# Global Exponential Stability for DCNNs with Impulses on Time Scales

Authors: Yongkun Li; Yuanhong Zhi
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

A class of delayed cellular neural networks (DCNNs) with impulses on time scales is considered. By using the topological degree theory, and the time scale calculus theory some sufficient conditions are derived to ensure the existence, uniqueness, and global exponential stability of equilibria for this class of neural networks. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and the discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new and complementary to the previously known results.


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## Body

## 1. Introduction

Chua and Yang [1] proposed a novel class of information-processing systems called cellular neural networks (CNNs) in 1988. The CNNs can be applied in signal processing and can also be used to solve some image processing and pattern recognition problems [2]. Since time delays are unavoidable due to finite switching speeds of the amplifiers, delayed cellular neural networks (DCNNs) have been widely studied and successfully applied to pattern recognition, associative memories, and signal processing and optimization, especially in image processing. The dynamic behavior of the networks plays an important role in such applications [3–8]. Therefore, there are many works on the stability of equilibrium point of delayed cellular neural networks (DCNNs) [5–13].Most neural networks can be classified into two types: continuous or discrete. However, many real-world systems and natural processes cannot be categorized into one of them. They display characteristics of both continuous and discrete styles. For instance, some biological neural networks in biology, bursting rhythm models in pathology, and optimal control models in economics are characterized by abrupt changes of state. These are the familiar impulsive phenomena. Other examples can also be found in information science, electronics, automatic control systems, computer networking, artificial intelligence, robotics, telecommunications, and so forth. Such a kind of phenomena, in which sudden and sharp changes often occur in a continuous process, cannot be well described by pure continuous or pure discrete models. Therefore, it is important and, in effect, necessary to study a new type of neural networks—impulsive neural networks—as an appropriate description of these phenomena of abrupt qualitative dynamical changes of essentially continuous systems. The fundamental theory of impulsive differential equations has been developed in [14]. Since delays and impulses can affect the dynamical behaviors of the system, it is necessary to investigate both delay and impulsive effects on the stability of neural networks. For more details, one can refer to [10, 13, 15–23].The theory of time scale was initiated by Hilger in 1988, which has recently received a lot of attention [24–26]. The field of dynamic equations on time scale contains links and extends the classical theory of differential and difference equations. It is well known that both continuous and discrete systems are very important in implementation and applications (see [27–30]). But it is troublesome to study the stability for continuous and discrete systems, respectively. Therefore, it is significant to study that on time scales which can unify the continuous and discrete situations [21, 31–40].Motivated by above, in this paper, we are concerned with the following impulsive DCNN on time scales:(1)xiΔ(t)=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijfj(xj(ζij(t,xj(t))))+Ii,t∈𝕋0+,t≠tk,xi(tk+)=xi(tk-)+Pi(xi(tk-)),k=1,2,…,i=1,2,…,n,
where n corresponds to the numbers of units in a neural network; xi(t) corresponds to the state of the ith unit at time t; fj(xj(t)) denotes the output of the jth unit at time t. 𝕋0+ is the 𝕋-interval {t∈𝕋,t≥0}, and 𝕋 denotes a time scale, which is an arbitrary nonempty closed subset of the real number ℝ and with bounded graininess μ. For the simplicity, we assume that 0∈𝕋 and 𝕋 is unbounded above; that is, sup𝕋=+∞. Further, bij, cij, ai, and Ii are constants. bij, cij denote the strength of the jth unit at time t and ζij(t,xj(t)), respectively. Ii denotes the external bias on the ith unit and ai represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs. tk, k=1,2,… are the moments of impulsive perturbations and satisfy 0=t0<t1<t2<⋯ and limk→∞=∞, μ(tk)=0 (see Definition 3). Pi(xi(tk)) represents the abrupt change of the state xi(t) at the impulsive moment tk. To the best of our knowledge, this is first paper to study DCNNs with impulses on time scales.Throughout this paper, we assume thatxi(tk)≡xi(tk-) and(H1)
functionsζij satisfy ζij:𝕋×ℝ→𝕋 for all t∈𝕋, i,j=1,2,…,n;(H2)
fj∈C(ℝ,ℝ)(i=1,2,…,n) and there exists a positive number Fi such that |fi(x)-fi(y)|≤Fi|x-y| for all x,y∈ℝ, i=1,2,…,n.Remark 1.
The neural network (1) is a system of differential equations with state-dependent deviating arguments and from (H1), one can see that deviating arguments in (1) may be delayed type, advanced type, or mixed type.Our main purpose of this paper is to study the existence and global exponential stability of the equilibria of (1) by using the topological degree theory and the time scale calculus theory. The results of this paper are completely new and complementary to the previously known results.The organization of this paper is as follows. In the next section, some notations, definitions, and lemmas are presented. Section3 addresses the existence and uniqueness of equilibria of system (1) by using the method of topological degree theory. In Section 4, we give the criteria of global exponential stability of the equilibrium point of system (1). In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections 3 and 4.


