# Periodicity of Nonautonomous Fuzzy Neural Networks with Reaction-Diffusion terms and Distributed Time Delays

Authors: Weiyi Hu; Kelin Li; Shuyue Zhao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, the periodicity of a class of nonautonomous fuzzy neural networks with impulses, reaction-diffusion terms, and distributed time delays are investigated. By establishing an integro-differential inequality with impulsive initial conditions and time-varying coefficients, employing theM-matrix theory, Poincar mappings, and fixed point theory, several new sufficient conditions to ensure the periodicity and global exponential stability of the formulated system are obtained. It is worthwhile to mention that our technical methods are practical, in the sense that all new stability conditions are stated in simple algebraic forms, and an optimization method is provided to estimate the exponential convergence rate, so their verification and applications are straightforward and convenient. The validity and generality of our methods are illustrated by two numerical examples.


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## Body

## 1. Introduction

In 1996, Yang et al. studied fuzzy cellular neural networks (FCNNs) [1–3] by combining fuzzy logic with traditional cellular neural networks based on the previous cellular neural networks [4]. It was shown that FCNNs play an important role in image processing problems and pattern recognition. These applications rely heavily on the dynamic behavior of FCNNs. Therefore, it is particularly important to analyze the dynamics of FCNNs. As we all know, neural networks often have delays in the process of information processing. The existence of time delays may cause the systems to oscillate, diverge or become unstable. Neural dynamics considering the delay problems are very important for the stability and balance of the neural networks. Some scholars have studied the stability of FCNNs with constant and time-varying delays [5, 6], and some have studied the stability of FCNNs with distributed time delays [7] and leaky time delays [8]. Furthermore, diffusion effects in neural networks are unavoidable when electrons move in asymmetric electromagnetic fields. Therefore, we must consider that the activation is different in time and space. A number of neural network models with reaction-diffusion terms and various delays have been developed and studied [9–11].On the other hand, in neural network systems, in addition to time delays and diffusion effects, there are impulse effects, which are because of the fact that many neural networks undergo abrupt changes at a given moment due to transient disturbances. These changes occur in the fields of physics, chemistry, population dynamics, etc. At the same time, impulse controls also have important theoretical significance and application values in the fields of optimal controls [12–14]. These changes occur in the fields of physics, chemistry, population dynamics, optimal control, etc. Some results about impulse effects have been obtained in time delays neural networks [12–20]. In particular, when we consider the long-term dynamic behaviours of systems, the parameters of the systems usually change over time and this nonautonomous phenomenons often occur in many practical systems. In [21], the authors studied the stability of a nonautonomous fuzzy neural network with reaction-diffusion terms without impulses. Long [22] studied the dynamic behaviors of nonautonomous cellular neural networks with time-varying delays. In [23], the authors studied the existence, uniqueness, and global stability of periodic solutions of general nonautonomous impulsive cellular neural networks and obtained some criteria.Based on what we know, there are no results on the exponential stability of FCNNs with impulses, distributed time delays, and reaction-diffusion terms at the same time, which is very important in theories and applications. In terms of mathematical models, FCNNs have not only fuzzy logic but also impulse effects between its template input and/or output, except the sum of product operations. The models include reaction-diffusion terms, fuzzy logic, and impulse characteristics, which have complex dynamic behaviors. It is therefore necessary to further investigate the dynamic behaviours of FCNNs. We have used the properties ofM-matrices and inequality tricks to establish a new differential inequality that yields a sufficient condition for global exponentially stable periodic solutions of the systems. Finally, the validity of the results are verified by means of arithmetic examples and numerical simulations using [24].Consider the non-autonomous FCNNs, which contain the reaction-diffusion terms, distributed time delays, and impulses.