# Fixed-Time Synchronization of Neural Networks with Discrete Delay

Authors: Shuai Liu; Chuan Chen; Haipeng Peng
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we establish a new fixed-time stability theorem, which provides a novel fixed-time stability criterion and a novel upper bound estimate formula for the settling time. Numerical simulations show that the upper bound estimate for the settling time in this paper is tighter than those given in the existing fixed-time stability theorems. By designing a simple feedback controller, the fixed-time synchronization of neural networks with discrete delay is investigated based on the fixed-time stability theorem established in this paper. A numerical example is included to validate the effectiveness of the obtained theoretical results.


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## Body

## 1. Introduction

Neural networks can be described by differential equations [1], and they usually present complex dynamic behaviors [2]. The stability and synchronization of neural networks can be applied in many fields, such as associative memory [3], secure communication [4], and image encryption [5]. In recent years, considerable results have been reported about the stability and synchronization of neural networks [6–12].Different from infinite-time synchronization (including asymptotic synchronization and exponential synchronization), finite-time synchronization [13] requires that the trajectories of the error system tend to the origin after some finite time, which is called the settling time. Considering the control cost, finite-time synchronization is more feasible and valuable than infinite-time synchronization in the engineering fields. Recently, the finite-time synchronization of neural networks has been intensively studied [14–22]. In [14], the finite-time synchronization of neural networks with mixed delays and perturbations was investigated. In [19], the finite-time synchronization of Cohen–Grossberg neural networks was studied via delayed feedback control.The drawback of finite-time synchronization is that the settling time varies with the initial conditions of the drive-response systems. Since different initial conditions lead to different settling times, the corresponding settling times will be confusing when we consider many different initial conditions. Moreover, if the initial conditions of the studied dynamical systems are unknown beforehand, it is even impossible to estimate the settling times. Therefore, it will be desirable that finite-time synchronization can be achieved in a fixed time interval, irrespective of the initial conditions of the studied systems.In 2012, a new concept named “fixed-time stability” was introduced by Polyakov [23]. Fixed-time stability is a kind of special finite-time stability actually, and its settling time is bounded by a fixed constant for any initial conditions of the studied systems. As we know, the synchronization problem of the drive-response systems can be converted into the stability problem of the error system, so the related fixed-time stability theorems [23, 24] can be used to investigate the fixed-time synchronization of nonlinear systems [25–28]. The fixed-time synchronization of nonlinear systems has important applications in secure communication and image encryption. Take secure communication for example: when can the ciphertext signals be decrypted correctly without considering the specific initial values of the drive-response systems? The solution to this problem must be attributed to the fixed-time synchronization control of the drive-response systems. That is to say, for the drive-response systems, the upper bound estimate for the settling time of the fixed-time synchronization can be used to adjust the start time of correct decryption.As typical nonlinear systems, neural networks usually present some strange dynamic behaviors, so the fixed-time synchronization of neural networks can also be applied in many fields, such as secure communication and image encryption. Based on the fixed-time stability theorems in [23, 24, 29], there have been some publications about the fixed-time synchronization of neural networks [29–34]. It should be pointed out that the upper bound estimate formulas for the settling time given in [23, 24, 29] are very inaccurate. Owing to the limitations of theoretical analysis, it is difficult to obtain the least upper bound estimate for the settling time. However, from an application point of view, it is important to derive a tighter upper bound estimate for the settling time.Inspired by the aforementioned discussion, in this paper, we establish a new fixed-time stability theorem, which provides a novel fixed-time stability criterion and a novel upper bound estimate formula for the settling time. Numerical simulations show that the upper bound estimate for the settling time in this paper is tighter than those given in [23, 24, 29]. By utilizing the established fixed-time stability theorem, the fixed-time synchronization of neural networks with discrete delay is studied. A numerical example is included to show the effectiveness of the obtained theoretical results.


