# Global Exponential Stability and Periodicity of Nonautonomous Impulsive Neural Networks with Time-Varying Delays and Reaction-Diffusion Terms

Authors: Weiyi Hu; Kelin Li
Journal: Complexity
 (2021)

---

## Abstract

In this paper, we investigate the global exponential stability and periodicity of nonautonomous cellular neural networks with reaction-diffusion, impulses, and time-varying delays. By establishing a new differential inequality for nonautonomous systems, using the properties of M-matrix and inequality techniques, some new sufficient conditions for the global exponential stability of the system are obtained. Moreover, sufficient conditions for the periodic solutions of the system are obtained by using the Poincare mapping and the fixed point theory. The validity and superiority of the main results are verified by numerical examples and simulations.


---

## Body

## 1. Introduction

Since Chua proposed the cellular neural networks (CNNs) in the 1980s [1], the neural network models have been widely studied and applied in the fields of signal recognition, image processing, pattern classification, and so on. Indeed, all these applications rely on the dynamic behaviors of neural networks. The key to using neural networks to solve these problems is that the neural network must be globally dynamic stable; that is to say, each of its loci must converge to a unique balance. In the practical application of neural networks, time delay is inevitable because of the limited signal transmission and switching speed. As we all know, the time delay may make the system lose stability or produce nonlinear oscillation, which makes the dynamic behaviors of the system more complicated. Under the circumstances, the neural network model with time delay is introduced to better reflect the actual state. Hence, how to ensure the stability and good performance of the system is particularly important in the case of time delay. The dynamic analysis of time-delay neural networks has received considerable attention; for example, see [2–7].In practical applications, neurons change over time and space. In order to better describe the state of neurons, a partial differential equation that considers diffusion is introduced in space. So, in 1995, Chua et al. [8] established the structure of reaction-diffusion CNNs. In [9,10], the authors have investigated the global exponential stability and the global exponential robust stability of reaction-diffusion neural networks with time-varying delays, by using partial differential equations to express the diffusion term of the CNNs, some interesting results were obtained. In [11], the authors used the properties and inequality analysis of the nonsingular M-matrix to study the exponential stability of CNNs with diffusion. In [12], considering the global exponential stability under the Neumann boundary conditions, using the properties of the M-matrix and algebraic inequality techniques, some sufficient conditions for the global exponential stability of the system are obtained. In [13], under Neumann boundary conditions, several sufficient conditions based on linear matrix inequalities for the global asymptotic stability of reactor-diffusion CNNs with distributed delay are given. In [14, 15], the Dirichlet boundary conditions of the rectangular region are introduced into the neural network with reaction-diffusion. In [16], under the Dirichlet boundary conditions, a diffusive-dependent Lyapunov functional related to the diffusion term is proposed to analyze the effect of the diffusion coefficient of neurons on the dynamics of the model, and the stabilization effect of the reaction-diffusion term is studied by using the extended Wirtinger inequality. In [17, 18], the authors considered the exponential stability of reaction-diffusion neural networks with time delays in general spatial regions and obtained some new sufficient conditions for system stability by using Poincare’s inequality technique.As we all know, the phenomenon of nonautonomy arises in a lot of real systems frequently. In particular, when a system is in a dynamic behavior for a long time, the parameters of the system often change with time. Compared