(1a)∂upt,x∂t=∑k=1m∂∂xkDpk∂upt,x∂xk−dptupt,x+∑q=1nhpqtfquqt,x+∑q=1nypqtvqt+Jpt+∧q=1napqt∫−∞tKpqt−sgquqs,xds+∧q=1nαpqtvqt+∨q=1nbpqt∫−∞tKpqt−sgquqs,xds+∨q=1nβpqtvqt,t≠ti,x∈X,(1b)upti+,x=φpiupti−,x,x∈X,i∈N≜0,1,2,…,(1c)upt,x=0,t≥0,x∈∂X,(1d)ups,x=γps,x,s∈−∞,0,where p=1,2,…,n, t∈0,+∞, x=x1,x2,…,xmT∈X⊂Rm, X=x=x1,x2,…,xmT‖xk|<Lk,k=1,2,…,m is a bounded compact set with smooth boundary ∂X and mes X>0 in space RmLk>0; Kpq⋅ represent the delay kernel function which is real-valued piecewise continuous. upt,x denotes the p th neuron in space x and at time t; f⋅ and g⋅ represent the signal activation function of the q th neuron; dpt>0 denotes the rate of potential recovery to the isolated state of the p th neuron at moment t. hpqt and ypqt denote the elements of the feedback template and the feed-forward template at moment t; ∨ and ∧ represent the fuzzy AND and fuzzy OR operations, respectively; apqt and bpqt, represent elements of fuzzy feedback MIN template and fuzzy feedback MAX template at time t, respectively; αpqt and βpqt represent elements of fuzzy feed-forward MIN template and fuzzy feed-forward MAX template at time t, respectively; Jpt and vpt represent the input and bias of the p th neuron at moment t; Dpk≥0 represents the transmission diffusion coefficient. In (1b), ti>0 satisfies ti<ti+1,limi⟶+∞ti=+∞;upti−,x and upti+,x represent the left and right limits at tk, respectively; φpi shows impulsive perturbation of the p th neuron at time ti. Let upti+,x=upti,x,i∈N. Equation (1c) denote the Dirichlet boundary conditions and (1d) denote the initial conditions.If the impulsive operatorφpiup=0,p=1,2,…,n,i∈N, we obtain the following systems (2a)–(2c):(2a)∂upt,x∂t=∑k=1m∂∂xkDpk∂upt,x∂xk−dptupt,x+∑q=1nhpqtfquqt,x+∑q=1nypqtvqt+Jpt+∧q=1napqt∫−∞tKpqt−sgquqs,xds+∧q=1nαpqtvqt+∨q=1nbpqt∫−∞tKpqt−sgquqs,xds+∨q=1nβpqtvqt,x∈X,(2b)upt,x=0,t≥0,x∈∂X,(2c)ups,x=γps,x,s∈−∞,0.Systems (2a)–(2c) are continuous forms of systems (1a)–(1d).The main contributions of this manuscript are as follows:(a)
A new neural network model is formulated, this model assembles nonautonomous fuzzy neural networks, and reaction-diffusion neural networks with distributed time delays, impulses, and Dirichlet boundary conditions. The systems formulated in this paper are more complex than previously considered models, but there is no doubt that some of the existing systems are also special cases of this system.(b)
A new integro-differential inequality with impulsive initial conditions and time-varying coefficients is established, and it may be a powerful tool for analyzing impulsive nonautonomous systems.(c)
Several new criteria to ensure the global exponential periodicity and stability of the studied system are obtained, these results are expressed in the form of simple algebraic inequalities which depend only on the formulated systems parameters, and their verification and applications are straightforward and convenient.


## 2. Preliminaries

In this section, we explain some of the necessary assumptions, associated notations, and definitions.(H1) There exist diagonal matricesF = diag F1,F2,…,Fn and G = diag G1,G2,…,Gn such that,(3)Fp=supz1≠z2fpz1−fpz2z1−z2,Gp=supz1≠z2gpz1−gpz2z1−z2.For allz1,z2∈Rz1≠z2,p=1,2…n.(H2) There exists a non-negative diagonal matrixΨi=diagφ1i,…,φni such that,(4)φpiz1−φpiz2⩽φpiz1−z2.For allz1,z2∈Rz1≠z2,p∈1,2,…,n,i∈N.(H3)dpt,hpqt,apqt and bpqt are continuous bounded function defined on t∈0,+∞.(H4) there exists a positive numberσ>0 such that,(5)Kpqλ=∫0+∞eλsKpqsds,is continuous forλ∈0,σ,p,q=1,2,…,n.Remark 1.
Assumptions (H1)–(H4) are widely used in the work on exponential stability of cellular neural networks (see, [1–11, 16, 17, 19–26]).
LetPCX≜μ:−∞,0×X⟶Rn|μs,x is bounded on −∞,0×X and μs+,x=μs,x for s∈−∞,0. μs−,x exists for s∈−∞,0 and μs−,x=μs,x for all but a finite number of points s∈−∞,0. For μs,x=μ1s,x,μ2s,x,…,μns,xT∈PCX , μ is defined as.(6)μ=sup−∞≤s≤0∑p=1nμps,x2,where upt,x2=∫Xupt,x2dx1/2, p=1,2,…,n.
LetPC≜μ:−∞,0⟶Rn|μs is bounded on −∞,0 and μs+=μs for s∈−∞,0. μs− exists for s∈−∞,0 and μs−=μs for all but a finite number of points s∈−∞,0.
LetC=cpqm×n, and B=bpqm×n, then the Schur product of C and B is defined by C⊗B=cpqbpqm×n.
e=1,1,…,1T∈Rn and E denotes a n×n identity matrix.Definition 1.
Ifut,xu:R×X⟶Rn satisfies,(i)
ut,x is piecewise continuous and right-continuous at every discontinuity point tii∈N which are the first kind of discontinuity points;(ii)
us,x=γs,xs∈−∞,0 satisfies systems (1a)–(1d) for all t⩾0.
Hence,ut,γ,x represents the special solution of systems (1a)–(1d) under initial condition γ∈PCX.Definition 2.
