + 1113954Ii(t) minus Ii1113872 1113873 + κi(t)

ei(s) φ0i(s) s isin [minusτ 0]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(60)

Corollary 3 Under Assumption 1 and Assumption 3 theglobal synchronization between (46) and (47) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t)

Dα1113954ci(t) ]iei(t)

Dα1113954Ii(t) minusϱiei(t)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where ℓi(0)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)1113957σ lowasti andσi minus2ci + (1113936

mj1 amax

ij pj + 1113936mj1 amax

ji pi)

Suppose that there is no time delay τ in the course ofcommunication then the error system (48) can be changedinto

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(62)

Corollary 4 Under Assumption 1 and Assumption 3 (46)and (47) can achieve global synchronization based on thefollowing adaptive controllers

8 Complexity

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ]iei(t x)

Dα1113954Ii(t x) minusϱiei(t x)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(63)

where ℓi(0 x)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)σlowasti and

σlowasti 21113936

mj1 amax

ij qj

4 Numerical Simulations

Several numerical simulations are presented to illustrate themain results+e first example is used to validate+eorem 1+e second one is used to text +eorem 2 Now let usconsider the 2D (q 1) reaction-diffusion FMNNs withtime delay and zero Direclet boundary values Here m 2n 1 and Ω [minus2 2]

Example 1 Consider error system (26) with suchparameters

A a11 minus002

013 minus0011113888 1113889

B 035 b12

036 0371113888 1113889

C 035 0

0 0231113888 1113889

I 060 0

0 0401113888 1113889

(64)

where the memristive coefficients are defined by

a11

019 δ lt 1

026 δ 1

055 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

minus013 δ lt 1

minus017 δ 1

minus024 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(65)

Suppose that time delay τ 11 and discontinuous activa-tion functions are f(δ) g(δ) 098δ + 005sgn(δ) Letinitial values of (24) and (25) be

u1(x t) 8 sin minus2 + x + t2( 1113857

u2(x t) 6 sin(minus2 + x + t)1113896

v1(x t) minus5 cos(minus2 + x + t)

v2(x t) minus6 cos(minus2 + x + t)1113896

(66)

where t isin [minus11 0] and x isin (minus2 2) Given the fractionalorder α 098 and d11 d21 085 Under these parame-ters the dynamical behaviors of the synchronization errors

between (24) and (25) are shown in Figure 1 From thefigure it can be observed that the synchronization errorsgradually increase as time goes on Notice that f(δ) andg(δ) have only one discontinuous point 0 Assumption 1 isvalid According to Assumption 2 we can choose p1 p2

r1 r2 098 and q1 q2 s1 s2 005 By calculationthe parameters of controllers are selected as ℓ1 070gt 039w1 006gt 005 ℓ2 060gt 036 w2 005gt 004ℓ1(0 x) ℓ2(0 x) 001gt 0 and ϖ1 ϖ2 001gt 0 It iseasy to verify that condition (9) in +eorem 1 is satisfiedFigure 2 suggests that the synchronization errors asymp-totically converge to zero

+eorem 1 is not only dependent on order but alsorelated with reaction-diffusion coefficients Figure 3 showsthe trajectories of the error system at spatial position x 55with different orders α 06 07 08 09 when the reaction-diffusion coefficients d 18 While if the order is fixedα 07 the trajectories of the error system at spatial positionx 55 with different reaction-diffusion coefficientsd 06 14 22 30 are shown in Figure 4 From the nu-merical results we could note that the error system wouldachieve the asymptotic stability faster with the higher orderIt also holds true with increasing reaction-diffusioncoefficient

Example 2 Consider the error system (48) with the fol-lowing parameters

A a11 minus095

058 0761113888 1113889

B 022 b12

015 0031113888 1113889

C 058 0

0 0491113888 1113889

I 023 0

0 0251113888 1113889

(67)

where

a11

019 δ lt 1

080 δ 1

026 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

005 δ lt 1

011 δ 1

006 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(68)

Assume α 098 d11 d21 085 and (47) with unknownparameters ci and Ii and initial values of compensators in(47) are given by

1113954C0 1 0

0 11113888 1113889

1113954I0 2 0

0 21113888 1113889

(69)

Complexity 9

100100

Space x50

Time t50

0 0

ndash20

ndash10

0

10

20

30

Erro

r 1(t

x)

ndash10

ndash5

0

5

10

15

20

(a)

100100

5050

0 0

ndash10

0

10

20

30

40

Erro

r 2(t

x)

ndash5

0

5

10

15

20

25

30

Space xTime t

(b)

Figure 1 Synchronization errors between systems (24) and (25) with no controller

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 1(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Space xTime t

(a)

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 2(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Space xTime t

(b)

Figure 2 Synchronization errors between systems (24) and (25) under controller (30)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 1 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(a)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(b)

Figure 3 Trajectories of error system (26) at spatial position x 55 under different orders

10 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

zαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus civi(t x) + 1113944

m

j1langaijfjrang(t x)

+ 1113944m

j1langbijgjrang(t minus τ x) + Ii + κi(t x)

(19)

where i 1 2 m and κi(t x) is the controller which willbe designed later

Similarly there are

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) s isin [minusτ 0] x isin Ω(20)

and Φ0 (Φ01Φ02 Φ0m)T which have the sameproperty to φ0

Since the right-hand sides of equations are discon-tinuous the solutions of the equations in the conven-tional sense have been shown to be invalid In this caseFilippov solutions [43] are introduced to deal with thediscontinuous equations in functional differential in-clusions framework +e solutionrsquos global existence forfractional differential inclusions with time delay is givenby [44]

Assumption 1 For each i isin N fi(middot) R⟶ R is piecewisecontinuous function Moreover fi(middot) has at most limited thenumber of discontinuous points on any compact interval ofR +e abovementioned properties are the same to gi(middot)Based on the definition of Filippov solution (15) can berewritten as

zui(t x)

ztαisin 1113944

n

q1diqΔui(t x) minus ciui(t x)

+ 1113944

m

j1K aij1113960 1113961(t)co fj uj(t x)1113872 11138731113960 1113961

+ 1113944

m

j1K bij1113960 1113961(t)co gj uj(t minus τ x)1113872 11138731113960 1113961 + Ii

(21)

where i 1 2 m ui(t x) are absolutely continuous onany compact subinterval of [t0infin) timesΩ

K aij1113960 1113961(t)

1113954aij uj(t middot)gt Taj

11138681113868111386811138681113868

11138681113868111386811138681113868

co 1113954aij aij1113966 1113967 uj(t middot) Taj

11138681113868111386811138681113868

11138681113868111386811138681113868

aij uj(t middot)lt Taj

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

K bij1113960 1113961(t)

1113954bij uj(t middot)gt Tbj

11138681113868111386811138681113868

11138681113868111386811138681113868

co 1113954bijbij1113966 1113967 uj(t middot) Tb

j

11138681113868111386811138681113868

11138681113868111386811138681113868

bij uj(t middot)lt Tbj

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(22)

Or equivalently there are measurable functions1113957aij(middot) isin K[aij](middot) 1113957bij(middot) isin K[bij](middot) 1113957f1j(middot) isinco[fj(uj(middot))]and 1113957g1j(middot) isinco[gj(uj(middot))] satisfyzαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x) + 1113944

m

j11113957aij(t)1113957f1j(t x)

+ 1113944m

j1

1113957bij(t)1113957g1j(t minus τ x) + Ii

(23)

for ae t isin [t0infin) x isin Ω i 1 2 mFor writing convenience we denote max |1113954aij| |aij|1113966 1113967 as

amaxij and max |1113954bij| |bij|1113966 1113967 as bmax

ij +e IVP of the driven system (15) can be written aszαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x) + 1113944

m

j1lang1113957aij

1113957f1jrang(t x)

+ 1113944m

j1lang1113957bij 1113957g1jrang(t minus τ x) + Ii

ui(t x) 0 t isin [minusτinfin) x isin zΩ

ui(s x) φ0i(s x) t isin [minusτ 0] x isin Ω

1113957f1j(s x) cj(s x) ae t isin [minusτ 0] x isin Ω

1113957g1j(s x) ηj(s x) ae t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

+e IVP of the response system (19) under the controllercould be given by

zαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus civi(t x) + 1113944

m

j1lang1113957aij

1113957f2jrang(t x)

+ 1113944m

j1lang1113957bij 1113957g2jrang(t minus τ x) + Ii + κi(t x)

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) t isin [minusτ 0] x isin Ω

1113957f2j(s x) ξj(s x) ae t isin [minusτ 0] x isin Ω

1113957g2j(s x) ωj(s x) ae t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)

where i j 1 2 m lang1113957aij1113957f1jrang(t x) 1113957aij(t)1113957f1j(t x)

lang1113957bij 1113957g1jrang(t minus τ x) 1113957bij(t)1113957g1j(t minus τ x) lang1113957aij1113957f2jrang(t x)

1113957aij(t)1113957f2j(t x) and lang1113957bij 1113957g2jrang(t minus τ x) 1113957bij(t)1113957g2j(t minus τ x)1113957f2j(middot) isinco[fj(vj(middot))] and 1113957g2j(middot) isin co[gj(vj(middot))]

4 Complexity

Now let ei(t x) ui(t x) minus vi(t x) be the synchroni-zation error between (15) and (19) +e IVP about the errorsystem is of the form

zαei(t x)

ztα 1113944

n

q1diqΔei(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang(t x)

+ 1113944m

j1langbij gjrang(t minus τ x) + κi(t x)

ei(t x) 0 t isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

where i j 1 2 m the symbol lang1113957aij fjrang(t x) is writtenas 1113957aij(t)[1113957f2j(t x) minus 1113957f1j(t x)] and langbij gjrang(t minus τ x) whichstands for 1113957bij(t)[1113957g2j(t minus τ x) minus 1113957g1j(t minus τ x)]

Denote ei(t x) [1113938Ω1113936mi1 e2i (t x)dx]12 as the norm

of vector e(t x) (e1(t x) e2(t x) em(t x))T isin Rm

Definition 3 +e drive system (15) and response system (19)is said to be globally synchronized if limt⟶infinei(t x) 0

Remark 1 Owing to the zero Dirichlet boundary conditionsof (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 zeroequilibrium of system (26)

3 Main Results

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

31 Global Synchronization for Systems with KnownParameters First of all we assume that the parameters in(15) are known +en the following adaptive controllers areproposed to implement global synchronization

Assumption 2 For i 1 2 m there are positive con-stants pi and qi such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

supαiisinco gi(μ)[ ]βiisinco gi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868le ri μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + si forallμ ϑ isin R

(27)

where for any θ isin R

co fi(θ)1113858 1113859 min fminusj (θ) f

+j (θ)1113966 1113967 max f

minusj (θ) f

+j (θ)1113966 11139671113960 1113961

co gi(θ)1113858 1113859 min gminusj (θ) g

+j (θ)1113966 1113967 max g

minusj (θ) g

+j (θ)1113966 11139671113960 1113961

(28)

For the convenience of description the following no-tations are introduced

σi 2 minusci + 1113944n

q1minus

diq

l2q

⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ + 1113944m

j1amaxij pj + a

maxji pi1113872 1113873 + 1113944

m

j1bijrj

σ lowasti 2 1113944m

j1amaxij qj + 1113944

m

j1bmaxij sj

⎛⎝ ⎞⎠

σlowastlowasti 1113944m

j1bmaxij ri

σi minus2ci + 1113944m

j1amaxij pj + a

maxji pi + bijrj1113872 1113873

1113957σi minus2 ci + 1113944n

q1

diq

l2q

⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ + 1113944m

j1amaxij pj + a

maxji pi1113872 1113873

σ lowasti 21113944m

j1amaxij qj

(29)

Theorem 1 Under Assumption 1 and Assumption 2 theglobal synchronization of (24) and (25) can be achieved basedon the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x) i 1 2 m

