# Adaptive Exponential Synchronization for Stochastic Competitive Neural Networks with Time-Varying Leakage Delays and Reaction-Diffusion Terms

Authors: Zhiqiang Wang; Lili Wang; Rui Xu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

We study the exponential synchronization problem for a class of stochastic competitive neural networks with different timescales, as well as spatial diffusion, time-varying leakage delays, and discrete and distributed time-varying delays. By introducing several important inequalities and using Lyapunov functional technique, an adaptive feedback controller is designed to realize the exponential synchronization for the proposed competitive neural networks in terms ofp-norm. According to the theoretical results obtained in this paper, the influences of the timescale, external stimulus constants, disposable scaling constants, and controller parameters on synchronization are analyzed. Numerical simulations are presented to show the feasibility of the theoretical results.


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## Body

## 1. Introduction

Neural networks are mathematical models that are inspired by the structure and functional aspects of biological neural networks. Meyer-Baese et al. [1] proposed competitive neural networks with different timescales, which describe the dynamics of cortical cognitive maps with unsupervised synaptic modifications. In the competitive neural networks model, there are two types of state variables: the short-term-memory (STM) variables describing the fast neural activity and the long-term-memory (LTM) variables describing the slow unsupervised synaptic modifications. Hence, there are two timescales in the competitive neural networks, one of which corresponds to the fast change of the state and the other to the slow change of the synapse by external stimuli. The above competitive neural networks are described by the following differential equations:(1)STM:εdyitdt=-ciyit+∑j=1naijfyjt+Hi∑l=1rmiltwl,LTM:dmiltdt=-milt+wlfyit,where i=1,2,…,n, yi(t) is the neuron current activity level, mil(t) is the synaptic efficiency, f(yj(t)) is the output of neurons, ci>0 is the time constant of the neuron, aij denotes the connection strength of the jth neuron on the ith neuron, Hi is the strength of the external stimulus, wl is the constant external stimulus, r is the number of the constant external stimuli, and ε>0 is the timescale of the STM state.Synchronization problems of neural networks have been widely researched because of their extensive applications in secure communication, information processing, and chaos generators design. Synchronization of competitive neural networks with different timescales has attracted a great interest [2–7]. In [7], Gan et al. studied the adaptive synchronization for a class of competitive neural networks with different timescales and stochastic perturbation by constructing a Lyapunov-Krasovskii functional:(2)STM:εdyitdt=-ciyit+∑j=1naijfjyjt+∑j=1nbijfjyjt-τt,x+∑j=1ndij∫t-τ∗ttfjyjsds+Hi∑l=1rmiltwl,LTM:dmiltdt=-Kimilt+wlfiyit,where τ(t) and τ∗(t) are the discrete time-varying delay and the distributed time-varying delay, respectively; bij and dij are, respectively, the discrete time-varying delay connection strength and the distributed time-varying delay connection strength of the jth neuron on the ith neuron; Ki is the disposable scaling constant.The first termyi(t) in each of the right sides of (2) is called leakage term corresponding to a stabilizing negative feedback of the system [8, 9]. In real world, the transmission delays often appear in leakage terms, which are called leakage delays [10]. It is well known that leakage delays have been incorporated into neural networks by many researchers [11–14]. However, leakage delays of neural networks in most bibliographies listed above are