In the initial condition ofυ∈PCX, ut,υ,x is any solution to the systems (1a)–(1d). If there exist two positive numbers λ>0 and M⩾1 such that,(7)ut,γ,x−ut,υ,x⩽Mγ−υe−λtfor allt⩾0.
Then, systems (1a)–(1d) are globally exponentially stable.Definition 3 (see [28]).
IfB=bpqn×n is a real matrix, suppose that,(i)
bpq⩽0,for allp,q=1,2,…,n,p≠q(ii)
All successive principal minors ofB are positive
Then,B is a nonsingular M-matrix.Lemma 1 (see [28]).
SettingB=bpqn×n with bpq⩽0p≠q for all p,q=1,2,…,n. Then, the necessary and sufficient conditions for B to be a nonsingular M-matrix are that there exists a vector ξ=ξ1,ξ2,…,ξn>0 such that Bξ>0 or BTξ>0.Lemma 2 (see [29]).
Letfx is a real-valued function and X=x=x1,x2,…,xmT‖xk|<Lk,k=1,2,…,m is a cube. If fx meets fx|∂X=0, that is fx is equal to zero at the boundary of X. Then,(8)f2∫Xxdx⩽Lk2∫X∂f∂xk2dx.Lemma 3 (see [28]).
Letμ=μ1,μ2,…,μn and ν=ν1,ν2,…,νn be two states of neural networks (1a)–(1d), and f⋅ be a real-valued function. Then, following inequalities hold:(9)∧q=1nαpqtfqμq−∧q=1nαpqtfqνq≤∑q=1nαpqtfqμq−fqνq,∨q=1nβpqtfqμq−∨q=1nβpqtfqνq≤∑q=1nβpqtfqμq−fqνq.Lemma 4.
Leta<b⩽+∞, If Vt=V1t,V2t,…,VntT∈Ca,b,Rn makes the following differential inequality hold:(10)D+Vt⩽PtVt+∫0+∞Rt⊗KsVt−sds,a⩽t<b,Va+s∈PC,−∞<s⩽0,where Pt=ppqtn×n with ppqt⩾0p≠q, Ks=Kpqsn×n, Rt=rpqtn×n with rpqt⩾0. If the initial condition meets,(11)Vt⩽κξe−λt−a,κ⩾0,t∈−∞,a,where λ>0 and ξ=ξ1,ξ2,…,ξnT>0 satisfy the following inequality:(12)λE+Pt+Rt⊗Kλξ<0,Kλ=Kpqλn×n,
Then,Vt⩽κξe−λt−a,t∈a,b.Proof 1.
Forp∈1,2,…,n, ∀ε>0, let ωpt≜κ+εξpe−λt−a. Then,(13)Vpt⩽ωpt=κ+εξpe−λt−a,t∈a,b,p=1,2,…,n.
If the above-given is false, that is there exist a numbert∗∈a,b and several integer l such that,(14)Vlt∗=ωlt∗,D+Vlt∗⩾ω˙lt∗,Vpt⩽ωpt,t∈a,t∗,p=1,2,…,n.
According to Lemma4 and (13), we obtain the following equation:(15)D+Vlt∗⩽∑q=1nplqt∗Vqt∗+∫0+∞rlqtKlqsVqt∗−sds⩽∑q=1nplqt∗κ+εξqe−λt∗−a+rlqt∗∫0+∞Klqsκ+εξqe−λt∗−s−ads=∑q=1nplqt∗κ+εξqe−λt∗−a+rlqt∗κ+εξqe−λt∗−a∫0+∞eλsKlqsds⩽∑q=1nplqt∗+rlqt∗Klqλκ+εξqe−λt∗−a.
FromλE+Pt+R⊗Kλξ<0,ppqt∗⩾0p≠q, rpqt∗⩾0 and Kpqλ⩾0, it follows that,(16)∑q=1nplqt∗+rlqtKlqλξq<−λξl<0.
From (15), we obtain the following equation:(17)D+Vlt∗<−λξlκ+εe−λt∗−l=ω˙lt∗.
That is,(18)D+Vlt∗<ω˙lt∗.
This is contradictory to (14). Therefore, the inequality (13) holds for all t∈a,b.
Now, lettingε⟶0 in (13), we have that,(19)Vpt⩽κξpe−λt−a,t∈a,b,p=1,2,…,n.
That is,(20)Vt⩽κξe−λt−afor allt∈a,b.


## 3. Main Results

In this section, we introduce the main results of systems (1a)–(1d) and their proof process.Theorem 1.