⎧⎪⎨

⎪⎩

(30)

where ℓi(0 x)ge 0 ϖi gt 0 and ℓi and ωi are constants whichsatisfy the following inequalities

ℓi gt12

σi + σlowastlowasti cscαπ2

1113874 11138751113874 1113875gt 0

ωi gt12σ lowasti

(31)

Proof Employ the following auxiliary function

V(t e(t middot)) 1113946Ω

12

1113944

m

i1e2i (t x)dx (32)

By the means of Lemma 1 the inequality can beestablished

dα

dtα1113946Ω

e2i (t x)dx1113874 1113875le 21113946

Ωei(t x)

zαei(t x)

ztαdx (33)

According to Lemma 4 taking the derivative of (32)yields

Complexity 5

DαV(t e(t middot))le 1113944

m

i11113946Ω

ei(t x) 1113944n

q1diqΔei(t x)⎡⎢⎢⎣ ⎤⎥⎥⎦ minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113944

m

j1langbij gjrang(t minus τ x)

+ κi(t x)dx

(34)

Using Greenrsquos formula and Lemma 6 it can be obtainedthat

1113944

n

q11113946Ω

ei(t x)diqΔei(t x)dxle minus 1113944n

q11113946Ω

diq

lqe2i (t x)dx

(35)

Recalling Assumption 1 and Assumption 2 the in-equalities hold

1113944

m

j11113946Ω

ei(t x)lang1113957aij fjrang(t x)dxle12

1113944

m

j1amaxij pj1113946

Ωe2i (t x)

+ e2j(t x)dx + 1113944

m

j1amaxij qj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

1113944

m

j11113946Ω

ei(t x)langbij gjrang(t minus τ x)dxle12

1113944

m

j1bmaxij rj1113946

Ωe2i (t x)

+ e2j(t minus τ x)dx + 1113944

m

j1bmaxij sj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

(36)

Combining controller (30) designed in +eorem 1 itfollows that

DαV(t e(t middot))le

12

1113944

m

i1σi1113946Ω

e2i (t x)dx +

12

1113944

m

i1σlowastlowasti

1113946Ω

e2i (t minus τ x)dx +

12

1113944

m

i1σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

minus 1113944m

i11113946Ω

li(t x) + ℓi( 1113857e2i (t x)dx

minus 1113944m

i1ωi1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

(37)

From Definition 2 and Property 2 one obtains

ℓi(t x) ℓi(0 x) +1Γ(α)

1113946t

0(t minus τ)

αϖie2i (t x)dx (38)

Substituting (38) into (37) and noting that ℓi and ωi meetinequality (31) it can be deduced that

DαV(t e(t middot))le minusλ1V(t e(t middot)) + ρ1V(t minus τ e(t middot))

(39)

where λ1 1113936mi1(2ℓi minus σi) and ρ1 1113936

mi1 σlowastlowasti

According to Lemma 2 and Lemma 3 it can be assertthat the global synchronization between (24) and (25) can berealized

+e proof of +eorem 1 completes

Remark 2 When α 1 the error system (26) be changedinto the integer form+e conclusion is the same as the resultin +eorem 31 in article [45] If the reaction-diffusion inspace Ω disappears then the states of the systemrsquos variableare only relevant to time but independent of space +us theerror system (26) can be converted to

Dαei(t) minusciei(t) + 1113944m

j1lang1113957aij fjrang(t) + 1113944

m

j1langbij gjrang(t minus τ) + κi(t)

ei(s) φ0i(s) s isin [minusτ 0] i 1 2 m

⎧⎪⎪⎨

⎪⎪⎩

(40)

For convenience the inequalities are given by

ℓi gt12

σi + σlowastlowasti cscαπ2

1113874 11138751113874 1113875gt 0

ωi gt12σ lowasti

(41)

Corollary 1 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t) i 1 2 m

⎧⎨

⎩ (42)

where ℓi(0)gt 0 ϖi gt 0 and ℓi and ωi are constants whichsatisfy inequality (41)

Remark 3 In fact synchronization of FMNNs with timedelay has been investigated in [20] +e results about syn-chronization of the systems with single delay in +eorem 3in [20] are the same as Corollary 1 obtained in this paperSuppose that there is no time delay τ in the course ofcommunication between the different neurons then errorsystem (26) can be reduced to

zαei(t x)

ztα 1113944

n

q1diqΔei(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + κi(t x)

ei(t x) 0 t isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

6 Complexity

For convenience the inequalities are given by

ℓi gt12

1113957σi gt 0

ωi gt12σ lowasti

(44)

Corollary 2 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be realizedbased on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

i 1 2 m

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(45)

where ℓi(0 x)ge 0ϖi gt 0 and ℓi and ωi satisfy inequality (44)

Remark 4 Comparing the coefficients of the adaptivecontrollers (31) and (44) it can be concluded that thecontrollers for global synchronization of the system withtime delay are more consumptive

32 Global Synchronization for Systems with UnknownParameters In real neural networks the parameters of thesystem may be unknown so it is difficult to achieve syn-chronization by the conventional adaptive control schemesIn the following text adaptive controllers with the dynamiccompensators are introduced

Here suppose that self connection weight ci and input Ii

in the drive system (15) are unknown +en the IVP of thedrive system (15) can be written as

zαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x)

+ 1113944m

j1lang1113957aij

1113957f1jrang(t x) + 1113944m

j1lang1113957bij 1113957g1jrang(t minus τ x) + Ii

ui(t x) 0 t isin [minusτinfin) x isin zΩ

ui(s x) φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

+e IVP of the response system (19) can be shown aszαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus 1113954ci(t x)vi(t x) + 1113944

m

j1lang1113957aij

1113957f2jrang

(t x) + 1113944m

j1lang1113957bij 1113957g2jrang(t minus τ x) + 1113954Ii(t x) + κi(t x)

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where i j 1 2 m and 1113954ci(t x) and 1113954Ii(t x) are treated asthe compensation of ci and Ii

+en for i j 1 2 m the IVP of the synchroni-zation error can be described as

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + 1113944m

j1langbij gjrang(t minus τ x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(48)

Assumption 3 +ere are positive constants pi qi and gi

such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

gi(θ)legi

(49)

where i 1 2 m θ isin R and the notation co[fi(θ)]

represents [min fminusj (θ) f+

j (θ)1113966 1113967 max fminusj (θ) f+

j (θ)1113966 1113967]

Theorem 2 Under Assumption 1 and Assumption 3 theglobal asymptotic synchronization between (46) and (47) canrealized based on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ϱiei(t x)vi(t x)

Dα1113954Ii(t x) minus]iei(t x)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(50)

where i 1 2 m ℓi(0 x) ϖi ϱi and ]i are positiveconstants and ℓi and ωi are constants which satisfy thefollowing inequalities

ℓi gt12

1113957σi gt 0

ωi gt12

1113957σ lowasti

(51)

with 1113957σi minus2[ci + 1113936nq1(diql2q)] + 1113936

mj1(amax

ij pj + amaxji pi) and

1113957σ lowasti 21113936mj1(amax

ij qj + bmaxij gj)

Proof Employ the following auxiliary function

V(t) V1(t) + V2(t) + V3(t) (52)

where

Complexity 7

V1(t) 1113946Ω

12

1113944

m

i1e2i (t x)dx

V2(t) 1113946Ω

12ϱi

1113944

m

i11113954ci(t x) minus ci( 1113857

2dx

V3(t) 1113946Ω

12]i

1113944

m

i1

1113954Ii(t x) minus Ii1113872 11138732dx

(53)

Similarly to the proof of +eorem 1 one obtains

DαV1(t)le 1113944

m

i1

12

1113957σi1113946Ω

e2i (t x)dx

+ 1113944m

i1

12

1113957σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

+ 1113944m

i11113946Ω

1113954Ii(t x) minus Ii1113872 1113873ei(t x)dx

minus 1113944m

i11113946Ω

1113954ci(t x) minus ci( 1113857vi(t x)ei(t x)dx

+ 1113944m

i11113946Ω

ei(t x)κi(t x)dx

DαV3(t)le minus 1113944

m

i11113946Ω

ei(t x) 1113954Ii(t x) minus Ii1113872 1113873dx

DαV2(t)le 1113944

m

i11113946Ω

ei(t x) 1113954ci(t x) minus ci( 1113857vi(t x)dx

(54)

Since ℓi and ωi fulfill condition (51) it follows

Dαt V(t)le minus λ2V1(t) (55)

where λ2 1113936mi1(2ℓi minus 1113957σi)

By Lemma 5 it can be concluded that

V(t) minus V(0)le minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (56)

According to the definition of V1 thus

V1(t)leV(t)leV(0) minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (57)

By Lemma 7 it reveals that

V1(t)leV(0)exp minusλΓ(α)

1113946t

0(t minus τ)

αminus 1dτ1113888 1113889

V(0)exp minusλΓ(α + 1)

tα

1113888 1113889

(58)

+erefore

limt⟶infin

V1(t) limt⟶infin

1113946Ω

12

1113944

m

i1e2i (t x)dx 0 (59)

From (59) it is easy to see that the global synchroni-zation between (46) and (47) can be realized

+e proof of +eorem 2 completes

Remark 5 +e result obtained in +eorem 2 is morepractical for considering the situation where the parametersof the drive system (24) are unknown +rough the proof of+eorem 2 it can be asserted that the dynamical compen-sator is necessary for the unknown system If the reaction-diffusion in space Ω disappears the error system (48) can beconverted to

Dαei(t) minus 1113954ci(t) minus ci( 1113857vi(t) minus ciei(t) + 1113944m

j1lang1113957aij fjrang(t)

+ 1113944m

j1langbij gjrang(t minus τ) + 1113954Ii(t) minus Ii1113872 1113873 + κi(t)

ei(s) φ0i(s) s isin [minusτ 0]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(60)

Corollary 3 Under Assumption 1 and Assumption 3 theglobal synchronization between (46) and (47) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t)

Dα1113954ci(t) ]iei(t)

Dα1113954Ii(t) minusϱiei(t)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where ℓi(0)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)1113957σ lowasti andσi minus2ci + (1113936

mj1 amax

ij pj + 1113936mj1 amax

ji pi)

Suppose that there is no time delay τ in the course ofcommunication then the error system (48) can be changedinto

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(62)

Corollary 4 Under Assumption 1 and Assumption 3 (46)and (47) can achieve global synchronization based on thefollowing adaptive controllers

8 Complexity

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ]iei(t x)

Dα1113954Ii(t x) minusϱiei(t x)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(63)

where ℓi(0 x)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)σlowasti and

σlowasti 21113936

mj1 amax

ij qj

4 Numerical Simulations

Several numerical simulations are presented to illustrate themain results+e first example is used to validate+eorem 1+e second one is used to text +eorem 2 Now let usconsider the 2D (q 1) reaction-diffusion FMNNs withtime delay and zero Direclet boundary values Here m 2n 1 and Ω [minus2 2]

Example 1 Consider error system (26) with suchparameters

A a11 minus002

013 minus0011113888 1113889

B 035 b12

036 0371113888 1113889

C 035 0

0 0231113888 1113889

I 060 0

0 0401113888 1113889

(64)

where the memristive coefficients are defined by

a11

019 δ lt 1

026 δ 1

055 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

minus013 δ lt 1

minus017 δ 1

minus024 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(65)

