# Existence, Uniqueness, and Stability Analysis of Impulsive Neural Networks with Mixed Time Delays

Authors: Qiang Xi; Jianguo Si
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study a class of impulsive neural networks with mixed time delays and generalized activation functions. The mixed delays include time-varying transmission delay, bounded time-varying distributed delay, and discrete constant delay in the leakage term. By using the contraction mapping theorem, we obtain a sufficient condition to guarantee the global existence and uniqueness of the solution for the addressed neural networks. In addition, a delay-independent sufficient condition for existence of an equilibrium point and some delay-dependent sufficient conditions for stability are derived, respectively, by using topological degree theory and Lyapunov-Krasovskii functional method. The presented results require neither the boundedness, monotonicity, and differentiability of the activation functions nor the differentiability (even differential boundedness) of time-varying delays. Moreover, the proposed stability criteria are given in terms of linear matrix inequalities (LMI), which can be conveniently checked by the MATLAB toolbox. Finally, an example is given to show the effectiveness and less conservativeness of the obtained results.


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## Body

## 1. Introduction

As we know, time delay in a system is a common phenomenon that describes the fact that the future state of the system depends not only on the present state but also on the past state and is always unavoidably encountered in many fields such as automatic control, biological chemistry, physical engineer, and neural networks [1–5]. Moreover, time delays can affect the stability of a neural network and create oscillatory and bad dynamic performance [3–5]. Hence, it is significant and necessary to take into account the delay effects on dynamics of neural networks, for example, existence, uniqueness and stability, and so on. To date, neural network models with two categories of time delays, namely, discrete and continuously distributed time delays, have been extensively investigated by many researchers, using some effective approaches; see [6–22] and references therein. For instance, in [6], Kharitonov and Zhabko studied the robust stability of time-delay systems via Lyapunov-Krasovskii functional approach. Wu et al. [9] introduced free-weighting matrix approach and investigated the robust stability problem for time-varying delay systems. Gu introduced the delay decomposition method in [10].Recently, a special type of time delay, namely, leakage delay (or forgetting delay), is identified and investigated due to its existence in many real systems. In 2007, Gopalsamy [11] proposed the bidirectional associative memory neural networks with constant delay in the leakage term and derived sufficient conditions for existence and stability of equilibrium. Based on this work, Li and Huang [12] investigated the stability of general nonlinear systems with leakage delay, by model transformation, contraction mapping theorem, and degenerate Lyapunov-Krasovskii functional. However, dynamical analysis of neural networks with time delay in leakage term has been little considered due to some theoretical and technical difficulties [23–30]. In fact, time delay in the stabilizing negative feedback term has a tendency to destabilize a system [11] and has great impact on the dynamics of neural networks.On the other hand, besides delay, impulses are also likely to exist in neural networks. In implementation of electronic networks, the state is subject to instantaneous perturbations and experiences abrupt change at certain moments, which may be caused by switching phenomenon, frequency change, or other sudden noises; that is, it does exhibit impulsive effects; see [13–21, 31–35]. Therefore, impulsive perturbations should be taken into account when studying the dynamics of neural networks. Since the existence of delays and impulses is frequently a source of instability, bifurcation, and chaos for dynamical systems, it is significant to study both delay and impulsive effects on dynamical systems [13–21, 25, 26, 32]. In [25], Li et al. investigated the existence, uniqueness, and stability problems of recurrent neural networks with discrete time delay and time delay in the leakage term under impulsive perturbations, while being without distributed delay. Since neural networks usually have a spatial extent, there is a distribution of propagation delays over a period of time. In these circumstances the signal propagation is not instantaneous and cannot be modelled only with discrete delays and a more appropriate way is to incorporate continuously distributed delays in neural network models. To the best of our knowledge, so far, there has been very little existing work on impulsive neural networks with time delay in the leakage term and discrete and time-varying distributed delays via LMI approach [27].Motivated by aforementioned discussion, in this paper, we consider a class of impulsive neural networks with mixed time delays and generalized activation functions which could be different from each other. The mixed time delays include time-varying transmission delay, bounded time-varying distributed delay, and constant delay in the leakage term. Firstly, by using the contraction mapping theorem, we obtain a delay-independent sufficient condition to guarantee the global existence and uniqueness of the solution for the addressed neural networks. Secondly, we present a delay-independent sufficient condition to guarantee the existence of an equilibrium point by using topological degree theory. Thirdly, some sufficient conditions which are dependent on the leakage delay, time-varying transmission delay, and distributed delay have been derived to guarantee the global asymptotic stability of the equilibrium point by using a new Lyapunov-Krasovskii functional and some analysis technique. The presented results require neither the boundedness, monotonicity, and differentiability of the activation functions nor the differentiability (even differential boundedness) of time-varying delays, which are more effective and less conservative than other existing literatures [27]. In the absence of leakage delay, the obtained results are also new ones. Moreover, the proposed stability criteria are given in terms of linear matrix inequalities (LMI) [36] and can be conveniently checked by the LMI toolbox in MATLAB. Finally, an example is given to show the effectiveness and less conservativeness of the obtained results.Notations. Let R  (R+) denote the set of (positive) real numbers, Z+ denote the set of positive integers, and Rn denote the n-dimensional real spaces equipped with the Euclidean norm ∥·∥. A>0 or A<0 denotes that the matrix A is a symmetric and positive definite or negative definite matrix. The notations AT and A-1 mean the transpose of A and the inverse of a square matrix. λmax⁡(A) or λmin⁡(A) denotes the maximum eigenvalue or the minimum eigenvalue of matrix A. I denotes the identity matrix with appropriate dimensions and Λ={1,2,…,n}. [·]* denotes the integer function. For any interval J⊆R, set V⊆Rk(1≤k≤n), C(J,V)={φ:J→Viscontinuous}, and PC1(J,V)={φ:J→V is continuously differentiable everywhere except at finite number of points t, at which φ(t+), φ(t-), φ˙(t+), and φ˙(t-) exist and φ(t+)=φ(t), φ˙(t+)=φ˙(t), where φ˙ denotes the derivative of φ}. For any x=(x1,x2,…,xn)∈Rn, [x]+=(|x1|,|x2|,…,|xn|)T and, for any Q=(qij)n×n∈Zn×n, [Q]+=(|qij|)n×n. For any t∈R+, xt is defined by xt=x(t+s), xt-=x(t-+s), s∈[-σ,0]. In addition, the notation ⋆ always denotes the symmetric block in one symmetric matrix.