with autonomous CNNs, the study of nonautonomous CNNs is equally important. In fact, many authors have explored the dynamic behavior of different types of nonautonomous CNNs with time delays and produced some interesting results by constructing some Lyapunov–Krasovskii functionals [19–26]. In addition, some functions in these methods must be differentiable, while other features require additional constraints in terms of delay. In [27], the author improved some of the earlier Halanay-type inequalities and obtained the exponential stability of CNNs and the existence of periodic solutions through these new inequalities. In [28], sufficient conditions for the global exponential stability of a special class of nonautonomous differential systems with infinite time delay are given, without proving the existence of equilibrium point, periodic solution, or almost periodic solution. In addition, during the physical simulation of the neural network, the neural network will sometimes be interrupted due to the transient disturbance, which is called the impulse effect. Impulse may cause system instability or other more complex dynamic behavior. The influence of pulses on neural networks has been considered in many literatures [29–33]. Therefore, it is of great significance to study the stability and periodicity of neural network models with dynamic impulses, time delays, and reaction-diffusion terms.Here, we introduce a class of new nonautonomous impulsive neural networks with time-varying delays and reaction-diffusion terms as follows:(1a)∂zpt,x∂t=∑k=1m∂∂xkDpk∂zpt,x∂xk−βptzpt,x+∑q=1napqtfqzqt,x+∑q=1nbpqtgqzqt−τpqt,x+Jpt,t≠tl,p=1,2,…,n,x∈Ξ,(1b)zptl+,x=Iplzptl−,x,p=1,2,…,n,x∈Ξ,l∈N≜0,1,2,…,(1c)zpt,x=0,t≥0,x∈∂Ξ,(1d)zps,x=μps,x,s∈−τ,0,p=1,2,…,n,where zpt,x is the state of the pth neuron in space x and time t. x=x1,x2,…,xmT∈Ξ⊂Rm. Ξ=x=x1,x2,…,xmT‖xk|<Lk,k=1,2,…,m is a bounded compact set with smooth boundary ∂Ξ and mes Ξ>0 in space Rm (Lk is a positive constant). zt,x=z1t,x,z2t,x,…,znt,xT∈Rn. βp∈CR+,R+ denotes the rate at which the pth neuron resets its potential to the isolated resting state when disconnected from the network and external input; apq∈CR+,R denotes the intensity of the pth element on the qth element at time t; bpq∈CR+,R represents the strength of the qth element due to delay at the pth element. τpqt represents the transmission delay from the pth element to the qth element and satisfies 0≤τpqt≤τpq≤ττ=max1≤p,q≤nτpq; fq∈CR,R and gq∈CR,R denote the activation function and activation function caused by delay, respectively; Jpt represents the external input of the pth neuron at time t. In (1b), tl>0 satisfies tl<tl+1,liml⟶+∞tl=+∞;zptl−,x and zptl+,x represent the left and right limits at tk, respectively; Ipk shows impulsive perturbation of the pth neuron at time tk. Let zptl+,x=zptl,x,l∈N. Equation (1c) denotes the Dirichlet boundary conditions and (1d) denotes the initial conditions.If impulsive operatorIplzp=0,p=1,2,…,n,l∈N, systems (1a)–(1d) will become a model of the following form:(2a)∂zpt,x∂t=∑k=1m∂∂xkDpk∂zpt,x∂xk−βptzpt,x+∑q=1napqtfqzqt,x+∑q=1nbpqtgqzqt−τpqt,x+Jpt,p=1,2,…,n,x∈Ξ,(2b)zpt,x=0,t≥0,x∈∂Ξ,(2c)zps,x=μps,x,s∈−τ,0,p=1,2,…,n.Systems (2a)–(2c) are called the continuous form of impulsive systems (1a)–(1d).The main contributions of this work are as follows:(I)
We have formulated a class of new neural network models which assembles nonautonomous neural networks, reaction-diffusion cellular neural networks with time-varying delays, impulses, and the Dirichlet boundary conditions.(II)
We have obtained several new criteria which guarantee the global exponential stability and periodicity of systems (1a)–(1d). These criteria are shown in simple algebraic inequality forms, and they rely only on the system parameters.(III)
An optimization method is proposed in order to estimate the exponential convergence rate of systems (1a)–(1d).The remainder of this paper is organized as follows. In Section2, some assumptions and the necessary knowledge are provided. We give our main results in Section 3. In Section 4, two examples are given to show the effectiveness of the results obtained here. Finally, we give the conclusion in Section 5.