constants. As pointed out in [15–18], the delays in neural networks are usually time-varying. Hence, the results about the neural networks with constant delays in the leakage term are imperfect.In addition, dynamic behaviors of neural networks derive from the interactions of neurons, which is dependent on not only the time of each neuron but also its space position [19, 20]. From this point, diffusion phenomena should not be ignored in neural networks. Many good results about reaction-diffusion neural networks have been obtained [21–25]. The boundary conditions in most literatures listed are assumed to be Dirichlet boundary conditions. In engineering applications, such as thermodynamics, Neumann boundary conditions need to be considered. As far as we know, there are few results concerning the synchronization of competitive neural networks with reaction-diffusion term under Neumann boundary conditions.Based on the above discussion, we are concerned with the combined effects of time-varying leakage delays, stochastic perturbation, and spatial diffusion on the synchronization of competitive neural networks with Neumann boundary conditions in terms ofp-norm via an adaptive feedback controller to improve the previous results. To this end, we discuss the following neural networks:(3)STM:ε∂yit,x∂t=DiΔyit,x-ciyit-δit,x+∑j=1naijfjyjt,x+∑j=1nbijfjyjt-τijt,x+∑j=1ndij∫t-τij∗ttfjyjs,xds+Hi∑l=1rmilt,xwl,LTM:∂milt,x∂t=-Kimilt,x+wlfiyit,x,where x=(x1,x2,…,xm)T∈Ω⊂Rm and Ω={x=(x1,x2,…,xm)T∣xk<mk} is a bound compact set with smooth boundary ∂Ω and mesΩ>0 in space Rm; y(t,x)=(y1(t,x),y2(t,x),…,yn(t,x)) with yi(t,x) denotes the state of the ith neuron at time t and in space x; Δ=∑k=1m∂2/∂xk2 is the Laplace operator; 0<τij(t)≤τ and 0<τij∗(t)≤τ∗ are the discrete time-varying delay and the distributed time-varying delay, respectively; 0<δi(t)≤δ is the time-varying leakage delay; Di>0 corresponds to the transmission diffusion coefficient along the ith neuron.Letsi(t,x)=∑l=1rmil(t,x)wl=miT(t,x)w, where mi(t,x)=(mi1(t,x),mi2(t,x),…,mir(t,x))T and w=(w1,w2,…,wr)T, and then then system (3) can be rewritten as(4)STM:ε∂yit,x∂t=DiΔyit,x-ciyit-δit,x+∑j=1naijfjyjt,x+∑j=1nbijfjyjt-τijt,x+∑j=1ndij∫t-τij∗ttfjyjs,xds+Hisit,x,LTM:∂sit,x∂t=-Kisit,x+w2fiyit,x,where |w|2=w12+w22+⋯+wr2. Without loss of generality, the input stimulus vector w is assumed to be normalized with magnitude |w|2=1. System (4) is simplified to(5)STM:ε∂yit,x∂t=DiΔyit,x-ciyit-δit,x+∑j=1naijfjyjt,x+∑j=1nbijfjyjt-τijt,x+∑j=1ndij∫t-τij∗ttfjyjs,xds+Hisit,x,LTM:∂sit,x∂t=-Kisit,x+fiyit,x.The boundary condition of system (5) takes the form(6)∂yit,x∂n≔∂yit,x∂x1,∂yit,x∂x2,…,∂yit,x∂xmT=0,t,x∈-τ¯,+∞×∂Ω,∂sit,x∂n≔∂sit,x∂x1,∂sit,x∂x2,…,∂sit,x∂xmT=0,t,x∈-τ¯,+∞×∂Ω.The initial value of system (5) takes the form(7)yθ,x=ϕyθ,x,sθ,x=ϕsθ,x,θ,x∈-τ¯,0×Ω,where τ¯=max⁡{δ,τ,τ∗}, s(θ,x)=(s1(θ,x),s2(θ,x),…,sn(θ,x))T, ϕy(θ,x)=(ϕ1y(θ,x),ϕ2y(θ,x),…,ϕny(θ,x))T, ϕs(θ,x)=(ϕ1s(θ,x),ϕ2s(θ,x),…,ϕns(θ,x))T∈C([-τ¯,0)×Ω,Rn), and C([-τ¯,0)×Ω,Rn) is the Banach space of continuous functions which maps [-τ¯,0)×Ω into Rn with the topology of uniform converge and p-norm (p is a positive integer) defined by (8)ϕyp=∫Ω∑i=1nsup-τ¯≤θ≤0⁡ϕiyθ,xpdx1/p,ϕsp=∫Ω∑i=1nsup-τ¯≤θ≤0⁡ϕisθ,xpdx1/p.In order to observe the exponential synchronization behavior of system (5), the response system with stochastic perturbation is designed as(9)STM:dzit,x=1εDiΔzit,x-cizit-δit,x+∑j=1naijfjzjt,x+∑j=1nbijfjzjt-τijt,x+∑j=1ndij∫t-τij∗ttfjzjs,xds+Hihit,x+uit,xdt+∑j=1nσijejt,x,ejt-δit,x,ejt-τijt,x,ejt-τij∗t,xdωjt,LTM:∂hit,x∂t=-Kihit,x+fizit,x,where z(t,x)=(z1(t,x),z2(t,x),…,zn(t,x))T and h(t,x)=(h1(t,x),h2(t,x),…,hn(t,x))T denote the state of the response system; e(t,x)=z(t,x)-y(t,x) and R(t,x)=h(t,x)-s(t,x) are the synchronization error system; σ=(σij)n×n is the noise intensity matrix and the stochastic disturbance ω(t)=[ω1(t),ω2(t),…,ωn(t)]T∈Rn