.If assumptions (H1)–(H4) are satisfied, suppose that,(C1) There existξ=ξ1,ξ2,…,ξnT>0 and λ>0 such that,(21)λE−Wt+HtF+Rt⊗Kλξ<0,t⩾0,whereWt=diagw1t,w2t,…,wnt with wpt=dpt+∑k=1mDpk/Lk2, Rt=At+BtGAt=|apqt|n×n,Bt=|bpqt|n×n, Ht=|hpqt|n×n, Rt=rpqtn×n, Kλ=Kpqλn×n, F= diag F1,F2,…,Fn, G= diag G1,G2,…,Gn;(C2) There exists a constantϕ>0 such that,(22)supi∈Nlnϕiti−ti−1⩽ϕ<λ,where ϕi=max1⩽p⩽n1,φpi,i∈N;
Then systems (1a)–(1d) are globally exponentially stable.Proof 2.
Forθ,ϑ∈PCX, let ut,θ,x=u1t,θ,x,u2t,θ,x,…,unt,θ,xT and ut,ϑ,x=\\u1t,ϑ,x,u2t,ϑ,x,…,unt,ϑ,xT be solutions of systems (1a)–(1d) through 0,θ and 0,ϑ , respectively. Define utθ,x=ut+s,θ,x,utϑ,x=ut+s,ϑ,x,−∞<s⩽0,t⩾0, that is utθ,x, utϑ,x∈PCX for all t⩾0.
LetUpt,γ,x=upt,θ,x−upt,ϑ,x,p=1,2,…,n,γ=θ−ϑ , then,(23)∂Upt,γ,x∂t=∑k=1m∂∂xkDpk∂Upt,γ,x∂xk−dptUpt,γ,x−∑q=1nhpqtfquqt,θ,x−fquqt,ϑ,x+∧q=1napqt∫−∞tKpqt−sgquqs,θ,xds−∧q=1napqt∫−∞tKpqt−sgquqs,ϑ,xds+∨q=1nbpqt∫−∞tKpqt−sgquqs,ϑ,xds−∨q=1nbpqt∫−∞tKpqt−sgquqs,ϑ,xds.
fort≠ti,x∈X,p=1,2,…,n.
Multiply both sides of (23) by Upt,γ,x and integrate it, one can obtain,(24)ddt∫XUpt,γ,x2dx=2∫XUpt,γ,x∑k=1m∂∂xkDpk∂Upt,γ,x∂xkdx−2∫XdptUp2t,γ,xdx+2∑q=1nhpqt∫XUpt,γ,xfquqt,θ,x−fquqt,ϑ,xdx+∫XUpt,γ,x∧q=1napqt∫−∞tKpqt−sgquqs,θ,xds−∧q=1napqt∫−∞tKpqt−sgquqs,ϑ,xdsdx.+∫XUpt,γ,x∨q=1nbpqt∫−∞tKpqt−sgquqs,θ,xds−∨q=1nbpqt∫−∞tKpqt−sgquqs,ϑ,xdsdx.
Due to Green formula and Dirichlet boundary conditions, one has.(25)∫XUpt,γ,x∑k=1m∂∂xkDpk∂Upt,γ,x∂xkdx=−∑k=1m∫XDpk∂Upt,γ,x∂xk2dx.
From Lemma2, we can obtain the following equation:(26)∫XUpt,γ,x∑k=1m∂∂xkDpk∂Upt,γ,x∂xkdx⩽−∑k=1mDpkLk2∫ΞUpt,γ,x2dx=−∑k=1mDpkLk2Upt,γ,x22.
According to (H1) and Hoder inequality, we have,(27)∑q=1nhpqt∫XUpt,γ,xfquqt,θ,x−fquqt,ϑ,xdx⩽∑q=1nhpqt∫XUpt,γ,xfquqt,θ,x−fquqt,ϑ,xdx⩽∑q=1nhpqt∫XUpt,γ,xUqt,γ,xFqdx⩽∑q=1nhpqtUpt,γ,x2FqUqt,γ,x2.
By assumption (H1), Lemma2 and Hoder inequality, one obtains,(28)∫XUpt,γ,x∧q=1napqt∫−∞tKpqt−sgquqs,θ,xds−∧q=1napqt∫−∞tKpqt−sgquqs,ϑ,xdsdx⩽∑q=1napqtGq∫−∞tKpqt−sUpt,γ,x2Uqs,γ,x2ds.
By the same way, we can obtain the following equation:(29)∫XUpt,γ,x∨q=1nbpqt∫−∞tKpqt−sgquqs,θ,xds−∨q=1nbpqt∫−∞tKpqt−sgquqs,ϑ,xdsdx⩽∑q=1nbpqtGq∫−∞tKpqt−sUpt,γ,x2Uqs,γ,x2ds.
Applying (24)–(29) to (23), we can obtain the following equation:(30)12ddtUpt,γ,x22⩽−dpt+∑k=1mDpkLk2Upt,γ,x22+∑q=1nhpqtUpt,γ,x2FqUqt,γ,x2+∑q=1napqtGq∫−∞tKpqt−sUpt,γ,x2Uqs,γ,x2ds+∑q=1nbpqtGq∫−∞tKpqt−sUpt,γ,x2Uqs,γ,x2ds,i.e.,(31)D+Upt,γ,x2≤−dpt+∑k=1mDpkLk2Upt,γ,x2+∑q=1nhpqtFqUqt,γ,x2+∑q=1napqt+bpqtGq∫0+∞KpqsUqt−s,γ,x2ds.