## 2. Preliminaries

Consider the following neural network with discrete delay:(1)ξ˙it=−ϱiξit+∑j=1naijfjξjt+∑j=1nbijfjξjt−τt+Ii,i=1,2,⋯,n, where ξit expresses the state of neuron; ϱi>0 denotes the rate of neuron self-inhibition; aij and bij denote connection weights; fj⋅ represents the activation function; discrete delay τt satisfies 0≤τt≤τ, where τ is a positive constant; and Ii stands for the external input. The initial condition of system (1) is assumed to be ξν=φν∈C−τ,0,Rn, where ξν=ξ1ν,ξ2ν,⋯,ξnνT.In this paper, system (1) is the drive system, and the corresponding response system is(2)η˙it=−ϱiηit+∑j=1naijfjηjt+∑j=1nbijfjηjt−τt+Ii+uit,i=1,2,⋯,n, where uit stands for the controller. Let ηt=η1t,η2t,⋯,ηntT; the initial condition of MNN (2) is ην=ϕν∈C−τ,0,Rn.The synchronization errors between systems (1) and (2) are defined by eit=ηit−ξit,i=1,2,⋯,n; then, the error system should be(3)e˙it=−ϱieit+Fit+uit,i=1,2,⋯,n, where(4)Fit=∑j=1naijfjηjt−∑j=1naijfjξjt+∑j=1nbijfjηjt−τt−∑j=1nbijfjξjt−τt.Letet=e1t,e2t,⋯,entT; then, the initial condition of system (3) should be eν=ϕν−φν, ν∈−τ,0.Throughout this paper, the following assumptions will be needed:A1. For ∀μ,ν∈R, there exist constants lj>0 such that(5)fjμ−fjν≤ljμ−ν,j=1,2,⋯,n.A2. For ∀j∈1,2,⋯,n, activation function fj⋅ satisfies fj⋅≤Mj, where Mj>0 is a constant.Lemma 1.
Fit≤∑j=1naij⋅ljejt+2∑j=1nbijMj, i=1,2,⋯,n.Proof.
Based on assumptionsA1 and A2, the proof is obvious.Lemma 2 (see [35]).
Ify1,y2,⋯,yn≥0,0<ρ≤1,σ>1, then we have(6)∑i=1nyiρ≥∑i=1nyiρ,∑i=1nyiσ≥n1−σ∑i=1nyiσ.Definition 1.
The origin of system (3) can achieve finite-time stability, if there is a constant Te0>0 such that limt⟶Te0et=0 and et≡0 for ∀t>Te0, whereTe0 is called the settling time.Definition 2.
The origin of system (3) can achieve fixed-time stability, if two conditions can be satisfied: (a) the origin of system (3) can achieve finite-time stability; (b) for any e0, there is a fixed constant Tmax>0 such that Te0≤Tmax.Lemma 3 (see [23]).
Suppose thatV⋅:Rn⟶R+∪0 is a continuous radically unbounded function and satisfies the following:(1)
Vz=0⟺z=0(2)
V˙et≤−αVρet−βVσet for any et, where α,β>0,0<ρ<1 and σ>1Then, the origin of system (3) can achieve fixed-time stability, and(7)Tmax1=1α1−ρ+1βσ−1.Lemma 4 (see [29]).
Suppose thatV⋅:Rn⟶R+∪0 is a continuous radically unbounded function and satisfies the following:(1)
Vz=0⟺z=0(2)
V˙et≤−αVρet−βVσet for any et, where α,β>0,0<ρ<1 and σ>1Then, the origin of system (3) can achieve fixed-time stability, and(8)Tmax2=1α⋅αβ1−ρ/σ−ρ11−ρ+1σ−1.Lemma 5 (see [24]).
Suppose thatV⋅:Rn⟶R+∪0 is a continuous radically unbounded function and satisfies the following(1)
Vz=0⟺z=0(2)
V˙et≤−αVρet−βVσet for any et, where α,β>0,ρ=1−1/2δ and σ=1+1/2δ (δ>1)Then, the origin of system (3) can achieve fixed-time stability, and(9)Tmax3=πδαβ.Lemma 6 (see [36]).
SupposeV⋅:Rn⟶R+∪0 is C-regular. If V⋅ satisfies(10)V˙et≤−αVρet,for anyet, where α>0and0<ρ<1 are two constants, then the origin of system (3) can achieve finite-time stability, and the settling time Te0 is given by(11)Te0=V1−ρe0α1−ρ.


