# Dynamic Analysis of Nonlinear Impulsive Neutral Nonautonomous Differential Equations with Delays

Authors: Jinxian Li
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A class of neural networks described by nonlinear impulsive neutral nonautonomous differential equations with delays is considered. By means of Lyapunov functionals and differential inequality technique, criteria on global exponential stability of this model are derived. Many adjustable parameters are introduced in criteria to provide flexibility for the design and analysis of the system. The results of this paper are new and they supplement previously known results. An example is given to illustrate the results.


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## Body

## 1. Introduction

Many evolution processes in nature exhibit abrupt changes of states at certain moments. That was the reason for the development of the theory of impulsive differential equations and impulsive delay differential equations; see the monographs [1, 2]. But the theory of impulsive neutral differential equations is not well developed due to some theoretical and technical difficulties. For impulsive neutral differential equations, some existence results and oscillation criteria are obtained in [3–5] and some stability conditions are derived in [6]; for neural networks described by impulsive neutral differential equations with delays, the exponential stability results are obtained in [7–11], but their work focuses on the autonomous system. So in this paper, the exponential stability for neural networks described by nonlinear impulsive neutral nonautonomous differential equations with delays is considered.The purpose of this paper is to study the stability of the following impulsive neural networks with variable coefficients and several time-varying delays:(1a)x˙i(t)=-bi(t)xi(t)+∑j=1n‍aij(t)fij(xj(t))+∑j=1n‍cij(t)gij(xj(t-τij(t)))+∑j=1n‍dij(t)hij(xj′(t-τ^ij(t)))+ki(t),a.e.t>0,t≠tk,(1b)xi(t+)=Iik(xi(t))+Jik(xi(t-ςi(t)))+Kik(t),t=tk,i=1,2,…,n;k=1,2,…,where n corresponds to the number of units in a neural network; for i,j=1,2,…,n, xi(t) denotes the potential of cell i at time t; 0≤τij(t),  τ^ij(t),  ςi(t)≤τ correspond to the transmission delays. (1a) (called continuous part) describes the continuous evolution processes of the neural networks. For i,j=1,2,…,n, aij(t), cij(t), and dij(t) denote the strengths of connectivity between cells i and j at time t, respectively; fij,  gij,  hij show how the ith neuron reacts to the input; ki(t) is the external bias on the ith at time t. (1b) (called discrete part) describes that the evolution processes experience abrupt change of states at the moments of tk (called impulsive moments); for i=1,2,…,n,  k=1,2,…, the fixed moment tk satisfies t1<t2<⋯<tk<⋯, and lim⁡k→∞tk=∞; Iik represents impulsive perturbations of ith unit at time tk; Jik represents impulsive perturbations of ith unit at time tk, which is caused by the transmission delays; Kik(tk) represents the external impulsive input at time tk.The theory on linear matrix inequality (LMI) orM-Matrix provides effective methods for the analysis of exponential stability of autonomous neural networks. See [7, 9, 10] and the reference therein. But for nonautonomous neural networks, it is invalid. Differential