Suppose that time delay τ 11 and discontinuous activa-tion functions are f(δ) g(δ) 098δ + 005sgn(δ) Letinitial values of (24) and (25) be

u1(x t) 8 sin minus2 + x + t2( 1113857

u2(x t) 6 sin(minus2 + x + t)1113896

v1(x t) minus5 cos(minus2 + x + t)

v2(x t) minus6 cos(minus2 + x + t)1113896

(66)

where t isin [minus11 0] and x isin (minus2 2) Given the fractionalorder α 098 and d11 d21 085 Under these parame-ters the dynamical behaviors of the synchronization errors

between (24) and (25) are shown in Figure 1 From thefigure it can be observed that the synchronization errorsgradually increase as time goes on Notice that f(δ) andg(δ) have only one discontinuous point 0 Assumption 1 isvalid According to Assumption 2 we can choose p1 p2

r1 r2 098 and q1 q2 s1 s2 005 By calculationthe parameters of controllers are selected as ℓ1 070gt 039w1 006gt 005 ℓ2 060gt 036 w2 005gt 004ℓ1(0 x) ℓ2(0 x) 001gt 0 and ϖ1 ϖ2 001gt 0 It iseasy to verify that condition (9) in +eorem 1 is satisfiedFigure 2 suggests that the synchronization errors asymp-totically converge to zero

+eorem 1 is not only dependent on order but alsorelated with reaction-diffusion coefficients Figure 3 showsthe trajectories of the error system at spatial position x 55with different orders α 06 07 08 09 when the reaction-diffusion coefficients d 18 While if the order is fixedα 07 the trajectories of the error system at spatial positionx 55 with different reaction-diffusion coefficientsd 06 14 22 30 are shown in Figure 4 From the nu-merical results we could note that the error system wouldachieve the asymptotic stability faster with the higher orderIt also holds true with increasing reaction-diffusioncoefficient

Example 2 Consider the error system (48) with the fol-lowing parameters

A a11 minus095

058 0761113888 1113889

B 022 b12

015 0031113888 1113889

C 058 0

0 0491113888 1113889

I 023 0

0 0251113888 1113889

(67)

where

a11

019 δ lt 1

080 δ 1

026 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

005 δ lt 1

011 δ 1

006 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(68)

Assume α 098 d11 d21 085 and (47) with unknownparameters ci and Ii and initial values of compensators in(47) are given by

1113954C0 1 0

0 11113888 1113889

1113954I0 2 0

0 21113888 1113889

(69)

Complexity 9

100100

Space x50

Time t50

0 0

ndash20

ndash10

0

10

20

30

Erro

r 1(t

x)

ndash10

ndash5

0

5

10

15

20

(a)

100100

5050

0 0

ndash10

0

10

20

30

40

Erro

r 2(t

x)

ndash5

0

5

10

15

20

25

30

Space xTime t

(b)

Figure 1 Synchronization errors between systems (24) and (25) with no controller

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 1(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Space xTime t

(a)

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 2(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Space xTime t

(b)

Figure 2 Synchronization errors between systems (24) and (25) under controller (30)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 1 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(a)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(b)

Figure 3 Trajectories of error system (26) at spatial position x 55 under different orders

10 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

Now let ei(t x) ui(t x) minus vi(t x) be the synchroni-zation error between (15) and (19) +e IVP about the errorsystem is of the form

zαei(t x)

ztα 1113944

n

q1diqΔei(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang(t x)

+ 1113944m

j1langbij gjrang(t minus τ x) + κi(t x)

ei(t x) 0 t isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

where i j 1 2 m the symbol lang1113957aij fjrang(t x) is writtenas 1113957aij(t)[1113957f2j(t x) minus 1113957f1j(t x)] and langbij gjrang(t minus τ x) whichstands for 1113957bij(t)[1113957g2j(t minus τ x) minus 1113957g1j(t minus τ x)]

Denote ei(t x) [1113938Ω1113936mi1 e2i (t x)dx]12 as the norm

of vector e(t x) (e1(t x) e2(t x) em(t x))T isin Rm

Definition 3 +e drive system (15) and response system (19)is said to be globally synchronized if limt⟶infinei(t x) 0

Remark 1 Owing to the zero Dirichlet boundary conditionsof (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 zeroequilibrium of system (26)

3 Main Results

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

31 Global Synchronization for Systems with KnownParameters First of all we assume that the parameters in(15) are known +en the following adaptive controllers areproposed to implement global synchronization

Assumption 2 For i 1 2 m there are positive con-stants pi and qi such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

supαiisinco gi(μ)[ ]βiisinco gi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868le ri μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + si forallμ ϑ isin R

(27)

where for any θ isin R

co fi(θ)1113858 1113859 min fminusj (θ) f

+j (θ)1113966 1113967 max f

minusj (θ) f

+j (θ)1113966 11139671113960 1113961

co gi(θ)1113858 1113859 min gminusj (θ) g

+j (θ)1113966 1113967 max g

minusj (θ) g

+j (θ)1113966 11139671113960 1113961

(28)

For the convenience of description the following no-tations are introduced

σi 2 minusci + 1113944n

q1minus

diq

l2q

⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ + 1113944m

j1amaxij pj + a

maxji pi1113872 1113873 + 1113944

m

j1bijrj

σ lowasti 2 1113944m

j1amaxij qj + 1113944

m

j1bmaxij sj

⎛⎝ ⎞⎠

σlowastlowasti 1113944m

j1bmaxij ri

σi minus2ci + 1113944m

j1amaxij pj + a

maxji pi + bijrj1113872 1113873

1113957σi minus2 ci + 1113944n

q1

diq

l2q

⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ + 1113944m

j1amaxij pj + a

maxji pi1113872 1113873

σ lowasti 21113944m

j1amaxij qj

(29)

Theorem 1 Under Assumption 1 and Assumption 2 theglobal synchronization of (24) and (25) can be achieved basedon the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x) i 1 2 m

⎧⎪⎨

⎪⎩

(30)

where ℓi(0 x)ge 0 ϖi gt 0 and ℓi and ωi are constants whichsatisfy the following inequalities

ℓi gt12

σi + σlowastlowasti cscαπ2

1113874 11138751113874 1113875gt 0

ωi gt12σ lowasti

(31)

Proof Employ the following auxiliary function

V(t e(t middot)) 1113946Ω

12

1113944

m

i1e2i (t x)dx (32)

By the means of Lemma 1 the inequality can beestablished

dα

dtα1113946Ω

e2i (t x)dx1113874 1113875le 21113946

Ωei(t x)

zαei(t x)

ztαdx (33)

According to Lemma 4 taking the derivative of (32)yields

Complexity 5

DαV(t e(t middot))le 1113944

m

i11113946Ω

ei(t x) 1113944n

q1diqΔei(t x)⎡⎢⎢⎣ ⎤⎥⎥⎦ minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113944

m

j1langbij gjrang(t minus τ x)

+ κi(t x)dx

(34)

Using Greenrsquos formula and Lemma 6 it can be obtainedthat

1113944

n

q11113946Ω

ei(t x)diqΔei(t x)dxle minus 1113944n

q11113946Ω

diq

lqe2i (t x)dx

(35)

Recalling Assumption 1 and Assumption 2 the in-equalities hold

1113944

m

j11113946Ω

ei(t x)lang1113957aij fjrang(t x)dxle12

1113944

m

j1amaxij pj1113946

Ωe2i (t x)

+ e2j(t x)dx + 1113944

m

j1amaxij qj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

1113944

m

j11113946Ω

ei(t x)langbij gjrang(t minus τ x)dxle12

1113944

m

j1bmaxij rj1113946

Ωe2i (t x)

+ e2j(t minus τ x)dx + 1113944

m

j1bmaxij sj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

(36)

Combining controller (30) designed in +eorem 1 itfollows that

DαV(t e(t middot))le

12

1113944

m

i1σi1113946Ω

e2i (t x)dx +

12

1113944

m

i1σlowastlowasti

1113946Ω

e2i (t minus τ x)dx +

12

1113944

m

i1σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

minus 1113944m

i11113946Ω

li(t x) + ℓi( 1113857e2i (t x)dx

minus 1113944m

i1ωi1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

(37)

From Definition 2 and Property 2 one obtains

ℓi(t x) ℓi(0 x) +1Γ(α)

1113946t

0(t minus τ)

αϖie2i (t x)dx (38)

Substituting (38) into (37) and noting that ℓi and ωi meetinequality (31) it can be deduced that

DαV(t e(t middot))le minusλ1V(t e(t middot)) + ρ1V(t minus τ e(t middot))

(39)

where λ1 1113936mi1(2ℓi minus σi) and ρ1 1113936

mi1 σlowastlowasti

According to Lemma 2 and Lemma 3 it can be assertthat the global synchronization between (24) and (25) can berealized

+e proof of +eorem 1 completes

Remark 2 When α 1 the error system (26) be changedinto the integer form+e conclusion is the same as the resultin +eorem 31 in article [45] If the reaction-diffusion inspace Ω disappears then the states of the systemrsquos variableare only relevant to time but independent of space +us theerror system (26) can be converted to

Dαei(t) minusciei(t) + 1113944m

j1lang1113957aij fjrang(t) + 1113944

m

j1langbij gjrang(t minus τ) + κi(t)

ei(s) φ0i(s) s isin [minusτ 0] i 1 2 m

⎧⎪⎪⎨

⎪⎪⎩

(40)

For convenience the inequalities are given by

ℓi gt12

σi + σlowastlowasti cscαπ2

1113874 11138751113874 1113875gt 0

ωi gt12σ lowasti

(41)

Corollary 1 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t) i 1 2 m

⎧⎨

⎩ (42)

where ℓi(0)gt 0 ϖi gt 0 and ℓi and ωi are constants whichsatisfy inequality (41)

Remark 3 In fact synchronization of FMNNs with timedelay has been investigated in [20] +e results about syn-chronization of the systems with single delay in +eorem 3in [20] are the same as Corollary 1 obtained in this paperSuppose that there is no time delay τ in the course ofcommunication between the different neurons then errorsystem (26) can be reduced to

zαei(t x)

ztα 1113944

n

q1diqΔei(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + κi(t x)

ei(t x) 0 t isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

6 Complexity

For convenience the inequalities are given by

ℓi gt12

1113957σi gt 0

ωi gt12σ lowasti

(44)

Corollary 2 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be realizedbased on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

i 1 2 m

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(45)

where ℓi(0 x)ge 0ϖi gt 0 and ℓi and ωi satisfy inequality (44)

Remark 4 Comparing the coefficients of the adaptivecontrollers (31) and (44) it can be concluded that thecontrollers for global synchronization of the system withtime delay are more consumptive

32 Global Synchronization for Systems with UnknownParameters In real neural networks the parameters of thesystem may be unknown so it is difficult to achieve syn-chronization by the conventional adaptive control schemesIn the following text adaptive controllers with the dynamiccompensators are introduced

Here suppose that self connection weight ci and input Ii

in the drive system (15) are unknown +en the IVP of thedrive system (15) can be written as

zαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x)

+ 1113944m

j1lang1113957aij

1113957f1jrang(t x) + 1113944m

j1lang1113957bij 1113957g1jrang(t minus τ x) + Ii

ui(t x) 0 t isin [minusτinfin) x isin zΩ

ui(s x) φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

+e IVP of the response system (19) can be shown aszαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus 1113954ci(t x)vi(t x) + 1113944

m

j1lang1113957aij

1113957f2jrang

(t x) + 1113944m

j1lang1113957bij 1113957g2jrang(t minus τ x) + 1113954Ii(t x) + κi(t x)

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where i j 1 2 m and 1113954ci(t x) and 1113954Ii(t x) are treated asthe compensation of ci and Ii

+en for i j 1 2 m the IVP of the synchroni-zation error can be described as

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + 1113944m

j1langbij gjrang(t minus τ x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(48)

Assumption 3 +ere are positive constants pi qi and gi

such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

gi(θ)legi

(49)

where i 1 2 m θ isin R and the notation co[fi(θ)]

represents [min fminusj (θ) f+

j (θ)1113966 1113967 max fminusj (θ) f+

j (θ)1113966 1113967]

Theorem 2 Under Assumption 1 and Assumption 3 theglobal asymptotic synchronization between (46) and (47) canrealized based on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ϱiei(t x)vi(t x)

