# Stability Analysis of a Class of Neural Networks with State-Dependent State Delay

Authors: Yue Chen; Jin-E Zhang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

The differential equations with state-dependent delay are very important equations because they can describe some problems in the real world more accurately. Due to the complexity of state-dependent delay, it also brings challenges to the research. The value of delay varying with the state is the difference between state-dependent delay and time-dependent delay. It is impossible to know exactly in advance how far historical state information is needed, and then the problem of state-dependent delay is more complicated compared with time-dependent delay. The dominating work of this paper is to solve the stability problem of neural networks equipped with state-dependent state delay. We use the purely analytical method to deduce the sufficient conditions for local exponential stability of the zero solution. Finally, a few numerical examples are presented to prove the availability of our results.


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## Body

## 1. Introduction

Neural network is an information processing system to simulate the structures and functions of the human brain. As far as we know, the functions and structures of the individual neuron are simple, but the dynamic behavior of the neural network is very rich and intricate [1, 2]. In the neural network, several simple processing units are connected to each other in a certain way. Neural network system is highly complex, which not only has the common characteristics of general nonlinear systems but also has its own unique features [3–6]. For example, it is proved that the neural network has the ability to approximate nonlinear mapping, and any continuous nonlinear function mapping can be approximated by the multilayer neural network with arbitrary precision. It shows that the neural network has a good application prospect in the challenging nonlinear control field [3]. Also, the parallel distributed integrated optimization processing of information makes the neural network very suitable for solving large-scale real-time computing problems in system control [4]. At the same time, some neural network models have the trait of automatically searching the extremum of energy function, which is very useful in adaptive control design [5]. In addition, the neural network also has the peculiarity of high fault-tolerant ability, generalization ability, and adaptive ability for learning behavior [6].Over that seventy years, neural networks have been involved in combinatorial optimization, pattern recognition, image processing, robot control, signal processing, and other scientific fields and achieved extensive success [7–10]. In the application of the neural network, on the one hand, information transmission between neurons needs time. On the other hand, due to the influence of hardware implementation in reality (such as limited switching speed), the time delay phenomenon is inevitable [11]. The existence of time delay may induce instability, oscillation, and poor performance, but we can also overcome the unfavorable effect about some types of time delays by developed control frameworks, such as based on the Halanay inequality framework [1, 2, 11, 12]. At present, many different delays have been applied to the neural network [2, 12–18], for instance, time-varying delay [12, 14], distributed delay [13, 15, 16], discrete constant delay [2, 13], proportional delay [17], and unbounded delay [18].State-dependent delay (SDD) is all around us [19–22]. With limited natural resources, Antarctic whales and seals tend to mature longer if their populations are large [19]. In the problem of car following, it is inevitable to encounter the phenomenon that the time delay changes with the state, which contains physiological time delay, mechanical time delay, and motion time delay [20, 21]. In addition, in the blood circulation system, the concentration of nutrients regulates the mitotic cycle of hematopoietic stem cells; thus, the mitotic cycle of stem cells is affected by the concentration of cells in the region [22]. In these cases, in order to describe change and evolution of things more accurately and make the research results more realistic and modest with nuanced understanding, we must adopt differential equations with SDD.Neurodynamics help to identify the highly complex and precise multilevel nonlinear brain system. The processing of neural information involves the coupling and cooperation of multiple levels and regions. In this way, the nerve activity about the structural neural network is worthy of study and discussion from the perspective of models and evolution, and then to some extent, the cognitive function of the specific functional neural network is realized by evolutive neurodynamics [13]. The work about evolutive neurodynamics will be helpful to understand the information processing mechanism and the neural energy coding rule in the nervous system and also provides the basis for the research of the potential mechanism of cognitive function.In this paper, the local evolution characteristics of neural networks with state-dependent state delay (SDSD) will be discussed. The topic introduced here may interest researchers in engaging the theory and application of the new neural network model having kinematics and dynamics feature. Prior to this, there have been many studies on the stability of nonlinear systems with SDSD. Hartung [23] described a type of nonlinear functional differential equation with SDSD and analyzed the stability conditions of periodic solutions based on the linearization method. Exponential stability conditions of nonlinear systems with SDSD were reported via comparing with time-dependent delay systems in [24]. Fiter and Fridman [25] developed the Lyapunov–Krasovskii functional method to discuss asymptotic stability about some particular linear systems with SDSD. Li and Wu [26] considered a class of nonlinear differential systems with SDD pulses. By pulse control theory, uniformly stable, uniformly asymptotically stable, and exponentially stable results were presented. To derive stability criteria of nonlinear systems with SDSD, Li and Yang [27] initially created purely analytical frame structure. To the best of the authors’ knowledge, although the neural networks are widely used and the theoretical results are abundant, the research on stability of the neural network with SDSD is still blank. Moreover, since time delay is an inevitable factor in neural networks, it is of great significance to study the problem of neural networks with SDSD. Then, the primary contributions of this paper are generalized as follows: (1) A general neural network model with SDSD is established. Neural network model with SDSD may interest all those professionals and academics in processing operations who would desire to utilize the capabilities of control systems about capturing rich history information for cost effective and yet robust events to be portrayed. Such research also contributes to reveal influences of neurodynamics evoked by SDSD characteristics. (2) Locally exponentially stable sufficient conditions of the neural networks with SDSD are obtained. The purely analytical method employed by us demonstrates that it is possible to analyze computational neurodynamics without a few additional restrictions. The purely analytical method itself intents to address the universality of analysis framework. Definitely, it may be extended to a more general class of nonlinear systems with SDSD.The rest of this paper is arranged as follows. In Section2, we will present a specific neural network model. The results of our work are described in Section 3. In Section 4, two numerical examples and simulation results are given to verify the validity of our results. Finally, in Section 5, the thesis is summarized, and the future work is prospected.