## 2. Preliminaries

Consider the following impulsive neural networks model:(1)x˙(t)=-Dx(t-σ)+Af(x(t))+Bg(x(t-τ(t)))+W∫t-ρ(t)tK(t-s)h(x(s))ds+I,t>0,t≠tk,Δx(tk)=x(tk)-x(tk-)=Ik(x(tk-),xtk-),k∈Z+,
where x(t)=(x1(t),…,xn(t))T is the neuron state vector of the neural network; D=diag⁡(d1,…,dn) is a diagonal matrix with di>0, i∈Λ, Λ={1,2,…,n};  A, B, and W are the connection weight matrix, the delayed weight matrix, and the distributively delayed connection weight matrix, respectively; I is an external input; σ≥0 is a constant which denotes the leakage delay; τ(t) is a time-varying transmission delay of the neural network; ρ(t) is a time-varying distributed delay of the neural network; f(x(·))=(f1(x1(·)),…,fn(xn(·)))T, g(x(·))=(g1(x1(·)),…,gn(xn(·)))T, and h(x(·))=(h1(x1(·)),…,hn(xn(·)))T represent the neuron activation functions; K(·)=diag⁡(k1(·),…,kn(·)) is the delay kernel function.Throughout this paper, we make the following assumptions.(H1)
τ(t) represents the discrete transmission delay with 0≤τ(t)≤τ; ρ(t) represents the time-varying distributed delay with 0≤ρ(t)≤ρ, where τ, ρ are two positive constants.(H2)
The delay kernelskj(·), j∈Λ, are some real valued continuous functions defined on [0,ρ] and satisfy
(2)|kj(s)|≤|k(s)|,∫0ρ|k(s)|ds=κ,whereκ is a positive constant.(H3)
The neuron activation functionsfj, gj, and hj, j∈Λ, are continuous on R and satisfy
(3)σj-≤fj(u)-fj(v)u-v≤σj+,δj-≤gj(u)-gj(v)u-v≤δj+,ζj-≤hj(u)-hj(v)u-v≤ζj+,for anyu, v∈R, u≠v, j∈Λ, where σj-, σj+, δj-, δj+, ζj-, and ζj+ are some real constants and they may be positive, zero, or negative.(H4)
Ik(·):R×Rn→Rn, k∈Z+, are some continuous functions.(H5)
The impulse timestk satisfy 0=t0<t1<⋯<tk→∞ and inf⁡k∈Z+{tk-tk-1}>0.We will consider model (1) with the initial condition
(4)x(s)=φ(s),s∈[-η,0],
where η=max⁡⁡{σ,τ,ρ}, φ(·)=(φ1,…,φn)T∈PC1([-η,0]), Rn, whose norm is defined by
(5)∥φ∥η2=max⁡⁡{max⁡-η≤θ≤0∑i=1n|φi2(θ)|,max⁡-τ≤θ≤0∑i=1n|φ˙i2(θ)|}.Definition 1 (see [37]).
Assume thatΩ∈Rn is a bounded and open set and F(u):Ω→Rn is a continuous and differentiable function. If p∈¯F(∂Ω) and JF(u)≠0 for any u∈F-1(p), where ∂Ω denotes the boundary of Ω and JF denotes the Jacobian determinant relative to F, then the topological degree relative to Ω and p is defined by
(6)deg⁡⁡(F,Ω,p)={∑u∈F-1(p)sgn⁡JF(u),F-1(p)≠∅,0,F-1(p)=∅.Remark 2.
Generally speaking, the topological degree ofF(u) relative to Ω and p can be regarded as the algebraic number of solutions of F(u)=p in Ω if F(∂Ω)≠0. For instance, deg⁡⁡(F,Ω,0)=±1 implies that F(u)=0 has at least one solution in Ω.Lemma 3 (see [38]).
Given any real matrixM=MT>0 of appropriate dimension and a vector function ω(·):[a,b]→Rn, such that the integrations concerned are well defined, then
(7)[∫abω(s)ds]TM[∫abω(s)ds]≤(b-a)∫abωT(s)Mω(s)ds.Lemma 4 (see [12]).
Given any real matricesΣ1, Σ2, Σ3 of appropriate dimensions and a scalar ϵ>0 such that 0<Σ3=Σ3T, then the following inequality holds:
(8)Σ1TΣ2+Σ2TΣ1≤ϵΣ1TΣ3Σ1+ϵ-1Σ2TΣ3-1Σ2.Lemma 5 (see [27]).
Supposing that℧, ℧ij≥0(i,j=1,2) are symmetric matrices of appropriate dimensions, α∈[0,1] and β∈[0,1], then ℧+[(1-α)℧11+α℧12]+[(1-β)℧21+β℧22]<0 holds if the following four inequalities ℧+℧11+℧21<0, ℧+℧11+℧22<0, ℧+℧12+℧21<0, and ℧+℧12+℧22<0 hold simultaneously.For presentation convenience, in the following, we denote(9)Σ1=diag⁡(σ1-σ1+,…,σn-σn+),Σ2=diag⁡(σ1-+σ1+2,…,σn-+σn+2),Σ3=diag⁡(δ1-δ1+,…,δn-δn+),Σ4=diag⁡(δ1-+δ1+2,…,δn-+δn+2),Σ5=diag⁡(ζ1-ζ1+,…,ζn-ζn+),Σ6=diag⁡(ζ1-+ζ1+2,…,ζn-+ζn+2).


