# Existence and Global Uniform Asymptotic Stability of Almost Periodic Solutions for Cellular Neural Networks with Discrete and Distributed Delays

Authors: Zongyi Hou; Hongying Zhu; Chunhua Feng
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper discusses the existence and global uniform asymptotic stability of almost periodic solutions for cellular neural networks (CNNS). By utilizing the theory of the almost periodic differential equation and the Lyapunov functionals
method, some sufficient conditions are obtained to ensure the existence and global
uniform asymptotic stability. An example is given to illustrate the effectiveness of
the main results.


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## Body

## 1. Introduction

Cellular neural networks (CNNS) are composed of a large number of simple processing units (called neurons), widely interconnected to form a complex network system. It reflects many basic features of the human brain functions. It is a highly complicated nonlinear dynamics system and has successful applications in many fields such as associative, signal, and image processing, pattern recognition, and optimization.In 1984, Hopfield proposed that the dynamic behavior of neurons should be described with a set of ordinary differential equations or functional differential equations. Since then, a lot of research achievements have been published in the world.Recently, many scholars have paid much attention to the research on the dynamics and applications of CNNS. Specially, some scholars have studied the existence and stability of almost periodic solution for neural networks, which can be seen from [1–10] and therein references.In [4], without product systems, by utilizing the generalized Halanay inequality technique and combining the theory of exponential dichotomy with fixed point method, Huang et al. study the existence and global exponential stability of almost periodic solutions for recurrent neural network with continuously distributed delays as follows:
(1)xi′(t)=-ai(t)xi(t)+∑j=1n‍bij(t)fj(xj(t))+∑j=1n‍cij(t)gj(xj(t-τij(t)))+∑j=1n‍dij(t)∫-∞t‍kij(t-s)hj(xj(s))ds+Ii(t),
where fj,gj,hj:R→R,j=1,2,…,n are incentive functions, which satisfy fj(0)=gj(0)=hj(0)=0.  ai(t),bij(t),cij(t) are all almost periodic functions.In [5], Xiang and Cao discuss the following system:
(2)xi′(t)=-ai(xi(t))+∑j=1n‍bij(t)fj(xj(t))+∑j=1n‍cij(t)gj(xj(t-τij(t)))+∑j=1n‍dij(t)∫-∞t‍kij(t-s)hj(xj(s))ds+Ii(t).
Without product systems, by using the Lyapunov functionals method and analytical skills, the results about the existence, attractivity, and exponential stability of almost periodic solutions for the system (2) are obtained.However, a more general system than the systems above is discussed in this paper. We consider the existence and global uniform asymptotic stability of almost periodic solutions to the CNNS with discrete and continuously distributed delays. The system is as follows:(3)xi′(t)=-ai(t,xi(t))xi(t)+∑j=1n‍bij(t)fj(xj(t))+∑j=1n‍cij(t)gj(xj(t-τij))+∑j=1n‍dij(t)∫-∞t‍kij(t-s)hj(xj(s))ds+Ii(t).By using Lemmas3 and 4 in the next section and under the less restrictive conditions, some sufficient conditions are obtained to ensure the existence and global uniform asymptotic stability of almost periodic solutions to the system (3). An example is given to illustrate the effectiveness of the main results at last.