## 2. Preliminaries

In this section, we make some assumptions and introduce some notions.(A1) For the activation functionsfpz and gpz, there exist positive diagonal matrices F=diagF1,F2,…,Fn and G=diagG1,G2,…,Gn such that(3)Fp=supz≠wfpz−fpwz−w,Gp=supz≠wgpz−gpwz−w,for allz,w∈Rz≠w,p=1,2,…,n.(A2) There exists a positive diagonal matrixℑl=diagI1l,…,Inl such that(4)Iplz−Iplz≤Iplz−v,for allz,v∈Rz≠v,p∈1,2,…,n,l∈N.(A3)apqt,bpqt,βpt,Jpt, and τpqt are continuous bounded functions defined on t∈0,+∞ for all p,q∈1,2,…,n.We definezpt,x2=∫Ξzpt,x2dx1/2, p=1,2,…,n. R+=0,+∞ and h+t=max0,ht, where ht is a continuous function. For any ωs,x=ω1s,x,ω2s,x,…,ωns,xT∈PCΞ, the norm on PCΞ is defined by(5)ω=sup−τ≤s≤0∑p=1nωps,x2.PCΞ≜ω:−τ,0×Ξ⟶Rn|ωs,x is bounded on −τ,0×Ξ and ωs+,x=ωs,x for s∈−τ,0.ωs−,x exists for s∈−τ,0 and ωs−,x=ωs,x for all but a finite number of points s∈−τ,0}.PC≜ω:−τ,0⟶Rn|ωs is bounded on −τ,0 and ωs+=ωs for s∈−τ,0. ωs− exists for s∈−τ,0 and ωs−=ωs for all but a finite number of points s∈−τ,0.ForC,D∈Rm×n, we define the Schur product by C⊗D=cpqdpqm×n, and C≥DC>D means that the inequality cpq≥bpqcpq>dpq holds. e=1,1,…,1T∈Rn and E is a n-dimensional identity matrix.Definition 1.
A functionzt,xz:−τ,+∞×Ξ⟶Rn is said to be the solution of system (1a)–(1d) if zt,x satisfies the following conditions:(1)
Fort, zt,x is piecewise continuous with the first kind of discontinuity at the points tl, l∈N, and zt,x is right-continuous at every discontinuity point(2)
zt,x satisfies (1a)–(1c) for all t≥0, and zt,x=μt,x for t∈−τ,0
Therefore,zt,x is a particular solution that satisfies the initial conditions for systems (1a)–(1d), and zt,μ,x represents the special solution of systems (1a)–(1d) under initial condition μ∈PCΞ.Definition 2.
zt,ω,x is a particular solution of systems (1a)–(1d) and satisfies the initial conditions ω∈PCΞ. In the initial condition of ϖ∈PCΞ, zt,ϖ,x is any solution to systems (1a)–(1d). If there is a positive number λ>0 and M≥1 such that(6)zt,ω,x−zt,ϖ,x≤Mω−ϖe−λt,for allt≥0,then systems (1a)–(1d) are said to be globally exponentially stable.Definition 3 (see [34]).
A real matrixD=dpqn×n is said to be a nonsingular M-matrix if dpq≤0,p,q=1,2,…,n,p≠q, and all successive principal minors of D are positive.Lemma 1 (see [34]).
LetD=dpqn×n with dpq≤0p≠q, then D is a nonsingular M-matrix if and only if the diagonal elements of D are all positive, and there exists a positive vector d such that Dd>0 or DTd>0.Lemma 2 (see [35]).