is a Brownian motion defined on (Ω,F,P) (where Ω is the sample, F is the σ-algebra of subsets of the sample space, and P is the probability measure on F), and(10)Edωt=0,Edω2t=0,where E{·} is the mathematical expectation operator with respect to the given probability measure P; u(t,x)=(u1(t,x),u2(t,x),…,un(t,x))T is a feedback controller of the following form:(11)ut,x=ϵet,x.The feedback strength ϵ=diag⁡(ϵ1,ϵ2,…,ϵn) is updated by the following law:(12)∂ϵi∂t=-νieit,xpeμt,where μ>0 and νi>0(i=1,2,…,n) are arbitrary positive constants.The boundary condition and initial condition for response system (9) are given in the following forms:(13)∂zit,x∂n≔∂zit,x∂x1,∂zit,x∂x2,…,∂zit,x∂xmT=0,t,x∈-τ¯,+∞×∂Ω,∂hit,x∂n≔∂hit,x∂x1,∂hit,x∂x2,…,∂hit,x∂xmT=0,t,x∈-τ¯,+∞×∂Ω,(14)zθ,x=ψzθ,x,hθ,x=ψhθ,x,θ,x∈-τ¯,0×Ω,where ψz(θ,x)=(ψ1z(θ,x),ψ2z(θ,x),…,ψnz(θ,x))T and ψh(θ,x)=(ψ1h(θ,x),ψ2h(θ,x),…,ψnh(θ,x))T∈C([-τ¯,0)×Ω,Rn).Subtracting (5) from (9) yields the error system as follows:(15)STM:deit,x=1εDiΔeit,x-cieit-δit,x+∑j=1naijfj∗ejt,x+∑j=1nbijfj∗ejt-τijt,x+∑j=1ndij∫t-τij∗ttfj∗ejs,xds+HiRit,x+uit,xdt+∑j=1nσijejt,x,ejt-δit,x,ejt-τijt,x,ejt-τij∗t,xdωjt,LTM:∂Rit,x∂t=-KiRit,x+fi∗eit,x,where fj∗(ej(·,x))=fj(zj(·,x))-fj(yj(·,x)).In this paper, we give the following hypotheses.(
H
1
) There exists a positive constant Li such that the neuron activation function fi satisfies the following conditions: (16)fiv^i-fivˇi≤Liv^i-vˇi,where v^i,vˇi∈R,i=1,2,…,n.(
H
2
) There exists a positive constant ηij such that (17)σijv~1,v^1,vˇ1,v˘1-σijv~2,v^2,vˇ2,v˘22≤ηijv~1-v~22+v^1-v^22+vˇ1-vˇ22+v˘1-v˘22,for all v~1,v~2,v^1,v^2,vˇ1,vˇ2,v˘1,v˘2∈R, and σij(0,0,0,0)=0,i,j=1,2,…,n.(
H
3
) There exist positive constants ρ′ and ρ′′ such that δ˙i(t)≤ρ′<1 or δ˙i(t)≥ρ′′>1 for all t, i=1,2,…,n.(
H
4
) There exist positive constants ϱ′ and ϱ′′ such that τ˙ij(t)≤ϱ′<1 or τ˙ij(t)≥ϱ′′>1 for all t,i,j=1,2,…,n.(
H
5
) There exist positive constants ϱ∗′ and ϱ∗′′ such that τ˙ij∗(t)≤ϱ∗′<1 or τ˙ij(t)≥ϱ∗′′>1 for all t,i,j=1,2,…,n.The paper is organized as follows. In the next section, we introduce some definitions and state several lemmas which will be essential to our proofs. In Section3, by constructing a suitable Lyapunov functional, some new criteria are obtained to ensure the exponential synchronization of systems (5) and (9) under the adaptive feedback controller (11) and (12). Numerical simulations are carried out in Section 4 to illustrate the feasibility of the main theoretical results. A brief conclusion is given in Section 5.


## 2. Preliminary

In this section, we introduce some notations and lemmas which will be useful in the next section.Definition 1.
The noise-perturbed response system (9) and the drive system (5) can be exponentially synchronized under the adaptive controller (11) and (12) based on p-norm, if there exist constants γ,γ∗>0 and M,M∗≥1 such that (18)Ezt,x-yt,xp+Eht,x-st,xp≤MEψz-ϕype-γt+M∗Eψh-ϕspe-γ∗t,t,x∈0,+∞×Ω,where z(t,x),h(t,x) and y(t,x),s(t,x) are solutions of systems (9) and (5) with differential initial functions (14) and (7), respectively, and (19)zt,x-yt,xp=∫Ω∑i=1nzit,x-yit,xpdx1/p,ht,x-st,xp=∫Ω∑i=1nhit,x-sit,xpdx1/p.Lemma 2 (Wang [26], Itô’s formula).
Letx(t)(t≥0) be Itô processes, and (20)dxt=ftdt+gtdBt,where f∈L1(R+,Rn)  (L1 is the space of absolutely integrable function) and g∈L2(R+,Rn×m) (L2 is the space of square integrable function). If V(x,t)∈C2,1(Rn×R+;R), (C2,1(Rn×R+;R) is the family of all nonnegative functions on R+×Rn which are continuously twice differentiable in x and once differentiable in t), then V(x(t),t) are still Ito^ processes, and(21)dVxt,t=Vtxt,t+Vxxt,tft+12trgTtVxxxt,tgtdt+Vxxt,tgtdBt,where(22)Vtxt,t=∂Vxt,t∂t,Vxxt,t=∂Vxt,t∂x1,…,∂Vxt,t∂xn,Vxxxt,t=∂2Vxt,t∂ei∂ejn×n.Lemma 3 (Mei et al. [27]).
Letp≥2 and let a,b,h>0. Then(23)ap-1b≤p-1happ+bpphp-1,(24)ap-2b2≤p-2happ+2bpphp-2/2.Lemma 4 (Mao [28]).