Dα1113954Ii(t x) minus]iei(t x)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(50)

where i 1 2 m ℓi(0 x) ϖi ϱi and ]i are positiveconstants and ℓi and ωi are constants which satisfy thefollowing inequalities

ℓi gt12

1113957σi gt 0

ωi gt12

1113957σ lowasti

(51)

with 1113957σi minus2[ci + 1113936nq1(diql2q)] + 1113936

mj1(amax

ij pj + amaxji pi) and

1113957σ lowasti 21113936mj1(amax

ij qj + bmaxij gj)

Proof Employ the following auxiliary function

V(t) V1(t) + V2(t) + V3(t) (52)

where

Complexity 7

V1(t) 1113946Ω

12

1113944

m

i1e2i (t x)dx

V2(t) 1113946Ω

12ϱi

1113944

m

i11113954ci(t x) minus ci( 1113857

2dx

V3(t) 1113946Ω

12]i

1113944

m

i1

1113954Ii(t x) minus Ii1113872 11138732dx

(53)

Similarly to the proof of +eorem 1 one obtains

DαV1(t)le 1113944

m

i1

12

1113957σi1113946Ω

e2i (t x)dx

+ 1113944m

i1

12

1113957σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

+ 1113944m

i11113946Ω

1113954Ii(t x) minus Ii1113872 1113873ei(t x)dx

minus 1113944m

i11113946Ω

1113954ci(t x) minus ci( 1113857vi(t x)ei(t x)dx

+ 1113944m

i11113946Ω

ei(t x)κi(t x)dx

DαV3(t)le minus 1113944

m

i11113946Ω

ei(t x) 1113954Ii(t x) minus Ii1113872 1113873dx

DαV2(t)le 1113944

m

i11113946Ω

ei(t x) 1113954ci(t x) minus ci( 1113857vi(t x)dx

(54)

Since ℓi and ωi fulfill condition (51) it follows

Dαt V(t)le minus λ2V1(t) (55)

where λ2 1113936mi1(2ℓi minus 1113957σi)

By Lemma 5 it can be concluded that

V(t) minus V(0)le minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (56)

According to the definition of V1 thus

V1(t)leV(t)leV(0) minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (57)

By Lemma 7 it reveals that

V1(t)leV(0)exp minusλΓ(α)

1113946t

0(t minus τ)

αminus 1dτ1113888 1113889

V(0)exp minusλΓ(α + 1)

tα

1113888 1113889

(58)

+erefore

limt⟶infin

V1(t) limt⟶infin

1113946Ω

12

1113944

m

i1e2i (t x)dx 0 (59)

From (59) it is easy to see that the global synchroni-zation between (46) and (47) can be realized

+e proof of +eorem 2 completes

Remark 5 +e result obtained in +eorem 2 is morepractical for considering the situation where the parametersof the drive system (24) are unknown +rough the proof of+eorem 2 it can be asserted that the dynamical compen-sator is necessary for the unknown system If the reaction-diffusion in space Ω disappears the error system (48) can beconverted to

Dαei(t) minus 1113954ci(t) minus ci( 1113857vi(t) minus ciei(t) + 1113944m

j1lang1113957aij fjrang(t)

+ 1113944m

j1langbij gjrang(t minus τ) + 1113954Ii(t) minus Ii1113872 1113873 + κi(t)

ei(s) φ0i(s) s isin [minusτ 0]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(60)

Corollary 3 Under Assumption 1 and Assumption 3 theglobal synchronization between (46) and (47) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t)

Dα1113954ci(t) ]iei(t)

Dα1113954Ii(t) minusϱiei(t)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where ℓi(0)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)1113957σ lowasti andσi minus2ci + (1113936

mj1 amax

ij pj + 1113936mj1 amax

ji pi)

Suppose that there is no time delay τ in the course ofcommunication then the error system (48) can be changedinto

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(62)

Corollary 4 Under Assumption 1 and Assumption 3 (46)and (47) can achieve global synchronization based on thefollowing adaptive controllers

8 Complexity

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ]iei(t x)

Dα1113954Ii(t x) minusϱiei(t x)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(63)

where ℓi(0 x)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)σlowasti and

σlowasti 21113936

mj1 amax

ij qj

4 Numerical Simulations

Several numerical simulations are presented to illustrate themain results+e first example is used to validate+eorem 1+e second one is used to text +eorem 2 Now let usconsider the 2D (q 1) reaction-diffusion FMNNs withtime delay and zero Direclet boundary values Here m 2n 1 and Ω [minus2 2]

Example 1 Consider error system (26) with suchparameters

A a11 minus002

013 minus0011113888 1113889

B 035 b12

036 0371113888 1113889

C 035 0

0 0231113888 1113889

I 060 0

0 0401113888 1113889

(64)

where the memristive coefficients are defined by

a11

019 δ lt 1

026 δ 1

055 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

minus013 δ lt 1

minus017 δ 1

minus024 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(65)

Suppose that time delay τ 11 and discontinuous activa-tion functions are f(δ) g(δ) 098δ + 005sgn(δ) Letinitial values of (24) and (25) be

u1(x t) 8 sin minus2 + x + t2( 1113857

u2(x t) 6 sin(minus2 + x + t)1113896

v1(x t) minus5 cos(minus2 + x + t)

v2(x t) minus6 cos(minus2 + x + t)1113896

(66)

where t isin [minus11 0] and x isin (minus2 2) Given the fractionalorder α 098 and d11 d21 085 Under these parame-ters the dynamical behaviors of the synchronization errors

between (24) and (25) are shown in Figure 1 From thefigure it can be observed that the synchronization errorsgradually increase as time goes on Notice that f(δ) andg(δ) have only one discontinuous point 0 Assumption 1 isvalid According to Assumption 2 we can choose p1 p2

r1 r2 098 and q1 q2 s1 s2 005 By calculationthe parameters of controllers are selected as ℓ1 070gt 039w1 006gt 005 ℓ2 060gt 036 w2 005gt 004ℓ1(0 x) ℓ2(0 x) 001gt 0 and ϖ1 ϖ2 001gt 0 It iseasy to verify that condition (9) in +eorem 1 is satisfiedFigure 2 suggests that the synchronization errors asymp-totically converge to zero

+eorem 1 is not only dependent on order but alsorelated with reaction-diffusion coefficients Figure 3 showsthe trajectories of the error system at spatial position x 55with different orders α 06 07 08 09 when the reaction-diffusion coefficients d 18 While if the order is fixedα 07 the trajectories of the error system at spatial positionx 55 with different reaction-diffusion coefficientsd 06 14 22 30 are shown in Figure 4 From the nu-merical results we could note that the error system wouldachieve the asymptotic stability faster with the higher orderIt also holds true with increasing reaction-diffusioncoefficient

Example 2 Consider the error system (48) with the fol-lowing parameters

A a11 minus095

058 0761113888 1113889

B 022 b12

015 0031113888 1113889

C 058 0

0 0491113888 1113889

I 023 0

0 0251113888 1113889

(67)

where

a11

019 δ lt 1

080 δ 1

026 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

005 δ lt 1

011 δ 1

006 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(68)

Assume α 098 d11 d21 085 and (47) with unknownparameters ci and Ii and initial values of compensators in(47) are given by

1113954C0 1 0

0 11113888 1113889

1113954I0 2 0

0 21113888 1113889

(69)

Complexity 9

100100

Space x50

Time t50

0 0

ndash20

ndash10

0

10

20

30

Erro

r 1(t

x)

ndash10

ndash5

0

5

10

15

20

(a)

100100

5050

0 0

ndash10

0

10

20

30

40

Erro

r 2(t

x)

ndash5

0

5

10

15

20

25

30

Space xTime t

(b)

Figure 1 Synchronization errors between systems (24) and (25) with no controller

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 1(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Space xTime t

(a)

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 2(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Space xTime t

(b)

Figure 2 Synchronization errors between systems (24) and (25) under controller (30)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 1 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(a)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(b)

Figure 3 Trajectories of error system (26) at spatial position x 55 under different orders

10 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

DαV(t e(t middot))le 1113944

m

i11113946Ω

ei(t x) 1113944n

q1diqΔei(t x)⎡⎢⎢⎣ ⎤⎥⎥⎦ minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113944

m

j1langbij gjrang(t minus τ x)

+ κi(t x)dx

(34)

Using Greenrsquos formula and Lemma 6 it can be obtainedthat

1113944

n

q11113946Ω

ei(t x)diqΔei(t x)dxle minus 1113944n

q11113946Ω

diq

lqe2i (t x)dx

(35)

Recalling Assumption 1 and Assumption 2 the in-equalities hold

1113944

m

j11113946Ω

ei(t x)lang1113957aij fjrang(t x)dxle12

1113944

m

j1amaxij pj1113946

Ωe2i (t x)

+ e2j(t x)dx + 1113944

m

j1amaxij qj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

1113944

m

j11113946Ω

ei(t x)langbij gjrang(t minus τ x)dxle12

1113944

m

j1bmaxij rj1113946

Ωe2i (t x)

+ e2j(t minus τ x)dx + 1113944

m

j1bmaxij sj1113946

Ωei(t x)

11138681113868111386811138681113868111386811138681113868dx

(36)

Combining controller (30) designed in +eorem 1 itfollows that

DαV(t e(t middot))le

12

1113944

m

i1σi1113946Ω

e2i (t x)dx +

12

1113944

m

i1σlowastlowasti

1113946Ω

e2i (t minus τ x)dx +

12

1113944

m

i1σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

minus 1113944m

i11113946Ω

li(t x) + ℓi( 1113857e2i (t x)dx

minus 1113944m

i1ωi1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

(37)

From Definition 2 and Property 2 one obtains

ℓi(t x) ℓi(0 x) +1Γ(α)

1113946t

0(t minus τ)

αϖie2i (t x)dx (38)

Substituting (38) into (37) and noting that ℓi and ωi meetinequality (31) it can be deduced that

DαV(t e(t middot))le minusλ1V(t e(t middot)) + ρ1V(t minus τ e(t middot))

(39)

where λ1 1113936mi1(2ℓi minus σi) and ρ1 1113936

mi1 σlowastlowasti

According to Lemma 2 and Lemma 3 it can be assertthat the global synchronization between (24) and (25) can berealized

+e proof of +eorem 1 completes

Remark 2 When α 1 the error system (26) be changedinto the integer form+e conclusion is the same as the resultin +eorem 31 in article [45] If the reaction-diffusion inspace Ω disappears then the states of the systemrsquos variableare only relevant to time but independent of space +us theerror system (26) can be converted to

Dαei(t) minusciei(t) + 1113944m

j1lang1113957aij fjrang(t) + 1113944

m

j1langbij gjrang(t minus τ) + κi(t)

ei(s) φ0i(s) s isin [minusτ 0] i 1 2 m

⎧⎪⎪⎨

⎪⎪⎩

(40)

For convenience the inequalities are given by

ℓi gt12

σi + σlowastlowasti cscαπ2

1113874 11138751113874 1113875gt 0

ωi gt12σ lowasti

(41)

Corollary 1 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t) i 1 2 m

⎧⎨

⎩ (42)

where ℓi(0)gt 0 ϖi gt 0 and ℓi and ωi are constants whichsatisfy inequality (41)

Remark 3 In fact synchronization of FMNNs with timedelay has been investigated in [20] +e results about syn-chronization of the systems with single delay in +eorem 3in [20] are the same as Corollary 1 obtained in this paperSuppose that there is no time delay τ in the course ofcommunication between the different neurons then errorsystem (26) can be reduced to

zαei(t x)

ztα 1113944

n

q1diqΔei(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + κi(t x)

ei(t x) 0 t isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

6 Complexity

For convenience the inequalities are given by

ℓi gt12

1113957σi gt 0

ωi gt12σ lowasti

(44)

Corollary 2 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be realizedbased on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

i 1 2 m

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(45)

where ℓi(0 x)ge 0ϖi gt 0 and ℓi and ωi satisfy inequality (44)