## 3. Main Results

First, we derive a new fixed-time stability theorem, which provides a novel fixed-time stability criterion and a novel upper bound estimate formula for the settling time.Theorem 1.
Suppose thatV⋅:Rn⟶R+∪0 is a continuous radically unbounded function and satisfies the following:(1)
Vz=0⟺z=0(2)
V˙et≤−αVρet−βVσet−γVet for any et, where α,β,γ>0,0<ρ<1 and σ>1Then, the origin of system (3) can achieve fixed-time stability, and(12)Tmax4=1γ1−ρln1+γα+1βσ−1.Proof.
It is obvious that(13)V˙et≤−αVρet−βVσet−γVet≤−αVρet−βVσet.
Based on Lemma3, the origin of system (3) can achieve fixed-time stability.
LetWs=V1−ρs; then, we have(14)V˙s=11−ρW˙sVρs.
Therefore,(15)11−ρW˙etVρet≤−αVρet−βVσet−γVet.
SinceVs=W1/1−ρs, it follows that(16)W˙et≤1−ρ−α−βVσ−ρet−γV1−ρet=1−ρ−α−βWσ−ρ/1−ρet−γWet.
This means(17)dtdWet≥11−ρ−α−βWσ−ρ/1−ρet−γWet.
So we can obtain that(18)Te0≤∫We0011−ρ−α−βwσ−ρ/1−ρ−γwdw=11−ρ∫0We01α+βwσ−ρ/1−ρ+γwdw.
Two cases will be considered separately:(1)
If0≤Ve0≤1, we have 0≤We0=V1−ρe0≤1. Then,(19)Te0≤11−ρ∫0We01α+γwdw=1γ1−ρln1+γαWe0≤1γ1−ρln1+γα.(2)
IfVe0>1, we have We0=V1−ρe0>1. Then,(20)Te0≤11−ρ∫011α+βwσ−ρ/1−ρ+γwdw+∫1We01α+βwσ−ρ/1−ρ+γwdw≤11−ρ∫011α+γwdw+∫1We01βwσ−ρ/1−ρdw=1γ1−ρln1+γα+1β1−ρ1−ρσ−1−1−ρW1−σ/1−ρx0σ−1≤1γ1−ρln1+γα+1βσ−1.
Then, we have(21)Tmax4=2γ1−ρln1+γα+2βσ−1.Remark 1.
Sinceln1+γ/α<γ/α, we have 1/γ1−ρln1+γ/α+1/βσ−1<1/α1−ρ+1/βσ−1. Therefore, compared with Lemma 3, Theorem 1 can give Tmax a more accurate estimate. Moreover, if α≪γ, ln1+γ/α≪γ/α.
Next, we investigate the fixed-time synchronization of systems (1) and (2) by means of the following controller:(22)uit=−kieit−qisigneit−risigneiteitξ−sisigneiteitη,i=1,2,⋯,n, where ki and qi will be determined later, ri and si are arbitrary positive constants, 0<ξ<1, and η>1.Theorem 2.
Suppose assumptionsA1 and A2 hold. If control gains ki and qi satisfy ki>∑j=1naijlj+ajili/2−ϱi, qi≥2∑j=1nbijMj, i=1,2,⋯,n, systems (1) and (2) can realize fixed-time synchronization under controller (22). In addition,(23)Tmax4=2γ1−ξln1+γα+2βη−1,where α=miniri2ξ+1/2, β=minisin1−η/2⋅2η+1/2, and γ=mini2ϱi+2ki−∑j=1naijlj+ajili.Proof.
We choose the following Lyapunov function:(24)Vet=12eTtet=12∑i=1neit2.
The derivative ofVet is(25)V˙et=∑i=1neiteit=∑i=1neit⋅−ϱieit+Fit+uit≤−∑i=1nϱieit2+∑i=1neit∑j=1naij⋅ljejt+2∑j=1nbijMj−∑i=1nkieit2−∑i=1nqieit−∑i=1nrieitξ+1−∑i=1nsieitη+1≤−∑i=1nϱi+ki−∑j=1naijlj+ajili2eit2+∑i=1n2∑j=1nbijMj−qieit−∑i=1nrieitξ+1−∑i=1nsieitη+1,where Lemma 1 has been used.