LetF(x),G(x):[a,b]→R be continuous functions. Suppose that positive constants p and q satisfy (25)1p+1q=1.Then (26)∫abFxGxdx≤∫abFxpdx1/p∫abGxqdx1/q.Lemma 5 (Gu et al. [29]).
Suppose thatΩ is a bound domain of Rm with a smooth boundary ∂Ω. u(x),v(x) are real-valued functions belonging to C2(Ω∪∂Ω). Then(27)∫ΩuxΔvxdx=∫∂Ωux∂vx∂ndx-∫Ω∇uxT∇vxdx,where ∇=(∂/∂x1,∂/∂x2,…,∂/∂xm)T is the gradient operator.Lemma 6.
Letp≥2 be a positive integer and let Ω be a bound domain of Rl∗ with a smooth boundary ∂Ω. φ(x)∈C1(Ω) is a real-valued function and ∂φ(x)/∂n|∂Ω=0. Then(28)∫Ωφxpdx≤p-1λ1∫Ωφxp-2∇φx2dx,where λ1 is the smallest positive eigenvalue of the Neumann boundary problem:(29)-Δϑx=λϑx,x∈Ω,∂ϑx∂n=0,x∈∂Ω.The proof of Lemma6 is attached in Appendix.Remark 7.
Ifp=2, the integral inequality (28) is the Poincaré integral inequality in [30]. The smallest eigenvalue λ1 of the Neumann boundary problem (29) is determined by the boundary of Ω [30]. If Ω={x=(x1,x2,…,xl∗)T∣mk-≤xk≤mk+,k=1,2,…,l∗}⊂Rl∗, then(30)λ1=min⁡πm1+-m1-2,πm2+-m2-2,…,πml∗+-ml∗-2.


## 3. Exponential Synchronization Criterion

In this section, the exponential synchronization criterion of the drive system (5) and the response system (9) is obtained under the adaptive feedback controller (11) and (12). For convenience, the following denotations are introduced.Denote(31)αi=1κ1-ρeμδ1ε1ξi∗∗p-1+p-1∑j=1nηjiςi∗p-2/2,βij=1eμτκ1-ϱp-1ηjiςi∗∗p-2/2+1εbjiLiϖi∗p-1,γij=p-1ηjieμτ∗κ1-ϱ∗ςi∗∗∗p-2/2,αi∗=αi-καisgn⁡1-ρ,βij∗=βij-κβijsgn⁡1-ϱ,γij∗=γij-κγijsgn⁡1-ϱ∗,γij∗∗=1ετ∗p-1Lidjiϖi∗∗p-1,li=εp-1εpλ1Di+μ+αieμδ+1εcip-1ξi∗∗+Liξip-1+1εHip-1ξi∗+eμτ∑j=1nβij+eμτ∗∑j=1nγij+τ∗eμτ∗∑j=1nγij∗∗+1ε∑j=1n1ϖip-1ajiLi+1εp-1∑j=1nbijLjϖj∗+1εp-1∑j=1ndijLjϖj∗∗+1εp-1∑j=1naijLjϖj+p-1p-22∑j=1nηijςj+p-1p-22∑j=1nηijςj∗+p-1p-22∑j=1nηijςj∗∗+p-1p-22∑j=1nηijςj∗∗∗+p-1∑j=1nηjiςip-2/2,where 0<κ<1, |1-ρ|=max⁡{|1-ρ′|,|1-ρ′′|}, |1-ϱ|=max⁡{|1-ϱ′|,|1-ϱ′′|}, |1-ϱ∗|=max⁡{|1-ϱ∗′|,|1-ϱ∗′′|}, and ξi, ξi∗, ξi∗∗, ςi, ςi∗, ςi∗∗, ςi∗∗∗, ϖi, ϖi∗, and ϖi∗∗ are nonnegative real numbers, respectively.Theorem 8.
Under assumptions(H1)–(H5), the nonlinear couple neural networks (9) and (5) can be exponentially synchronized under the adaptive feedback controller (11) and (12) based on p-norm, if the following condition is also satisfied.
(
H
6
)
μ
-
p
K
i
+
L
i
(
p
-
1
)
ξ
i
+
1
/
ε
|
H
i
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/
ξ
i
∗
p
-
1
≤
0.Proof.
Define(32)Vt=∫Ω∑i=1nVit,x+αieμδ∫t-δittVis,xds+αi∗eμδ∫t-δt-δitVis,xds+eμτ∑j=1nβij∫t-τijttVis,xds+eμτ∑j=1nβij∗∫t-τt-τijtVis,xds+eμτ∗∑j=1nγij∫t-τij∗ttVis,xds+eμτ∗∑j=1nγij∗∫t-τ∗t-τij∗tVis,xds+eμτ∗∑j=1nγij∗∗∫-τ∗0∫t+stViη,xdηds+eμtRit,xp+p2ενiϵi+li2dx,where Vi(t,x)=eμt|ei(t,x)|p.