## 2. Preliminaries and Model Description

### 2.1. Notations

Letℛ and ℛ+ represent the sets of real numbers and nonnegative real numbers, respectively. ℛn denotes the n-dimension Euclidean space. For a matrix E, λmaxE is used to denote its maximum eigenvalue. PE stands for the minimum value of all elements of matrix E. The vector 1-norm and 2-norm are severally expressed by ⋅1 and ⋅2.


### 2.2. Some Preliminaries and Problem Formulation

Based on the work of Li and Yang [27], we put forward the following neural network model with SDSD, which is described by(1)x˙it=−aixit+∑j=1nbijgjxjt+∑j=1ndijfjxjt−τt,X,i=1,2,…,n,t≥t0,for the sake of presentation; we also give the compact form of system (1) as follows:(2)X˙t=−AXt+BgXt+DfXt−τt,X,where n stands for the number of neurons in the network, X˙t denotes the upper right derivative of Xt, X=Xt=x1t,x2t,…,xntT, and xit represents the state of the ith neuron. A is a diagonal matrix, for i=1,2,…,n, ai>0, and B and D are constant matrices with corresponding dimensions. gXt=g1x1t,g2x2t,…,gnxntT and fXt−τt,X=f1x1t−τt,X,f2x2t−τt,X,…,fnxnt−τt,XT are the excitation functions of the ith neuron at time t and t−τt,X, respectively.Furthermore, we useXs=Ψs,s∈t0−η,t0 to denote the initial value of system (2), where Ψ=Ψs=ψ1s,ψ2s,…,ψnsT∈Ct0−η,t0,ℛn. Ct0−η,t0,ℛn is a Banach space whose elements are continuous vector-valued functions. These continuous functions map the interval t0−η,t0 into ℛn. Let Ψα=supt0−η≤s≤t0Ψs stand for the norm of a function Ψ⋅∈Ct0−η,t0,ℛn, where ⋅ is the vector norm matching with the content of the paper.Remark 1.
X
t is right-upper derivable, which implies that the solution of system (2) can be continuous but not smooth. The state delay τt,X is related to the state of each neuron.
For subsequent analysis, we need the following assumptions for system (1) and (2).Assumption 1.
Functionsg⋅,f⋅∈ℛn satisfy f0=0,g0=0.
Through Assumption1, this ensures that X=0 is a constant solution of systems (1) and (2).Assumption 2.
g
⋅
,
f
⋅
∈
ℛ
n are locally Lipschitz continuous; in other words, ∀β1,β2∈ℛ, and we have(3)giβ1−giβ2≤ℓiβ1−β2,∀i∈1,2,…,n,fiβ1−fiβ2≤ℓ˜iβ1−β2,∀i∈1,2,…,n,where ℓi>0 and ℓ˜i>0.
According to Assumption2, we can get two constant sets ℓ1,ℓ2,…,ℓn and ℓ˜1,ℓ˜2,…,ℓ˜n. Let ℓ1,ℓ2,…,ℓn and lf=maxℓ˜1,ℓ˜2,…,ℓ˜n.Assumption 3.
The state delayτt,X∈Cℛ+×ℛn,0,η is locally Lipschitz continuous, namely, for any Γ1,Γ2∈ℛn, there always exists a constant ℓτ>0 such that(4)τt,Γ1−τt,Γ2≤ℓτΓ1−Γ2.Assumption 4.
WhenX=0, τt,X has supremum equipped with τ0≤η, supτt,0,t≥t0=τ0.
For ease of expression, let(5)π1=max1≤i≤n−ai+∑j=1nbjiℓi,π2=max1≤i≤n∑j=1ndjiℓ˜i.Definition 1 (see [28]).
The zero solution of system (2) is said to be locally exponentially stable (LES) in region ℳ; if there exist constants γ>0 and Lyapunov exponent ζ>0, for any t≥t0, we have(6)Xt;t0,Ψ≤γΨαe−ζt−t0,where Xt;t0,Ψ is a solution of system (2) with the initial condition Ψ∈Ct0−η,t0,ℳ, ℳ⊂ℛn, and ℳ is called a local exponential attraction set of the zero solution.Lemma 1 (see [29]).
LetΓ1,Γ2∈ℛn, and we have(7)Γ1TΓ2+Γ2TΓ1≤ϖΓ1TΓ1+ϖ−1Γ2TΓ2,for any ϖ>0.