## 3. Global Existence and Uniqueness of Solution

In this section, by using the contraction mapping theorem, we give a delay-independent sufficient condition to guarantee the global existence and uniqueness of the solution for models (1) and (4).Theorem 6.
Assume that the assumptions(H1)–(H5) hold; then the solution x=x(t,0,φ) of models (1) and (4) exists uniquely on [-η,∞).Proof.
Transform the global existence and uniqueness of solution of the models (1) and (4) into a fixed point problem. Let ∥·∥* be the norm in C([0,t1],Rn) defined by
(10)∥u∥*≐max⁡t∈[0,t1]{e-λt·max⁡s∈[0,t]{∥u(s)∥}},u∈C([0,t1],Rn),
where
(11)λ=∑j=1ndj2+σ·∑i=1n∑j=1naij2+δ·∑i=1n∑j=1nbij2+ζ·κ·∑i=1n∑j=1nwij2+1,(12)σ=max⁡j∈Λ{|σj-|,|σj+|},δ=max⁡j∈Λ{|δj-|,|δj+|},ζ=max⁡j∈Λ{|ζj-|,|ζj+|}.
Then it is easy to see that C([0,t1],Rn) is a Banach space endowed with the norm ∥·∥*. Let u∈C([0,t1],Rn) and consider the operator L1:C([0,t1],Rn)→C([0,t1],Rn) defined by
(13)(L1u)(t)=φ(0)+∫0t{W∫0ρ(t)K(θ)h(u(s-θ))dθ+I}-Du(s-σ)+Af(u(s))hhhhhhhhhhhhi+Bg(u(s-τ(s)))hhhhhhhhhhhhi+W∫0ρ(t)K(θ)h(u(s-θ))dθ+I∫0ρ(t)K(θ)h(u(s-θ))dθ}ds,
where u(s)=φ(s), s∈[-η,0].
First we show thatL1 is a contraction on C([0,t1],Rn). Let u,v∈C([0,t1],Rn); we have
(14)∥(L1u)(t)-(L1v)(t)∥=∥-∫0tD[u(s-σ)-v(s-σ)]ds+∫0tA[f(u(s))-f(v(s))]ds+∫0tB[f(u(s-τ(s)))-f(v(s-τ(s)))]ds+∫0tW∫0ρ(t)K(θ)[h(u(s-θ))-h(v(s-θ))]dθds∥≤∫0t∥D[u(s-σ)-v(s-σ)]∥ds+∫0t∥A[f(u(s))-f(v(s))]∥ds+∫0t∥B[f(u(s-τ(s)))-f(v(s-τ(s)))]∥ds+∫0t∥W∫0ρ(t)K(θ)[h(u(s-θ))-h(v(s-θ))]dθ∥ds≤∑j=1ndj2∫0t∥u(s-σ)-v(s-σ)∥ds+σ·∑i=1n∑j=1naij2∫0t∥u(s)-v(s)∥ds+δ·∑i=1n∑j=1nbij2∫0t∥u(s-τ(s))-v(s-τ(s))∥ds+∫0t∥W∫0ρ(t)K(θ)[h(u(s-θ))-h(v(s-θ))]dθ∥ds.
In view of (H2), we get
(15)∫0t∥W∫0ρ(t)K(θ)[h(u(s-θ))-h(v(s-θ))]dθ∥ds≤ζ·∑i=1n∑j=1nwij2∫0t∥∫0ρ(t)K(θ)[u(s-θ)-v(s-θ)]dθ∥ds≤ζ·∑i=1n∑j=1nwij2∫0t∫0ρ(t)∥K(θ)[u(s-θ)-v(s-θ)]∥dθds≤ζ·∑i=1n∑j=1nwij2∫0t∫0ρ(t)|k(θ)|∥u(s-θ)-v(s-θ)∥dθds≤ζ·∑i=1n∑j=1nwij2∫0t∫0ρ(t)|k(θ)|dθmax⁡r∈[0,s]⁡∥u(r)-v(r)∥ds≤ζ·κ·∑i=1n∑j=1nwij2∫0tmax⁡r∈[0,s]⁡∥u(r)-v(r)∥ds.
Substituting the above inequality to (14), we obtain
(16)∥(L1u)(t)-(L1v)(t)∥≤∑j=1ndj2∫0tmax⁡r∈[0,s]∥u(r)-v(r)∥ds+σ·∑i=1n∑j=1naij2∫0tmax⁡r∈[0,s]∥u(r)-v(r)∥ds+δ·∑i=1n∑j=1nbij2∫0tmax⁡r∈[0,s]∥u(r)-v(r)∥ds+ζ·κ·∑i=1n∑j=1nwij2∫0tmax⁡r∈[0,s]∥u(r)-v(r)∥ds=(λ-1)∫0tmax⁡r∈[0,s]∥u(r)-v(r)∥ds=(λ-1)∫0teλse-λsmax⁡r∈[0,s]∥u(r)-v(r)∥ds≤(λ-1)∫0teλdsmax⁡s∈[0,t1]{e-λsmax⁡r∈[0,s]∥u(r)-v(r)∥}ds=(λ-1)∫0teλsds∥u-v∥*≤λ-1λeλt∥u-v∥*.
Since eλt is increasing in t,
(17)max⁡s∈[0,t]{∥(L1u)(s)-(L1v)(s)∥}≤λ-1λeλt∥u-v∥*.
Then
(18)e-λtmax⁡s∈[0,t]{∥(L1u)(s)-(L1v)(s)∥}≤λ-1λ∥u-v∥*.
Note the definition of ∥·∥*; we have
(19)∥L1u-L1v∥*≤λ-1λ∥u-v∥*.
Thus L1 is a contraction on C([0,t1],Rn), and it has a unique fixed point u1*∈C([0,t1],Rn). Thus we get the fact that u1*(t1) exists finitely. It implies that u1*(t1)+J1(u1*(t1),u1*t1) also exists finitely, since assumption (H4) holds. Then we replace u1*(t1) with u1*(t1)+I1(u1*(t1),u1*t1) and define η1=u1*(t1)+I1(u1*(t1),u1*t1) for later use.
Next we show thatL2 is a contraction on C([t1,t2],Rn). For u∈C([t1,t2],Rn), let
(20)∥u∥*≐max⁡t∈[t1,t2]{e-λ(t-t1)·max⁡s∈[t1,t]{∥u(s)∥}},
where λ is defined in (11). Let u∈C([t1,t2],Rn) and consider the operator L2:C([t1,t2],Rn)→C([t1,t2],Rn) defined by
(21)(L2u)(t)=η1+∫t1t‍{W∫0ρ(t)‍K(θ)h(u(s-θ))dθ+I}-Du(s-σ)+Af(u(s))hhhhhhhhhhhh+Bg(u(s-τ(s)))hhhhhhhhhhhh+W∫0ρ(t)‍K(θ)h(u(s-θ))dθ+I}ds,
where η1=u1*(t1)+I1(u1*(t1)) and
(22)u(s)={φ(s),s∈[-η,0],u1*(s),s∈[0,t1).
By virtue of the definition of L2, similar to the proof of (19), we get, for u,v∈C([t1,t2],Rn),
(23)∥L2u-L2v∥*≤λ-1λ∥u-v∥*.
Thus L2 is a contraction on C([t1,t2],Rn), and it has a unique fixed point u2*∈C([t1,t2],Rn). Moreover, we know that u2*(t2) exists finitely, which implies that u2*(t2)+I2(u2*(t2),u2*t2) exists finitely in view of assumption (H4). Then we replace u2*(t2) with u2*(t2)+I2(u2*(t2),u2*t2) and define η2=u2*(t2)+I2(u2*(t2),u2*t2) for later use.
Finally we show thatLn+1 is a contraction on C([tn,tn+1],Rn). For u∈C([tn,tn+1],Rn), let
(24)∥u∥*≐max⁡t∈[tn,tn+1]{e-λ(t-tn)·max⁡s∈[tn,t]{∥u(s)∥}},
where λ is defined in (11). Let u∈C([tn,tn+1],Rn); then we can similarly consider the operator Ln+1:C([tn,tn+1],Rn)→C([tn,tn+1],Rn) defined by
(25)(Ln+1u)(t)=ηn+∫tnt‍{∫0ρ(t)K(θ)h(u(s-θ))dθ+I}-Du(s-σ)+Af(u(s))hhhhhhhhhhh+Bg(u(s-τ(s)))hhhhhhhhhhh+W∫0ρ(t)K(θ)h(u(s-θ))dθ+I}ds,
where ηn=un*(tn)+In(un*(tn)) and
(26)u(s)={φ(s),s∈[-η,0],u1*(s),s∈[0,t1),u2*(s),s∈[t1,t2),⋮un*(s),s∈[tn-1,tn).
Then repeating the argument with Ln+1 replacing L1, similar to the proof of (19), we see that, for u,v∈C([tn,tn+1],Rn),
(27)∥Ln+1u-Ln+1v∥*≤λ-1λ∥u-v∥*.
Thus Ln+1 is a contraction on C([tn,tn+1],Rn), and it has a unique fixed point un+1*∈C([tn,tn+1],Rn).
Continuing in this manner, we construct(28)u*(t)={φ(t),t∈[-η,0],u1*(t),t∈[0,t1),u2*(t),t∈[t1,t2),⋮un+1*(t),t∈[tn,tn+1),⋮
Then u*(t) is the global solution of models (1) and (4). If v*(t) is another solution of models (1) and (4), then it is easy to check from the above argument that u*(t)=v*(t). Hence, the solution u*(t)=u*(t,0,φ) of models (1) and (4) exists uniquely on [-η,∞). This completes the proof.