By (10), Itô’s differential formula, and Dini derivation, it can be deduced that(33)D+EVt=E∫Ω∑i=1nμVit,x+αieμδVit,x-Vit-δit,x1-δ˙it+αi∗eμδVit-δit,x1-δ˙it-Vit-δ,x+eμτ∑j=1nβijVit,x-Vit-τijt,x1-τ˙ijt+eμτ∑j=1nβij∗Vit-τijt,x1-τ˙ijt-Vit-τ,x+eμτ∗∑j=1nγijVit,x-Vit-τij∗t,x1-τ˙ij∗t+eμτ∗∑j=1nγij∗Vit-τij∗t,x1-τ˙ij∗t-Vit-τ∗,x+eμτ∗∑j=1nγij∗∗∫-τ∗0Vit,x-Vit+s,xds+μeμtRit,xp+peμtRit,xp-1-KiRit,x+fi∗eit,x-pεϵi+lieit,xpeμt+1εpeμteit,xp-1DiΔeit,x-cieit-δit,x+∑j=1naijfj∗ejt,x+∑j=1nbijfj∗ejt-τijt,x+HiRit,x+ϵieit,x+pp-1eμteit,xp-22·∑j=1nσij2ejt,x,ejt-δit,x,∫Ω∑i=1nμVit,x+αieμδVit,x-Vit-δit,x1-δ˙itejt-τijt,x,ejt-τij∗t,xdx≤E∫Ω∑i=1nμVit,x+αieμδVit,x-καi1-ρeμδVit-δit,x+eμτ∑j=1nβijVit,x-eμτ∑j=1nκβij1-ϱVit-τijt,x+eμτ∗∑j=1nγijVit,x-eμτ∗∑j=1nκγij1-ϱ∗Vit-τij∗t,x+τ∗eμτ∗∑j=1nγij∗∗Vit,x-eμτ∗∑j=1nγij∗∗∫t-τ∗tVis,xds+μ-pKieμtRit,xp+peμtRit,xp-1Lieit,x-pεϵi+lieit,xpeμt+1εpeμteit,xp-1DiΔeit,x+cieit-δit,x+∑j=1naijLjejt,x+∑j=1nbijLjejt-τijt,x+∑j=1ndij∫t-τij∗ttLjejs,xds+HiRit,x+ϵieit,x+pp-1eμteit,xp-22∑j=1nηijejt,x2+ejt-δit,x2+ejt-τijt,x2+ejt-τij∗t,x2∫Ω∑i=1nμVit,x+αieμδVit,xdx∫Ω∑i=1nμVit,x+αieμδVit,x.
From the boundary conditions (6) and (13) and Lemma 6, we get(34)p∫Ωeit,xp-1DiΔeit,xdx=p∫∂Ωeit,xp-1Di∑k=1m∂eit,x∂xkcos⁡xk,nds-∫Ω∑k=1mDi∂eit,x∂xk·∂eit,xp-1∂xkdx=-pp-1Di∫Ωeit,xp-2∑k=1m∂eit,x∂xk·∂eit,x∂xkdx=-pp-1Di∫Ωeit,xp-2∣∇eit,x2dx≤-pλ1Di∫Ωeit,xpdx.
By Lemma4, we obtain(35)∫t-τ∗tVis,xds≥eμte-μτ⋆∫t-τij∗tteis,xpds≥eμte-μτ⋆∫t-τij∗tteis,xdspτij∗tp/q≥eμte-μτ⋆τ∗1-p∫t-τij∗tteis,xdsp.
It follows from (23) that(36)pRit,xp-1eit,x≤p-1ξiRit,xp+eit,xpξip-1,peit,xp-1Rit,x≤p-1ξi∗eit,xp+Rit,xpξi∗p-1,peit,xp-1eit-δit,x≤p-1ξi∗∗eit,xp+eit-δit,xpξi∗∗p-1,peit,xp-1∑j=1naijLjejt,x≤p-1eit,xp∑j=1naijLjϖj+∑j=1nLjaijejt,xpϖjp-1,peit,xp-1∑j=1nbijLjejt-τijt,x≤p-1eit,xp∑j=1nbijLjϖj∗+∑j=1nLjbijejt-τijt,xpϖj∗p-1,peit,xp-1∑j=1ndijLj∫t-τij∗ttejs,xds≤p-1eit,xp∑j=1ndijLjϖj∗∗+∑j=1ndijLjϖj∗∗p-1∫t-τij∗ttejs,xdsp.
It follows from (24) that(37)peit,xp-2∑j=1nηijejt,x2≤p-2eit,xp∑j=1nηijςj+∑j=1n2ηijejt,xpςjp-2/2,peit,xp-2∑j=1nηijejt-δit2≤p-2eit,xp∑j=1nηijςj∗+∑j=1n2ηijejt-δitpςj∗p-2/2,peit,xp-2∑j=1nηijejt-τijt2≤p-2eit,xp∑j=1nηijςj∗∗+∑j=1n2ηijejt-τijtpςj∗∗p-2/2,peit,xp-2∑j=1nηijejt-τij∗t2≤p-2eit,xp∑j=1nηijςj∗∗∗+∑j=1n2ηijejt-τij∗tpςj∗∗∗p-2/2.