## 2.1. Notations

Letℛ and ℛ+ represent the sets of real numbers and nonnegative real numbers, respectively. ℛn denotes the n-dimension Euclidean space. For a matrix E, λmaxE is used to denote its maximum eigenvalue. PE stands for the minimum value of all elements of matrix E. The vector 1-norm and 2-norm are severally expressed by ⋅1 and ⋅2.


## 2.2. Some Preliminaries and Problem Formulation

Based on the work of Li and Yang [27], we put forward the following neural network model with SDSD, which is described by(1)x˙it=−aixit+∑j=1nbijgjxjt+∑j=1ndijfjxjt−τt,X,i=1,2,…,n,t≥t0,for the sake of presentation; we also give the compact form of system (1) as follows:(2)X˙t=−AXt+BgXt+DfXt−τt,X,where n stands for the number of neurons in the network, X˙t denotes the upper right derivative of Xt, X=Xt=x1t,x2t,…,xntT, and xit represents the state of the ith neuron. A is a diagonal matrix, for i=1,2,…,n, ai>0, and B and D are constant matrices with corresponding dimensions. gXt=g1x1t,g2x2t,…,gnxntT and fXt−τt,X=f1x1t−τt,X,f2x2t−τt,X,…,fnxnt−τt,XT are the excitation functions of the ith neuron at time t and t−τt,X, respectively.Furthermore, we useXs=Ψs,s∈t0−η,t0 to denote the initial value of system (2), where Ψ=Ψs=ψ1s,ψ2s,…,ψnsT∈Ct0−η,t0,ℛn. Ct0−η,t0,ℛn is a Banach space whose elements are continuous vector-valued functions. These continuous functions map the interval t0−η,t0 into ℛn. Let Ψα=supt0−η≤s≤t0Ψs stand for the norm of a function Ψ⋅∈Ct0−η,t0,ℛn, where ⋅ is the vector norm matching with the content of the paper.Remark 1.
X
t is right-upper derivable, which implies that the solution of system (2) can be continuous but not smooth. The state delay τt,X is related to the state of each neuron.
For subsequent analysis, we need the following assumptions for system (1) and (2).Assumption 1.
Functionsg⋅,f⋅∈ℛn satisfy f0=0,g0=0.
Through Assumption1, this ensures that X=0 is a constant solution of systems (1) and (2).Assumption 2.
g
⋅
,
f
⋅
∈
ℛ
n are locally Lipschitz continuous; in other words, ∀β1,β2∈ℛ, and we have(3)giβ1−giβ2≤ℓiβ1−β2,∀i∈1,2,…,n,fiβ1−fiβ2≤ℓ˜iβ1−β2,∀i∈1,2,…,n,where ℓi>0 and ℓ˜i>0.
According to Assumption2, we can get two constant sets ℓ1,ℓ2,…,ℓn and ℓ˜1,ℓ˜2,…,ℓ˜n. Let ℓ1,ℓ2,…,ℓn and lf=maxℓ˜1,ℓ˜2,…,ℓ˜n.Assumption 3.
The state delayτt,X∈Cℛ+×ℛn,0,η is locally Lipschitz continuous, namely, for any Γ1,Γ2∈ℛn, there always exists a constant ℓτ>0 such that(4)τt,Γ1−τt,Γ2≤ℓτΓ1−Γ2.Assumption 4.
WhenX=0, τt,X has supremum equipped with τ0≤η, supτt,0,t≥t0=τ0.
For ease of expression, let(5)π1=max1≤i≤n−ai+∑j=1nbjiℓi,π2=max1≤i≤n∑j=1ndjiℓ˜i.Definition 1 (see [28]).
The zero solution of system (2) is said to be locally exponentially stable (LES) in region ℳ; if there exist constants γ>0 and Lyapunov exponent ζ>0, for any t≥t0, we have(6)Xt;t0,Ψ≤γΨαe−ζt−t0,where Xt;t0,Ψ is a solution of system (2) with the initial condition Ψ∈Ct0−η,t0,ℳ, ℳ⊂ℛn, and ℳ is called a local exponential attraction set of the zero solution.Lemma 1 (see [29]).
LetΓ1,Γ2∈ℛn, and we have(7)Γ1TΓ2+Γ2TΓ1≤ϖΓ1TΓ1+ϖ−1Γ2TΓ2,for any ϖ>0.