LetΞ be a cube xk<Lkk=1,2,…,m, and let hx be a real-valued function belonging to C1Ξ which vanishes on the boundary an of Ξ, i.e., hx|∂Ξ=0. Then,(7)∫Ξh2xdx≤Lk2∫Ξ∂h∂xk2dx.Lemma 3.
Ifax and bx are continuous on x0,+∞ with ax−bx≥ρ>0 for all x∈x0,+∞ and there is a positive number L such that 0≤bx≤L for all x≥x0, then λ∗≡infx≥x0λ>0:λ−ax+bxeλτx=0>0, where τx is nonnegative function (i.e., τx≥0 for all x≥0).Proof.
SetFλ=λ−ax+bxeλτx. Note that(8)F0=−ax+bx<0,dFdλ=1+τxbxeλτx>0andF+∞>0,then we know that Fλ is a strictly monotone increasing function. Hence, for any x≥x0, there exists a unique positive λx such that(9)λx−ax+bxeλxτx=0.
Therefore,λ∗ exists and λ∗≥0. We will prove λ∗>0 in the following.
Whenτx=0, we can obtain that λ∗=ax−bx≥ρ>0.
Whenτx>0, λ∗<0. Here, for any σ<minρ/2,1/τxn1+ρ/2L, there exists x∗>0 such that λ∗x∗<σ and satisfies(10)λ∗x∗−ax∗+bx∗eλ∗x∗τx∗=0.
It follows that(11)0=λ∗x∗−ax∗+bx∗eλ∗x∗τx∗<σ−ax∗+bx∗1+ρ2L<ρ2−ax∗−bx∗+bx∗ρ2L<ρ2−ρ+ρ2=0,and this is a contradiction. So, λ∗>0. The proof is completed.Remark 1.
Lemma3 is a generalization of Lemma 4.1 in [36].Lemma 4.
Letτ>0,a<b≤+∞, suppose that Zt=Z1t,Z2t,…,ZntT∈Ca,b,Rn satisfies the following differential inequality:(12)D+Zt≤PtZt+Qt⊗Z¯te,a≤t<b,Za+s∈PC,−τ≤s<0,where Pt=ppqtn×n with ppqt≥0p≠q,Qt=qpqtn×n with qpqt≥0, Z¯t=Zt−τpqtn×n. If the initial condition satisfies(13)Zt≤κξe−λt,κ≥0,t∈a−τ,a,in which the scalar λ>0 and ξ=ξ1,ξ2,…,ξnT>0 are determined by(14)λE+Pt+Qt⊗ζλξ<0,ζλ=eλτpqtn×n,then Zt≤κξe−λt−a,fort∈a,b.Proof.
Forp∈1,2,…,n and arbitrary ε>0, set wpt≜κ+εξpe−λt−a. We prove that(15)Zpt≤wpt=κ+εξpe−λt−a,t∈a,b,p=1,2,…,n.
If this is not true, then there exist a numbert∗∈a,b and some integer r such that(16)Zrt∗=wrt∗,D+Zrt∗≥w˙rt∗,Zpt≤wpt,t∈a,t∗,p=1,2,…,n.
From Lemma4 and (15), we have(17)D+Zrt∗≤∑q=1nprqt∗Zqt∗+qrqt∗Zqt∗−τrqt∗≤∑q=1nprqt∗κ+εξqe−λt∗−a+qrqt∗κ+εξqe−λt∗−τrqt∗−a=∑q=1nprqt∗κ+εξqe−λt∗−a+qrqt∗eλτrqt∗κ+εξqe−λt∗−a≤∑q=1nprqt∗+qrqt∗eλτrqt∗κ+εξqe−λt∗−a.
According toλE+Pt+Qt⊗ζλξ<0,ppqt∗≥0p≠q, qpqt∗≥0, and eλτpqt∗>0, it follows that(18)∑q=1nprqt∗+qrqt∗eλτrqt∗ξq<−λξr<0.
From (17), we have(19)D+Zrt∗<−λξrκ+εe−λt∗−l=w˙rt∗,that is,(20)D+Zrt∗<w˙rt∗.
It is clear that this inequality and (16) contradict each other, and this implies that inequality (15) holds for all t∈a,b.
Lettingε⟶0, we obtain that(21)Zpt≤κξpe−λt−a,p=1,2,…,n,t∈a,b.
The proof is completed.