Sinceqi≥2∑j=1nbijMj, i=1,2,⋯,n, we obtain that(26)V˙et≤−miniri∑i=1neit2ξ+1/2−minisi∑i=1neit2η+1/2−miniϱi+ki−∑j=1naijlj+ajili2∑i=1neit2≤−miniri∑i=1neit2ξ+1/2−minisin1−η/2∑i=1neit2η+1/2−miniϱi+ki−∑j=1naijlj+ajili2∑i=1neit2=−miniri2ξ+1/2Vξ+1/2et−minisin1−η/2⋅2η+1/2Vη+1/2et−2miniϱi+ki−∑j=1naijlj+ajili2Vet,where Lemma 2 has been used.
Letα=miniri2ξ+1/2, β=minisin1−η/2⋅2η+1/2, and γ=mini2ϱi+2ki−∑j=1naijlj+ajili. Then, we have(27)V˙et≤−αVξ+1/2et−βVη+1/2et−γVet.
It is obvious that0<ξ+1/2<1 and η+1/2>1. Since ki>∑j=1naijlj+ajili/2−ϱi, we have γ>0. Based on Theorem 1, the origin of system (3) can achieve fixed-time stability. In addition,(28)Tmax4=1γ1−ξ+1/2ln1+γα+1βη+1/2−1=2γ1−ξln1+γα+2βη−1.Corollary 1.
Suppose assumptionsA1 and A2 hold. If control gains ki and qi satisfy ki>∑j=1naijlj+ajili/2−ϱi, qi≥2∑j=1nbijMj, i=1,2,⋯,n, systems (1) and (2) can realize fixed-time synchronization under controller (22). In addition,(29)Tmax1=2α1−ξ+2βη−1,where α=miniri2ξ+1/2 and β=minisin1−η/2⋅2η+1/2.Proof.
Similarly, we can prove that(30)V˙et≤−αVξ+1/2et−βVη+1/2et−γVet≤−αVξ+1/2et−βVη+1/2et,where α=miniri2ξ+1/2, β=minisin1−η/2⋅2η+1/2, and γ=mini2ϱi+2ki−∑j=1naijlj+ajili>0.
Based on Lemma3, the origin of system (3) can achieve fixed-time stability. In addition,(31)Tmax1=1α1−ξ+1/2+1βη+1/2−1=2α1−ξ+2βη−1.Corollary 2.
Suppose assumptionsA1 and A2 hold. If control gains ki and qi satisfy ki>∑j=1naijlj+ajili/2−ϱi, qi≥2∑j=1nbijMj, i=1,2,⋯,n, systems (1) and (2) can realize fixed-time synchronization under controller (22). In addition,(32)Tmax2=1α⋅αβ1−ξ/η−ξ21−ξ+2η−1,where α=miniri2ξ+1/2 and β=minisin1−η/2⋅2η+1/2.Proof.
Similarly, we can prove that(33)V˙et≤−αVξ+1/2et−βVη+1/2et,where α=miniri2ξ+1/2 and β=minisin1−η/2⋅2η+1/2.
Based on Lemma4, the origin of system (3) can achieve fixed-time stability. In addition,(34)Tmax2=1α⋅αβ1−ξ/η−ξ21−ξ+2η−1.
Now, we consider the following controller:(35)uit=−kieit−qisigneit−risigneiteitξ−sisigneiteitη,i=1,2,⋯,n, where ki and qi will be determined later, ri and si are arbitrary positive constants, ξ=1−1/δ, and η=1+1/δ, δ>1.Corollary 3.
Suppose assumptionsA1 and A2 hold. If control gains ki and qi satisfy ki>∑j=1naijlj+ajili/2−ϱi, qi≥2∑j=1nbijMj, i=1,2,⋯,n, systems (1) and (2) can realize fixed-time synchronization under controller (35). In addition,(36)Tmax3=πδαβ,where α=miniri2ξ+1/2 and β=minisin1−η/2⋅2η+1/2.Proof.