## 2. Notations and Preliminaries

In this section, we will first recall some basic definitions and lemmas which will be useful for the proof of our main results.Definition 2 (see [33, 34]).
A time scale𝕋 is arbitrary nonempty closed subset of the real set ℝ with the topology and ordering inherited from ℝ.Definition 3 (see [33, 34]).
On any time scale𝕋, we define the forward and backward jump operators by
(2)σ(t)≔inf{s∈𝕋:s>t},ρ(t)≔sup{s∈𝕋:s<t}.
A point t is said to be left-dense if t>inf𝕋 and ρ(t)=t, right-dense if t<sup𝕋 and σ(t)=t, left-scattered if ρ(t)<t, and right-scattered if σ(t)>t. The graininess function μ for a time scale 𝕋 is defined by μ(t)≔σ(t)-t. If 𝕋 has a left-scattered maximum m, then we defined 𝕋k to be 𝕋∖{m}. Otherwise, 𝕋k=𝕋.Definition 4 (see [33, 34]).
For a functionf:𝕋→ℝ (the range ℝ of f may be actually replaced by Banach space), the (delta) derivative is defined by
(3)fΔ=f(σ(t))-f(t)σ(t)-t
if f is continuous at t and t is right-scattered. If t is not right-scattered, then the derivative is defined by
(4)fΔ=lims→t,s∈𝕋f(σ(t))-f(s)σ(t)-s=lims→t,s∈𝕋f(t)-f(s)t-s,
provided this limit exists.Lemma 5 (see [33, 34]).
Iff, g are differential at t∈𝕋, one has(1)
f(σ(t))=f(t)+μ(t)fΔ(t);(2)
(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t).Definition 6 (see [33, 34]).
A functionF:𝕋k→ℝ is called a delta-antiderivative of f:𝕋→ℝ provided FΔ=f holds for all t∈𝕋k. In this case, we define the integral of f by
(5)∫at‍f(s)Δs=F(t)-F(a)fort∈𝕋,
and we have the following formula:
(6)∫tσ(t)‍f(s)Δs=μ(t)f(t)fort∈𝕋k.Definition 7 (see [33, 34]).
A functionf:𝕋→ℝ is called right-dense continuous (rd-continuous) provided it is continuous at right-dense points of 𝕋 and the left-sided limit exists (finite) at left-dense point of 𝕋. The set of all right-dense continuous functions on 𝕋 is defined by Crd=Crd(𝕋)=Crd(𝕋,ℝ). If f is continuous at each right-dense point and each left-dense point, then f is said to be continuous function on 𝕋. We define C(J,ℝ)={f(t)iscontinuousonJ}.Lemma 8 (see [33, 34]).
Ifa,b∈𝕋, α,β∈ℝ and f,g∈C(𝕋,ℝ), then one has (1)
∫ab‍[αf(t)+βg(t)]Δt=α∫ab‍f(t)Δt+β∫ab‍g(t)Δt;(2)
iff(t)≥0 for all a≤t<b, then ∫ab‍f(t)Δt≥0;(3)
if|f(t)|≤g(t) on [a,b):={t∈𝕋:a≤t<b}, then |∫ab‍f(t)Δt|≤∫ab‍g(t)Δt.Definition 9 (see [33, 34]).
A functionp:𝕋→ℝ is called regressive if 1+μ(t)p(t)≠0 for all t∈𝕋. If p is regressive function, then the generalized exponential function ep is defined by
(7)ep(t,s)=exp{∫stξμ(τ)(r(τ))Δτ},fors,t∈𝕋,
with the cylinder transformation
(8)ξh(z)={log(1+hz)h,ifh≠0,z,ifh=0.Letp,q:𝕋→ℝ be two regressive functions; we define
(9)p⊕q≔p+q+μpq,⊖p≔-p1+μp,p⊖q≔p⊕(⊖q).
Then, the generalized exponential function has the following properties.Lemma 10 (see [33, 34]).
Assume thatp,q:𝕋→ℝ are two regressive functions; then (1)
e0(t,s)≡1 and ep(t,t)≡1;(2)
ep(σ(t),s)=(1+μ(t)p(t))ep(t,s);(3)
ep(t,σ(s))=ep(t,s)/(1+μ(s)p(s));(4)
1/ep(t,s)=e⊖p(t,s);(5)
ep(t,s)=1/ep(s,t)=e⊖p(s,t);(6)
ep(t,s)ep(s,r)=ep(t,r);(7)
ep(t,s)eq(t,s)=ep⊕q(t,s);(8)
ep(t,s)/eq(t,s)=ep⊖q(t,s).Definition 11.
A pointx*=(x1*,x2*,…,xn*)T∈ℝn is called an equilibrium point of model (1) if x(t)=x* is a solution of (1).Throughout this paper, we always assume that the impulsive jump vectorP satisfies
(10)P(x*)=(P1(x1*),P2(x2*),…,Pn(xn*))T=0.
That is, if x* is an equilibrium point of the following non-impulsive system:
(11)xiΔ(t)=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijfj(xj(ζij(t,xj(t))))+Ii,i=1,2,…,n,
then it is also the equilibrium point of impulsive system (1).Definition 12 (see [41]).
A real matrixD=(dij)n×n is said to be a nonsingular M-matrix if dij≤0(i,j=1,2,…,n,i≠j), and all successive principal minors of D are positive.Lemma 13 (see [41]).
LetD=(dij)n×n with dij≤0(i,j=1,2,…,n,i≠j); 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.