## 3. Main Results

In this part, we establish a series of sufficient conditions to ensure the global exponential stability of systems (1a)–(1d).Theorem 1.
Under assumptions (A1)–(A3), suppose that(C1) There exist a vectorξ=ξ1,ξ2,…,ξnT>0 and a constant λ>0 such that(22)λE−Dt+AtF+BtG⊗ζλξ<0,t≥0,whereDt=diagd1t,d2t,…,dnt with dpt=βpt+∑k=1mDpk/Lk2, ζλ=eλτpqtn×n, At=apq+tn×n,Bt=bpq+tn×n, F=diagF1,F2,…,Fn, and G=diagG1,G2,…,Gn;(C2) There is a positive numberγ>0 such that(23)supl∈Nlnγltl−tl−1≤γ<λ,where γl=max1≤p≤n1,Ipl,l∈N; then, systems (1a)–(1d) are globally exponentially stable.Proof.
Forω,ϖ∈PCΞ, we denote, respectively, the solutions of systems (1a)–(1d) through 0,ω and 0,ϖ:(24)zt,ω,x=z1t,ω,x,z2t,ω,x,…,znt,ω,xT,zt,ϖ,x=z1t,ϖ,x,z2t,ϖ,x,…,znt,ϖ,xT.
Defineztt,ω,x=zt+s,ω,x,ztϖ,x=zt+s,ϖ,x,−τ<s≤0,t≥0, then ztω,x, ztϖ,x∈PCΞ for t≥0.
LetZpt,ψ,x=zpt,ω,x−zpt,ϖ,x,p=1,2,…,n,ψ=ω−ϖ, we have(25)∂Zpt,ψ,x∂t=∑k=1m∂∂xkDpk∂Zpt,ψ,x∂xk−βptZpt,ψ,x−∑q=1napqtfqzqt,ω,x−fqzqt,ϖ,x−∑q=1nbpqtgqzqt−τpqt,ω,x−gqzqt−τpqt,ϖ,x,for t≠tl,x∈Ξ,p=1,2,…,n.
When we multiply both sides of (11) by Zpt,ψ,x and integrate them, we can get(26)ddt∫ΞZpt,ψ,x2dx=2∫ΞZpt,ψ,x∑k=1m∂∂xkDpk∂Zpt,ψ,x∂xkdx−2∫ΞβptZp2t,ψ,xdx+2∑q=1napqt∫ΞZpt,ψ,xfqzqt,ω,x−fqzqt,ϖ,xdx+2∑q=1nbpqt∫ΞZpt,ψ,xgqzqt−τpqt,ω,x−gqzqt−τpqt,ϖ,xdx.
From the Dirichlet boundary conditions and Greens formula, we can obtain(27)∫ΞZpt,ψ,x∑k=1m∂∂xkDpk∂Zpt,ψ,x∂xkdx=−∑k=1m∫ΞDpk∂Zpt,ψ,x∂xk2dx.
By Lemma2, we have(28)∫ΞZpt,ψ,x∑k=1m∂∂xkDpk∂Zpt,ψ,x∂xkdx≤−∑k=1mDpkLk2∫ΞZpt,ψ,x2dx=−∑k=1mDpkLk2Zpt,ψ,x22.
From (A1)–(A3) and Hoder inequality, we can get(29)∑q=1napqt∫ΞZpt,ψ,xfqzqt,ω,x−fqzqt,ϖ,xdx≤∑q=1napq+t∫ΞZpt,ψ,xfqzqt,ω,x−fqzqt,ϖ,xdx≤∑q=1napq+t∫ΞZpt,ψ,xZqt,ψ,xFqdx≤∑q=1napq+tZpt,ψ,x2FqZqt,ψ,x2.
In a similar way, we have(30)∑q=1nbpqt∫ΞZpt,ψ,xgqzqt−τpqt,ω,x−gqzqt−τpqt,ϖ,xdx≤∑q=1nbpq+tZqt−τpqt,ψ,x2GqZpt,ψ,x2.
Applying (26)–(30) to (25), we can obtain(31)12ddtZpt,ψ,x22≤−βpt+∑k=1mDpkLk2Zpt,ψ,x22+∑q=1napq+tZpt,ψ,x2FqZpt,ψ,x2+∑q=1nbpq+tZqt−τpqt,ψ,x2GqZpt,ψ,x2,i.e.,(32)D+Zpt,ψ,x2≤−βpt+∑k=1mDpkLk2Zpt,ψ,x2+∑q=1napq+tFqZqt,ϑ,x2+∑q=1nbpq+tZqt−τpqt,ψ,x2Gq.
LetZpt=Zpt,ψ,x2,Zt=Z1t,Z2t,…,ZntT,Z¯t=Zpt−τpqtn×n,dpt=βpt+∑k=1mDpk/Lk2, p=1,2,…,n,Dt=diagd1t,d2t,…,dnt, and Pt=−Dt+AtF,Qt=BtG. Then, (31) can be reduced to the following form:(33)D+Zt≤PtZt+Qt⊗Z¯te.
From condition (C1), there exist a vectorξ=ξ1,ξ2,…,ξnT>0 and a constant λ>0 such that(34)λE−Dt+AtF+BtG⊗ζλξ<0.
Takingκ=ψ/min1≤p≤nξp, obviously(35)Zt≤κξe−λt,t∈−τ,t0,t0=0.
By Lemma4, we have(36)Zt≤κξe−λt,t∈t0,t1.
Assume that the following inequality holds form≤l:(37)Zt≤κγ0,…,γm−1ξe−λt,t∈tm−1,tm,γ0=1.
Whenm=l+1, we can obtain(38)Ztl=ztl,ω,x−ztl,ϖ,x2=Ilztl−,ω,x−Ilztl−,ϖ,x2≤ℑlZtl−,ψ,x2=ℑlZtl−≤κγ0,…,γl−1ℑlξlimt⟶tl−e−λtl≤κγ0,…,γl−1γlξe−λtl.
By (37) and (38) and γl≥1, we obtain(39)Zt≤κγ0,…,γl−1γlξe−λt,tl−τ≤t≤tl.
Combining (34), (35), and (39) and Lemma 4, we get(40)Zt≤κγ0,…,γl−1γlξe−λt,tl≤t<tl+1.
Applying mathematical induction, we have(41)Zt≤κγ0,…,γl−1ξe−λt,tl−1≤t<tl,l∈N.
FromC2 and (41), we obtain(42)Zt≤κeγt1eγt2−t1,…,eγtl−1−tl−2ξe−λt≤κξeγte−λt=κξe−λ−γt,for all l∈N,tl−1≤t<tl.
This implies that(43)zt,ω,x−zt,ϖ,x=∑p=1nzpt,ω,x−zpt,ϖ,x2=∑p=1nZpt≤∑p=1nκξpe−λ−γt=∑p=1nξpmin1≤p≤nξpω−ϖe−λ−γt,for all t≥0.