## 3. Main Results

Theorem 1.
Under Assumptions1–4, the zero equilibrium of system (2) is LES if(8)π1+π2<0,and Lyapunov exponent ζ>0 satisfies(9)ζ+π1+π2eζℓτΨα+τ0≤0.Proof.
We assume thatXt;t0,Ψ is a trajectory of system (2) with initial value t0,Ψ, where Ψ∈Ct0−η,t0,ℛn and Ψ≠0. For the sake of convenience, let Vt=Vt,X=Xt1=∑i=1nxit and V0=supVs,s∈t0−η,t0. Then, for any ϵ∈0,ζ, we claim that(10)eζ−ϵt−t0Vt≤V0,∀t≥t0.
Firstly, whent=t0, (10) holds. Next, we prove that (10) holds on t0,+∞. In contrast to (10), there are some instants on t0,+∞ to make (10) untenable, and then we can find an instant tq≥t0; the following three events will happen:(1)
eζ−ϵtq−t0Vtq=V0.(2)
eζ−ϵt−t0Vt≤V0, for ∀t∈t0−η,tq.(3)
There exists a right neighbor oftqU+0tq,ξ such that ∀tξ∈U+0tq,ξ and eζ−ϵtξ−t0Vtξ>V0.
On the contrary, by Assumptions1–4 and combining (2), the derivative of eζ−ϵt−t0Vt at time tq is as follows:(11)ddteζ−ϵt−t0Vtt=tq=ζ−ϵeζ−ϵtq−t0Vtq+eζ−ϵt−t0V˙tt=tq≤ζ−ϵV0+eζ−ϵtq−t0∑i=1nsgnxitqx˙itq=ζ−ϵV0+eζ−ϵtq−t0∑i=1nsgnxitq−aixitq+∑j=1nbijgjxjtq+∑j=1ndijfjxjtq−τtq,Xtq≤ζ−ϵV0+eζ−ϵtq−t0∑i=1n−aixitq+eζ−ϵtq−t0∑i=1n∑j=1nbijℓjxjtq+eζ−ϵtq−t0∑i=1n∑j=1ndijℓ˜jxjtq−τtq,Xtq≤ζ−ϵV0+eζ−ϵtq−t0max1≤i≤n−ai+∑j=1nbjiℓiXtq+eζ−ϵtq−t0max1≤i≤n∑j=1ndjiℓ˜iXtq−τtq,X=ζ−ϵ+π1V0+eζ−ϵtq−τtq,X−t0Xtq−τtq,X×π2eζ−ϵτtq,X≤ζ−ϵ+π1+π2eζ−ϵτtq,XV0=ζ−ϵ+π1+π2eζ−ϵτtq,X−τtq,0eζ−ϵτtq,0V0≤ζ−ϵ+π1+π2eζ−ϵℓτXtq1eζ−ϵτtq,0V0≤ζ−ϵ+π1+π2eζ−ϵℓτXtq1+τ0V0.
Together with the definition ofV0,Vt,tq and condition (1), we have(12)Xtq1=Vtq≤V0=Ψα,and then from (9) and (11), we obtain(13)ddteζ−ϵt−t0Vtt=tq≤ζ−ϵ+π1+π2eζ−ϵℓτΨα+τ0Ψα<0,which is a contradiction with condition (9), and thus, (10) holds.
Consider the arbitrariness ofϵ, let ϵ⟶0, and then we obtain(14)eζt−t0Vt≤V0,∀t≥t0,i.e.,(15)Xt1=Vt≤Ψαe−ζt−t0,∀t≥t0.
The reasoning process of Theorem1 is completed.Theorem 2.
B and D are symmetric matrices; then, under Assumptions 1–4, the zero equilibrium of system (2) is LES if(16)−2PA+λmaxB2+lg2+λmaxD2+lf2<0,and μ=2ζ>0 satisfies(17)μ−2PA+λmaxB2+lg2+λmaxD2+lf2eμℓτΨα+τ0≤0,where ζ is the Lyapunov exponent.Proof.
SupposeXt;t0,Ψ is a solution of system (2) with initial state t0,Ψ, where Ψ∈Ct0−η,t0,ℛn and Ψ≠0. Let Vt,X=Xt22=XTtXt. For convenience, set Vt=Vt,X and V0=supVs,s∈t0−η,t0. Then, for any ϵ∈0,μ, we claim that(18)eμ−ϵt−t0Vt≤V0,∀t≥t0.