## 4. Existence of an Equilibrium Point

In previous sections, we have showed the global existence and uniqueness of solution for models (1) and (4). In this section, without requiring the boundedness, differentiability, or monotonicity of the activation functions, we establish a delay-independent sufficient condition for the existence of an equilibrium point of model (1). As usual, we denote an equilibrium point of the model (1) by the constant vector x*∈Rn, where x* satisfies
(29)-Dx*+Af(x*)+Bg(x*)+WKh(x*)+I=0,Ik(x*,x*)=0,k∈Z+,
where K=K(t)=∫0ρ(t)K(s)ds. In this paper, it is assumed that the impulsive function Ik satisfies Ik(x*,x*)=0 for k∈Z+. Hence, to prove the existence of solution of (29), it suffices to show that the following has a solution:
(30)x*-D-1Af(x*)-D-1Bg(x*)-D-1WKh(x*)-D-1I=0,
in view of D>0.Theorem 7.
Assume that the assumption(H3) holds. Then model (1) has at least one equilibrium point if, for any t>0, D-[A]+Σ-[B]+Δ-[WK]+Θ is an M-matrix, where Σ=diag⁡(σ1,…,σn), σj=max⁡⁡{|σj-|,|σj+|}, Δ=diag⁡(δ1,…,δn), δj=max⁡⁡{|δj-|,|δj+|}, and Θ=diag⁡(ζ1,…,ζn), ζj=max⁡⁡{|ζj-|,|ζj+|}.Proof.
From (30), we note that it suffices to prove that the following has at least one solution:
(31)p(x)=x-W1f(x)-W2g(x)-W3h(x)-J′=0,
where W1=D-1A, W2=D-1B, W3=D-1WK, J′=D-1J. In order to use topological degree theory, we consider the following homotopic mapping:
(32)P(x,λ)=λp(x)+(1-λ)x,λ∈[0,1].
Note that D-[A]+Σ-[B]+Δ-[WK]+Θ is an M-matrix; it can be deduced that I-[W1]+Σ-[W2]+Δ-[W3]+Θ is also an M-matrix. This implies that (I-[W1]+Σ-[W2]+Δ-[W3]+Θ)-1≥0 and there exists a positive vector X0∈Rn such that (I-[W1]+Σ-[W2]+Δ-[W3]+Θ)X0>0. It then follows that
(33)[P(x,λ)]+=[λp(x)+(1-λ)x]+=[x-λW1f(x)-λW2g(x)-λW3h(x)-λJ′]+≥[x]+-λ[W1f(x)]+-λ[W2g(x)]+-λ[W3h(x)]+-λ[J′]+≥[x]+-λ[W1]+[f(x)]+-λ[W2]+[g(x)]+-λ[W3]+[h(x)]+-λ[J′]+≥[x]+-λ[W1]+Σ[x]+-λ[W2]+Δ[x]+-λ[W3]+Θ[x]+-λ[J′]+≥(1-λ)[x]++λ(I-[W1]+Σ-[W2]+Δ-[W3]+Θ)·{[x]+-(I-[W1]+Σ-[W2]+Δ-[W3]+Θ)-1[J′]+}.
Let
(34)Ω={-[W3]+Θ)-1[J′]++X0}x∣[x]+≤(I-[W1]+Σ-[W2]+Δhhhhhhhhhhhh-[W3]+Θ)-1[J′]++X0}.
It is obvious that set Ω is not empty and, for any x∈∂Ω, we have
(35)[P(x,λ)]+≥(1-λ)[x]++λ(I-[W1]+Σ-[W2]+Δ-[W3]+Θ)X0>0,λ∈[0,1],
which implies that P(x,λ)≠0 for all x∈∂Ω and λ∈[0,1]. By topological degree invariance theory, we obtain
(36)deg⁡⁡(p(x),Ω,0)=deg⁡⁡(P(x,λ),Ω,0)=deg⁡⁡(P(x,0),Ω,0)=1.
Therefore, from Remark 2, we know that p(x)=0 has at least one solution in Ω. This completes the proof.