That is,(44)ztω,x−ztϖ,x≤Mω−ϖe−λ−γt,t≥0,where M=∑p=1nξp/min1≤p≤nξp. The proof is completed.Remark 2.
In Theorem1, the condition (C2) γ=supl∈Nlnγl/tl−tl−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 systems (1a)–(1d):(45)OPmaxλs.t.C1holds.
Obviously,λ is associated with all time delays, the diffusion coefficients, the Dirichlet boundary conditions, and system parameters. When γ∈0,λ, Theorem 1 shows that systems (1a)–(1d) are globally exponentially stable and its exponential convergence rate equals λ−γ. About the exponential convergence rate λ, we have the following result.Theorem 2.
If condition (C1) holds, thenmaxλ=λ∗, where(46)λ∗=inft≥0λt>0:λtE−Dt+AtF+BtG⊗ζtξ=0>0,ζt=eλtτpqtn×n.Proof.
By Lemma3, λ∗ exists and λ∗>0. Let λ=λ∗−ɛ for all ɛ∈0,λ∗, then(47)λ∗−ɛE−Dt+AtF+BtG⊗ζλ∗−ɛξ<0,for all t≥0. That is,(48)λE−Dt+AtF+BtG⊗ζtξ<0,for all t≥0. Therefore, maxλ=λ∗. The proof is completed.Remark 3.
Some existing neural network models become special cases of systems (2a)–(2c) formulated in this paper; for example, see [15, 16]. It is not hard to find out that all criteria in [15] are improved. Comparing with LMIs method in [16], our results are more concise, and these results do require the differentiability on time-varying delays, so our method has different advantages and disadvantages with the method in [16]. In addition, although those models in [37, 38] are different from systems (2a)–(2c) because of the difference in reaction-diffusion terms and boundary conditions, all criteria in [37] are independent of diffusion effect. In [38], the authors had, respectively, obtained global exponential stability criteria of nonautonomous neural networks with time-varying delays and reaction-diffusion terms under the Neumann boundary conditions and the Dirichlet boundary conditions. However, our approach is similar to the one in [38] when the Dirichlet boundary conditions are considered.Remark 4.
LetΔt=Dt−AtF−BtG, then condition (C1) is equivalent to that Δt is a nonsingular M-matrix for any t≥0. In fact, if Δt is a nonsingular M-matrix for any t≥0, from Lemma 1, there exists a vector ξ=ξ1,ξ2,…,ξnT>0 such that(49)Dt−AtF−BtGξ>0.
Because of the uniform continuity, there exists a real numberλ>0 such that(50)λE−Dt+AtF+BtG⊗θλξ<0,and this shows that condition (C1) holds. Reversely, let λ=0 in (C1); it is easy to obtain that Δt is a nonsingular M-matrix for t≥0.
Based on the above discussion, we have the following conclusions.Corollary 1.
Under assumptionsA1−A3, suppose thatC1′ There exist some numbers ξp>0p=1,2,…,n and a number λ>0 such that(51)βpt+∑k=1mDpkLk2−∑q=1nξqξpapq+tFq−∑q=1nξqξpbpq+teλτpqtGq>λ>0,for allt≥0,p=1,2,…,n.(C2) There is a positive numberγ>0 such that(52)supl∈Nlnγltl−tl−1≤γ<λ,where γl=max1≤p≤n1,Ipl,l∈N; then, systems (1a)–(1d) are globally exponentially stable.Proof.
Note thatC1 and (C1’) are equivalent; it follows from Theorem 1 that Corollary 1 holds.Corollary 2.
Under assumptionsA1 and A3, if condition C1 holds, then systems (2a)–(2c) are globally exponentially stable.Corollary 3.