inequalities are important tools for investigating the stability of impulsive differential equations. See [7, 8, 12, 13] and the reference therein. The method in this paper is partially motivated by the work in [7].In this paper, we will investigate the global exponential stability of the nonautonomous neural networks and focus on the effect of impulse on the dynamic behavior of (1a) and (1b). The results do not require the boundedness of {tk-tk-1} and the differentiability of τij. So they are new and complement previously known results.For a continuous functiona(t), we denote
(2)a+(t)=max⁡{0,a(t)},a-(t)=min⁡{0,a(t)},a(t+)=lim⁡s→t+a(s),a(t-)=lim⁡s→t-a(s).Define(3)R+=[0,∞),N={1,2,…,n},N*={1,2,…},C(Ω,R)={ψ:Ω⟶R∣ψiscontinous,Ω⊂R},CB(Ω,R)={ψ∈C(Ω,R)∣ψisbounded},PC([-τ,0],R)={ψ:[-τ,0]⟶R∣ψ(t-)=ψ(t),fort∈[-τ,0],ψ(t+)existson⁡Randψ(t+)=ψ(t)forallbutatmostafinitenumberofpointson⁡[-τ,0].},PC1([-τ,0],R)={ψ∈PC([-τ,0],R)∣ψ′(t+)andψ′(t-)exist,ψ′(t)=ψ′(t-)fort∈[-τ,0]andψ′(t+)=ψ′(t)forallbutatmostafinitenumberofpointson⁡[-τ,0].},PC([-τ,0],Rn)={ψ^=(ψ1,ψ2,…,ψn)T∣ψi∈PC([-τ,0],R),i∈N.},PC1([-τ,0],Rn)={ψ^=(ψ1,ψ2,…,ψn)T∣ψi∈PC1([-τ,0],R),i∈N.}.For anyϕ∈PC([-τ,0],R),  ϕ^∈PC1([-τ,0],R),  ψ=(ψ1,ψ2,…,ψn)T∈PC([-τ,0],Rn), and ψ^=(ψ^1,ψ^2,…,ψ^n)T∈PC1([-τ,0],Rn), define ∥·∥τ, ∥·∥1τ, ∥·∥τn, and ∥·∥1τn as
(4)∥ϕ∥τ=sup⁡-τ≤s≤0|ϕ(s)|,∥ϕ^∥1τ=max⁡{∥ϕ^∥τ,∥ϕ^′∥τ},∥ψ∥τn=max⁡1≤i≤n∥ψi∥τ,∥ψ^∥1τn=max⁡1≤i≤n∥ψ^i∥1τ,
respectively.For convenience, the following conditions are listed.(H1)
Fori,j∈N,bi∈C(R+,R+),aij∈C(R+,R), and cij,dij∈CB(R+,R),fij,gij,hij∈C(R,R).(H2)
There are positive constantsFij,Gij,Hij, i,j∈N, such that
(5)|fij(u)-fij(v)|≤Fij|u-v|,|gij(u)-gij(v)|≤Gij|u-v|,|hij(u)-hij(v)|≤Hij|u-v|,for allu,v∈R.(H3)
There exist positive constantsIik* and Jik*, i∈N,k∈N*, such that
(6)|Iik(u)-Iik(v)|≤Iik*|u-v|,|Jik(u)-Jik(v)|≤Jik*|u-v|,max⁡i∈N,k∈N*Iik*+max⁡i∈N,k∈N*Jik*<1,for allu,v∈R.(H4)
There exist positive constantspi,qi,i∈N and σ such that
(7)pibi(t)-∑j=1n‍pjFijaij+(t)-∑j=1n‍(pjGijcij+(t)+qjHijdij+(t))≥σ>0,qi-pibi(t)-∑j=1n‍pjFijaij+(t)-∑j=1n‍(pjGijcij+(t)+qjHijdij+(t))≥σ>0,fort∈[0,∞),i∈N.We assume that (1a) and (1b) are with the following initial conditions:
(8)x(s)=ϕ(s),s∈[-τ,0],
where ϕ∈PC([-τ,0],Rn). According to [13], the initial value problems (1a), (1b), and (8) have the unique solution x(t,ϕ) under assumptions (H2) and (H3).Definition 1.
A functionx(t)=(x1(t),x2(t),…,xn(t))T is said to be a solution of (1a) and (1b) on [-τ,∞) if for i∈N,(i)
xi(t) is absolutely continuous on each interval (0,t1) and (tk,tk+1), k∈N*;(ii)
for anytk,k∈N*, xi(tk+) and xi(tk-) exist and xi(tk-)=xi(tk);(iii)
x(t) satisfies (1a) for almost everywhere in [0,∞) and satisfies (1b) for every t=tk,  k∈N*.
Obviously, a solutionX(t)=(x1(t),x2(t),…,xn(t))T of (1a) and (1b) is continuous at t≠tk and discontinuous at t=tk. Furthermore, X′(t)=(x1′(t),x2′(t),…,xn′(t))T has discontinuities of the first kind at the fixed impulsive moments tk and some moments t¯∈(tk,tk+1),  k∈N*. Denote X′(tk)=X′(tk-),  X′(t¯)=X′(t¯-).Definition 2.
LetX(t)=(x1(t),x2(t),…,xn(t))T and Y(t)=(y1(t),y2(t),…,yn(t))T be two solutions of (1a), (1b), and (8) with ϕ=φ and ϕ=ψ, respectively, where φ and ψ∈PC([-τ,0],Rn). If there exist α>0 and M>1 such that
(9)|xi(t)-yi(t)|≤M∥φ-ψ∥1τne-αt,∀t>0,i∈N,
then (1a) and (1b) are said to be globally exponentially stable.