## 5. Global Asymptotic Stability

It should be noted that Theorem7 can guarantee the existence of an equilibrium point but not the uniqueness. In this section, we will derive some sufficient conditions to guarantee not only the global asymptotic stability but also the uniqueness of the equilibrium point. For this purpose, the impulsive function Ik which is viewed as a perturbation of the equilibrium point x* of models (1) and (4) without impulses is defined by
(37)x(tk)-x(tk-)=Ik(x(tk-),xtk-)=-Jk{x(tk-)-x*-D∫tk-σtk‍(x(u)-x*)du},k∈Z+,
where Jk,k∈Z+ are some n×n real matrices. It is clear that Ik(x*,x*)=0, k∈Z+. Such type of impulses describes the fact that the encountered instantaneous perturbations depend on not only the state of neurons at impulse times tk but also the state of neurons in its recent history, which reflects more realistic dynamics. The similar nonlinear impulsive perturbations, which include linear impulsive perturbations and nonimpulsive perturbations as their special cases, have also been investigated by some researchers recently [17–21, 39].Lety(t)=x(t)-x*; then we rewrite the models (1) and (4) as follows:
(38)y˙(t)=-Dy(t-σ)+Af^(y(t))+Bg^(y(t-τ(t)))+W∫t-ρ(t)tK(t-s)h^(y(s))ds,hhhhhhihhhhhhhit>0,t≠tk,Δy(tk)=y(tk)-y(tk-)=-Jk{y(tk-)-D∫tk-σtky(u)du},k∈Z+,y(s)=φ(s)-x*,s∈[-η,0],
where f^(y(·))=f(y(·)+x*)-f(x*), g^(y(·))=g(y(·)+x*)-g(x*) and h^(y(·))=h(y(·)+x*)-h(x*). For convenience in our discussion, in the following, we replace f^ with f, replace g^ with g, and replace h^ with h. Then using a simple transformation, model (38) has an equivalent form as follows:
(39)ddt[y(t)-D∫t-σty(u)du]=-Dy(t)+Af(y(t))+Bg(y(t-τ(t)))+W∫t-ρ(t)tK(t-s)h(y(s))ds,t>0,t≠tk,Δy(tk)=y(tk)-y(tk-)=-Jk{y(tk-)-D∫tk-σtky(u)du},k∈Z+,y(s)=φ(s)-x*,s∈[-η,0].Theorem 8.
Under the conditions in Theorem7, model (1) has a unique equilibrium point which is globally asymptotically stable if there exist a constant γ>0, an n×n inverse matrix Q1, six n×n matrices P>0, Q2>0, Q3>0, Z>0, T11>0, and T22>0, three n×n diagonal matrices U1>0, U2>0, and U3>0, and real matrices T12, Xl(l=1,…,6) with appropriate dimension such that
(40)[T11T12⋆T22]>0,(41)[P(I-Jk)TP⋆P]≥0,k∈Z+,(42)[Ξχj⋆-Z]<0,j=1,2,
where (43)Ξ=[Π11-Q1γT12T-Q1DX1DPDΠ17Π18U3Σ6Π1,10⋆Π220-γQ1DX20γQ1AγQ1B0Π2,10⋆⋆Π330000U2Σ400⋆⋆⋆-Q3000000⋆⋆⋆⋆Π5500000⋆⋆⋆⋆⋆-Q2-DPA-DPB0-DPW⋆⋆⋆⋆⋆⋆-U1000⋆⋆⋆⋆⋆⋆⋆-U200⋆⋆⋆⋆⋆⋆⋆⋆Π990⋆⋆⋆⋆⋆⋆⋆⋆⋆Π10,10],χ1=col{X1,X2,O,O,X3,O,O,O,O,O},χ2=col{X4,X5,O,O,O,O,O,O,O,X6}, with
(44)Π11=-PD-DP+σ2Q2+Q3-U1Σ1-U3Σ5,Π17=PA+Q1A+U1Σ2,Π18=PB+Q1B,Π1,10=PW+Q1W+X4,Π22=-γQ1-γQ1T+γ2τT22,Π2,10=γQ1W+X5,Π33=τT11-γT12T-γT12-U2Σ3,Π55=X3+X3T,Π99=ρ2K(0)ZK(0)-U3,Π10,10=X6+X6T.Proof.
Consider the following Lyapunov-Krasovskii functional as(45)V(t,y(t))=V1(t,y(t))+V2(t,y(t))+V3(t,y(t))+V4(t,y(t))+V5(t,y(t)),
where
(46)V1(t,y(t))=[y(t)-D∫t-σty(u)du]Thhhhhhhhhh×P[y(t)-D∫t-σty(u)du],V2(t,y(t))=σ∫t-σt∫styT(u)Q2y(u)dudshhhhhhhhhh+∫t-σtyT(s)Q3y(s)ds,V3(t,y(t))=∫0t∫u-τ(u)u(y(u-τ(u))γy˙(s))T(T11T12⋆T22)hhhhhhhhhhhhhhi×(y(u-τ(u))γy˙(s))dsdu,V4(t,y(t))=γ2∫-τ0∫t+uty˙T(s)T22y˙(s)dsdu,V5(t,y(t))=ρ∫t-ρt∫uthT(y(s))K(t-s)ZK(t-s)hhhhhhhhhhhhhhhhh×h(y(s))dsdu.
Calculating the upper right derivative of V along the trajectories of model (39) at the continuous interval [tk-1,tk),k∈Z+, we obtain
(47)D+V1=2[y(t)-D∫t-σty(u)du]T×P[+W∫t-ρ(t)t‍K(t-s)h(y(s))ds]-Dy(t)+Af(y(t))+Bg(y(t-τ(t)))hhhhh+W∫t-ρ(t)tK(t-s)h(y(s))ds]=-2yT(t)PDy(t)+2yT(t)PAf(y(t))+2yT(t)PBg(y(t-τ(t)))+2yT(t)PW∫t-ρ(t)tK(t-s)h(y(s))ds+2yT(t)DPD∫t-σty(u)du-2[∫t-σty(u)du]TDPAf(y(t))-2[∫t-σty(u)du]TDPBg(y(t-τ(t)))-2[∫t-σty(u)du]TDPW∫t-ρ(t)tK(t-s)h(y(s))ds.
It follows from Lemma 3 that