LetVpt=Upt,γ,x2,Vt=V1t,V2t,…,VntT,wpt=dpt+∑k=1mDpk/Lk2,, p=1,2,…,n,Wt=diagw1tw2t,…,wnt, Pt=−Wt+HtF,Rt=At+BtG,Ks=Kpqsn×n. Then, (31) can be simplified into the following form:(32)D+Vt⩽PtVt+∫0+∞Rt⊗KsVt−sds.
From condition (C1), There existξ=ξ1,ξ2,…,ξnT>0 and λ>0, then,(33)λE−Wt+HtF+Rt⊗Kλξ<0.
Here, takingκ=γ/min1⩽p⩽nξp, we have the following equation:(34)Vt⩽κξe−λt,t∈−∞,t0,t0=0.
From Lemma4, one can obtain,(35)Vt⩽κξe−λt,t∈t0,t1.
If the following inequality is true form⩽i(36)Vt⩽κϕ0…ϕm−1ξe−λt,t∈tm−1,tm,ϕ0=1.
Whenm=i+1, we can obtain the following equation:(37)Vti=uti,θ,x−uti,ϑ,x2=φiuti−,θ,x−φiuti−,ϑ,x2⩽ΨiVti−,γ,x2=ΨiVti−⩽κϕ0⋯ϕi−1ϕiξlimt⟶ti−e−λti⩽κϕ0⋯ϕi−1ϕiξe−λti.
By (36) and (37) and ϕi⩾1, we have the following equation:(38)Vt⩽κϕ0⋯ϕi−1ϕiξe−λt,−∞⩽t⩽ti.
Combining (33), (34) and (38) and Lemma 3, one has,(39)Vt⩽κϕ0⋯ϕi−1ϕiξe−λt,ti⩽t<ti+1.
According to mathematical induction, then we have,(40)Vt⩽κϕ0⋯ϕi−1ξe−λt,ti−1⩽t<ti,i∈N,
applying the conditionC2 and (40), one can obtain,(41)Vt⩽κeϕt1eϕt2−t1⋯eϕti−1−ti−2ξe−λt⩽κξeϕte−λt=κξe−λ−ϕt,
for alli∈N,ti−1⩽t<ti.
This implies that,(42)ut,θ,x−ut,ϑ,x=∑p=1nupt,θ,x−upt,ϑ,x2=∑p=1nVpt⩽∑p=1nκξpe−λ−ϕt=∑p=1nξpmin1⩽p⩽nξpθ−ϑe−λ−ϕt.
That is,(43)utθ,x−utϑ,x⩽Mθ−ϑe−λ−ϕt,t⩾0,where M=∑p=1nξp/min1⩽p⩽nξp.Remark 2.
Condition (C1) is equivalent to thatΥt=Wt−HtF−Rt⊗K0 is a nonsingular M-matrix for all t⩾0. As a matter of fact, if Υt is a nonsingular M-matrix for any t⩾0, by using Lemma 1, there exist ξ=ξ1,ξ2,…,ξnT>0 such that,(44)Wt−HtF−Rt⊗K0ξ>0.
By the uniform continuity, there existsλ>0 satisfies:(45)λE−Wt+HtF+Rt⊗Kλξ<0.
It tells us that (C1) is true. Reversely, settingλ=0 in (C1), we can easily get that Υt=Wt−HtF−Rt⊗K0 is a non-singular M-matrix for all t⩾0.Corollary 1.
If assumptionsH1,H3, and H4 are satisfied, suppose that condition C1 holds. Then, systems (2a)–(2c) are globally exponentially stable.Corollary 2.
When the coefficient of system (1a)–(1d) are constants, they degenerate into the following autonomous FCNNs with reaction-diffusion terms and distribution delays:(46a)∂upt,x∂t=∑k=1m∂∂xkDpk∂upt,x∂xk−dpupt,x+∑q=1nhpqfquqt,x+∑q=1nypqvq+Jpt+∧q=1napq∫−∞tKpqt−sgquqs,xds+∧q=1nαpqvq+∨q=1nbpq∫−∞tKpqt−sgquqs,xds+∨q=1nβpqvq,t≠ti,x∈X,(46b)upti+,x=ϕpiupti−,x,x∈X,i∈N,(46c)upt,x=0,x∈∂X,(46d)ups,x=γps,x,s∈−∞,0.