Substituting (34)–(37) into (33), it follows from (31) and (H5) that(38)D+EVt≤E∫Ω∑i=1nμ+αieμδ+eμτ∑j=1nβij+eμτ∗∑j=1nγij+τ∗eμτ∗∑j=1nγij∗∗+Liξip-1-pεli-1εpλ1Di+1εcip-1ξi∗∗+1εp-1∑j=1naijLjϖj+1εp-1∑j=1nbijLjϖj∗+1εp-1∑j=1ndijLjϖj∗∗+1εHip-1ξi∗+1ε∑j=1n1ϖip-1ajiLi+12p-1p-2∑j=1nηijςj+12p-1p-2∑j=1nηijςj∗+12p-1p-2∑j=1nηijςj∗∗+12p-1p-2∑j=1nηijςj∗∗∗+p-1∑j=1nηjiςip-2/2Vit,x+1ε1ξi∗∗p-1+p-1∑j=1nηjiςi∗p-2/21ε1ξi∗∗p-1+p-1∑j=1nηjiςi∗p-2/2-καi1-ρeμδVit-δit,x+p-1∑j=1nηjiςi∗∗p-2/2+1ε∑j=1nbjiLiϖi∗p-1-eμτ∑j=1nκβij1-ϱVit-τijt,x+p-1∑j=1nηjiςi∗∗∗p-2/2-eμτ∗∑j=1nκγij1-ϱ∗Vit-τij∗t,x+1ε∑j=1ndjiLiϖi∗∗p-1-∑j=1nγij∗∗τ∗1-p·eμt∫t-τij∗tteis,xdsp+1εHiξi∗p-1+μ-pKi+Lip-1ξi∫Ω∑i=1nμ+αieμδ+eμτ∑j=1nβijeμtRit,xpdx=E∫Ω∑i=1n1εHiξi∗p-1+μ-pKi+Lip-1ξieμtRit,xpdx≤0which implies that (39)EVt≤EV0.
Note that(40)EV0≤E∫Ω∑i=1nei0,xp+Ri0,xp+p2ενiϵi0,x+li2+αieμδ∫-δi00eμseis,xpds+αi∗eμδ∫-δ-δi0eμseis,xpds+eμτ∑j=1nβij∫-τij00eμseis,xpds+eμτ∑j=1nβij∗∫-τ-τij0eμseis,xpds+eμτ∗∑j=1nγij∫-τij∗00eμseis,xpds+eμτ∗∑j=1nγij∗∫-τ∗-τij∗0eμseis,xpds+eμτ∗∑j=1nτ∗γij∗∗∫-τ∗0eμseis,xpds∫Ω∑i=1nei0,xp+Ri0,xp+p2ενiϵi0,x+li2dx≤E∫Ω∑i=1nei0,xp+Ri0,xp+p2ενiϵi0,x+li2+αieμδ∫-δi00eμseis,xpds+αi∗eμδ∫-δ-δi0eμseis,xpds+eμτmaxi=1,2,…,n⁡∑j=1nβij∑j=1n∫-τij00eμseis,xpds+eμτmaxi=1,2,…,n⁡∑j=1nβij∗∑j=1n∫-τ-τij0eμseis,xpds+eμτ∗maxi=1,2,…,n⁡∑j=1nγij∑j=1n∫-τij∗00eμseis,xpds+eμτ∗maxi=1,2,…,n⁡∑j=1nγij∗∑j=1n∫-τ∗-τij∗0eμseis,xpds+τ∗eμτ∗maxi=1,2,…,n⁡∑j=1nγij∗∗∑j=1n∫-τ∗0eμseis,xpds∫Ω∑i=1nei0,xp+Ri0,xp+p2ενiϵi0,x+li2dx.
Since(41)ψz-ϕyp=∫Ω∑i=1nsup-τ¯≤θ≤0⁡ψzθ,x-ϕiyθ,xpdx1/p,then(42)∫Ω∑i=1n∫-τij00eμseis,xpdsdx≤∫Ω∑i=1n∫-τ0eis,xpdsdx≤∫Ω∑i=1nτsup-τ¯≤s≤0⁡eis,xpdx=τψz-ϕypp.
Similarly,(43)∫Ω∑i=1nei0,xpdx≤ψz-ϕypp,∫Ω∑i=1nRi0,xpdx≤ψh-ϕspp,∫Ω∑i=1n∫-τ∗0eμseis,xpdsdx≤τ∗ψz-ϕypp,∫Ω∑i=1n∫-δ-δi0eμseis,xpdsdx≤δψz-ϕypp,∫Ω∑i=1n∫-δi00eμseis,xpdsdx≤δψz-ϕypp,∫Ω∑i=1n∫-δ-δi0eμseis,xpdsdx≤δψz-ϕypp,∫Ω∑i=1n∫-τ-τij0eμseis,xpdsdx≤τψz-ϕypp,∫Ω∑i=1n∫-τij∗00eμseis,xpdsdx≤τ∗ψz-ϕypp,∫Ω∑i=1n∫-τ∗-τij∗0eμseis,xpdsdx≤τ∗ψz-ϕypp.