Remark 4 Comparing the coefficients of the adaptivecontrollers (31) and (44) it can be concluded that thecontrollers for global synchronization of the system withtime delay are more consumptive

32 Global Synchronization for Systems with UnknownParameters In real neural networks the parameters of thesystem may be unknown so it is difficult to achieve syn-chronization by the conventional adaptive control schemesIn the following text adaptive controllers with the dynamiccompensators are introduced

Here suppose that self connection weight ci and input Ii

in the drive system (15) are unknown +en the IVP of thedrive system (15) can be written as

zαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x)

+ 1113944m

j1lang1113957aij

1113957f1jrang(t x) + 1113944m

j1lang1113957bij 1113957g1jrang(t minus τ x) + Ii

ui(t x) 0 t isin [minusτinfin) x isin zΩ

ui(s x) φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

+e IVP of the response system (19) can be shown aszαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus 1113954ci(t x)vi(t x) + 1113944

m

j1lang1113957aij

1113957f2jrang

(t x) + 1113944m

j1lang1113957bij 1113957g2jrang(t minus τ x) + 1113954Ii(t x) + κi(t x)

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where i j 1 2 m and 1113954ci(t x) and 1113954Ii(t x) are treated asthe compensation of ci and Ii

+en for i j 1 2 m the IVP of the synchroni-zation error can be described as

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + 1113944m

j1langbij gjrang(t minus τ x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(48)

Assumption 3 +ere are positive constants pi qi and gi

such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

gi(θ)legi

(49)

where i 1 2 m θ isin R and the notation co[fi(θ)]

represents [min fminusj (θ) f+

j (θ)1113966 1113967 max fminusj (θ) f+

j (θ)1113966 1113967]

Theorem 2 Under Assumption 1 and Assumption 3 theglobal asymptotic synchronization between (46) and (47) canrealized based on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ϱiei(t x)vi(t x)

Dα1113954Ii(t x) minus]iei(t x)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(50)

where i 1 2 m ℓi(0 x) ϖi ϱi and ]i are positiveconstants and ℓi and ωi are constants which satisfy thefollowing inequalities

ℓi gt12

1113957σi gt 0

ωi gt12

1113957σ lowasti

(51)

with 1113957σi minus2[ci + 1113936nq1(diql2q)] + 1113936

mj1(amax

ij pj + amaxji pi) and

1113957σ lowasti 21113936mj1(amax

ij qj + bmaxij gj)

Proof Employ the following auxiliary function

V(t) V1(t) + V2(t) + V3(t) (52)

where

Complexity 7

V1(t) 1113946Ω

12

1113944

m

i1e2i (t x)dx

V2(t) 1113946Ω

12ϱi

1113944

m

i11113954ci(t x) minus ci( 1113857

2dx

V3(t) 1113946Ω

12]i

1113944

m

i1

1113954Ii(t x) minus Ii1113872 11138732dx

(53)

Similarly to the proof of +eorem 1 one obtains

DαV1(t)le 1113944

m

i1

12

1113957σi1113946Ω

e2i (t x)dx

+ 1113944m

i1

12

1113957σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

+ 1113944m

i11113946Ω

1113954Ii(t x) minus Ii1113872 1113873ei(t x)dx

minus 1113944m

i11113946Ω

1113954ci(t x) minus ci( 1113857vi(t x)ei(t x)dx

+ 1113944m

i11113946Ω

ei(t x)κi(t x)dx

DαV3(t)le minus 1113944

m

i11113946Ω

ei(t x) 1113954Ii(t x) minus Ii1113872 1113873dx

DαV2(t)le 1113944

m

i11113946Ω

ei(t x) 1113954ci(t x) minus ci( 1113857vi(t x)dx

(54)

Since ℓi and ωi fulfill condition (51) it follows

Dαt V(t)le minus λ2V1(t) (55)

where λ2 1113936mi1(2ℓi minus 1113957σi)

By Lemma 5 it can be concluded that

V(t) minus V(0)le minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (56)

According to the definition of V1 thus

V1(t)leV(t)leV(0) minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (57)

By Lemma 7 it reveals that

V1(t)leV(0)exp minusλΓ(α)

1113946t

0(t minus τ)

αminus 1dτ1113888 1113889

V(0)exp minusλΓ(α + 1)

tα

1113888 1113889

(58)

+erefore

limt⟶infin

V1(t) limt⟶infin

1113946Ω

12

1113944

m

i1e2i (t x)dx 0 (59)

From (59) it is easy to see that the global synchroni-zation between (46) and (47) can be realized

+e proof of +eorem 2 completes

Remark 5 +e result obtained in +eorem 2 is morepractical for considering the situation where the parametersof the drive system (24) are unknown +rough the proof of+eorem 2 it can be asserted that the dynamical compen-sator is necessary for the unknown system If the reaction-diffusion in space Ω disappears the error system (48) can beconverted to

Dαei(t) minus 1113954ci(t) minus ci( 1113857vi(t) minus ciei(t) + 1113944m

j1lang1113957aij fjrang(t)

+ 1113944m

j1langbij gjrang(t minus τ) + 1113954Ii(t) minus Ii1113872 1113873 + κi(t)

ei(s) φ0i(s) s isin [minusτ 0]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(60)

Corollary 3 Under Assumption 1 and Assumption 3 theglobal synchronization between (46) and (47) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t)

Dα1113954ci(t) ]iei(t)

Dα1113954Ii(t) minusϱiei(t)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where ℓi(0)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)1113957σ lowasti andσi minus2ci + (1113936

mj1 amax

ij pj + 1113936mj1 amax

ji pi)

Suppose that there is no time delay τ in the course ofcommunication then the error system (48) can be changedinto

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(62)

Corollary 4 Under Assumption 1 and Assumption 3 (46)and (47) can achieve global synchronization based on thefollowing adaptive controllers

8 Complexity

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ]iei(t x)

Dα1113954Ii(t x) minusϱiei(t x)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(63)

where ℓi(0 x)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)σlowasti and

σlowasti 21113936

mj1 amax

ij qj

4 Numerical Simulations

Several numerical simulations are presented to illustrate themain results+e first example is used to validate+eorem 1+e second one is used to text +eorem 2 Now let usconsider the 2D (q 1) reaction-diffusion FMNNs withtime delay and zero Direclet boundary values Here m 2n 1 and Ω [minus2 2]

Example 1 Consider error system (26) with suchparameters

A a11 minus002

013 minus0011113888 1113889

B 035 b12

036 0371113888 1113889

C 035 0

0 0231113888 1113889

I 060 0

0 0401113888 1113889

(64)

where the memristive coefficients are defined by

a11

019 δ lt 1

026 δ 1

055 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

minus013 δ lt 1

minus017 δ 1

minus024 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(65)

Suppose that time delay τ 11 and discontinuous activa-tion functions are f(δ) g(δ) 098δ + 005sgn(δ) Letinitial values of (24) and (25) be

u1(x t) 8 sin minus2 + x + t2( 1113857

u2(x t) 6 sin(minus2 + x + t)1113896

v1(x t) minus5 cos(minus2 + x + t)

v2(x t) minus6 cos(minus2 + x + t)1113896

(66)

where t isin [minus11 0] and x isin (minus2 2) Given the fractionalorder α 098 and d11 d21 085 Under these parame-ters the dynamical behaviors of the synchronization errors

between (24) and (25) are shown in Figure 1 From thefigure it can be observed that the synchronization errorsgradually increase as time goes on Notice that f(δ) andg(δ) have only one discontinuous point 0 Assumption 1 isvalid According to Assumption 2 we can choose p1 p2

r1 r2 098 and q1 q2 s1 s2 005 By calculationthe parameters of controllers are selected as ℓ1 070gt 039w1 006gt 005 ℓ2 060gt 036 w2 005gt 004ℓ1(0 x) ℓ2(0 x) 001gt 0 and ϖ1 ϖ2 001gt 0 It iseasy to verify that condition (9) in +eorem 1 is satisfiedFigure 2 suggests that the synchronization errors asymp-totically converge to zero

+eorem 1 is not only dependent on order but alsorelated with reaction-diffusion coefficients Figure 3 showsthe trajectories of the error system at spatial position x 55with different orders α 06 07 08 09 when the reaction-diffusion coefficients d 18 While if the order is fixedα 07 the trajectories of the error system at spatial positionx 55 with different reaction-diffusion coefficientsd 06 14 22 30 are shown in Figure 4 From the nu-merical results we could note that the error system wouldachieve the asymptotic stability faster with the higher orderIt also holds true with increasing reaction-diffusioncoefficient

Example 2 Consider the error system (48) with the fol-lowing parameters

A a11 minus095

058 0761113888 1113889

B 022 b12

015 0031113888 1113889

C 058 0

0 0491113888 1113889

I 023 0

0 0251113888 1113889

(67)

where

a11

019 δ lt 1

080 δ 1

026 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

005 δ lt 1

011 δ 1

006 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(68)

Assume α 098 d11 d21 085 and (47) with unknownparameters ci and Ii and initial values of compensators in(47) are given by

1113954C0 1 0

0 11113888 1113889

1113954I0 2 0

0 21113888 1113889

(69)

Complexity 9

100100

Space x50

Time t50

0 0

ndash20

ndash10

0

10

20

30

Erro

r 1(t

x)

ndash10

ndash5

0

5

10

15

20

(a)

100100

5050

0 0

ndash10

0

10

20

30

40

Erro

r 2(t

x)

ndash5

0

5

10

15

20

25

30

Space xTime t

(b)

Figure 1 Synchronization errors between systems (24) and (25) with no controller

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 1(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Space xTime t

(a)

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 2(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Space xTime t

(b)

Figure 2 Synchronization errors between systems (24) and (25) under controller (30)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 1 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(a)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(b)

Figure 3 Trajectories of error system (26) at spatial position x 55 under different orders

10 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

For convenience the inequalities are given by

ℓi gt12

1113957σi gt 0

ωi gt12σ lowasti

(44)

Corollary 2 Under Assumption 1 and Assumption 2 theglobal synchronization between (24) and (25) can be realizedbased on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

i 1 2 m

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(45)

where ℓi(0 x)ge 0ϖi gt 0 and ℓi and ωi satisfy inequality (44)

Remark 4 Comparing the coefficients of the adaptivecontrollers (31) and (44) it can be concluded that thecontrollers for global synchronization of the system withtime delay are more consumptive

32 Global Synchronization for Systems with UnknownParameters In real neural networks the parameters of thesystem may be unknown so it is difficult to achieve syn-chronization by the conventional adaptive control schemesIn the following text adaptive controllers with the dynamiccompensators are introduced

Here suppose that self connection weight ci and input Ii

in the drive system (15) are unknown +en the IVP of thedrive system (15) can be written as

zαui(t x)

ztα 1113944

n

q1diqΔui(t x) minus ciui(t x)

+ 1113944m

j1lang1113957aij

1113957f1jrang(t x) + 1113944m

j1lang1113957bij 1113957g1jrang(t minus τ x) + Ii

ui(t x) 0 t isin [minusτinfin) x isin zΩ

ui(s x) φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

+e IVP of the response system (19) can be shown aszαvi(t x)

ztα 1113944

n

q1diqΔvi(t x) minus 1113954ci(t x)vi(t x) + 1113944

m

j1lang1113957aij

1113957f2jrang

(t x) + 1113944m

j1lang1113957bij 1113957g2jrang(t minus τ x) + 1113954Ii(t x) + κi(t x)

vi(t x) 0 t isin [minusτinfin) x isin zΩ

vi(s x) Φ0i(s x) t isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where i j 1 2 m and 1113954ci(t x) and 1113954Ii(t x) are treated asthe compensation of ci and Ii