Similarly, we can prove that(37)V˙et≤−αVξ+1/2et−βVη+1/2et,where α=miniri2ξ+1/2, β=minisin1−η/2⋅2η+1/2, ξ+1/2=1−1/2δ, η+1/2=1+1/2δ, and δ>1.
Based on Lemma5, the origin of system (3) can achieve fixed-time stability. In addition,(38)Tmax3=πδαβ.
To study the finite-time synchronization of systems (1) and (2), we design the following controller:(39)uit=−kieit−qisigneit−risigneiteitξ,i=1,2,⋯,n, where ki and qi will be determined later, ri is an arbitrary positive constant, and 0<ξ<1.Corollary 4.
Suppose assumptionsA1 and A2 hold. If control gains ki and qi satisfy ki>∑j=1naijlj+ajili/2−ϱi, qi≥2∑j=1nbijMj, i=1,2,⋯,n, systems (1) and (2) can realize finite-time synchronization under controller (36). In addition,(40)Te0=1α1−ξ12∑i=1nei021−ξ/2,where α=miniri2ξ+1/2.Proof.
Similarly, we can prove that(41)V˙et≤−αVξ+1/2et,where α=miniri2ξ+1/2, 0<ξ+1/2<1.
Based on Lemma6, the origin of system (3) can achieve finite-time stability. In addition,(42)Te0=V1−ξ+1/2e0α1−ξ+1/2=2V1−ξ/2e0α1−ξ=1α1−ξ12∑i=1nei021−ξ/2.Remark 2.
Based on the fixed-time stability theorems in [23, 24, 29], there have been some publications about the fixed-time synchronization of neural networks [29–34]. However, the upper bound estimate formulas for the settling time given in [23, 24, 29] are very inaccurate. In this paper, we establish a new fixed-time stability theorem, which provides a novel upper bound estimate formula for the settling time. Numerical simulations show that the upper bound estimate for the settling time in this paper is tighter than those given in [23, 24, 29].Remark 3.
In the controllers designed in this paper, the switching itemsigneit is used to deal with the sign problems in theoretical derivation. In theoretical derivation, eit is desirable, while eit is not desirable. In this case, since signeiteit=eit, signeit can play a key role. It should be pointed out that signeit is a discontinuous term, which may lead to chattering phenomenon. Alternatively, the continuous term eit/eit+γ can be used to replace signeit in engineering applications, where γ>0 is a sufficiently small constant. However, in strict theoretical analysis, we need not do this kind of replacement, and this kind of replacement usually leads to problems.