Applying (42)-(43) into (40), we have(44)EV0≤1+αiδeμδ+αi∗δeμδ+nτeμτmaxi=1,2,…,n⁡∑j=1nβij+nτeμτmaxi=1,2,…,n⁡∑j=1nβij∗+nτ∗eμτ∗maxi=1,2,…,n⁡∑j=1nγij+nτ∗eμτ∗maxi=1,2,…,n⁡∑j=1nγij∗+nτ∗2eμτ∗maxi=1,2,…,n⁡∑j=1nγij∗∗+Nψz-ϕyppEψz-ϕypp+Eψh-ϕspp=MEψz-ϕypp+M∗Eψh-ϕspp,where (45)M∗=1,N=∫Ω∑i=1np2ενiϵi0,x+li2dx,M=1+αiδeμδ+αi∗δeμδ+nτeμτmaxi=1,2,…,n⁡∑j=1nβij+nτeμτmaxi=1,2,…,n⁡∑j=1nβij∗+nτ∗eμτ∗maxi=1,2,…,n⁡∑j=1nγij+nτ∗eμτ∗maxi=1,2,…,n⁡∑j=1nγij∗+nτ∗2eμτ∗maxi=1,2,…,n⁡∑j=1nγij∗∗+Nψz-ϕypp≥1.
Therefore,(46)EVt≤EV0≤MEψz-ϕypp+M∗Eψh-ϕspp.
Further, we obtain(47)EVt≥E∫Ω∑i=1nVit,x+eμtRit,xpdx=eμtE∫Ω∑i=1nzit,x-yit,xp+hit,x-sit,xpdx=eμtEzt,x-yt,xp+eμtEht,x-st,xp.From (46) and (47), we have(48)Ezt,x-yt,xp+Eht,x-st,xp≤MEψz-ϕyppe-μt+M∗Eψh-ϕsppe-μt.Hence, the nonlinear couple neural networks (9) and (5) can be exponentially synchronized under the adaptive feedback controller (11) and (12) based on p-norm. The proof of Theorem 8 is complete.Remark 9.
It is the first time to consider the combined effects of time-varying leakage delays, the discrete time-varying delay, the distributed time-varying delay, stochastic perturbation, and spatial diffusion on the exponential synchronization of competitive neural networks under an adaptive feedback controller. The neural networks discussed in [6, 7, 31] are the special cases of the model in this paper. From this point, our results are more general.Remark 10.
In Theorem8, the sufficient conditions are derived to achieve the adaptive synchronization for the proposed competitive neural networks. Compared with the adaptive synchronization criteria given in [7], the conditions obtained in Theorem 8 depend on not only the timescale ε but also the controller parameter μ. It is beneficial to design an adaptive controller to realize the adaptive synchronization for the neural networks. Therefore, the criteria derived in this paper have wider application.


## 4. Numerical Simulations

In this section, some numerical simulation examples demonstrate the main results in Theorem8.In system (5), we choose n=2. Then system (5) takes the form(49)STM:ε∂yit,x∂t=DiΔyit,x-ciyit-δt,x+∑j=1naijfjyjt,x+∑j=1nbijfjyjt-τt,x+∑j=1ndij∫t-τ∗ttfjyjs,xds+Hisit,x,LTM:∂sit,x∂t=-Kisit,x+fiyit,x,where fj(uj(t,x))=tanh⁡(uj(t,x)), δ(t)=0.4+0.3cos⁡t, τ(t)=0.5+0.4cos⁡t, and τ∗(t)=0.3+0.2sin⁡t. The parameters of (49) are assumed as follows: ε=0.5, D1=0.025, D2=0.025, c1=0.35, c2=0.25, a11=0.625, a12=-0.3, a21=-0.225, a22=-0.3, b11=-0.475, b12=-0.425, b21=0.375, b22=0.475, d11=-0.375, d12=-0.325, d21=0.6, d22=0.8, H1=0.375, H2=0.375, K1=1, K2=1, and x∈Ω=[-5,5]. The initial conditions of system (49) are chosen as(50)y1s,x=0.1cos⁡x+510π,y2s,x=0.2cos⁡x+510π,s1s,x=0.3cos⁡x+510π,s2s,x=0.4cos⁡x+510π,where (s,x)∈[-0.9,0]×Ω.Numerical simulation illustrates that the reaction-diffusion neural network (49) with boundary condition (6) and the initial condition (50) exhibits a chaotic behavior (see Figure 1).Figure 1
Chaotic behaviors of competitive neural networks (49).The noise-perturbed response system is described by(51)STM:dzit,x=1εDiΔzit,x-cizit-δt,x+∑j=1naijfjzjt,x+∑j=1nbijfjzjt-τt,x+∑j=1ndij∫t-τ∗ttfjzjs,xds+Hihit,x+uit,xdt+∑j=1nσijejt,x,ejt-δit,x,ejt-τijt,x,ejt-τij∗t,xdωjt,LTM:∂hit,x∂t=-Kihit,x+fizit,x,where (52)σ11=0.1e1t,x+0.2e1t-τt,x+0.1e1t-δt,x+0.3e1t-τ∗t,x,σ12=0,σ21=0,σ22=0.2e2t,x+0.1e2t-τt,x+0.3e2t-δt,x+0.1e2t-τ∗t,x.The adaptive controller is (53)uit,x=ϵieit,x,∂ϵi∂t=-eit,x2ei0.1t.The initial conditions for the response system (51) are chosen as (54)z1s,x=0.1cos⁡x+510π,z2s,x=0.2cos⁡x+510π,h1s,x=0.3cos⁡x+510π,h2s,x=0.4cos⁡x+510π,where (s,x)∈[-0.9,0]×Ω.Evidently,L1=L2=1, ρ′=0.3<1, ϱ′=0.4<1, and ϱ∗′=0.2<1. Let p=2, ε=0.5, ξ1=ξ2=0.1, and ξ1∗=ξ2∗=1. By simple computation, it is easy to verify that assumptions (H1)–(H6) are satisfied. According to Theorem 8, the drive system (49) and the response system (51) are exponentially synchronized based on p-norm. Numerical simulation illustrates our results (see Figure 2).Figure 2
Asymptotical behaviors of the synchronization errors between systems (49) and (51).Remark 11.