## 3. Existence and Uniqueness of Equilibrium Point

In this section, we will discuss the existence and uniqueness of equilibria of the DCNN with impulses on time scales and give their proofs.Theorem 14.
Under assumptions (H1) and (H2), if the following condition is satisfied(H)
ai-∑j=1nFj(|bij|+|cij|)>0, i=1,2,…,n,
then there is exactly one equilibrium point of model (1).Remark 15.
From Lemma13, we can easily prove that (H) holds implying that the following condition is true:(H0)
there exists a vectorξ=(ξ1,ξ2,…,ξn)T>0 such that
(12)aiξi-Fi∑j=1n(|bji|+|cji|)ξj>0,i=1,2,…,n.For convenience, we setA=diag(a1,a2,…,an), B=(|bij|)n×n, C=(|cij|)n×n, and F=diag(F1,F2,…,Fn). Let η=(1,1,…,1︸n)T, D=A-(B+C)F. From assumption (H), we have
(13)Dη=[A-(B+C)F]η=(a1-∑j=1n‍Fj(|b1j|+|c1j|)a2-∑j=1nFj(|b2j|+|c2j|)⋮an-∑j=1nFj(|bnj|+|cnj|))>0,
which implies that D is a nonsingular M-matrix. So we know that DT is a nonsingular M-matrix. Hence, there exists a vector ξ=(ξ1,ξ2,…,ξn)T>0 such that
(14)DTξ=[AT-FT(BT+CT)]ξ=(a1ξ1-F1∑j=1n(|bj1|+|cj1|)ξja2ξ2-F2∑j=1n(|bj2|+|cj2|)ξj⋮anξn-Fn∑j=1n(|bjn|+|cjn|)ξj)>0.
It follows that (H0) holds.Now, we prove our theorem.Proof.
Letx*=(x1*,x2*,…,xn*)T be an equilibrium point of system (1); then, we have
(15)-aixi*+∑j=1nbijfj(xj*)+∑j=1ncijfj(xj*)+Ii=0,i=1,2,…,n.
We denote h(x1,x2,…,xn)=(h1,h2,…,hn)T, where
(16)hi=aixi-∑j=1nbijfj(xj)-∑j=1ncijfj(xj)-Ii,i=1,2,…,n.
Obviously, the equilibrium points of model (1) are solutions of system
(17)hi=0,i=1,2,…,n.
Define the following homotopic mapping:
(18)H(x1,x2,…,xn)=λh(x1,x2,…,xn)+(1-λ)(x1,x2,…,xn)T,
where λ∈[0,1]. Let Hk(k=1,2,…,n) denote the kth component of H(x1,x2,…,xn); then, we can get
(19)|Hi|=|(1-λ)xi+λaixi-λ∑j=1nbijfj(xj)-λ∑j=1ncijfj(xj)-λIi|≥(1-λ)|xi|+λai|xi|-λ∑j=1n|bij||fj(xj)|-λ∑j=1n|cij||fj(xj)|-λ|Ii|≥λai|xi|-λ∑j=1n(|bij|+|cij|)|fj(xj)-fj(0)|-λ∑j=1n(|bij|+|cij|)|fj(0)|-λ|Ii|≥λai|xi|-λ∑j=1nFj(|bij|+|cij|)|xj|-λ∑j=1n(|bij|+|cij|)|fj(0)|-λ|Ii|.