+en for i j 1 2 m the IVP of the synchroni-zation error can be described as

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x) + 1113944

m

j1lang1113957aij fjrang

(t x) + 1113944m

j1langbij gjrang(t minus τ x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [minusτinfin) x isin zΩ

ei(s x) φ0i(s x) s isin [minusτ 0] x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(48)

Assumption 3 +ere are positive constants pi qi and gi

such that

supαiisinco fi(μ)[ ]βiisinco fi(ϑ)[ ]

αi minus βi

11138681113868111386811138681113868111386811138681113868lepi μi minus ϑi

11138681113868111386811138681113868111386811138681113868 + qi forallμ ϑ isin R

gi(θ)legi

(49)

where i 1 2 m θ isin R and the notation co[fi(θ)]

represents [min fminusj (θ) f+

j (θ)1113966 1113967 max fminusj (θ) f+

j (θ)1113966 1113967]

Theorem 2 Under Assumption 1 and Assumption 3 theglobal asymptotic synchronization between (46) and (47) canrealized based on the following adaptive controllers

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ϱiei(t x)vi(t x)

Dα1113954Ii(t x) minus]iei(t x)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(50)

where i 1 2 m ℓi(0 x) ϖi ϱi and ]i are positiveconstants and ℓi and ωi are constants which satisfy thefollowing inequalities

ℓi gt12

1113957σi gt 0

ωi gt12

1113957σ lowasti

(51)

with 1113957σi minus2[ci + 1113936nq1(diql2q)] + 1113936

mj1(amax

ij pj + amaxji pi) and

1113957σ lowasti 21113936mj1(amax

ij qj + bmaxij gj)

Proof Employ the following auxiliary function

V(t) V1(t) + V2(t) + V3(t) (52)

where

Complexity 7

V1(t) 1113946Ω

12

1113944

m

i1e2i (t x)dx

V2(t) 1113946Ω

12ϱi

1113944

m

i11113954ci(t x) minus ci( 1113857

2dx

V3(t) 1113946Ω

12]i

1113944

m

i1

1113954Ii(t x) minus Ii1113872 11138732dx

(53)

Similarly to the proof of +eorem 1 one obtains

DαV1(t)le 1113944

m

i1

12

1113957σi1113946Ω

e2i (t x)dx

+ 1113944m

i1

12

1113957σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

+ 1113944m

i11113946Ω

1113954Ii(t x) minus Ii1113872 1113873ei(t x)dx

minus 1113944m

i11113946Ω

1113954ci(t x) minus ci( 1113857vi(t x)ei(t x)dx

+ 1113944m

i11113946Ω

ei(t x)κi(t x)dx

DαV3(t)le minus 1113944

m

i11113946Ω

ei(t x) 1113954Ii(t x) minus Ii1113872 1113873dx

DαV2(t)le 1113944

m

i11113946Ω

ei(t x) 1113954ci(t x) minus ci( 1113857vi(t x)dx

(54)

Since ℓi and ωi fulfill condition (51) it follows

Dαt V(t)le minus λ2V1(t) (55)

where λ2 1113936mi1(2ℓi minus 1113957σi)

By Lemma 5 it can be concluded that

V(t) minus V(0)le minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (56)

According to the definition of V1 thus

V1(t)leV(t)leV(0) minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (57)

By Lemma 7 it reveals that

V1(t)leV(0)exp minusλΓ(α)

1113946t

0(t minus τ)

αminus 1dτ1113888 1113889

V(0)exp minusλΓ(α + 1)

tα

1113888 1113889

(58)

+erefore

limt⟶infin

V1(t) limt⟶infin

1113946Ω

12

1113944

m

i1e2i (t x)dx 0 (59)

From (59) it is easy to see that the global synchroni-zation between (46) and (47) can be realized

+e proof of +eorem 2 completes

Remark 5 +e result obtained in +eorem 2 is morepractical for considering the situation where the parametersof the drive system (24) are unknown +rough the proof of+eorem 2 it can be asserted that the dynamical compen-sator is necessary for the unknown system If the reaction-diffusion in space Ω disappears the error system (48) can beconverted to

Dαei(t) minus 1113954ci(t) minus ci( 1113857vi(t) minus ciei(t) + 1113944m

j1lang1113957aij fjrang(t)

+ 1113944m

j1langbij gjrang(t minus τ) + 1113954Ii(t) minus Ii1113872 1113873 + κi(t)

ei(s) φ0i(s) s isin [minusτ 0]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(60)

Corollary 3 Under Assumption 1 and Assumption 3 theglobal synchronization between (46) and (47) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t)

Dα1113954ci(t) ]iei(t)

Dα1113954Ii(t) minusϱiei(t)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where ℓi(0)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)1113957σ lowasti andσi minus2ci + (1113936

mj1 amax

ij pj + 1113936mj1 amax

ji pi)

Suppose that there is no time delay τ in the course ofcommunication then the error system (48) can be changedinto

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(62)

Corollary 4 Under Assumption 1 and Assumption 3 (46)and (47) can achieve global synchronization based on thefollowing adaptive controllers

8 Complexity

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ]iei(t x)

Dα1113954Ii(t x) minusϱiei(t x)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(63)

where ℓi(0 x)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)σlowasti and

σlowasti 21113936

mj1 amax

ij qj

4 Numerical Simulations

Several numerical simulations are presented to illustrate themain results+e first example is used to validate+eorem 1+e second one is used to text +eorem 2 Now let usconsider the 2D (q 1) reaction-diffusion FMNNs withtime delay and zero Direclet boundary values Here m 2n 1 and Ω [minus2 2]

Example 1 Consider error system (26) with suchparameters

A a11 minus002

013 minus0011113888 1113889

B 035 b12

036 0371113888 1113889

C 035 0

0 0231113888 1113889

I 060 0

0 0401113888 1113889

(64)

where the memristive coefficients are defined by

a11

019 δ lt 1

026 δ 1

055 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

minus013 δ lt 1

minus017 δ 1

minus024 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(65)

Suppose that time delay τ 11 and discontinuous activa-tion functions are f(δ) g(δ) 098δ + 005sgn(δ) Letinitial values of (24) and (25) be

u1(x t) 8 sin minus2 + x + t2( 1113857

u2(x t) 6 sin(minus2 + x + t)1113896

v1(x t) minus5 cos(minus2 + x + t)

v2(x t) minus6 cos(minus2 + x + t)1113896

(66)

where t isin [minus11 0] and x isin (minus2 2) Given the fractionalorder α 098 and d11 d21 085 Under these parame-ters the dynamical behaviors of the synchronization errors

between (24) and (25) are shown in Figure 1 From thefigure it can be observed that the synchronization errorsgradually increase as time goes on Notice that f(δ) andg(δ) have only one discontinuous point 0 Assumption 1 isvalid According to Assumption 2 we can choose p1 p2

r1 r2 098 and q1 q2 s1 s2 005 By calculationthe parameters of controllers are selected as ℓ1 070gt 039w1 006gt 005 ℓ2 060gt 036 w2 005gt 004ℓ1(0 x) ℓ2(0 x) 001gt 0 and ϖ1 ϖ2 001gt 0 It iseasy to verify that condition (9) in +eorem 1 is satisfiedFigure 2 suggests that the synchronization errors asymp-totically converge to zero

+eorem 1 is not only dependent on order but alsorelated with reaction-diffusion coefficients Figure 3 showsthe trajectories of the error system at spatial position x 55with different orders α 06 07 08 09 when the reaction-diffusion coefficients d 18 While if the order is fixedα 07 the trajectories of the error system at spatial positionx 55 with different reaction-diffusion coefficientsd 06 14 22 30 are shown in Figure 4 From the nu-merical results we could note that the error system wouldachieve the asymptotic stability faster with the higher orderIt also holds true with increasing reaction-diffusioncoefficient

Example 2 Consider the error system (48) with the fol-lowing parameters

A a11 minus095

058 0761113888 1113889

B 022 b12

015 0031113888 1113889

C 058 0

0 0491113888 1113889

I 023 0

0 0251113888 1113889

(67)

where

a11

019 δ lt 1

080 δ 1

026 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

005 δ lt 1

011 δ 1

006 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(68)

Assume α 098 d11 d21 085 and (47) with unknownparameters ci and Ii and initial values of compensators in(47) are given by

1113954C0 1 0

0 11113888 1113889

1113954I0 2 0

0 21113888 1113889

(69)

Complexity 9

100100

Space x50

Time t50

0 0

ndash20

ndash10

0

10

20

30

Erro

r 1(t

x)

ndash10

ndash5

0

5

10

15

20

(a)

100100

5050

0 0

ndash10

0

10

20

30

40

Erro

r 2(t

x)

ndash5

0

5

10

15

20

25

30

Space xTime t

(b)

Figure 1 Synchronization errors between systems (24) and (25) with no controller

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 1(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Space xTime t

(a)

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 2(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Space xTime t

(b)

Figure 2 Synchronization errors between systems (24) and (25) under controller (30)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 1 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(a)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(b)

Figure 3 Trajectories of error system (26) at spatial position x 55 under different orders

10 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

V1(t) 1113946Ω

12

1113944

m

i1e2i (t x)dx

V2(t) 1113946Ω

12ϱi

1113944

m

i11113954ci(t x) minus ci( 1113857

2dx

V3(t) 1113946Ω

12]i

1113944

m

i1

1113954Ii(t x) minus Ii1113872 11138732dx

(53)

Similarly to the proof of +eorem 1 one obtains

DαV1(t)le 1113944

m

i1

12

1113957σi1113946Ω

e2i (t x)dx

+ 1113944m

i1

12

1113957σ lowasti 1113946Ω

ei(t x)1113868111386811138681113868

1113868111386811138681113868dx

+ 1113944m

i11113946Ω

1113954Ii(t x) minus Ii1113872 1113873ei(t x)dx

minus 1113944m

i11113946Ω

1113954ci(t x) minus ci( 1113857vi(t x)ei(t x)dx

+ 1113944m

i11113946Ω

ei(t x)κi(t x)dx

DαV3(t)le minus 1113944

m

i11113946Ω

ei(t x) 1113954Ii(t x) minus Ii1113872 1113873dx

DαV2(t)le 1113944

m

i11113946Ω

ei(t x) 1113954ci(t x) minus ci( 1113857vi(t x)dx

(54)

Since ℓi and ωi fulfill condition (51) it follows

Dαt V(t)le minus λ2V1(t) (55)

where λ2 1113936mi1(2ℓi minus 1113957σi)

By Lemma 5 it can be concluded that

V(t) minus V(0)le minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (56)

According to the definition of V1 thus

V1(t)leV(t)leV(0) minusλΓ(α)

1113946t

0(t minus τ)

αminus 1V1(τ)dτ (57)

By Lemma 7 it reveals that

V1(t)leV(0)exp minusλΓ(α)

1113946t

0(t minus τ)

αminus 1dτ1113888 1113889

V(0)exp minusλΓ(α + 1)

tα

1113888 1113889

(58)

+erefore

limt⟶infin

V1(t) limt⟶infin

1113946Ω

12

1113944

m

i1e2i (t x)dx 0 (59)

From (59) it is easy to see that the global synchroni-zation between (46) and (47) can be realized

+e proof of +eorem 2 completes

Remark 5 +e result obtained in +eorem 2 is morepractical for considering the situation where the parametersof the drive system (24) are unknown +rough the proof of+eorem 2 it can be asserted that the dynamical compen-sator is necessary for the unknown system If the reaction-diffusion in space Ω disappears the error system (48) can beconverted to

Dαei(t) minus 1113954ci(t) minus ci( 1113857vi(t) minus ciei(t) + 1113944m

j1lang1113957aij fjrang(t)

+ 1113944m

j1langbij gjrang(t minus τ) + 1113954Ii(t) minus Ii1113872 1113873 + κi(t)

ei(s) φ0i(s) s isin [minusτ 0]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(60)

Corollary 3 Under Assumption 1 and Assumption 3 theglobal synchronization between (46) and (47) can be achievedbased on the following adaptive controllers