## 2. Preliminaries

In order to facilitate the following section description, we introduce some marks and basic definitions in this section.Iff(t,φ) is an almost periodic in t uniformly for x∈Ω, where Ω⊆Rn is an open set, then the equation
(4)x′(t)=f(t,xt)
is called lagging-type almost periodic differential equation. The following system is defined as the product systems of (4):
(5)x′(t)=f(t,xt(t)),y′(t)=f(t,yt(t)).Definition 1 (see [11]).
If there is a constantM=M(σ,φ) for (σ,φ)∈Ω⊆R×C such that the solution x(t,σ,φ) of (5) through (σ,φ) when t≥σ-r satisfies |x(t,σ,φ)|<M, then the solution x(t,σ,φ) is bound.Definition 2 (see [11]).
Lyapunov functionalsV(t,φ):R×C→R,C=C([-r,0],Rn). Suppose that the solution of (5) through (σ,φ) is x(t,σ,φ), xt(σ,φ) is defined as x(t+θ,σ,φ),θ∈[-r,0]. The total derivative is defined as follows:
(6)V′(t,φ)|(5)=limh→0+-1h[V(t+h,xt+h(t,φ))-V(t,φ)].
Then V(5)′(t,φ) is the right derivative of functionals V(t,φ) along (5).Lemma 3 (see [11]).
Lagging-type almost periodic differential equation (5) has an asymptotically almost periodic solution |x¯(t)|, which satisfies |x¯(t)|≤a<H,H∈R or H=+∞ for all t≥0 defined in R+; then (5) has an almost periodic solution.Lemma 4 (see [11]).
There is a continuousV functional of V(t,φ,ψ) for t≥0,φ,ψ∈CH,CH={φ:φ∈C,|φ|<H},C=C[-r,0],|φ|=supθ∈[-r,0]|φ(θ)|=supθ∈[-r,0]∑i=1n|φi(θ)| such that(Ha)
u(|φ-ψ|)≤V(t,φ,ψ)≤v(|φ-ψ|),(Hb)
|V(t,φ1,ψ1)-V(t,φ2,ψ2)|≤k(|φ1-φ2|+|ψ1-ψ2|),(Hc)
V(5)′(t,φ,ψ)≤-aV(t,φ,ψ),where a is a positive constant, u(s) and v(s) are continuous and non-decreasing, when s→0, u(s)→0, and k is a positive constant. At this time, if (5) has a bounded solution x(t,σ,φ) such that |x(t,σ,φ)|≤H1, where t≥σ≥0,H>H1>0, then (5) in CH has a unique almost periodic solution which is global uniform asymptotic stability.Throughout this paper, we make the following assumptions.(H2.1)
ai(t,xi) is uniform almost periodic continuous function to xi about t. ai(t,xi)>0 for all (t,xi), and we denote min1≤i≤n{inf(t,xi)∈R×Rai(t,xi)}=2a0>0. In addition, ai(t,xi) also satisfies the Lipschitz condition as follows:(7)|ai(t,xi)-ai(t,yi)|≤Lia|xi-yi|,whereLia<a0.
(H2.2)
b
i
j
(
t
), cij(t), dij(t) and Ii(t) are almost periodic continuous functions. we denote constants, respectively, bijl,bijm,cijl,cijm,dijl,dijm,Iil,Iim(i,j=1,2,…,n) as follows:
(8)
inf
t
∈
R
b
i
j
(
t
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=
b
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j
l
,
sup
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∈
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j
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=
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i
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>
0
,
inf
t
∈
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c
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j
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=
c
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l
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sup
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=
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0
,
inf
t
∈
R
d
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=
d
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sup
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∈
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d
i
j
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=
d
i
j
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>
0
,
inf
t
∈
R
I
i
(
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=
I
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l
,
sup
t
∈
R
I
i
(
t
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=
I
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>
0
.(H2.3)
Functionsfj(xj), gj(xj), and hj(xj) are bounded continuous functions, and they satisfy the following Lipschitz conditions:
(9)
|
f
j
(
x
j
)
-
f
j
(
y
j
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|
≤
L
j
f
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x
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-
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≤
L
j
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x
j
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.(H2.4)
Delay kernel functions satisfying
(10)
∫
0
∞
‍
k
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d
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=
1
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<
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hhl
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j
=
1,2
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)
.