For assumptionH1,H2 and H4, Theorem 1 can be expressed in the following form:C′1 There exist ξ=ξ1,ξ2,…,ξnT>0 and λ>0 such that(47)λE−W+HF+R⊗Kλξ<0,whereW=diagw1,w2,…,wn with wp=dp+∑k=1mDpk/Lk2, R=A+BG, A=apqn×n,B=bpqn×n, H=hpqn×n, R=rpqn×n, Kλ=Kpqλn×n, F= diag F1,F2,…,Fn, G= diag G1,G2,…,Gn;C2 There exists a constant ϕ>0 such that,(48)supi∈Nlnϕiti−ti−1⩽ϕ<λ,where ϕi=max1⩽p⩽n1,φpi,i∈N; then systems (46a)–(46d) are globally exponentially stable.Remark 3.
Some existing neural network models (see, [24, 30]) are special cases such as systems (2a)–(2c) and systems (46a)–(46d). Compared with the methods of constructing Lyapunov functional in [24], our results are more concise, and it is not difficult to find that some of the standards have been improved. Moreover, in [30], the authors gave sufficient conditions for the existence of uniqueness and global exponential stability of the equilibrium point of impulsive FCNNs with distributed time delays and reaction-diffusion terms, but we have to say that the method we using is similar.
Next, in order to consider the periodic solution of the systems (1a)–(1d), we add the following two assumptions.(H5)dpt,hpqt,apqt,bpqt,αpqt,βpqt,ypqt,vqt and Jpt are periodic continuous functions with a common period ϖ>0 for all t⩾0.(H6) ForΨi=diagφ1i,…,φni and the impulsive time tii∈N, there exists a positive integer l such that,(49)φpi+l=φpi,ti+l=ti+ϖ.Remark 4.
Assumptions (H5) and (H6) indicate that both the system parameters and the impulse function are periodic, which is a necessary prerequisite for solving the periodic solution of the systems. These assumptions can be seen in many studies on the periodicity of cellular neural networks (see, [15, 18, 26]).
Combined with assumptions (H5) and (H6), we have the following results for periodic systems (1a)–(1d), based on the discussion of the global exponential stability of the systems in Theorem 1.Theorem 2.
If assumptions (H1)–(H6) are satisfied, suppose that,(C1) There existξ=ξ1,ξ2,…,ξnT>0 and λ>0 such that,(50)λE−Wt+HtF+Rt⊗Kλξ<0,t⩾0,whereWt=diagw1t,w2t,…,wnt with wpt=dpt+∑k=1mDpk/Lk2, Rt=At+BtGAt=|apqt|n×n,Bt=|bpqt|n×n, Ht=|hpqt|n×n, Rt=rpqtn×n, Kλ=Kpqλn×n, F=diag F1,F2,…,Fn, G=diag G1,G2,…,Gn;(C2) There exists a constantϕ>0 such that,(51)supi∈Nlnϕiti−ti−1⩽ϕ<λ,where ϕi=max1⩽p⩽n1,φpi,i∈N; then systems (1a)–(1d) have exactly one globally exponentially stable ϖ-periodic solution.Proof 3.
To choose a positive integerη>0 such that Me−λ−ϕηϖ⩽1/2 and we define a Poincare mapping.(52)Γ:PCX⟶PCXbyΓθ=uϖ.
It follows thatΓηθ=uηϖθ,x. Setting t=ηϖ, we obtain the following equation:(53)Γηθ−Γηϑ≤12θ−ϑ.
Obviously,Γη is a contraction mapping, therefore there exists one unique fixed point θ∗∈PCX such that,(54)Γηθ∗=θ∗.
Hence, we obtain the following equation:(55)ΓηΓθ∗=ΓΓηθ∗=Γθ∗.
This implies thatΓθ∗∈PCX is also a fixed point of Γη. Then,(56)Γθ∗=θ∗,i.e.uθθ∗,x=θ∗.
Hence, ifut,θ∗,x is a solution of system (1) through 0,θ∗, then ut+ϖ,θ∗,x is also a solution to systems (1a)–(1d). Distinctly,(57)ut+ϖθ∗,x=utuϖθ∗,x=utθ∗,x,t⩾0,
i.e.,(58)ut+ϖ,θ∗,x=ut,θ∗,x.
This shows thatut,θ∗,x has one solution for systems (1a)–(1d) with ϖ-period and all other solution of systems (1a)–(1d) converge exponentially to it as t⟶+∞.Remark 5.
In Theorem1 and Theorem 2, the condition (C2) (ϕ=supi∈Nlnϕi/ti−ti−1 ) describes the influence of the impulsive intensity and the impulsive interval on the global exponential stability of systems (1a)–(1d). In the absence of impulses, the following optimization problem can be solved in order to estimate the exponential convergence rate of the systems (1a)–(1d).(59)OPmaxλ,s.t.C1holds.
Obviously,λ is related to delay kernel function, diffusion coefficients, Dirichlet boundary conditions, and system parameters. Theorem 1 shows that when ϕ∈0,λ, systems (1a)–(1d) is globally exponentially stable and its exponential convergence rate equals λ−ϕ.Corollary 3.