(48)D+V2=σ2yT(t)Q2y(t)-σ∫t-σt‍yT(u)Q2y(u)du+yT(t)Q3y(t)-yT(t-σ)Q3y(t-σ)≤σ2yT(t)Q2y(t)-[∫t-σty(u)du]TQ2[∫t-σty(u)du]+yT(t)Q3y(t)-yT(t-σ)Q3y(t-σ),D+V3=∫t-τ(t)t[y(t-τ(t))γy˙(s)]T[T11T12⋆T22][y(t-τ(t))γy˙(s)]ds=τ(t)yT(t-τ(t))T11y(t-τ(t))+2γyT(t)T12Ty(t-τ(t))-2γyT(t-τ(t))T12Ty(t-τ(t))+γ2∫t-τ(t)ty˙T(s)T22y˙(s)ds≤yT(t-τ(t))[τT11-2γT12T]y(t-τ(t))+2γyT(t)T12Ty(t-τ(t))+γ2∫t-τty˙T(s)T22y˙(s)ds,D+V4=γ2τy˙T(t)T22y˙(t)-γ2∫-τ0y˙T(t+u)T22y˙(t+u)du=γ2τy˙T(t)T22y˙(t)-γ2∫t-τty˙T(s)T22y˙(s)ds,D+V5=ρ2hT(y(t))K(0)ZK(0)h(y(t))-ρ∫t-ρthT(y(s))K(t-s)ZK(t-s)h(y(s))ds=ρ2hT(y(t))K(0)ZK(0)h(y(t))-ρ[∫t-ρt-ρ(t)hT(y(s))K(t-s)ZK(t-s)h(y(s))ds+∫t-ρ(t)thT(y(s))K(t-s)ZK(t-s)h(y(s))ds[∫t-ρt-ρ(t)hT(y(s))K(t-s)ZK(t-s)h(y(s))ds]≤ρ2hT(y(t))K(0)ZK(0)h(y(t))-ρρ-ρ(t)(∫t-ρt-ρ(t)K(t-s)h(y(s))ds)T×Z(∫t-ρt-ρ(t)K(t-s)h(y(s))ds)-ρρ(t)(∫t-ρ(t)tK(t-s)h(y(s))ds)T×Z(∫t-ρ(t)tK(t-s)h(y(s))ds).
In addition, we note that
(49)0=2[y(t)+γy˙(t)]TQ1[-y˙(t)+y˙(t)]=2[y(t)+γy˙(t)]TQ1×[+W∫t-ρ(t)t‍K(t-s)h(y(s))ds]-y˙(t)-Dy(t-σ)+Af(y(t))+Bg(y(t-τ(t)))+W∫t-ρ(t)tK(t-s)h(y(s))ds]=-2yT(t)Q1y˙(t)-2yT(t)Q1Cy(t-σ)+2yT(t)Q1Af(y(t))+2yT(t)Q1Bg(y(t-τ(t)))+2yT(t)Q1W∫t-ρ(t)tK(t-s)h(y(s))ds-2γy˙T(t)Q1y˙(t)-2γy˙T(t)Q1Cy(t-σ)+2γy˙T(t)Q1Af(y(t))+2γy˙T(t)Q1Bg(y(t-τ(t)))+2γy˙T(t)Q1W∫t-ρ(t)tK(t-s)h(y(s))ds.
For simplicity, we denote
(50)ξ1(t)=∫t-ρ(t)tK(t-s)h(y(s))ds,ξ2(t)=∫t-ρt-ρ(t)K(t-s)h(y(s))ds.
By applying Lemma 4, for any t>0 with 0≤τ(t)≤τ, 0≤ρ(t)≤ρ, we have
(51)-ρρ-ρ(t)ξ2(t)TZξ2(t)-ρρ(t)ξ1(t)TZξ1(t)≤2ξT(t)χ1ξ2(t)+ρ-ρ(t)ρξT(t)χ1Z-1χ1Tξ(t)+ρ(t)ρξT(t)χ2Z-1χ2Tξ(t)+2ξT(t)χ2ξ1(t),
where
(52)ξ(t)=(∫t-σty(s)ds,y(t),y˙(t),y(t-τ(t)),y(t-σ),ξ2(t),∫t-σty(s)ds,f(y(t)),g(y(t-τ(t))),h(g(t)),ξ1(t)ds∫t-σty(s)ds,)T.
Moreover, for any n×n diagonal matrices, U1>0, U2>0, and U3>0, the following inequality holds by the methods in [40]:
(53){[y(t-τ(t))g(y(t-τ(t)))]T[y(t)f(y(t))]T[-U1Σ1U1Σ2⋆-U1][y(t)f(y(t))]+[y(t-τ(t))g(y(t-τ(t)))]T[-U2Σ3U2Σ4⋆-U2]·[y(t-τ(t))g(y(t-τ(t)))]+[y(t)h(y(t))]T×[-U3Σ5U3Σ6⋆-U3][y(t)h(y(t))][y(t-τ(t))g(y(t-τ(t)))]T}≥0.
Combining (45)–(53), one may deduce that
(54)D+V≤ξT(t)Ξ(t)ξ(t),t∈[tk-1,tk),k∈Z+,
where(55)Ξ(t)=Ξ+ρ-ρ(t)ρχ1Z-1χ1T+ρ(t)ρχ2Z-1χ2T,Ξ=[Π11-Q1γT12T-Q1DX1DPDΠ17Π18U3Σ6Π1,10⋆Π220-γQ1DX20γQ1AγQ1B0Π2,10⋆⋆Π330000U2Σ400⋆⋆⋆-Q3000000⋆⋆⋆⋆Π5500000⋆⋆⋆⋆⋆-Q2-DPA-DPB0-DPW⋆⋆⋆⋆⋆⋆-U1000⋆⋆⋆⋆⋆⋆⋆-U200⋆⋆⋆⋆⋆⋆⋆⋆Π990⋆⋆⋆⋆⋆⋆⋆⋆⋆Π10,10],ξ(t)=(y(t),y˙(t),y(t-τ(t)),y(t-σ),ξ2(t),∫t-σty(s)ds,f(y(t)),g(y(t-τ(t))),h(g(t)),ξ1(t)ds)T.From Lemma 5 we obtain that Ξ(t)<0 if the following inequalities hold simultaneously:
(56)Ξ+χ1Z-1χ1T<0,Ξ+χ2Z-1χ2T<0.
Based on the well-known Schur complements [36], we get the fact that (56) is equivalent to (42). Therefore, Ξ(t)<0, t∈[tk-1,tk), k∈Z+.
For arbitraryt>0, without loss of generality, we set t∈[tn-1,tn), for some n∈Z+. Then integrating inequality (54) at each interval [tk-1,tk), 1≤k≤n-1, and [tn-1,t), we derive
(57)V(t1-)≤V(0)+∫0t1ξT(u)Ξ(t)ξ(u)du,V(t2-)≤V(t1)+∫t1t2ξT(u)Ξ(t)ξ(u)du,⋮V(tn-1-)≤V(tn-2)+∫tn-2tn-1ξT(u)Ξ(t)ξ(u)du,V(t)≤V(tn-1)+∫tn-1tξT(u)Ξ(t)ξ(u)du,
which implies that
(58)V(t)≤V(0)+∫0tξT(u)Ξ(t)ξ(u)du+∑0<tk≤t[V(tk)-V(tk-)],t≥0.
Now, in order to analyze (58), we consider the change of V at impulse times tk, k∈Z+.
Firstly, it follows from (41) that
(59)[P(I-Jk)P⋆P]≥0⟺[I-(I-Jk)T0I][P(I-Jk)P⋆P][I0-(I-Jk)I]≥0⟺[P-(I-Jk)TP(I-Jk)0⋆P]≥0⟺P-(I-Jk)TP(I-Jk)>0.
Secondly, from model (39), it can be obtained that
(60)y(tk)-D∫tk-σtk‍y(u)du=y(tk-)-Jk[y(tk-)-D∫tk-σtky(u)du]-D∫tk-σtky(u)du=(I-Jk)[y(tk-)-D∫tk-σtky(u)du],
which together with (59) yields
(61)V1(tk)=[y(tk)-D∫tk-σtky(u)du]T×P[y(tk)-D∫tk-σtky(u)du]=[y(tk-)-D∫tk-σtky(u)du]T(I-Jk)TP(I-Jk)×[y(tk-)-D∫tk-σtky(u)du]≤[y(tk-)-D∫tk-σtky(u)du]T×P[y(tk-)-D∫tk-σtky(u)du]=V1(tk-).
Thus, we can deduce that
(62)V(tk)≤V(tk-),k∈Z+.
Substituting the above inequality to (58), it yields
(63)V(t)-∫0tξT(u)Ξ(t)ξ(u)du≤V(0),t≥0.
Applying Lemma 3 and (63), we have
(64)∥D∫t-σty(u)du∥2=[D∫t-σty(u)du]T[D∫t-σty(u)du]≤λmax⁡(D2)[∫t-σty(s)ds]T[∫t-σty(s)ds]≤λmax⁡(D2)λmin⁡(Q3)[∫t-σty(s)ds]TQ3[∫t-σty(s)ds]≤σλmax⁡(D2)λmin⁡(Q3)∫t-σtyT(s)Q3y(s)ds≤σλmax⁡(D2)λmin⁡(Q3)V2(t)≤σλmax⁡(D2)λmin⁡(Q3)V(t)≤σλmax⁡(D2)λmin⁡(Q3)V(0),t≥0.
Similarly,
(65)∥y(u)-D∫t-σty(u)du∥2=[y(u)-D∫t-σty(u)du]T[y(u)-D∫t-σty(u)du]≤V1(t)λmin⁡(P)≤V(t)λmin⁡(P)≤V(0)λmin⁡(P),t≥0.
Hence, it can be obtained that
(66)∥y(t)∥≤∥D∫t-σty(u)du∥+∥y(u)-D∫t-σty(u)du∥≤σλmax⁡(D2)λmin⁡(Q3)V(0)+V(0)λmin⁡(P)<∞,t≥0,
where