## 3. Main Results

Theorem 5.
Assume that (H2.1)–(H2.4) hold; then all solutions of system (3) are bounded.Proof.
Let|fj(xj)|≤Pj, |gj(xj)|≤Qj, |hj(xj)|≤Rj and set N=∑j=1nbijmPj+∑j=1ncijmQj+∑j=1ndijmRj+Iim. From system (3) and the assumption (H2.3), we get
(11)xi′(t)=-ai(t,xi(t))xi(t)+∑j=1n‍bij(t)fj(xj(t))+∑j=1n‍cij(t)gj(xj(t-τij))+∑j=1n‍dij(t)∫-∞t‍kij(t-s)hj(xj(s))ds+Ii(t)≤-ai(t,xi(t))xi(t)+∑j=1n‍bijmPj+∑j=1n‍cijmQj+∑j=1n‍dijmRj+Iim=-ai(t,xi(t))xi(t)+N,
for i=1,2,…,n. By (11), we obtain
(12)xi(t)≤N2a0(1-e-2a0t)+xi(t0)e-2a0t.
This shows that(13)|xi(t)|≤N2a0,i=1,2,…,n.
This completes the proof of the theorem.From Theorem5, all solutions of system (3) are bounded. In order to investigate the globally uniform asymptotic stability of the almost periodic solutions, we assume that ∑i=1n(N/2a0)Lia≤a0.Theorem 6.
Assume that (H2.1)–(H2.4) hold, and suppose further that
(14)σ=min1≤i≤n{a0-∑j=1n‍bijmLif-∑j=1n‍cijmLjg-∑j=1n‍dijmLjh}>0.
Then, in system (3), there exists an almost periodic solution, which is global uniform asymptotic stable.Proof.
From the condition (H2.1), we rewrite system (3) as follows:
(15)xi′(t)=-ai(t,xi(t))xi(t)+∑j=1n‍bij(t)fj(xj(t))+∑j=1n‍cij(t)gj(xj(t-τij))+∑j=1n‍dij(t)∫-∞t‍kij(t-s)hj(xj(s))ds+Ii(t).
The product system of the system (15) is the following form:
(16)xi′(t)=-ai(t,xi(t))xi(t)+∑j=1n‍bij(t)fj(xj(t))+∑j=1n‍cij(t)gj(xj(t-τij))+∑j=1n‍dij(t)∫-∞t‍kij(t-s)hj(xj(s))ds+Ii(t),yi′(t)=-ai(t,yi(t))yi(t)+∑j=1n‍bij(t)fj(yj(t))+∑j=1n‍cij(t)gj(yj(t-τij))+∑j=1n‍dij(t)∫-∞t‍kij(t-s)hj(yj(s))ds+Ii(t).
In order to apply Lemma 4, we construct the Lyapunov functionals about the product system (16) as follows:
(17)V(t,x(·),y(·))=∑i=1n‍|xi(t)-yi(t)|+∑i=1n‍∑j=1n‍cijmLjg∫t-τijt‍|xj(s)-yj(s)|ds+∑i=1n‍∑j=1n‍dijmLjh×∫0+∞‍kij(s)∫t-st‍|xj(u)-yj(u)|duds.
For convenience sake, we denote(18)V(t,x(·),y(·))=V1(t)+V2(t)+V3(t),
where
(19)V1(t)=∑i=1n‍|xi(t)-yi(t)|,V2(t)=∑i=1n‍∑j=1n‍cijmLjg∫t-τijt‍|xj(s)-yj(s)|ds,V3(t)=∑i=1n‍∑j=1n‍dijmLjh∫0+∞‍kij(s)∫t-st‍|xj(u)-yj(u)|duds.
Using (H2.1) and∑i=1n(N/2a0)Lia≤a0, by the triangle inequality, we have
(20)sgn∑i=1n‍(-ai(t,xi(t))xi(t)+ai(t,yi(t))yi(t))=sgn∑i=1n‍{-ai(t,xi(t))xi(t)+ai(t,yi(t))xi(t)=sgn∑i=1n‍h-ai(t,yi(t))xi(t)+ai(t,yi(t))yi(t)}≤sgn∑i=1n‍{(xi(t)-yi(t))N2a0Lia-2a0(xi(t)-yi(t))}≤-a0sgn∑i=1n‍(xi(t)-yi(t)).
Calculating the upright derivative ofV1(t) along system (16) as follows:
(21)D+V1(t)|(16)≤-a0∑i=1n‍|xi(t)-yi(t)|+∑i=1n‍∑j=1n‍|bij(t)||fj(xj(t))-fj(yj(t))|+∑i=1n‍∑j=1n‍|cij(t)||gj(xj(t-τij))-gj(xj(t-τij))|+∑i=1n‍∑j=1n‍|dij(t)|∫-∞t‍kij(t-s)|hj(xj(s))-hj(yj(s))|ds.
Similarly, we calculate the upright derivatives ofV2(t) and V3(t) along system (16), respectively, as follows:
(22)D+V2(t)|(16)=∑i=1n‍∑j=1n‍cijmLjg|xj(t)-yj(t)|-∑i=1n‍∑j=1n‍cijmLjg|xj(t-τij)-yj(t-τij)|;D+V3(t)|(16)=∑i=1n‍∑j=1n‍dijmLjh∫0+∞‍kij(s)|xj(t)-yj(t)|ds-∑i=1n‍∑j=1n‍dijmLjh∫0+∞‍kij(s)|xj(t-s)-yj(t-s)|ds.
Note that(23)∑i=1n‍∑j=1n‍|cij(t)||gj(xj(t-τij))-gj(xj(t-τij))|≤∑i=1n‍∑j=1n‍cijmLjg|xj(t-τij)-yj(t-τij)|,∑i=1n‍∑j=1n‍|dij(t)|∫-∞t‍kij(t-s)|hj(xj(s))-hj(yj(s))|ds≤∑i=1n‍∑j=1n‍dijmLjh∫0+∞‍kij(s)|xj(t)-yj(t)|ds.
Combining with (21) and (22) and the assumptions of Theorem 5, we get
(24)D+V(t,x(·),y(·))|(16)≤-a0∑i=1n‍|xi(t)-yi(t)|+∑i=1n‍∑j=1n‍|bij(t)||fj(xj(t))-fj(yj(t))|+∑i=1n‍∑j=1n‍|cij(t)||gj(xj(t-τij))-gj(xj(t-τij))|+∑i=1n‍∑j=1n‍|dij(t)|∫-∞t‍kij(t-s)+∑i=1n‍∑j=1n‍|dij(t)|∫-∞t‍×|hj(xj(s))-hj(yj(s))|ds+∑i=1n‍∑j=1n‍cijmLjg|xj(t)-yj(t)|-∑i=1n‍∑j=1n‍cijmLjg|xj(t-τij)-yj(t-τij)|+∑i=1n‍∑j=1n‍dijmLjh∫0+∞‍kij(s)|xj(t)-yj(t)|ds-∑i=1n‍∑j=1n‍dijmLjh∫0+∞‍kij(s)|xj(t-s)-yj(t-s)|ds≤-a0∑i=1n‍|xi(t)-yi(t)|+∑i=1n‍∑j=1n‍|bij(t)||fj(xj(t))-fj(yj(t))|+∑i=1n‍∑j=1n‍cijmLjg|xj(t)-yj(t)|+∑i=1n‍∑j=1n‍dijmLjh∫0+∞‍kij(s)|xj(t)-yj(t)|ds≤-a0∑i=1n‍|xi(t)-yi(t)|+∑i=1n‍∑j=1n‍bijmLjf|xj(t)-yj(t)|+∑i=1n‍∑j=1n‍cijmLjg|xj(t)-yj(t)|+∑i=1n‍∑j=1n‍dijmLjh|xj(t)-yj(t)|≤-a0∑i=1n‍|xi(t)-yi(t)|+∑i=1n‍∑j=1n‍bjimLif|xi(t)-yi(t)|+∑i=1n‍∑j=1n‍cjimLig|xi(t)-yi(t)|+∑i=1n‍∑j=1n‍djimLih|xi(t)-yi(t)|=∑i=1n‍{-a0+∑j=1n‍bjimLif+∑j=1n‍cjimLig+∑j=1n‍djimLih}×|xi(t)-yi(t)|≤-σ∑i=1n‍|xi(t)-yi(t)|.
By Theorem 5 and Lemmas 3 and 4, there exists an almost periodic solution of system (3), which is global uniform asymptotic stable. This completes the proof of Theorem 6.