It follows that
(20)∑i=1n|Hi|≥∑i=1n[λai|xi|-λ∑j=1nFj(|bij|+|cij|)|xj|-λ∑j=1n(|bij|+|cij|)|fj(0)|-λ|Ii|]≥λ∑i=1n[ai-∑j=1nFj(|bij|+|cij|)]|xj|-λ∑i=1n[∑j=1n(|bij|+|cij|)|fj(0)|+|Ii|].
Let
(21)θ=min1≤i≤n{ai-∑j=1nFj(|bij|+|cij|)},γ=max1≤i≤n{(|bij|+|cij|)|fj(0)|},∥x∥1=∑i=1n|xi|.
From the assumption of the theorem, we can easily see that θ>0. Let
(22)Ω={x∣∥x∥1≤n(γ+1)θ}.
Then, for any x∈∂Ω, we have
(23)∑i=1n|Hi|≥λθ∑i=1n|xi|-λnγ=λθn(γ+1)θ-λnγ>0,λ∈(0,1].
As λ=0, we have
(24)H(x1,x2,…,xn)=(x1,x2,…,xn)T≠0,x∈∂Ω.
Hence, all the above conclusions mean that
(25)H(x1,x2,…,xn)≠0,foranyx∈∂Ω,λ∈[0,1].
From the homotopy invariance theorem, we obtain
(26)deg(h,Ω,0)=deg(H,Ω,0)=deg(I,Ω,0)=1,
where I is the identity operator. By topological degree theory, we can easily know that system (11) has at least one solution in Ω. That means model (1) has at least an equilibrium point.
In order to prove the uniqueness of the equilibrium point, letx*=(x1*,x2*,…,xn*)T and y*=(y1*,y2*,…,yn*)T be two equilibrium points of system (1). So, we have
(27)-aixi*+∑j=1nbijfj(xj*)+∑j=1ncijfj(xj*)+Ii=0,i=1,2,…,n,-aiyi*+∑j=1nbijfj(yj*)+∑j=1ncijfj(yj*)+Ii=0,i=1,2,…,n.
Then,
(28)ai(xi*-yi*)=∑j=1n(bij+cij)[fj(xj*)-fj(yj*)],i=1,2,…,n.
By using assumption (H2), we get
(29)ai|xi*-yi*|≤∑j=1nFj(|bij|+|cij|)|xj*-yj*|,i=1,2,…,n.
It follows that
(30)aiξi|xi*-yi*|≤ξi∑j=1nFj(|bij|+|cij|)|xj*-yj*|,i=1,2,…,n.
Hence,
(31)∑i=1naiξi|xi*-yi*|≤∑i=1nξi∑j=1nFj(|bij|+|cij|)|xj*-yj*|=∑i=1n∑j=1nξiFj(|bij|+|cij|)|xj*-yj*|=∑i=1n∑j=1nξjFi(|bji|+|cji|)|xi*-yi*|=∑i=1n[Fi∑j=1n(|bji|+|cji|)ξj]|xi*-yi*|.
So, we get
(32)∑i=1n(aiξi-Fi∑j=1n(|bji|+|cji|)ξj)|xi*-yi*|≤0.
From the assumption (H0), we getxi*=yi*, i=1,2,…,n. Therefore, system (1) has one unique equilibrium point. The proof is complete.