Under assumptionsA1 and A3, if there exist some numbers ξp>0p=1,2,…,n and a number λ>0 such that(53)ξpβpt+∑k=1mDpkLk2−∑q=1nξqapq+tFq−∑q=1nξqbpq+tGq>λ>0,for allt≥0,p=1,2,…,n,then systems (2a)–(2c) are globally exponentially stable.Corollary 4.
Under assumptionsA1−A3, suppose thatC1″ There exist a vector ξ=ξ1,ξ2,…,ξnT>0 and a positive number λ>0 such that(54)λE−Dt+AtF+BtG⊗ζ¯λξ<0,ζ¯λ=eλτn×n,t≥0,whereDt=diagd1td2t,…,dnt with dpt=βpt+∑k=1mDpk/Lk2, At=apq+tn×n,Bt=bpq+tn×n, F=diagF1,F2,…,Fn, and G=diagG1,G2,…,Gn;(C2) There is a positive numberγ>0 such that(55)supl∈Nlnγltl−tl−1≤γ<λ,where γl=max1≤p≤n1,Ipl,l∈N; then, systems (1a)–(1d) are globally exponentially stable.Proof.
Note thatζ¯λ≥ζλ, it follows that(56)BtG⊗ζ¯λ≥BtG⊗ζλ,t≥0.
This shows thatC1″ ensures (C1).
In this part, we consider the case of the periodic solution of systems (1a)–(1d). Let us make the following assumptions:(A4) Forp,q∈1,2,…,n, βpt,apqt,bpqt,τpqt, and Jpt are periodic continuous functions and has a common period Ω>0 for all t≥0.(A5) Forℑl=diagI1l,…,Inl and the impulsive moments tll∈N, there exists an integer r>0 such that(57)Ipl+r=Ipl,tl+r=tl+Ω.
For the existence and the global exponential stability of periodic solutions for periodic systems (1a)–(1d), we have the following results.Theorem 3.
Under assumptionsA1−A5, suppose that(C1) There exist a vectorξ=ξ1,ξ2,…,ξnT>0 and a constant λ>0 such that(58)λE−Dt+AtF+BtG⊗ζλξ<0,t≥0,whereDt=diagd1t,d2t,…,dnt with dpt=βpt+∑k=1mDpk/Lk2, ζλ=eλτpqtn×n, At=apq+tn×n,Bt=bpq+tn×n, F=diagF1,F2,…,Fn, and G=diagG1,G2,…,Gn;(C2) There is a positive numberγ>0 such that(59)supl∈Nlnγltl−tl−1≤γ<λ,where γl=max1≤p≤n1,Ipl,l∈N; then, systems (1a)–(1d) have exactly one globally exponentially stable Ω-periodic solution.Proof.
On the basis of Theorem1, there exists a integer K>0 such that Me−λ−γKΩ≤1/2, and we define a Poincare mapping as follows:(60)Γ:PCΞ⟶PCΞ,byΓω=zΩω,x,and it follows that ΓKω=zKΩω,x. Setting t=KΩ, we get(61)ΓKω−ΓKϖ≤12ω−ϖ.
Obviously,ΓK is a contraction mapping, and then there exists a unique fixed point ω∗∈PCΞ such that(62)ΓKω∗=ω∗.
And then, we have(63)ΓKΓω∗=ΓΓKω∗=Γω∗,and this implies that Γω∗∈PCΞ is also a fixed point of ΓK. So, we have(64)Γω∗=ω∗,i.e.zΩω∗,x=ω∗.
Hence, ifzt,ω∗,x is a solution of systems (1a)–(1d) through 0,ω∗, then zt+Ω,ω∗,x is also a solution to systems (1a)–(1d). It follows that(65)zt+Ωω∗,x=ztzΩω∗,x=ztω∗,x,t≥0,that is,(66)zt+Ω,ω∗,x=zt,ω∗,x.
This demonstrates thatzt,ω∗,x has a solution for systems (1a)–(1d) with Ω-period, and all other solutions of systems (1a)–(1d) converge exponentially to it as t⟶+∞. The proof is completed.Corollary 5.
Under assumptionsA1−A5,C1′ There exist constants ξp>0p=1,2,…,n and a number λ>0 such that(67)βpt+∑k=1mDpkLk2−∑q=1nξqξpapq+tFq−∑q=1nξqξpbpq+teλτpqtGq>λ>0,for allt≥0,p=1,2,…,n.(C2) There exists a positive numberγ>0 such that(68)supl∈Nlnγltl−tl−1≤γ<λ,where γl=max1≤p≤n1,Ipl,l∈N; then, systems (1a)–(1d) have exactly one globally exponentially stable Ω-periodic solution.Corollary 6.
Under assumptionsA1 and A4, if condition C1 holds, then systems (2a)–(2c) have exactly one globally exponentially stable Ω-periodic solution.