## 2. The Main Result

To study the exponential stability of (1a) and (1b), we need the following lemma.Lemma 3.
Assume that (H1) and (H4) hold and there exist nonnegative vector functions (V1(t),V2(t),…,Vn(t))Tand (W1(t),W2(t),…,Wn(t))T∈PC([-τ,0],Rn), where Vi(t) is continuous at t≠tk   (k∈N*), such that(10a)D-Vi(t-)≤-bi(t)Vi(t-)+∑j=1n‍aij+(t)FijVj(t-)+∑j=1n‍cij+(t)Gij∥Vjt-∥τ+∑j=1n‍dij+(t)Hij∥Wjt-∥τ,(10b)Wi(t+)≤bi(t)Vi(t+)+∑j=1n‍aij+(t)FijVj(t+)+∑j=1n‍cij+(t)Gij∥Vjt+∥τ+∑j=1n‍dij+(t)Hij∥Wjt+∥τ,(10c)Vi(tk+)≤Iik*Vi(tk)+Jik*Vi(tk-ςi(tk)),for t>0,  i∈N,  k∈N*. Then for all t≥0 and i∈N, there exists a positive constant L such that
(11)Vi(t)≤L∑l=1n‍max⁡{∥Vl0∥τ,∥Wl0∥τ}e-(λ*-μ)t,
where λ* and μ are defined, respectively, as
(12)λ*=min⁡{λi*,λ^i*∣i∈N},λ*+1τln⁡max⁡i∈N,k∈N*Jik*1-max⁡i∈N,k∈N*Iik*≤μ≤λ*,(13)λi*=inf⁡t≥0{eλ(t)τλ(t)>0,λ(t)-[bi(t)-1pi∑j=1n‍pjFijaij+(t)]+1pi∑j=1n‍(pjGijcij+(t)+qjHijdij+(t))×eλ(t)τ=0}>0,λ^i*=inf⁡t≥0{eλ(t)τλ(t)>0,-[1-piqibi(t)-1qi∑j=1n‍pjFijaij+(t)]+1qi∑j=1n‍(pjGijcij+(t)+qjHijdij+(t))×eλ(t)τ=0}>0.Proof.
By the similar analysis in [14, Lemma 4.1], we can deduce that λi* and λ^i* exist uniquely and λi*>0,  λ^i*>0 under the assumption of (H1) and (H4). Consequently, λ*>0. Choose a positive constant θ such that min⁡{pi,qi∣i∈N}θ>1. Let
(14)Φi(t)=max⁡⁡{1piVi(t),1qiWi(t)},Ψ(t)=θ∑l=1nmax⁡{∥Vl0∥τ,∥Wl0∥τ}e-(λ*-μ)t,kkkkkkkkkkkkkkkkkkkkkkkkkkkki∈N.
Then for all t∈[-τ,0] and γ>1, we have
(15)γΨ(t)=γθ∑l=1n‍max⁡{∥Vl0∥τ,∥Wl0∥τ}e-(λ*-μ)t>Φi(t).
Then
(16)Φi(t)<γΨ(t),∀t∈[0,∞),i∈N.
For the sake of contradiction, assume that there exist i∈N and t¯>0 such that
(17)Φi(t¯+)≥γΨ(t¯),Φj(t)<γΨ(t),hhhhhhhhhhhhfort∈[0,t¯),j∈N.
From (17), we have
(18)∥Vjt¯∥τ=pjsup⁡-τ≤θ≤01pjVj(t¯+θ)≤pjsup⁡-τ≤θ≤0γΨ(t¯+θ)≤γpjΨ(t¯-τ);
similarly,
(19)∥Wjt¯∥τ≤γqjΨ(t¯-τ).
Then we have the following cases.(I)
(1/pi)Vi(t¯+)≥γΨ(t¯); then we have the following subcases.(i)
t¯≠tk, k∈N*. So Vi(t) is continuous at t¯. By (17), we have
(20)1piVi(t¯)=γΨ(t¯),1piD-Vi(t¯)>γΨ′(t¯). From (H4), (17)–(19), and the definition of λ*, we have
(21)1piD-Vi(t¯)-γΨ′(t¯)≤-γbi(t¯)Ψ(t¯)+∑j=1n‍pjpiγaij+(t¯)FijΨ(t¯)+∑j=1n‍γ(pjpicij+(t¯)Gij+qjpidij+(t¯)Hij)×Ψ(t¯-τ)+γλ*Ψ(t¯)<0,
which is a contradiction with (20). (ii)
There exists ak0∈N* such that t¯=tk0. By (17), we have
(22)1piVi(t¯)≤γΨ(t¯)≤1piVi(t¯+).Noting (1/pi)Vi(t¯+)≠(1/pi)Vi(t¯-), we have (1/pi)Vi(t¯-)<γΨ(t¯) or γΨ(t¯)<(1/pi)Vi(t¯+). Without loss of generality, we assume that γΨ(t¯)<(1/pi)Vi(t¯+). From (10c) and (22), we get that
(23)γΨ(t¯)<1piVi(t¯+)≤γ(Iik0*+Jik0*e(λ*-μ)τ)Ψ(t¯).