## 4. Global Exponential Stability of the Equilibrium Point

In this section, we consider the following DCNN system with impulses of the type(33)xiΔ(t)=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijfj(xj(t-τij))+Ii,t∈𝕋0+,t≠tk,Δxi(tk)=xi(tk+)-xi(tk-)=Pi(xi(tk))Δxi(tk)=-γik(xi(tk)-xi*),k=1,2,…,i=1,2,…,n,
where ai, bij, cij, Ii, and fj(i,j=1,2,…,n) are defined as those in (1) and τij(i,j=1,2,…,n) are positive constants which satisfy t-τij∈𝕋 for all t∈𝕋, i,j=1,2,…,n. Let τ=max1≤i,j≤n(τij). Then, the initial conditions associated with (33) are of the form
(34)xi(s)=ϕi(s),s∈[-τ,0]∩𝕋,
where ϕi∈Crd([-τ,0]∩𝕋,ℝ), i=1,2,…,n are rd-continuous.Definition 16.
Letx*=(x1*,x2*,…,xn*)T be an equilibrium point of (33) with initial value ϕ*=(φ1*,φ2*,…,φn*)T. If there exists a positive constant λ with -λ∈ℛ+ such that for t0∈[-τ,0]𝕋, there exists M>1 such that for an arbitrary solution x(t)=(x1(t),x2(t),…,xn(t))T of (33) with initial value ϕ(s)=(ϕ1(s),ϕ2(s),…,ϕn(s))T satisfies
(35)|x(t)-x*|1≤M∥ϕ-ϕ*∥e-λ(t,t0),t∈[-τ,∞)𝕋,t≥t0,
where |x(t)-x*|1=max1≤i≤n{|xi(t)-xi*|}, ∥ϕ-ϕ*∥=max1≤i≤nsups∈[-τ,0]𝕋{|φi(s)-φi*|}. Then the equilibrium point x* is said to be exponentially stable.Now, we study the global exponential stability of the unique equilibrium to (33) on time scales by using Lyapunov method. We have the following.Theorem 17.
Let (H2) and (H) hold. Suppose further that(H3)
0<γik<2, i=1,2,…,n, k=1,2,….Then, the equilibrium x*=(x1*,x2*,…,xn*)T of system (33) is globally exponentially stable.Remark 18.
We denote the𝕋-interval [a,b]𝕋 as [a,b]𝕋≔{t∈𝕋∣a≤t≤b}.Now, we prove Theorem17.Proof.
Letyi(t)=xi(t)-xi*(i=1,2,…,n). Then, we can rewrite (33) as
(36)yiΔ(t)=-aiyi(t)+∑j=1nbij[fj(yj(t)+xj*)-fj(xj*)]+∑j=1ncij[fj(yj(t-τij)+xj*)-fj(xj*)],t≠tk,t∈𝕋0+,yi(tk+)=yi(tk)-γikyi(tk),i=1,2,…,n,k=1,2,….
Multiplying both sides of the first equation of (4.2) bye-ai(t,σ(s)) and integrating on [t0,t]𝕋, where t0∈[-τ,0]𝕋, we get
(37)yi(t)=yi(t0)e-ai(t,t0)+∫t0te-ai(t,σ(s))×{∑j=1nbij[fj(yj(s)+xj*)-fj(xj*)]∑j=1ncij[fj(yj(s-τij)+xj*)-fj(xj*)]}+∑j=1ncij[fj(yj(s-τij)+xj*){∑j=1ncij[fj(yj(s-τij)+xj*)-fj(xj*)]}∑j=1nbij[fj(yj(s)+xj*)-fj(xj*)]-fj(xj*)]}Δs,i=1,2,…,n.
For positive constant α<min1≤i≤nai with -α∈ℛ+, we have e⊖α(t,t0)>1, where t∈[-τ,t0]𝕋. Take
(38)M>max1≤i≤n{aiai-∑j=1mFj(|bij|+|cij|)}.
In view of (H), we have M>1. Hence, it is obvious that
(39)|y(t)|1≤Me⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈[-τ,t0]𝕋.
We claim that
(40)|y(t)|1≤Me⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈(t0,t1]𝕋.
To prove this claim, we show that for any p>1, the following inequality holds
(41)|y(t)|1<pMe⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈(t0,t1]𝕋.
By way of contradiction, assume that (41) does not hold. Then, there exist ρ∈(t0,t1]𝕋 and i0∈{1,2,…,n} such that
(42)|yi0(ρ)|≥pMe⊖α(ρ,t0)∥ϕ-ϕ*∥,|yi0(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,ρ)𝕋,|yl(t)|<pMe⊖α(t,t0)∥ϕ-ϕ*∥,forl≠i0,t∈(t0,ρ]𝕋,l=1,2,…,n.