(67)V(0)=[y(0)-D∫-σ0y(u)du]TP[y(0)-D∫-σ0y(u)du]+σ∫-σ0∫s0yT(u)Q2y(u)duds+∫-σ0yT(s)Q3y(s)ds+γ2∫-τ0∫u0y˙T(s)T22y˙(s)dsdu+ρ∫-ρ0∫u0hT(y(s))K(-s)ZK(-s)h(y(s))dsdu≤{+12ρ3λmax⁡(Z)}2λmax⁡(P)(1+σ2max⁡i∈Λ⁡di)+12σ3λmax⁡(Q2)+σλmax⁡(Q3)+12γ2τ2λmax⁡(T22)+12ρ3λmax⁡(Z)}∥φ∥η2<∞.
So the solution y(t) of models (1) and (4) is uniformly bounded on [0,∞). Thus, considering the continuity of activation function f (i.e., (H3)), it can be deduced from system (38) that there exists some constant M>0 such that ∥y˙(t)∥≤M, t∈[tk-1,tk), k∈Z+. It implies that, |y˙i(t)|≤M, t∈[tk-1,tk), k∈Z+, and i∈Λ, where y˙ denotes the right hand derivative of y at impulsive times.
Finally, we can prove that∥y(t)∥→0 as t→∞, which is similar to the corresponding proof in the literature [25]. Here we omit it. Therefore, the zero solution of (38) or (39) is globally asymptotically stable, which implies that models (1) and (4) have a unique equilibrium point which is globally asymptotically stable. This completes the proof.When there is no leakage delay, that is,σ=0, models (1) and (4) become
(68)x˙(t)=-Dx(t)+Af(x(t))+Bg(x(t-τ(t)))hh+W∫t-ρ(t)tK(t-s)h(x(s))ds+I,hhhhhhhhhhhhhhhhhhiit>0,t≠tk,Δx(tk)=x(tk)-x(tk-)=-Ik(x(tk-)-x*),k∈Z+,x(s)=φ(s),s∈[-η,0].
For model (68), we have the following result by Theorem 8.Corollary 9.
Under the conditions in Theorem7, model (68) has a unique equilibrium point which is globally asymptotically stable if there exist a constant γ>0, an n×n inverse matrix Q1, four n×n matrices P>0, Z>0, T11>0, and T22>0, three n×n diagonal matrices U1>0, U2>0, and U3>0, and real matrices T12, Xl(l=1,…,6) with appropriate dimension such that
(69)[T11T12⋆T22]>0,[P(I-Jk)TP⋆P]≥0,k∈Z+,[Ξχj⋆-Z]<0,j=1,2,
where
(70)Ξ=[Π11-Q1γT12TX1Π15Π16U3Σ6Π18⋆Π220X2γQ1AγQ1B0Π28⋆⋆Π3300U2Σ400⋆⋆⋆Π440000⋆⋆⋆⋆-U1000⋆⋆⋆⋆⋆-U200⋆⋆⋆⋆⋆⋆Π770⋆⋆⋆⋆⋆⋆⋆Π88],χ1=col{X1,X2,O,X3,O,O,O,O},χ2=col{X4,X5,O,O,O,O,O,X6},
with
(71)Π11=-PD-DP-U1Σ1-U3Σ5,Π15=PA+Q1A+U1Σ2,Π16=PB+Q1B,Π18=PW+Q1W+X4,Π22=-γQ1-γQ1T+γ2τT22,Π28=γQ1W+X5,Π33=τT11-γT12T-γT12-U2Σ3,Π44=X3+X3T,Π77=ρ2K(0)ZK(0)-U3,Π88=X6+X6T.Remark 10.
In [25], Li et al. have investigated a class of recurrent neural networks with discrete time-varying delay and constant delay in the leakage term under impulsive perturbations. However, the bounded time-varying distributed delays were not taken into account in network models. In fact, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, and hence there is a distribution of propagation delays over a period of time. Thus, the time-varying distributed delays should be taken into account to reflect more realistic dynamics. The models studied in the present paper have wider range of applications. In the future, we will further investigate the relevant models with such mixed time delays by other current approaches, such as constructing Lyapunov-Krasovskii functional with triple integral terms and introducing free-weighting matrices.Remark 11.
In [27], Wang et al. have dealt with the global asymptotic stability of neural networks with mixed time delays under impulsive perturbations. But there is a derivative restriction of transmission delay τ˙(t)≤c<1, which is a conservative condition. In our results, based on a superior Lyapunov-Krasovskii functional, the criteria do not require the differentiability even differential boundedness of time-varying transmission delay τ(t). In addition, we have considered the basic properties, such as existence and uniqueness of solutions and equilibrium. So our results are less conservative and may be applied effectively in a wider range.Remark 12.
Recently, more researchers have begun to take into account the effect of time-varying leakage delay, which has essential difference from constant leakage delay, on dynamics of models. We would like to think that it may lead to more technical difficulties. How to improve the dynamics, especially the stability properties of neural networks with (time-varying) leakage delay, may be an interesting problem and requires further research.