Simplifying (23), we obtain μ<λ*+(1/τ)ln⁡(Jik0*/(1-Iik0*)), which contradict (12).
If (I) does not hold, then
(II)(24)1qiWi(t¯+)≥γΨ(t¯),1qjWj(t)<γΨ(t),1pjVj(t)<γΨ(t)fort∈[0,t¯),j∈N.
Then from (10b) and (17)–(19), we have
(25)0≤-Wi(t¯+)+bi(t¯)Vi(t¯+)+∑j=1n‍aij+(t¯)FijVj(t¯+)+∑j=1n‍cij+(t¯)Gij∥Vjt¯+∥τ+∑j=1n‍dij+(t¯)Hij∥Wjt¯+∥τ,≤γΨ(t¯)[-qi+pibi(t¯)+∑j=1n‍pjaij+(t¯)Fij+∑j=1n‍(pjcij+(t¯)Gij+qjdij+(t¯)Hij)eλ*τ]<0,
which is a contradiction.
From (I) and (II), (16) holds. Letting γ→1+ in (16), we have
(26)Φi(t)≤Ψ(t),∀t∈[0,∞),i∈N.
So (1/pi)Vi(t)≤Ψ(t) for all t∈[0,∞),  i∈N. Let L=max⁡i∈N{θpi}; then for t≥0 and i∈N, we have
(27)Vi(t)≤L∑l=1n‍max⁡{∥Vl0∥τ,∥Wl0∥τ}e-(λ*-μ)t.
The proof of Lemma 3 is complete.Theorem 4.
Assume that (H1)–(H4) hold. Then systems (1a) and (1b) are globally exponentially stable.Proof.
LetX(t)=(x1(t),x2(t),…,xn(t))T and Y(t)=(y1(t),y2(t),…,yn(t))T be solutions of (1a), (1b), and (8) with ϕ=φ and ϕ=ψ, respectively. Let
(28)Vi(t)=|xi(t)-yi(t)|,Wi(t)=|xi′(t)-yi′(t)|,hhhhhhhhhhhhhhhhhhhhhhhhhit∈R+,i∈N.
By (1a) and (1b), for i∈N, we have
(29)D-Vi(t-)≤-bi(t)Vi(t-)+∑j=1n‍aij+(t)FijVj(t-)+∑j=1n‍cij+(t)Gij∥Vjt-∥τ+∑j=1n‍dij+(t)Hij∥Wjt-∥τ,t>0,(30)Wi(t+)≤bi(t)Vi(t+)+∑j=1n‍aij+(t)FijVj(t+)+∑j=1n‍cij+(t)Gij∥Vjt+∥τ+∑j=1n‍dij+(t)Hij∥Wjt+∥τ,t>0.
By (1b) and (H3), we have
(31)Vi(tk+)=|xi(tk+)-yi(tk+)|≤Iik*Vi(tk)+Jik*Vi(tk-ςi(tk)).
By (29)–(31) and Lemma 3, there exists a positive constant M such that
(32)Vi(t)≤M∑l=1n‍max⁡{∥Vl0∥τ,∥Wl0∥τ}e-(λ*-μ)t≤Mn∥ϕ-ψ∥1τne-(λ*-μ)t,
where λ* and μ are defined in (12).Remark 5.
For autonomous system, the exponential stability of the zero solution of (1a) with xi(tk+)=Iik(x1(tk),…,xn(tk)),k∈N*, is considered in [7]. But the results require that {tk-tk-1} is bounded.
When there is no impulse in systems (1a) and (1b), (1a) and (1b) reduce to the following model which has been studied in [9, 10]:
(33)x˙i(t)=-bi(t)xi(t)+∑j=1n‍aij(t)fij(xj(t))+∑j=1n‍cij(t)gij(xj(t-τij(t)))+∑j=1n‍dij(t)hij(xj′(t-τ^ij(t)))+ki(t),t>0,i∈N.Corollary 6.
Assume that(H1), (H2), and (H4) hold. (33) is globally exponentially stable.Remark 7.
For autonomous system, the stability of (33) with hij(x)=x, fij=gij, is considered in [10]. However, the authors assume that fij,i,j=1,2,⋯,n, are monotonic, bounded and τij,i,j=1,2,⋯,n,
are constants.Remark 8.
The stability results about the zero solution ofx′(t)=-b(t)x(t)+c(t)x(t-τ(t))+d(t)x′(t-τ(t)) are obtained by the fixed-point theory in [15]. But the differentiability of τ is needed.