Under assumptionsH1,H2,H4 and H5, suppose that,(C1) There existξ=ξ1,ξ2,…,ξnT>0 and λ>0 such that,(60)λE−Wt+HtF+Rt⊗Kλξ<0,t⩾0,where Wt=diagw1t,w2t,…,wnt with wpt=dpt+∑k=1mDpk/Lk2, Rt=At+BtGAt=|apqt|n×n,Bt=|bpqt|n×n, Ht=|hpqt|n×n, Rt=rpqtn×n, Kλ=Kpqλn×n, F=diag F1,F2,…,Fn, G=diag G1,G2,…,Gn; then systems (2a)–(2c) have exactly one globally exponentially stable ϖ-periodic solution.Remark 6.
IfDpk=0 or Dpk≪Lk, that is, when the diffusion effect is negligible, systems (1a)–(1d) will degenerate into a nonautonomous FCNN with distributed delays. By Theorem 2, We can obtain a corollary for the global exponentially stable periodic solution of the systems (1a)–(1d).Corollary 4.
If assumptions (H1)-(H6) are satisfied, suppose that,C1″ There exist ξ=ξ1,ξ2,…,ξnT>0 and λ>0 such that,(61)λE−Dt+HtF+Rt⊗Kλξ<0,t⩾0,whereDt=diagd1t,d2t,…,dnt, Rt=At+BtG, At=|apqt|n×n,Bt=|bpqt|n×n, Ht=|hpqt|n×n, Rt=rpqtn×n, Kλ=Kpqλn×n, F=diag F1,F2,…,Fn, G=diag G1,G2,…,Gn;C2 There exists a constant ϕ>0 such that,(62)supi∈Nlnϕiti−ti−1⩽ϕ<λ,where ϕi=max1⩽p⩽n1,φpi,i∈N; then systems (1a)–(1d) have exactly one globally exponentially stable ϖ-periodic solution.


## 4. Illustrative Examples

Finally, two examples are given to verify the validity and universality of our results.Example 1.
Consider the following two-neuron impulsive system:(63a)∂upt,x∂t=∑k=11∂∂xkDpk∂upt,x∂xk−dptupt,x+∑q=1nhpqtfquqt,x+∑q=1nypqtvqt+Jpt+∧q=1napqt∫−∞tKpqt−sgquqs,xds+∧q=1nαpqtvqt+∨q=1nbpqt∫−∞tKpqt−sgquqs,xds+∨q=1nβpqtvqt,t≠ti,x∈X,(63b)upti+,x=2.5upti−,x,x∈X,i∈N,(63c)upt,x=0,t≥0,x∈∂X,(63d)ups,x=1,s∈−∞,0.For p=1,2, where X=0,1,ti=1.5πi , i=1,2,….
In systems (63a)–(63d), choosing,(64)D11=D21=1,d1t=5.8+sint,d2t=6.9+sint,h11t=y11t=y12t=y21t=y22t=0.25sint,h12t=−0.2−0.3cost,h21t=0.5+0.1sint,h22t=0.6+0.2cost,a11t=b11t=0.1+0.5cost,a12t=b12t=0.6cost,a22t=b22t=0.7sint,a21t=b21t=0.8sint,K11s=K12s=K21s=K22s=e−s,v1t=v2t=1+0.25sint,α11t=α12t=cost,α21t=α22t=sint,β11t=β12t=sint+0.8,β21t=β22t=2cost−0.5,J1t=0.3cost,J2t=0.6sint,f1u=f2u=g1u=g2u=tanhu.
Distinctly, assumptions (H1)–(H6) are satisfied, and then we have the following equation:(65)Wt=6.8+sint007.9+sint,F=G=1001,Rt=0.2+cost1.2cost1.6sint1.4sint,Ht=0.25sint−0.2−0.3cost0.5+0.1sint0.6+0.2cost,Kλ=11−λ11−λ11−λ11−λ.
λ∈0,1,ϕi=max1,2.5=2.5,γ=supi∈Nlnγi/ti−ti−1=ln2.5/1.5π≈0.1944. By solving the optimization problem:(66)OPmaxλ,s.t.C1holds.
Ones can obtain thatλ≈0.4931 and ξ=2745063,3715964>0. Thus, we know that the systems (63a)–(63d) have exactly one globally exponentially stable 2π-periodic solution and the estimation of its exponential convergence rate is λ−ϕ≈0.4931−0.1944≈0.2987. The numerical simulation of systems (63a)–(63d) is shown in Figure 1.Figure 1
2π-periodic solutions of impulsive systems (63a)–(63d) in x∈0,1 and t∈−5,50.Remark 7.