Therefore, there must be a constant δ1≥1 such that
(43)|yi0(ρ)|=δ1pMe⊖α(ρ,t0)∥ϕ-ϕ*∥,|ui0(t)|<δ1pMe⊖α(t,t0)∥ϕ-ϕ*∥,t∈(t0,ρ)𝕋,|ul(t)|<δ1pMe⊖α(t,t0)∥ϕ-ϕ*∥,forl≠i0,t∈(t0,ρ]𝕋,l=1,2,…,n.
Note that, in view of (37), we have
(44)|yi0(ρ)|=|yi0(t0)e-ai0(ρ,t0)fj(xj*)]{∑j=1n}Δs|+∫t0ρe-ai0(ρ,σ(s))×{∑j=1nbi0j[fj(yj(s)+xj*)-fj(xj*)]+∑j=1nci0j[fj(xj*)]fj(yj(s-τij)+xj*){∑j=1n-fj(xj*)]}Δs|≤e-ai0(ρ,t0)∥ϕ-ϕ*∥+∫t0ρe-ai0(ρ,σ(s))×(∑j=1m|bi0j|Fj|yj(s)|+∑j=1n|ci0j|Fj|yj(s-τi0j)|)Δs≤e-ai0(ρ,t0)∥ϕ-ϕ*∥+∫t0ρe-ai0(ρ,σ(s))×(∑j=1m|bi0j|Fjδ1pMe⊖α(s,t0)(∑j=1m+∑j=1n|ci0j|Fjδ1pMe⊖α(s-τi0j,t0))Δs=∥ϕ-ϕ*∥e⊖α(ρ,t0)e-ai0⊕α(ρ,t0)+δ1pMe⊖α(ρ,t0)∥ϕ-ϕ*∥×∫t0ρe-ai0⊕α(ρ,σ(s))×(∑j=1m|bi0j|Fje⊖α(ρ,σ(s))+∑j=1n|ci0j|Fje⊖α(σ(s),s-τi0j))Δs≤∥ϕ-ϕ*∥e⊖α(ρ,t0)e-ai0+α(ρ,t0)+δ1pMe⊖α(ρ,t0)∥ϕ-ϕ*∥×∫t0ρe-ai0(ρ,σ(s))×(∑j=1m|bi0j|Fj+∑j=1n|ci0j|Fj)Δs≤∥ϕ-ϕ*∥e⊖α(ρ,t0)+δ1pMe⊖α(ρ,t0)∥ϕ-ϕ*∥1-ai0×∫t0ρ(-ai0)e-ai0(ρ,σ(s))Δs×(∑j=1m|bi0j|Fj+∑j=1n|ci0j|Fj)=∥ϕ-ϕ*∥e⊖α(ρ,t0)-1ai0δ1pMe⊖(-α)(ρ,t0)∥ϕ-ϕ*∥×(e-ai0(ρ,t0)-1)(∑j=1m|bi0j|Fj+∑j=1n|ci0j|Fj)<δ1pM∥ϕ-ϕ*∥e⊖α(ρ,t0)×(1δ1pM+1ai0∑j=1mFj(|bi0j|+|ci0j|))<δ1pM∥ϕ-ϕ*∥e⊖α(ρ,t0)×(1M+1ai0∑j=1mFj(|bi0j|+|ci0j|))<δ1pMe⊖α(ρ,t0)∥ϕ-ϕ*∥.
Thus, we get a contradiction. Hence, (41) holds. Let p→1; then (40) holds. From (40), we have that |yi(t1)|≤Me⊖α(t,t0)∥ϕ-ϕ*∥, i=1,2,…,n. Since |yi(t1+)|=|1-γi1||yi(t1)|<|yi(t1)|, i=1,2,…,n, it follows that
(45)|yi(t1+)|≤Me⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈[-τ,t1]𝕋,i=1,2,…,n.
Thus, for t∈[t1,t2]𝕋, we may repeat the above procedure and obtain
(46)|yi(t)|≤Me⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈[t1,t2]𝕋,i=1,2,…,n.
Similarly, we have
(47)|yi(t)|≤Me⊖α(t,t0)∥ϕ-ϕ*∥,∀t∈[-τ,∞)𝕋,i=1,2,…,n.
Take -λ=⊖α; then λ>0 and -λ∈ℛ+. Hence, we have that
(48)|y(t)|1≤M∥ϕ-ϕ*∥e-λ(t,t0),t∈[-τ,∞)𝕋,t≥t0,
which means that the equilibrium point x* of (33) is exponentially stable. This completes the proof.Remark 19.
If the time scale𝕋=ℝ, then μ(t)=0 and system (33) becomes the following model:
(49)dxi(t)dt=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijfj(xj(t-τij))+Ii,t≠tk,xi(tk+)=xi(tk)-γik(xi(tk)-xi*),k=1,2,…,i=1,2,…,n.
From Theorem 17, we can immediately derive the following result.Corollary 20.
Suppose that system (49) satisfies condition (H2) and (H), and the following assumptions hold: (1)
-ai+∑j=1nFj(|bij|+|cij|)<0;(2)
0<γik<2.Then, the equilibrium of system (49) is globally exponentially stable.Remark 21.
In [42], by utilizing the time scale calculus theory, topological degree theory, and Hölder’s inequality on time scales, authors studied the existence and the global exponential stability of equilibrium point to a class of impulsive BAM neural networks with distributed delays on time scales. But, results obtained in [42] cannot be applied to (1). Also, for establishing the global exponential stability of equilibrium point to (1), our method used in this paper is totally different from that used in [42].


