# Robustness Analysis of Control Laws in Complex Dynamical Networks Evoked by Deviating Argument

Authors: Biwen Lia; Jingjing Huang; Donglun Wang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In recent years, robust performance of the system has been broadly studied as a trending topic among a vast array of scholars. This paper discusses the robustness of control laws for complex dynamic networks (CDNs) with a deviation argument. We design two categories of control laws (linear control law and nonlinear control law) for the undisturbed CDNs to achieve exponential synchronization. It is intractable to ascertain the range of the deviation function exactly. Hence, some corresponding sufficient criteria are put forward to ensure exponential synchronization of CDNs with deviation argument when control laws are not changed. By adopting the Gronwall–Bellman lemma and solving the transcendental equation, we can obtain the admissible upper limits of the deviating function, to keep the corresponding control laws. In comparison with previous research findings, robustness, deviating argument, and control laws are all considered in this study, which enhances the previous findings. Finally, two emulation examples verify the validity of the analysis.


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## Body

## 1. Introduction

More recently, as a vital component of the nonlinear system, CDNs have been broadly investigated and have found many potential applications in various domains, including biology, sociology, physics, network science, engineering, automatic control, and so on [1–3]. In particular, the dynamical behavior of complex systems has flourished vigorously in the field of control engineering, which has aroused tremendous interest among scholars [4].Synchronization is a fairly vital nonlinear phenomenon widely existing in nature. It has wide-ranging implications for numerous applications, such as secret communication, facial recognition, artificial intelligence, and associative memory. Recently, the synchronization of CDNs has been extensively studied, for instance, quasi-synchronization [5], cluster synchronization [6], master-slave synchronization [7], exponential synchronization [8, 9], and so on.At present, the control law is indispensable in achieving the synchronization of CDNs. To guarantee synchronization, many effective synchronization control laws have been proposed. On the basis of existing literature [10–12], there are two main categories of control laws of the systems: linear control law and nonlinear control law. As we all know, linear control law is the basis of the control strategy, and the structure of the controller is relatively uncomplicated. Compared with linear control law, nonlinear control law has become an increasingly interesting research topic for researchers from diverse backgrounds. In [8], the exponential synchronization for a kind of CDN with stochastic perturbations and delays was well investigated by utilizing the time-delayed impulsive controller. In [4], based on the event-triggered strategy, quasi-synchronization of CDNs was obtained. In [12], the author addressed robust synchronization between fractional-order CDNs involving parameter uncertainty and applying the nonlinear control law. By means of a sliding mode control scheme, global asymptotical synchronization for CDNs was discussed in [13].As a result of imperfect measurements as well as the finite switching frequency of amplifiers, delays are always inherent in real applications of machine learning, which can exacerbate performance and derail the stability of the models. Generally speaking, time delays are inevitable, and we can never be too concerned about them. Especially, deviation argument can have substantial consequences in the running process of the dynamical system model. As a matter of fact, numerous biomedical models are inscribed by differential equations with a deviation argument. In the process of motion, the deviation function alters the relevant deviation characteristics, so the differential equation with deviation argument unites delay and advanced equation. More precisely, with respect to the study of economics, biology, and physics, past and future events are crucial to current decisions. Therefore, it is of great significance to discuss the models of retarded and advanced alternating differential equations. A system with a deviation argument is a mixture of retarded and advanced systems. Sufficient conditions on the globally exponentially stable for recursive systems containing deviating functions were obtained by virtue of Lyapunov functions in [14]. The author utilized algebraic inequalities to present novel analytical findings from Mittag-Leffler stability about the fractional-order model involving deviating arguments in [15]. On the basis of the comparison principle, valuable theoretic results on the stability and robustness of interval fuzzy Cohen-Grossberg networks were presented in [16]. Additionally, a useful impulsive control law was considered to accomplish the synchronization in the presence of the deviating argument of CDN in [9]. In order to elaborate on the convenience and completeness of CDNs, we will analyze CDNs with deviating arguments here. Among the above studies, only one investigated the synchronization of CDNs with deviating arguments. At present, there are few research studies on robust synchronization control schemes for CDNs with deviating arguments. The objective of our research is to fill the gap.To our knowledge, some work has been performed on the stabilization of dynamical systems equipped with deviating arguments [14, 17]. Some work has been conducted on stabilization and synchronization from the perspective of control [13, 18, 19]. Others have also studied research on complex dynamic networks with deviating arguments [9], but their purpose was to study stability rather than robustness. However, the robustness of control laws in complex dynamic networks is rarely studied by setting appropriate parameters for deviation argument.Motivated by the above-mentioned discussion, we will find that the presence of deviating arguments will exacerbate difficulties in achieving synchronization of controlled complex dynamical networks (CDNs). In view of the literature [20, 21], it is worth considering the influence of the deviation argument on the control method. A new issue arises: can linear control law (nonlinear law) still be kept if a deviation function occurs in the system? As the deviation function involves information about the past and future, it is worthwhile to investigate: under the constraints of linear or nonlinear control laws, how much is the deviation argument intensity to allow CDNs to maintain exponential synchronization?Therefore, the following is concretely scheduled for this paper. Some mathematical preliminaries are provided in Section2. In view of the two categories of control laws, several crucial lemmas and theorems are further elaborated in Section 3. Two emulation examples are mentioned to validate the feasibility of the analytical results in Section 4. Finally, Section 5 of this paper carries a concise summary and outlook for the future research direction.