## 6. Illustrative Example

In this section, an example is given to demonstrate the effectiveness of our results.Example 1.
Consider the following recurrent neural networks model:(72)x˙(t)=-Dx(t-σ)+Af(x(t))+Bg(x(t-τ(t)))+W∫t-ρ(t)tK(t-s)h(x(s))ds+I,hhhhhhhhhhhhhhhhhiit>0,t≠tk,Δx(tk)=x(tk)-x(tk-)=Ik(x(tk-),xtk-),k∈Z+,
where f=g=h=|s|, σ=0.08, τ(t)=0.09-0.01[sin⁡t]*, ρ(t)=0.08-0.02[sin⁡t]*, K(s)=e-s, I=(0,0)T, tk=0.1k, and Ik=diag⁡(0.5,0.5), k∈Z+, and parameter matrices D,A,B, and W are given as follows:
(73)D=[9008],A=[1.5-0.3-0.10.2],B=[-0.60.41.7-1.8],W=[0.20.3-0.50.15].
In this case, we know thatσj-=δj-=ζj-=-1, σj+=δj+=ζj+=1, σ=0.08, τ=0.1, and ρ=0.1. Let γ=2.6; via Matlab LMI toolbox, the feasible solution for the LMIs in Theorem 8 is derived as follows:
(74)P=[0.0017000.0026],Q1=[0.00300.00010.00010.0037],Q2=[0.80110.00020.00020.8061],Q3=[0.59820.00090.00090.6026],Z=[0.8887000.8887],U1=[1.0046001.0046],U2=[0.8793000.8793],U3=[1.0117001.0117],T11=[1.3262-0.0001-0.00011.3260],T12=[0.0388000.0387],T22=[0.00960.00020.00020.0113],X1=10-3×[0.05340.0807-0.13110.0400],X2=10-3×[0.11170.1773-0.35540.1106],X3=[-0.317600-0.3176],X4=[-0.0004-0.00070.0012-0.0004],X5=[-0.0014-0.00220.0044-0.0014],X6=[-0.317700-0.3177].
Hence, from Theorem 8, the unique equilibrium point x*=(0,0)T of system (72) is globally asymptotically stable.


## 7. Conclusion

In this paper, we have investigated a class of impulsive neural networks with mixed time delays and generalized activation functions. Firstly, by using the contraction mapping theorem, we have given a sufficient condition to guarantee the global existence and uniqueness of the solution for the addressed neural networks. Then, a delay-independent sufficient condition for existence of the equilibrium point and some delay-dependent sufficient conditions for stability have been derived, respectively, by using topological degree theory and suitable Lyapunov-Krasovskii functional. The obtained results require neither the boundedness, monotonicity, and differentiability of the activation functions nor the differentiability of time-varying delay. Finally, an example has been given to show the effectiveness and less conservativeness of the obtained results. In the future, we will do some further research on impulsive neural network models with leakage time-varying delay and continuously distributed delay.



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*Source: 327070-2014-05-15.xml*