## 4. Illustrative Examples

In this section, we give two examples to illustrate the conditions required for our theorems.Example 1.
Consider a two-neuron impulsive system with reaction-diffusion terms:(69a)∂zpt,x∂t=∑k=11∂∂xkDpk∂zpt,x∂xk−βptzpt,x+∑q=1napqtfqzqt,x+∑q=1nbpqtgqzqt−τpqt,x,t≥0,t≠tl,x∈Ξ,(69b)zptl+,x=1.4zptl−,x,x∈Ξ,l∈N,(69c)zpt,x=0,t≥0,x∈∂Ξ,(69d)zps,x=μps,x,s∈−τ,0,for p=1,2, where Ξ=−1,1,tl=1.5πl, and(70)Dp1=0.150.15,βt=sinπt00cos3t,At=sinπtcos4tsintcos3t,Bt=1251−e−tcos2tsint1−e−t301+e−t,τt2×2=sintsintcostcost,fqz=gqz=z+1−z−12.
It is easy to verify that assumptions (A1)–(A3) are satisfied, and we can obtain the following results:(71)ζλ=eλsinteλsinteλcosteλcost,Dt=0.15+sinπt000.15+cos3t,F=G=1001,λ∈0,1,γl=max1,1.4=1.4≥1,γ=supl∈Nlnγl/tl−tl−1=ln1.4/1.5π≈0.0714.
By solving the following optimization problem:(72)OPmaxλs.t.C1holds,we can obtain that λ≈0.1054>0.0714=γ and ξ=233240509,227642892>0. Thus, the conditions in Theorem 1 hold, then the solution of systems (69a)–(69d) is globally exponentially stable, and the exponential convergence rate is estimated as λ−γ≈0.1054−0.0714=0.0340. The numerical simulation of systems (69a)–(69d) is shown in Figure 1.Figure 1
Computer simulation of the neuronal statesz1t,x and z2t,x.Example 2.
Consider a two-neuron periodic impulsive system:(73a)∂zpt,x∂t=∑k=11∂∂xkDpk∂zpt,x∂xk−βptzpt,x+∑q=1napqtfqzqt,x+∑q=1nbpqtgqzqt−τpqt,x,t≥0,t≠tl,x∈Ξ,(73b)zptl+,x=1.5zptl−,x,x∈Ξ,l∈N,(73c)zpt,x=0,t≥0,x∈∂Ξ,(73d)zps,x=μps,x,s∈−τ,0,for p=1,2, where Ξ=−1,1,tl=1.5πl, and(74)Dp1=0.150.15,βt=sint00cost,At=sintcostsintcost,Bt=120sintcostsint130cost,τt2×2=sintsintcostcost,fqz=gqz=z+1−z−12.
It is easy to verify that assumptions (A1)-(A5) are satisfied, and we can obtain the following results:(75)ζλ=eλsinteλsinteλcosteλcost,Dt=0.15+sint000.15+cost,F=G=1001,λ∈0,1,γl=max1,1.5=1.5≥1,γ=supl∈Nlnγl/tl−tl−1=ln1.5/1.5π≈0.0860.
By solving the following optimization problem:(76)OPmaxλs.t.C1holds,we can obtain that λ≈0.0950>0.0860=γ and ξ=106069006,130884350>0. Thus, the conditions in Theorem 3 hold, then systems (73a)–(73d) have exactly one globally exponentially stable 2π-periodic solution, and the exponential convergence rate is estimated as λ−γ≈0.0950−0.0860=0.0090. The numerical simulation of systems (73a)–(73d) is shown in Figure 2.Figure 2
Computer simulation of the neuronal statesz1t,x and z2t,x.


## 5. Conclusion

We have formulated and investigated a class of new neural network models which assembles nonautonomous neural networks, reaction-diffusion cellular neural networks with time-varying delays, impulses, and the Dirichlet boundary conditions. Several new sufficient conditions have been obtained to ensure the global exponential stability and periodicity of systems (1a)–(1d), and these criteria are shown in simple algebraic inequality forms. In particular, an optimization method is proposed in order to estimate the exponential convergence rate of systems (1a)–(1d), and this method depends on the diffusion coefficient, the Dirichlet boundary conditions, the delays, system parameters, and impulses. Comparing with the method of Lyapunov functional in many previous publications, our method is simpler and more effective for stability and periodicity analysis of nonautonomous impulsive neural networks with time-varying delays and reaction-diffusion terms. Two examples have shown that our results improve and generalize previously known criteria. In near future, we will continue to study global exponential stability and periodicity of nonautonomous impulsive neural networks with distributed delays or leakage delays under the Dirichlet boundary conditions.



---

*Source: 3495545-2021-08-16.xml*