## 2. Preliminaries

### 2.1. Notation

On the basis of this paper,ℕ represents the sets of natural numbers. ℝn and ℝn×m be made up of all n real vectors, all n×m real matrices, respectively. In is denoted as the n-order identity matrix. Moreover, 0n refers to n×n zero matrix. ⋅ means the Euclidean vector norm or the induced matrix norm. Denote E⊗F as the Kronecker product of matrices E and F.Generally, a graphG=ν,ε,A˜ has three basic elements. ν=1,…,N signifies the set of nodes; ε=eij, i,j∈ν is the set of edges; and the coupling matrix A˜=aijN×N, where aij stands for coupling weight between i th CDN and node j th CDN. If the message is delivered from j th CDN to i th i≠j CDN, then aij≠0; otherwise, aij=0.Provide two real-valued sequencesρq, ηq, q∈ℕ, such that ρq<ρq+1, ρq≤ηq≤ρq+1 for all ρq⟶∞ as q⟶∞.


### 2.2. Model

Consider a kind of CDN with a deviating argument consisting ofN coupled nodes(1)y˙it=fyit+c∑j=1Naijyjt+c∑j=1Nbijyjδt+uit,yt0=y0∈ℝn.where yit=yi1,⋯yinT∈ℝn is the state vector of the i th node; the nonlinear f:ℝn⟶ℝn is called continuous vector-valued function; c stands for the coupling strength; uit∈ℝn is defined as the control input vectors of node i; and aij and bij denote the i,j-th term of coupling matrix A˜ and B˜ satisfying aii=−∑j=1,j≠iNaij and bii=−∑j=1,j≠iNbij, respectively. δt is a deviating argument satisfying δt=ηq∈ρq,ρq+1, if t∈ρq,ρq+1.Remark 1.
Obviously, whent∈ρq,ρq+1, if δt>t, then the coupling term of system (1) is advanced, and if δt<t, then the coupling term of system (1) is retarded. Hence, system (1) is a hybrid CDN, which integrates alternately advanced and retarded argument.
The dynamical equation for the isolated node of CDN (1) is(2)s˙t=fst,where st allows to be defined by an arbitrary desired state, that is, the equilibrium point, the periodic orbit, and so on.Remark 2.
Further, in contrast to (1), in case of δt=t, the CDN (1) turns into the following CDN:(3)x˙it=fxit+c∑j=1Naijxjt+c∑j=1Nbijxjt+uit,xt0=x0∈ℝn.
Similarly,(4)s˙t=fst,
Also referred to as the dynamical equation of the isolated node of CDN (3).
Define the synchronization error aseit=xit−st, and subtracting (4) from (3) to obtain the following system:(5)e˙it=feit+c∑j=1Naijejt+c∑j=1Nbijejt+uit,et0=e0∈ℝn,where feit=fxit−fst.