## 5. An Example

In this section, an example is given to show the effectiveness of the result obtained in the previous section. Because the condition (4.2) is not dependent on the impulses, we just need to check it with the nonimpulsive system.Consider the following simple DCNN on time scale𝕋:
(50)xiΔ(t)=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijfj(xj(t-τij))+Ii,t∈𝕋0+,
where (a1,a2)T=(1,1)T, Ii=2, τij=2(i,j=1,2),
(51)(b11b12b21b22)=(0.010.010.010.01),(c11c12c21c22)=(0.020.030.030.02).
Taking f1(x)=f2(x)=(1/2)(|x+1|+|x-1|), we can easily see that F1=F2=1.Let𝕋=⋃k=0∞[2k,2k+1], then for t∈𝕋, t≠2k+1(k=0,1,2,…), we have μ(t)=0, and for t=2k+1(k=0,1,2,…), we have μ(t)=1.We have that(52)a1-∑j=12Fj(|b1j|+|c1j|)=0.93>0,a2-∑j=12Fj(|b2j|+|c2j|)=0.93>0,
which imply that the assumption (H) of Theorem 14 holds. Thus, it follows from Theorems 14 and 17 that system (50) has a unique equilibrium point which is globally exponentially stable (see Figure 1).Figure 1
Transient responses of statesx1, x2 in Example when 𝕋=⋃k=1∞‍[2k,2k+1].Sinceμ(t)≡0 for t∈𝕋=ℝ and μ(t)≡1 for t∈𝕋=ℤ, from the discussion above one can easily see that for 𝕋=ℝ or 𝕋=ℤ, (50) always has a unique equilibrium point which is globally exponentially stable. That is, the following continuous-time system
(53)xi′(t)=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijfj(xj(t-τij))+Ii,t∈ℝ+
and its discrete-time analogue
(54)Δxi(t)=-aixi(t)+∑j=1nbijfj(xj(t))+∑j=1ncijfj(xj(t-τij))+Ii,t∈ℤ+
have the same dynamical properties, where ai, Ii, τij, and fj are the same as those in (50) (see Figures 2 and 3).Figure 2
Transient responses of statesx1, x2 in Example when 𝕋=ℝ.Figure 3
Transient responses of statesx1, x2 in Example when 𝕋=ℤ.


## 6. Conclusion

Using the topological degree theory and the time scale calculus theory, some sufficient conditions are obtained to ensure the existence and the global exponential stability of equilibria for DCNNs neural networks with impulses on time scales. This is the first time to apply the time scale calculus theory to unify the study of the stability of the equilibrium for DCNNs with impulses on time scales under the same framework. The results obtained in this paper possess highly important significance and are easily checked in practice. In addition, the method in this paper may be applied to some other systems such as the BAM and Cohen-Grossberg systems with impulses and so on.



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*Source: 934592-2014-01-02.xml*