## 4. An Example

Example 1.
Consider the following cellular neural network which consists of two neurons:(25)xi′(t)=-ai(t,xi(t))xi(t)+∑j=1n‍bij(t)fj(xj(t))+∑j=1n‍cij(t)gj(xj(t-τij))+∑j=1n‍dij(t)∫-∞t‍kij(t-s)hj(xj(s))ds+Ii(t),+∑j=1n‍dij(t)∫-∞t‍kij(t-s)hj(xj(s))dshi=1,2,
where
(26)(ai(t,xi(t))xi(t))2×2=(11-e-|x1(t)|+sinπt2-sinx2(t)x1(t)0011-e-|x2(t)|-cost2+sinx1(t)x2(t)),(bij(t))2×2=(0.2sinπt0.1cost0.1sint0.2cosπt),(cij(t))2×2=(0.2sint0.1sint0.2sin2t0.1cosπt),(dij(t))2×2=(0.3sin2t0.1cost0.2sinπt0.1cosπt),(Ii(t))1×2=(0.1sin2t0.1cost).
We select the functionsfj(x)=gj(x)=hj(x)=sh(x)/ch(x) and the kernel functions kij(s)=e-s. Then, Ljf=Ljg=Ljh=1. Because the periods of sinπt and cost are 2 and 2π, respectively. The quotient of 2π and 2 is irrational. Then system (25) is an almost periodic system. In addition, a0=1.5 and
(27)∑j=12‍bijmLif+∑j=12‍cijmLjg+∑j=12‍dijmLjh≤1.2<1.5.
From Pj=Qj=Rj=1, we have N=∑j=1nbijmPj+∑j=1ncijmQj+∑j=1ndijmRj+Iim≤1.2<a0; then we get ∑i=1n(N/2a0)Lia=0.2<1.5.
It is easy for us to verify that the conditions (H2.1)–(H2.4) in Theorem5 hold. Therefore, in the system (25), there exists an almost periodic solution, which is global uniform asymptotic stable.



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*Source: 516293-2013-12-30.xml*