To acquire synchronization, the linear controller is arranged as(6)uit=W˜eit,i=1,…,N,where W˜∈ℝn×n represents the feedback controller gain matrix.
Combining (5) and (6), one has that(7)e˙it=feit+c∑j=1Naijejt+c∑j=1Nbijejt+W˜eit,et0=e0∈ℝn.
With a view to simplifying writing, the following notations:
et=e1Tt,…eNTtT, eδt=e1Tδt,…eNTδtT, fet=fTe1t,…,fTeNtT, A˜=aijN×N, B˜=bijN×N, A=In⊗A˜, B=In⊗B˜, W=In⊗W˜, are put forward to describe the system. Incorporating these simplifications, the error system (7) is further characterized as a compact representation:(8)e˙t=fet+cAet+cBet+Wet,et0=e0∈ℝn.
In the same way, we can define errorzit=yit−st, and the following error dynamic equation can be obtained by subtracting (2) from (1)(9)z˙t=fzt+cAzt+cBzδt+Wzt,zt0=z0∈ℝn.
As a starting point, we will provide some essential definitions and required assumptions for this paper.Definition 1.
If the error system (8) is exponential stability, then the CDN (3) with (4) is described as exponential synchronization, namely, for any initial value e0∈ℝn, there exist two scalars α>0 and β>0 such that for any t∈ℝ+, satisfying et≤αe0e−βt−t0:(D1)
Assume that the existence of nonnegative scalarp, satisfying fmit−fnit≤pmit−nit and f0n=0 for any mit∈ℝn, nit∈ℝn(D2) There is a positive constantρ that has the property ρq+1−ρq≤ρ, for all q∈ℕ(D3)ρh1+2h2exph1ρ<1


## 2.1. Notation

On the basis of this paper,ℕ represents the sets of natural numbers. ℝn and ℝn×m be made up of all n real vectors, all n×m real matrices, respectively. In is denoted as the n-order identity matrix. Moreover, 0n refers to n×n zero matrix. ⋅ means the Euclidean vector norm or the induced matrix norm. Denote E⊗F as the Kronecker product of matrices E and F.Generally, a graphG=ν,ε,A˜ has three basic elements. ν=1,…,N signifies the set of nodes; ε=eij, i,j∈ν is the set of edges; and the coupling matrix A˜=aijN×N, where aij stands for coupling weight between i th CDN and node j th CDN. If the message is delivered from j th CDN to i th i≠j CDN, then aij≠0; otherwise, aij=0.Provide two real-valued sequencesρq, ηq, q∈ℕ, such that ρq<ρq+1, ρq≤ηq≤ρq+1 for all ρq⟶∞ as q⟶∞.