Firstly, whent=t0, (18) holds. Next, we prove that (18) holds on t0,+∞. In contrast to (18), there are some instants on t0,+∞ to make (18) untenable, and then we can find an instant tq≥t0 to satisfy the following three conditions:(1)
eμ−ϵtq−t0Vtq=V0.(2)
eμ−ϵt−t0Vt≤V0, for ∀t∈t0−η,tq.(3)
There exists a right neighbor oftqU+0tq,ξ such that ∀tξ∈U+0tq,ξ,eμ−ϵtξ−t0Vtξ>V0.
On the contrary, by Assumptions1–4 and Lemma 1, we compute the derivative of eμ−ϵt−t0Vt at time tq:(19)ddteμ−ϵt−t0Vtt=tq=μ−ϵeμ−ϵtq−t0Vtq+eμ−ϵt−t0V˙tt=tq≤μ−ϵV0+eμ−ϵtq−t02XTtqX˙tq=μ−ϵV0+eμ−ϵtq−t0×2XTtq×−AXtq+BgXtq+DfXtq−τtq,X=μ−ϵV0+eμ−ϵtq−t0−2XTtqAXtq+eμ−ϵtq−t02XTtqBgXtq+eμ−ϵtq−t02XTtqDfXtq−τtq,X≤μ−ϵV0+eμ−ϵtq−t0−2PAXTtqXtq+eμ−ϵtq−t0λmaxB2+lg2XTtqXtq+eμ−ϵtq−t0λmaxD2XTtqXtq+eμ−ϵtq−t0lf2XTtq−τtq,XXtq−τtq,X=μ−ϵ−2PA+λmaxB2+lg2+λmaxD2V0+eμ−ϵtq−τtq,X−t0lf2XTtq−τtq,XXtq−τtq,X×eμ−ϵτtq,X≤μ−ϵ−2PA+λmaxB2+lg2+λmaxD2V0+lf2eμ−ϵτtq,XV0=μ−ϵ−2PA+λmaxB2+lg2+λmaxD2V0+lf2eμ−ϵτtq,X−τtq,0eμ−ϵτtq,0V0≤μ−ϵ−2PA+λmaxB2+lg2+λmaxD2+lf2eμ−ϵℓτXtq2+τ0V0.
Combining the definition ofV0,Vt,tq and condition (1), we have(20)Xtq2=Vtq1/2≤V01/2=Ψα,and then from (17) and (19), we obtain(21)ddteμ−ϵt−t0Vtt=tq≤μ−ϵ−2PA+λmaxB2+lg2+λmaxD2+lf2eμ−ϵℓτΨα+τ0Ψα2<0.
This is in contradiction with (9), so (18) holds.
Consider the arbitrariness ofϵ, let ϵ⟶0, and then we could be capable of getting(22)eμt−t0Vt≤V0,∀t≥t0,i.e.,(23)Xt2=Vt1/2≤Ψαe−μ/2t−t0,∀t≥t0.
The reasoning process of Theorem2 is completed.Remark 2.
In Theorem2, by taking the value of ϖ in Lemma 1 to be 1, we can get the following inequality:(24)2XTtqBgXtq=XTtqBgXtq+XTtqBgXtq=XTtqBTgXtq+gTXtqBXtq=BXtqTgXtq+gTXtqBXtq≤BXtqTBXtq+gTXtqgXtq=XTtqBTBXtq+gTXtqgXtq=XTtqB2Xtq+gTXtqgXtq≤λmaxB2XTtqXtq+lg2XTtqXtq=λmaxB2+lg2XTtqXtq.
We give a particular case of system (2), considering the following one-dimensional system:(25)x˙t=axt+bgxt+dfxt−τt,x,t≥0,τt,x=δ+λxt,where a,b,d,δ,λ∈ℛ.Corollary 1.
If system (25) satisfies the following conditions,(1)
f0=0,g0=0. g⋅,f⋅ are locally Lipschitz continuous functions whose Lipschitz constants are l1,l2, respectively,(2)
a<0 and a+bl1+dl2<0,(3)
δ>0, and the initial condition of system (25) satisfies ψ⋅≤M<δ/λ, where ψ⋅=max−η≤t≤0ψ⋅,then we can obtain(26)xt≤Me−ζt,t≥0,where xt=xt;0,ψ represents the trajectory of system (25) with initial state 0,ψ, and Lyapunov exponent ζ>0 satisfies(27)ζ+a+bl1+dl2eζλM+δ≤0.Proof.