κi(t) minus ℓi(t) + ℓi( 1113857ei(t) minus ωisgn ei(t)( 1113857

Dαℓi(t) ϖie2i (t)

Dα1113954ci(t) ]iei(t)

Dα1113954Ii(t) minusϱiei(t)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where ℓi(0)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)1113957σ lowasti andσi minus2ci + (1113936

mj1 amax

ij pj + 1113936mj1 amax

ji pi)

Suppose that there is no time delay τ in the course ofcommunication then the error system (48) can be changedinto

zαei(t x)

ztα minus 1113954ci(t x) minus ci( 1113857vi(t x) minus ciei(t x)

+ 1113944m

j1lang1113957aij fjrang(t x) + 1113954Ii(t x) minus Ii1113872 1113873 + κi(t x)

ei(s x) 0 s isin [0infin) x isin zΩ

ei(s x) φ0i(s x) s 0 x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(62)

Corollary 4 Under Assumption 1 and Assumption 3 (46)and (47) can achieve global synchronization based on thefollowing adaptive controllers

8 Complexity

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ]iei(t x)

Dα1113954Ii(t x) minusϱiei(t x)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(63)

where ℓi(0 x)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)σlowasti and

σlowasti 21113936

mj1 amax

ij qj

4 Numerical Simulations

Several numerical simulations are presented to illustrate themain results+e first example is used to validate+eorem 1+e second one is used to text +eorem 2 Now let usconsider the 2D (q 1) reaction-diffusion FMNNs withtime delay and zero Direclet boundary values Here m 2n 1 and Ω [minus2 2]

Example 1 Consider error system (26) with suchparameters

A a11 minus002

013 minus0011113888 1113889

B 035 b12

036 0371113888 1113889

C 035 0

0 0231113888 1113889

I 060 0

0 0401113888 1113889

(64)

where the memristive coefficients are defined by

a11

019 δ lt 1

026 δ 1

055 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

minus013 δ lt 1

minus017 δ 1

minus024 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(65)

Suppose that time delay τ 11 and discontinuous activa-tion functions are f(δ) g(δ) 098δ + 005sgn(δ) Letinitial values of (24) and (25) be

u1(x t) 8 sin minus2 + x + t2( 1113857

u2(x t) 6 sin(minus2 + x + t)1113896

v1(x t) minus5 cos(minus2 + x + t)

v2(x t) minus6 cos(minus2 + x + t)1113896

(66)

where t isin [minus11 0] and x isin (minus2 2) Given the fractionalorder α 098 and d11 d21 085 Under these parame-ters the dynamical behaviors of the synchronization errors

between (24) and (25) are shown in Figure 1 From thefigure it can be observed that the synchronization errorsgradually increase as time goes on Notice that f(δ) andg(δ) have only one discontinuous point 0 Assumption 1 isvalid According to Assumption 2 we can choose p1 p2

r1 r2 098 and q1 q2 s1 s2 005 By calculationthe parameters of controllers are selected as ℓ1 070gt 039w1 006gt 005 ℓ2 060gt 036 w2 005gt 004ℓ1(0 x) ℓ2(0 x) 001gt 0 and ϖ1 ϖ2 001gt 0 It iseasy to verify that condition (9) in +eorem 1 is satisfiedFigure 2 suggests that the synchronization errors asymp-totically converge to zero

+eorem 1 is not only dependent on order but alsorelated with reaction-diffusion coefficients Figure 3 showsthe trajectories of the error system at spatial position x 55with different orders α 06 07 08 09 when the reaction-diffusion coefficients d 18 While if the order is fixedα 07 the trajectories of the error system at spatial positionx 55 with different reaction-diffusion coefficientsd 06 14 22 30 are shown in Figure 4 From the nu-merical results we could note that the error system wouldachieve the asymptotic stability faster with the higher orderIt also holds true with increasing reaction-diffusioncoefficient

Example 2 Consider the error system (48) with the fol-lowing parameters

A a11 minus095

058 0761113888 1113889

B 022 b12

015 0031113888 1113889

C 058 0

0 0491113888 1113889

I 023 0

0 0251113888 1113889

(67)

where

a11

019 δ lt 1

080 δ 1

026 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

005 δ lt 1

011 δ 1

006 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(68)

Assume α 098 d11 d21 085 and (47) with unknownparameters ci and Ii and initial values of compensators in(47) are given by

1113954C0 1 0

0 11113888 1113889

1113954I0 2 0

0 21113888 1113889

(69)

Complexity 9

100100

Space x50

Time t50

0 0

ndash20

ndash10

0

10

20

30

Erro

r 1(t

x)

ndash10

ndash5

0

5

10

15

20

(a)

100100

5050

0 0

ndash10

0

10

20

30

40

Erro

r 2(t

x)

ndash5

0

5

10

15

20

25

30

Space xTime t

(b)

Figure 1 Synchronization errors between systems (24) and (25) with no controller

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 1(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Space xTime t

(a)

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 2(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Space xTime t

(b)

Figure 2 Synchronization errors between systems (24) and (25) under controller (30)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 1 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(a)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(b)

Figure 3 Trajectories of error system (26) at spatial position x 55 under different orders

10 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

κi(t x) minus ℓi(t x) + ℓi( 1113857ei(t x) minus ωisgn ei(t x)( 1113857

Dαℓi(t x) ϖie2i (t x)

Dα1113954ci(t x) ]iei(t x)

Dα1113954Ii(t x) minusϱiei(t x)

i 1 2 m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(63)

where ℓi(0 x)ge 0 ϖi gt 0 ℓi gt (12)σi gt 0 ωi gt (12)σlowasti and

σlowasti 21113936

mj1 amax

ij qj

4 Numerical Simulations

Several numerical simulations are presented to illustrate themain results+e first example is used to validate+eorem 1+e second one is used to text +eorem 2 Now let usconsider the 2D (q 1) reaction-diffusion FMNNs withtime delay and zero Direclet boundary values Here m 2n 1 and Ω [minus2 2]

Example 1 Consider error system (26) with suchparameters

A a11 minus002

013 minus0011113888 1113889

B 035 b12

036 0371113888 1113889

C 035 0

0 0231113888 1113889

I 060 0

0 0401113888 1113889

(64)

where the memristive coefficients are defined by

a11

019 δ lt 1

026 δ 1

055 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

minus013 δ lt 1

minus017 δ 1

minus024 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(65)

Suppose that time delay τ 11 and discontinuous activa-tion functions are f(δ) g(δ) 098δ + 005sgn(δ) Letinitial values of (24) and (25) be

u1(x t) 8 sin minus2 + x + t2( 1113857

u2(x t) 6 sin(minus2 + x + t)1113896

v1(x t) minus5 cos(minus2 + x + t)

v2(x t) minus6 cos(minus2 + x + t)1113896

(66)

where t isin [minus11 0] and x isin (minus2 2) Given the fractionalorder α 098 and d11 d21 085 Under these parame-ters the dynamical behaviors of the synchronization errors

between (24) and (25) are shown in Figure 1 From thefigure it can be observed that the synchronization errorsgradually increase as time goes on Notice that f(δ) andg(δ) have only one discontinuous point 0 Assumption 1 isvalid According to Assumption 2 we can choose p1 p2

r1 r2 098 and q1 q2 s1 s2 005 By calculationthe parameters of controllers are selected as ℓ1 070gt 039w1 006gt 005 ℓ2 060gt 036 w2 005gt 004ℓ1(0 x) ℓ2(0 x) 001gt 0 and ϖ1 ϖ2 001gt 0 It iseasy to verify that condition (9) in +eorem 1 is satisfiedFigure 2 suggests that the synchronization errors asymp-totically converge to zero

+eorem 1 is not only dependent on order but alsorelated with reaction-diffusion coefficients Figure 3 showsthe trajectories of the error system at spatial position x 55with different orders α 06 07 08 09 when the reaction-diffusion coefficients d 18 While if the order is fixedα 07 the trajectories of the error system at spatial positionx 55 with different reaction-diffusion coefficientsd 06 14 22 30 are shown in Figure 4 From the nu-merical results we could note that the error system wouldachieve the asymptotic stability faster with the higher orderIt also holds true with increasing reaction-diffusioncoefficient

Example 2 Consider the error system (48) with the fol-lowing parameters

A a11 minus095

058 0761113888 1113889

B 022 b12

015 0031113888 1113889

C 058 0

0 0491113888 1113889

I 023 0

0 0251113888 1113889

(67)

where

a11

019 δ lt 1

080 δ 1

026 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

b12

005 δ lt 1

011 δ 1

006 δ gt 1

⎧⎪⎪⎨

⎪⎪⎩

(68)

Assume α 098 d11 d21 085 and (47) with unknownparameters ci and Ii and initial values of compensators in(47) are given by

1113954C0 1 0

0 11113888 1113889

1113954I0 2 0

0 21113888 1113889

(69)

Complexity 9

100100

Space x50

Time t50

0 0

ndash20

ndash10

0

10

20

30

Erro

r 1(t

x)

ndash10

ndash5

0

5

10

15

20

(a)

100100

5050

0 0

ndash10

0

10

20

30

40

Erro

r 2(t

x)

ndash5

0

5

10

15

20

25

30

Space xTime t

(b)

Figure 1 Synchronization errors between systems (24) and (25) with no controller

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 1(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Space xTime t

(a)

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 2(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Space xTime t

(b)

Figure 2 Synchronization errors between systems (24) and (25) under controller (30)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 1 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(a)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(b)

Figure 3 Trajectories of error system (26) at spatial position x 55 under different orders

10 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

100100

Space x50

Time t50

0 0

ndash20

ndash10

0

10

20

30

Erro

r 1(t

x)

ndash10

ndash5

0

5

10

15

20

(a)

100100

5050

0 0

ndash10

0

10

20

30

40

Erro

r 2(t

x)

ndash5

0

5

10

15

20

25

30

Space xTime t

(b)

Figure 1 Synchronization errors between systems (24) and (25) with no controller

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 1(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Space xTime t

(a)

100100

5050

0 0

ndash10

ndash5

0

5

10

Erro

r 2(t

x)

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Space xTime t

(b)

Figure 2 Synchronization errors between systems (24) and (25) under controller (30)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 1 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(a)

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

(b)

Figure 3 Trajectories of error system (26) at spatial position x 55 under different orders

10 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

ndash9

ndash8

ndash7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

1Er

ror 1

(t x

= 5

5)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(a)

ndash9

ndash8

-7

ndash6

ndash5

ndash4

ndash3

ndash2

ndash1

0

Erro

r 2 (t

x =

55)

0 10 20 30 40 50 60 70 80 90Time t

d = 06d = 14

d = 22d = 30

(b)

Figure 4 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

100100

50 500 0

ndash60

ndash40

ndash20

0

20

40

Erro

r 1(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

Space xTime t

(a)

100100

5050

0 0

Space xTime t

ndash60

ndash40

ndash20

0

20

40

Erro

r 2(t

x)

ndash40

ndash30

ndash20

ndash10

0

10

20

30

(b)

Figure 5 Synchronization errors between systems (46) and (47) with no controller

100100

5050

0 0

ndash4

ndash2

0

2

4

6

Erro

r 1(t

x)

ndash3

ndash2

ndash1

0

1

2

3

4

5

Space xTime t

(a)

ndash4

ndash2

0

2

4

6

Erro

r 2(t

x)

100

50

0

100

50

0

Space xTime t ndash3

ndash2

ndash1

0

1

2

3

4

5

(b)

Figure 6 Synchronization errors between systems (46) and (47) with controller (50)

Complexity 11

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

Activation functions are f(δ) 098δ + 005sgn(δ) andg(δ) 05sgn(δ) and time delay τ 35 Initial values of(46) and (47) are chosen as

u1(x t) 17 sin minus2 + x + t2( 1113857

u2(x t) minus21 sin(minus2 + x + t)1113896

v1(x t) 15 cos(minus2 + x + 8t)

v2(x t) minus19 cos minus2 + x + 8t2( 11138571113896

(70)

where t isin [minus34 0] and x isin (minus2 2)