## 4. Numerical Simulations

In this section, a numerical example is provided to validate the obtained theoretical results.Example 1.
Consider the following neural network:(43)ξ˙it=−ϱiξit+∑j=12aijfjξjt+∑j=12bijfjξjt−τt+Ii,i=1,2, where ϱ1=0.9, ϱ2=1.1, a11=−0.5, a12=2, a21=0.1, a22=1.5, b11=−1.2, b12=1, b21=0.1, b22=−1.5, I1=sint, and I2=cost.
Letf1μ=f2μ=μ+1−μ−1/2 and τt=1.2; we have M1=M2=1, l1=l2=1, and τ=1.2. The initial condition of system (43) is φν=1.5,−0.5T, where ν∈−1.2,0.
The corresponding response system is(44)η˙it=−ϱiηit+∑j=12aijfjηjt+∑j=12bijfjηjt−τt+Ii+uit,i=1,2, where uit,i=1,2, are controllers. Suppose the initial conditions of system (44) are ϕν=−0.6,1.8T,−8,9T,−50,60T, and ν∈−1.2,0, respectively. The synchronization errors between systems (43) and (44) without control inputs are given in Figure 1.
Choosek1=3, k2=4, q1=4.4, q2=3.2, r1=r2=1, s1=s2=1, ξ=2/3, and η=4/3; then, we have α=25/6, β=2, γ=2.7. It can be verified that ki>∑j=1naijlj+ajili/2−ϱi and qi≥2∑j=1nbijMj, i=1,2.
Based on Theorem2, systems (43) and (44) can realize fixed-time synchronization under controller (22) and Tmax4≈4.6486. The synchronization errors between systems (43) and (44) under controller (22) are presented in Figure 2. Although there are three kinds of different synchronization errors, one can see that all the synchronization errors tend to 0 within Tmax4.Figure 1
The synchronization errors between systems (43) and (44) without control inputs, where the initial condition of system (43) is φν=1.5,−0.5T and the initial conditions of system (44) are ϕν=−0.6,1.8T,−8,9T,−50,60T, and ν∈−1.2,0, respectively.Figure 2
The synchronization errors between system (22) under controller (22), where the initial condition of system (43) is φν=1.5,−0.5T, and the initial conditions of system (44) are ϕν=−0.6,1.8T,−8,9T,−50,60T, and ν∈−1.2,0, respectively.Remark 4.
Based on Corollary1 and Corollary 2, systems (43) and (44) can realize fixed-time synchronization under controller (22), and Tmax1≈6.3674 and Tmax2≈6.3568. Furthermore, systems (43) and (44) can realize fixed-time synchronization under controller (35) according to Corollary 3 (δ=3), and Tmax3≈4.9926. In fact, if ξ=2/3 and η=4/3, controller (35) is the same as controller (22). Obviously, numerical example shows that Tmax4 is smaller than Tmax1, Tmax2, and Tmax3 (see Figure 2).Remark 5.
Based on Corollary4, by choosing the same parameters as those in Example 1, systems (43) and (44) can realize finite-time synchronization under controller (36). It should be pointed out that the settling time Te0 is related to e0, which depends on the initial values of systems (43) and (44). Therefore, the settling time Te0 cannot be bounded by a unified upper bound. If e0 is large enough, the settling time Te0 will be larger than Tmax4. In contrast, for arbitrary e0, systems (43) and (44) can realize fixed-time synchronization under controller (22) within Tmax4.Remark 6.
In Example1, if we choose some larger ki and keep the other controller parameters unchanged, the obtained Tmax4 can be smaller. However, Tmax1, Tmax2, and Tmax3 still remain unchanged in this case.


## 5. Conclusions

In this paper, the fixed-time synchronization of neural networks with discrete delay is investigated by utilizing the newly developed fixed-time stability theorem, which can give the settling time a tighter upper bound estimate compared with the existing fixed-time stability theorems. The settling time of fixed-time synchronization/stability is bounded by a fixed constant, irrespective of the initial conditions of the considered systems. The obtained fixed-time synchronization criteria can be verified easily, and numerical simulations are provided to demonstrate the validity of the theoretical results. In the future, we will study the application of neural networks in associative memory.



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