The conclusions given in Theorem8 show that the adaptive synchronization criteria for competitive neural networks are dependent on the timescale ε, the disposable scaling constant Ki, and the external stimulus Hi. When ε increases, and Ki increases or Hi decreases, respectively, assumption (H6) can be satisfied more easily, and the adaptive synchronization of the competitive neural networks is more easily realized. Dynamical behaviors of the synchronization errors between systems (49) and (51) with the differential timescale, disposable scaling constant, and external stimulus, respectively, are shown in Figures 3–5.Figure 3
Asymptotical behaviors of the synchronization errors with differential timescaleε.Figure 4
Asymptotical behaviors of the synchronization errors with differential disposable scaling constantsKi.Figure 5
Asymptotical behaviors of the synchronization errors with differential external stimulus constantsHi.Remark 12.
By (48), it is clear to see that the controller parameter μ denotes the rate of the synchronization. That is, the larger the controller parameter μ is, the faster systems (49) and (51) realize synchronization. Hence, our results are consistent with the practical situation. Dynamical behaviors of the synchronization errors between systems (49) and (51) with differential controller parameter μ are shown in Figure 6.Figure 6
Asymptotical behaviors of the synchronization errors with differential controller parameterμ.The parameterνi is another controller parameter in the feedback controller (11). By numerical simulations, we can see that it is beneficial for competitive neural networks to realize the synchronization by increasing controller parameter νi. Dynamical behaviors of the synchronization errors between systems (49) and (51) with differential controller parameter νi are shown in Figure 7. However, we cannot prove it. It is an interesting open problem to research.Figure 7
Asymptotical behaviors of the synchronization errors with differential controller parameterνi.Remark 13.
In many cases, two-neuron networks show the same behavior as large-size networks and many research methods used in two-neuron networks can be applied to large-size networks. Therefore, a two-neuron network can be used as an example to improve our understanding of our theoretical results. In addition, the parameter values are selected randomly to ensure that neural networks (49) exhibit a chaotic behavior.


## 5. Conclusion

In this paper, an adaptive feedback controller was designed to achieve the exponential synchronization for stochastic competitive neural networks with spatial diffusion, time-varying leakage delays, and discrete and distributed time-varying delays based onp-norm. Evidently, the model discussed in this paper is more general than those correspondent models when the delays are constant delays. By constructing the Lyapunov functional and using and the stochastic analysis theory, the novel exponential synchronization criteria dependent on the timescale ε, external stimulus constants Hi, disposable scaling constants Ki, and controller parameter μ were obtained. By theory analysis, it was shown that competitive neural networks can achieve exponential synchronization more easily by increasing the timescale and disposable scaling constants or reducing disposable scaling constants, respectively. Numerical examples and their simulations are given to show the effectiveness of the obtained results.Figures8 and 9 show that it is beneficial for competitive neural networks with reaction-diffusion terms to realize the synchronization by increasing diffusion coefficients Di or decreasing diffusion space x, respectively. However, the exponential synchronization criteria obtained in this paper are independent of the diffusion coefficients and the diffusion space. They cannot reflect the influence of the diffusion coefficients and diffusion space on synchronization, which limits the application scopes of the results. Therefore, we will investigate that in our future work.Figure 8
Asymptotical behaviors of the synchronization errors with differential diffusion coefficientsDi.Figure 9
Asymptotical behaviors of the synchronization errors with differential diffusion space.



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