To prove Corollary1, we only need to prove that Corollary 1 satisfies Assumptions 1–4. Obviously, from condition (1), we can know that system (25) satisfies Assumptions 1 and 2. Next, we prove that system (25) satisfies Assumptions 3 and 4.
We assume thatxt;0,ψ is a trajectory of system (25) with initial value 0,ψ, where ψ≠0. From τt,x=δ+λxt, we could know that τt,x is continuous about t. Owing to ψ≤M and M<δ/λ, there exists ρ>0 such that ∀t∈0,ρ and τt,x≥0. In the circumstances, set Vt=xt. Next, under conditions (1)–(3) and by Theorem 1, we get xt≤Me−ζt, for ∀t∈0,ρ, where Lyapunov exponent ζ satisfies (27).
Then, for∀t∈0,+∞, we declare that τt,x≥0. In contrast to the claim, there must be some instant on ρ,+∞ to make τt,xt<0. Then, we could find an instant tq∈ρ,+∞ to meet the following three conditions:(1)
τtq,xtq=δ+λxtq=0.(2)
∀t∈0,tq, τt,xt≥0.(3)
There exists a right neighbor oftqU+0tq,ξ such that τtξ,xtξ<0, for any tξ∈U+0tq,ξ.
It follows from Theorem1, for any t∈0,tq, we have xt≤Me−ζt, which indicates that xtq≤Me−ζtq<δ/λ. On the basis of the continuity of xt, there exists ξ0>0 such that xt<δ/λ, for any t∈tq,tq+ξ0. It means that ∀t∈tq,tq+ξ0, and τt,xt=δ+λxt>0holds, which contradicts with (9); therefore, ∀t∈0,+∞ and τt,xt≥0. It ulteriorly indicates that Assumptions 3 and 4 are all satisfied. Therefore, it can be known from Theorem 1 that Corollary 1 is valid. The proof is completed.Remark 3.
Actually, in Theorem1, when π1+π2<0, we can definitely get ζ+π1+π2eζℓτΨ+τ0≤0 as long as ζ is small enough. Theorem 2 has a similar conclusion.Remark 4.
In Theorem1, by taking tq−τtq,X as a whole and then from condition (2), we are able to get eζ−ϵtq−τtq,X−t0Xtq−τtq,X≤V0. Theorem 2 has a similar conclusion.Remark 5.
The proofs of Theorems1 and 2 also provide an estimate of the locally exponentially convergent rate ζ which could be obtained by solving transcendental equation (9) or (17).Remark 6.
Remarkably, the Lyapunov exponent in Theorems1 and 2 is state-dependent, so only when the initial value is bounded can we find common ζ to make eζ−ϵVt≤V0. Furthermore, due to the effect of SDSD, the results in our paper are local features, not global features.