For Example1, the system (63a) can be viewed as an integral-differential equation of the following form:(67)∂u∂tx,t=D1∂2u∂x2x,t+D2∫0te−t−sux,sds,+ft,ux,t,where,(68)ft,ux,t=D2∫−∞0e−t−sux,sds−dptupt,x+∑q=1nhpqtfquqt,x+∑q=1nypqtvqt+Jpt+∧q=1nαpqtvqt+∨q=1nβpqtvqt.
We divide the interval ofxa,xb with a uniform grid, where x0=xa=0,xN=xb=1 and space stepsize h=xb−xa/N, where N=10. For the second derivative of u with respect to x, we discretize it by applying centered differences operator Dx, defined by,(69)Dxuhxp=uhxp+1−2uhxp+uhxp−1h2,p=1,2,…,9.
Let time stepsizei=τ/m with m as an integer, and uhnxp denote the numerical approximation to uxp,tn. The systems (63a)–(63d) can be approximately discretized as follows:(70)uhn+1xp=uhnxp+iD1Dxuhn+1xp+i2D2∑l=1n+1e−tn+1−tluhlxp+iftn+1,uhn+1xp,n=0,1,2,…
The Dirichlet boundary conditions discretization can be written as follows:(71)uhnx0=uhnxN=0,n=0,1,2,….
The initial condition discretization can be written as follows:(72)uhnxp=u0xp,tn=1,n=−m,−m+1,−m+2,…,0.p=1,2,…,9.
More details and verification on simulation can be found in the work on integro-differential equations [27, 31].Example 2.
Consider the reaction-diffusion two-neuron system without impulses:(73a)∂upt,x∂t=∑k=11∂∂xkDpk∂upt,x∂xk−dptupt,x+∑q=1nhpqtfquqt,x+∑q=1nypqtvqt+Jpt+∧q=1napqt∫−∞tKpqt−sgquqs,xds+∧q=1nαpqtvqt+∨q=1nbpqt∫−∞tKpqt−sgquqs,xds+∨q=1nβpqtvqt,t≠ti,x∈X(73b)upt,x=0,t≥0,x∈∂X,(73c)ups,x=essinπx,s∈−∞,0.For p=1,2, X=0,1.
In system (73a)–(73c), choosing,(74)D11=D21=1,d1t=3+sint,d2t=2+cost,h11t=y11t=y12t=y21t=y22t=sint,h21t=h12t=0h22t=cost,a11t=b11t=0.15costa12t=b12t=0.1cost,a22t=b22t=0.1sint,a21t=b21t=0.1sint,K11s=K12s=K21s=K22s=e−sv1t=v2t=sint,α11t=α12t=cost,α21t=α22t=sint,β11t=β12t=sint,β21t=β22t=cost,J1t=cost,J2t=sint,f1u=f2u=g1u=g2u=u+1−u−12.
Distinctly, assumptions (H1) and (H3)–(H5) are satisfied, and then we have the following equation:(75)Wt=4+sint003+sint,F=G=1001,Rt=0.3cost0.2cost0.2sint0.2sint,Ht=sint00cost,Kλ=11−λ11−λ11−λ11−λ.
Ones can obtain thatλ≈0.6659 and ξ=15354274,18850909>0. Thus, the systems (73a)–(73c) have exactly one globally exponentially stable 2π-periodic solution. The numerical simulation of systems (73a)–(73c) is presented in Figure 2.Figure 2
2π-periodic solutions of system (73a)–(73c) without impulses in x∈0,1 and t∈−5,50.Remark 8.
Consider a continuous systems (73a)–(73c), which has been studied in [24]. If the parameters in [24] are taken, we get λ≈0.4669 and ξ=9218617,9668710>0. This indicates that the systems (73a)–(73c) are globally exponentially stable. If the impulsive condition of Example 1 is added to solve the above optimization problem, from Theorem 1, systems (63a)–(63d) are globally exponentially stable and the exponential convergence rate is estimated as λ−ϕ≈0.4669−0.1944≈0.2725. We use our conclusions to calculate systems (73a)–(73c) and parameters in [24], and the results are consistent with the results of [24], and our method can estimate the convergence rate of the models. This is what the method of [24] does not have.


## 5. Conclusion

We have developed and studied a new class of neural network models that bring together non-autonomous neural networks, fuzzy neural networks, reaction-diffusion terms, distributed time delays, impulses, and Dirichlet boundary conditions. In the form of a simple algebraic inequality, several new sufficient conditions are obtained to guarantee the global exponential stability and periodicity of the systems (1a)–(1d). In particular, in order to estimate the exponential convergence rate of the systems (1a)–(1d), an optimization method is proposed which relies on diffusion coefficients, Dirichlet boundary conditions, distributed time delays, system parameters, and impulses. The method is simpler and more effective than the Lyapunov generalized method used in much of the previous literature for the stability and periodicity analysis of complex systems. Two examples show that our results improve and generalize previously known criteria. In the near future, we will continue to investigate the global exponential stability and periodicity of non-autonomous impulsive neural networks with leakage time delays.



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