## 3. An Illustrative Example

To show the effectiveness of Theorem4, consider the following nonautonomous neural networks with impulse:(34a)x˙i(t)=-bi(t)xi(t)+∑j=12‍aij(t)fij(xj(t))+∑j=12‍cij(t)gij(xj(t-τij(t)))+∑j=12‍dij(t)hij(xj′(t-τ^ij(t)))+ki(t),a.e.t>0,(34b)xi(tk+)=gixi(tk)+Ii,tk=5k,i=1,2;k=1,2,…,where
(35)(b1(t)b2(t))=(7+sint5-cos⁡t),(k1(t)k2(t))=(e-te-2t),(g1g2)=(0.60.3),(I1I2)=(0.3-0.1),(aij(t))2×2=(0,13cos⁡3tcos⁡2t2,0),(cij(t))2×2=(sin2t,00,cos⁡t2),(dij(t))2×2=(16sin3t,18sint19cos⁡t,110cos⁡2t),(fij(x))2×2=(0,|x+1|-|x-1|2|x+1|+|x-1|2,0),(gij(x))2×2=(|x+1|+|x-1|3,00,|x+1|-|x-1|3),(hij(x))2×2=(sinx,cos⁡xcos⁡x,sinx),(τij(t))2×2=(2sin2t,00,2|cos⁡t|),(τ^ij(t))2×2=(0,1-sint21+cos⁡t2,0).
Obviously, (Fij)2×2=(0,11,0),  (Gij)2×2=(2/3,00,2/3), and (Hij)2×2=(1,11,1).Letp1=p2=1 and q1=18,  q2=10. From the above assumption, the conditions of Theorem 4 are satisfied. Therefore, (34a) and (34b) are globally exponentially stable. (x1(t),x2(t))T and (u1(t),u2(t))T are the solutions of (34a) and (34b) with x1(0)=0.5,  x2(0)=-0.8 and u1(0)=-0.5,  u2(0)=0.8, respectively. Figures 1(a) and 1(b) depict time response of state variables x1,  u1 without and with impulse effects; Figures 2(a) and 2(b) depict time response of state variables x2,  u2 without and with impulse effects; Figures 3(a) and 3(b) depict the phase plot in the space (t,x1,x2),  (t,u1,u2) without and with impulse effects.(a) Time response of state variablesx1, u1 without impulsive effects. (b) Time response of state variables x1,u1 with impulsive effects.
(a)
(b)(a) Time response of state variablesx2, u2 without impulsive effects. (b) Time response of state variables x2, u2 with impulsive effects.
(a)
(b)(a) Phase plot in space(t,x1,x2), (t,u1,u2) without impulsive effects. (b) Phase plot in space (t,x1,x2), (t,u1,u2) with impulsive effects.
(a)
(b)



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