## 4. Illustrative Examples

To prove the effectiveness of our results, two numerical examples will be given in this section.Example 1.
Consider a 2-dimensional neural network with SDSD, which is described by(28)x˙1tx˙2t=−1001x1tx2t+0.250.250.020.01g1x1tg2x2t+0.30.20.010.02f1x1t−τt,Xf2x2t−τt,X,where t0=0 and(29)gi⋅=xit+1+xit−1,i=1,2,fi⋅=sinxit−sinx1t+x2t,i=1,2,τt,X=sinx1t+x2t.
Evidently,ℓ1=ℓ2=2,ℓ˜1=ℓ˜2=1,ℓτ=1,τt,0=0,τt,X∈0,1. By calculating,(30)π1=max1≤i≤n−ai+∑j=1nbjiℓi=−0.46,π2=max1≤i≤n∑j=1ndjiℓ˜i=0.31.
Then, from Theorem1, system (28) is LES. The trajectories of the solution from a random initial value are shown in Figure 1. As shown in Figure 1, x1t and x2t in neural network model (28) are convergent. Figure 2 shows the phase diagram of system (28) evolving with time.Figure 1
Transient behavior of (a)x1t and (b) x2t in system (28).
(a)
(b)Figure 2
Transient behavior ofx1t,x2t in system (28).Example 2.
Consider another 2-dimensional neural network with SDSD, which is depicted by(31)x˙1tx˙2t=−2002x1tx2t+0.50.20.20.5g1x1tg2x2t+0.30.20.20.3f1x1t−t,Xf2x2t−t,X,where t0=0 and(32)gi⋅=cosxit+π/22,i=1,2,fi⋅=tanhxit−sinx1t+x2t,i=1,2,τt,X=sinx1t+x2t.
Obviously,lg=1/2,lf=1,ℓτ=1,τt,0=0,t,X∈0,1,PA=2. By computing, λmaxB2=0.49,λmaxC2=0.25, and −2PA+lg2+lf2+λB2+λC2=−4+1/4+1+0.49+0.25=−2.01<0. Then, from Theorem 2, system (31) is LES. The trajectories of the solution from a random initial value are shown in Figure 3. As shown in Figure 3, x1t and x2t in neural network model (31) are convergent. Figure 4 shows the phase diagram of system (31) evolving with time.Figure 3
Transient behavior of (a)x1t and (b) x2t in system (31).
(a)
(b)Figure 4
Transient behavior ofx1t,x2t in system (31).


## 5. Concluding Remarks

We devote to resolve the problem of local dynamic property for neural networks with SDSD in this paper. Through pure analysis method and technique of reduction to absurdity, we obtain a certain number of sufficient conditions for local exponential stability of neural networks with SDSD. Based on our results, we know that the Lyapunov exponent relies on the state on account of the effect of SDSD. It also indicates that the exponential stability results derived are local rather than global. Consequently, we can take the global dynamics as a topic in the prospective research. In addition, we can also develop SDSD system methods, for instance, event-triggered control.



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