As seen in Figure 5 the error state is instable Choosingp1 p2 098 q1 q2 005 and g1 g2 05 we obtainℓ1 110gt 106 w1 011gt 010 ℓ2 100gt 097 and w2

008gt 007 and ℓi(0 x) 001gt 0 ϖi 001gt 0ϱi 0001gt 0 and ]i 002gt 0 i 1 2 Condition (51) in+eorem 2 is content clear Figure 6 shows that the syn-chronization errors achieve stability rapidly under theadaptive controller (50)

For the error system (48) under different orders α

06 07 08 09 with the same reaction-diffusion coefficientsd 06 and different reaction-diffusion coefficients

0 10 20 30 40 50 60 70 80 90Time t

ndash4

ndash2

0

2

4

6

8Er

ror 1

(t x

= 5

5)

α = 06α = 07

α = 08α = 09

(a)

0 10 20 30 40 50 60 70 80 90Time t

α = 06α = 07

α = 08α = 09

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Erro

r 2 (t

x =

55)

(b)

Figure 7 Trajectories of error system (26) at spatial position x 55 under different orders

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash2

0

2

4

6

8

Erro

r 1(t

x =

55)

(a)

0 10 20 30 40 50 60 70 80 90Time t

d = 08d = 11

d = 15d = 20

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Erro

r 2(t

x =

55)

(b)

Figure 8 Trajectories of error system (26) at spatial position x 55 under different diffusion coefficients

12 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

d 08 11 15 20 with the same order α 07 simulationsare carried out Corresponding error state trajectories aredepicted in Figures 7 and 8 It reveals that the asymptoticstability of the system varies with orders and reaction-dif-fusion coefficients Compared with the former case thefigures show little differences with different parameters

5 Conclusion and Discussion

+e global synchronization problem for reaction-diffusionFMNNs with time delay was emphasized firstly in this paperBoth the known and unknown parameters cases were in-vestigated separately Based on fractional comparisontheorem and GronwallndashBellman inequality two kinds ofadaptive control schemes were proposed Numerical sim-ulations were implemented to examine the effectiveness ofthe obtained results

Conventional adaptive controllers have mostly beenused for controlling neural networks system while theycannot guarantee the synchronization of unknown systemsAn adaptive controller with a dynamic compensator such asthe one described in this paper can eliminate this problemHowever the true values of the unknown parameters couldnot be obtained Further research is still in progress to designan adaptive controller with dynamic compensator to achievethe 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

Acknowledgments

+is work was supported by the National Nature ScienceFoundation of China (no 61772063) and Beijing NaturalScience Foundation (no Z180005)

References

[1] A A A Kilbas H M Srivastava and J J Trujillo4eory andApplications of Fractional Differential Equations Vol 204Elsevier Science Limited Amsterdam Netherlands 2006

[2] B B Mandelbrot and J A Wheeler ldquo+e fractal geometry ofnaturerdquo American Journal of Physics vol 51 no 3pp 286-287 1983

[3] A Boroomand and M B Menhaj ldquoFractional-order hopfieldneural networksrdquo in Proceedings of the International Con-ference on Neural Information Processing pp 883ndash890Auckland New Zealand November 2008

[4] H Wei R Li C Chen and Z Tu ldquoStability analysis offractional order complex-valued memristive neural networkswith time delaysrdquo Neural Processing Letters vol 45 no 2pp 379ndash399 2017

[5] X Yang C Li T Huang Q Song and X Chen ldquoQuasi-uniform synchronization of fractional-order memristor-based neural networks with delayrdquo Neurocomputing vol 234pp 205ndash215 2017

[6] Q Zou Q Jin and R Zhang ldquoDesign of fractional orderpredictive functional control for fractional industrial pro-cessesrdquo Chemometrics and Intelligent Laboratory Systemsvol 152 pp 34ndash41 2016

[7] P Arena R Caponetto L Fortuna and D Porto ldquoBifurcationand chaos in noninteger order cellular neural networksrdquoInternational Journal of Bifurcation and Chaos vol 8 no 7pp 1527ndash1539 1998

[8] H Zhang X-Y Wang and X H Lin ldquoTopology identifi-cation and modulendashphase synchronization of neural networkwith time delayrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 6 pp 885ndash892 2016

[9] H L Trentelman K Takaba and N Monshizadeh ldquoRobustsynchronization of uncertain linear multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 58 no 6pp 1511ndash1523 2013

[10] A Abdurahman H Jiang and Z Teng ldquoFinite-time syn-chronization for memristor-based neural networks with time-varying delaysrdquo Neural Networks vol 69 pp 20ndash28 2015

[11] J Gao P Zhu W Xiong J Cao and L Zhang ldquoAsymptoticsynchronization for stochastic memristor-based neural net-works with noise disturbancerdquo Journal of the Franklin In-stitute vol 353 no 13 pp 3271ndash3289 2016

[12] L Chen R Wu J Cao and J-B Liu ldquoStability and syn-chronization of memristor-based fractional-order delayedneural networksrdquo Neural Networks vol 71 pp 37ndash44 2015

[13] T Hu Z He X Zhang and S Zhong ldquoGlobal synchroni-zation of time-invariant uncertainty fractional-order neuralnetworks with time delayrdquo Neurocomputing vol 339pp 45ndash58 2019

[14] P Liu Z Zeng and J Wang ldquoGlobal synchronization ofcoupled fractional-order recurrent neural networksrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 8 pp 2358ndash2368 2019

[15] T Hu X Zhang and S Zhong ldquoGlobal asymptotic syn-chronization of nonidentical fractional-order neural net-worksrdquo Neurocomputing vol 313 pp 39ndash46 2018

[16] L Chua ldquoMemristor-the missing circuit elementrdquo IEEETransactions on Circuit 4eory vol 18 no 5 pp 507ndash519 1971

[17] D B Strukov G S Snider D R Stewart and R S Williamsldquo+e missing memristor foundrdquo Nature vol 453 no 7191pp 80ndash83 2008

[18] G Velmurugan R Rakkiyappan and J Cao ldquoFinite-timesynchronization of fractional-order memristor-based neuralnetworks with time delaysrdquo Neural Networks vol 73pp 36ndash46 2016

[19] L Zhang Y Yang and F Wang ldquoLag synchronization forfractional-order memristive neural networks via period in-termittent controlrdquo Nonlinear Dynamics vol 89 no 1pp 367ndash381 2017

[20] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[21] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[22] I D Landau R Lozano M MrsquoSaad et al Adaptive ControlVol 51 Springer Berlin Germany 1998

[23] S Chang and T Peng ldquoAdaptive guaranteed cost control ofsystems with uncertain parametersrdquo IEEE Transactions onAutomatic Control vol 17 no 4 pp 474ndash483 1972

Complexity 13

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity

[24] C Wang M Liang and Y Chai ldquoAn adaptive control offractional-order nonlinear uncertain systems with inputsaturationrdquo Complexity vol 2019 Article ID 564329817 pages 2019

[25] H Liu Y Pan and J Cao ldquoComposite learning adaptivedynamic surface control of fractional-order nonlinear sys-temsrdquo IEEE Transactions on Cybernetics vol 50 no 6pp 2557ndash2567

[26] M F Arevalo Castiblanco D Tellez Castro J Sofrony andE Mojica Nava ldquoAdaptive control for unknown heteroge-neous vehicles synchronization with unstructured uncer-taintyrdquo in Proceeding of the 2019 IEEE 4th ColombianConference on Automatic Control (CCAC) pp 1ndash6 MedellınColombia October 2019

[27] Z Xiong S Qu and J Luo ldquoAdaptive multi-switchingsynchronization of high-order memristor-based hyperchaoticsystemwith unknown parameters and its application in securecommunicationrdquo Complexity vol 2019 Article ID 382720118 pages 2019

[28] L Chen R Wu Z Chu Y He and L Yin ldquoPinning syn-chronization of fractional-order delayed complex networkswith non-delayed and delayed couplingsrdquo InternationalJournal of Control vol 90 no 6 pp 1245ndash1255 2017

[29] FWang Z Zheng and Y Yang ldquoSynchronization of complexdynamical networks with hybrid time delay under event-triggered control the threshold function methodrdquo Com-plexity vol 2019 Article ID 7348572 17 pages 2019

[30] Z Yang W Zhou and T Huang ldquoInput-to-state stability ofdelayed reaction-diffusion neural networks with impulsiveeffectsrdquo Neurocomputing vol 333 pp 261ndash272 2019

[31] X Lu W-H Chen Z Ruan and T Huang ldquoA new methodfor global stability analysis of delayed reaction-diffusionneural networksrdquo Neurocomputing vol 317 pp 127ndash1362018

[32] I Stamova and G Stamov ldquoMittag-Leffler synchronization offractional neural networks with time-varying delays and re-action-diffusion terms using impulsive and linear control-lersrdquo Neural Networks vol 96 pp 22ndash32 2017

[33] I Stamova and S Simeonov ldquoDelayed reaction-diffusioncellular neural networks of fractional order mittag-lefflerstability and synchronizationrdquo Journal of Computational andNonlinear Dynamics vol 13 no 1 Article ID 011015 2018

[34] H Liu S Li HWang and Y Sun ldquoAdaptive fuzzy control fora class of unknown fractional-order neural networks subjectto input nonlinearities and dead-zonesrdquo Information Sciencesvol 454-455 pp 30ndash45 2018

[35] S Li X Peng Y Tang and Y Shi ldquoFinite-time synchroni-zation of time-delayed neural networks with unknown pa-rameters via adaptive controlrdquo Neurocomputing vol 308pp 65ndash74 2018

[36] Y Gu Y Yu and H Wang ldquoSynchronization-based pa-rameter estimation of fractional-order neural networksrdquoPhysica A Statistical Mechanics and Its Applications vol 483pp 351ndash361 2017

[37] W F Ames ldquoMathematics in science and engineeringrdquo inFractional Differential Equations I Podlubny Ed Vol 198Elsevier Amsterdam Netherlands 1999

[38] HWang Y Yu GWen S Zhang and J Yu ldquoGlobal stabilityanalysis of fractional-order hopfield neural networks withtime delayrdquo Neurocomputing vol 154 pp 15ndash23 2015

[39] Z Ouyang ldquoExistence and uniqueness of the solutions for aclass of nonlinear fractional order partial differential equa-tions with delayrdquo Computers amp Mathematics with Applica-tions vol 61 no 4 pp 860ndash870 2011

[40] W Ma C Li Y Wu and Y Wu ldquoAdaptive synchronizationof fractional neural networks with unknown parameters andtime delaysrdquo Entropy vol 16 no 12 pp 6286ndash6299 2014

[41] J G Lu ldquoGlobal exponential stability and periodicity of re-action-diffusion delayed recurrent neural networks withDirichlet boundary conditionsrdquo Chaos Solitons amp Fractalsvol 35 no 1 pp 116ndash125 2008

[42] J-J E SlotineW Li et al Applied Nonlinear Control Vol 199Prentice Hall Englewood Cliffs NJ USA 1991

[43] P Phung and L Truong ldquoOn a fractional differential inclusionwith integral boundary conditions in banach spacerdquo Frac-tional Calculus and Applied Analysis vol 16 no 3pp 538ndash558 2013

[44] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential in-clusions with infinite delay and applications to control the-oryrdquo Fractional Calculus and Applied Analysis vol 11 no 1pp 35ndash56 2008

[45] Z Tu N Ding L Li Y Feng L Zou and W ZhangldquoAdaptive synchronization of memristive neural networkswith time-varying delays and reaction-diffusion termrdquo Ap-plied Mathematics and Computation vol 311 pp 118ndash1282017

14 Complexity