## 2.2. Model

Consider a kind of CDN with a deviating argument consisting ofN coupled nodes(1)y˙it=fyit+c∑j=1Naijyjt+c∑j=1Nbijyjδt+uit,yt0=y0∈ℝn.where yit=yi1,⋯yinT∈ℝn is the state vector of the i th node; the nonlinear f:ℝn⟶ℝn is called continuous vector-valued function; c stands for the coupling strength; uit∈ℝn is defined as the control input vectors of node i; and aij and bij denote the i,j-th term of coupling matrix A˜ and B˜ satisfying aii=−∑j=1,j≠iNaij and bii=−∑j=1,j≠iNbij, respectively. δt is a deviating argument satisfying δt=ηq∈ρq,ρq+1, if t∈ρq,ρq+1.Remark 1.
Obviously, whent∈ρq,ρq+1, if δt>t, then the coupling term of system (1) is advanced, and if δt<t, then the coupling term of system (1) is retarded. Hence, system (1) is a hybrid CDN, which integrates alternately advanced and retarded argument.
The dynamical equation for the isolated node of CDN (1) is(2)s˙t=fst,where st allows to be defined by an arbitrary desired state, that is, the equilibrium point, the periodic orbit, and so on.Remark 2.
Further, in contrast to (1), in case of δt=t, the CDN (1) turns into the following CDN:(3)x˙it=fxit+c∑j=1Naijxjt+c∑j=1Nbijxjt+uit,xt0=x0∈ℝn.
Similarly,(4)s˙t=fst,
Also referred to as the dynamical equation of the isolated node of CDN (3).
Define the synchronization error aseit=xit−st, and subtracting (4) from (3) to obtain the following system:(5)e˙it=feit+c∑j=1Naijejt+c∑j=1Nbijejt+uit,et0=e0∈ℝn,where feit=fxit−fst.
To acquire synchronization, the linear controller is arranged as(6)uit=W˜eit,i=1,…,N,where W˜∈ℝn×n represents the feedback controller gain matrix.
Combining (5) and (6), one has that(7)e˙it=feit+c∑j=1Naijejt+c∑j=1Nbijejt+W˜eit,et0=e0∈ℝn.
With a view to simplifying writing, the following notations:
et=e1Tt,…eNTtT, eδt=e1Tδt,…eNTδtT, fet=fTe1t,…,fTeNtT, A˜=aijN×N, B˜=bijN×N, A=In⊗A˜, B=In⊗B˜, W=In⊗W˜, are put forward to describe the system. Incorporating these simplifications, the error system (7) is further characterized as a compact representation:(8)e˙t=fet+cAet+cBet+Wet,et0=e0∈ℝn.
In the same way, we can define errorzit=yit−st, and the following error dynamic equation can be obtained by subtracting (2) from (1)(9)z˙t=fzt+cAzt+cBzδt+Wzt,zt0=z0∈ℝn.
As a starting point, we will provide some essential definitions and required assumptions for this paper.Definition 1.
If the error system (8) is exponential stability, then the CDN (3) with (4) is described as exponential synchronization, namely, for any initial value e0∈ℝn, there exist two scalars α>0 and β>0 such that for any t∈ℝ+, satisfying et≤αe0e−βt−t0:(D1)
Assume that the existence of nonnegative scalarp, satisfying fmit−fnit≤pmit−nit and f0n=0 for any mit∈ℝn, nit∈ℝn(D2) There is a positive constantρ that has the property ρq+1−ρq≤ρ, for all q∈ℕ(D3)ρh1+2h2exph1ρ<1


## 3. Main Results

Under (D1), (D2), and (D3), system (9) has a unique state zt;t0,z0 on t>t0 for any initial state t0,z0. Obviously, z=0 is the equilibrium point of system (9). Synchronization, as a highly representative subject of complex system, has been paid attention to and researched by many academics, for example [5–8, 10, 15]. For now, a fundamental point requires consideration. While keeping the original controller unchanged, if the deviation argument is added to the CDN, can synchronization of the system still hold? Obviously, synchronization is not established. How much is the deviation argument intensity in order to allow the CDNs (1) to maintain exponential synchronization when the control law is as valid as before? On the basis of this fact, we are going to investigate the robustness of the controller of CDNs (1) with a deviating argument when the control scheme of the CDNs (2) is fixed.As we all know, among all kinds of control laws, the most efficient and concise control law is the linear feedback controller. What should be noted is that the linear control law is the foundation of the control system. Compared with the nonlinear controller, its structure is relatively uncomplicated. However, nonlinear control law has become an increasingly interesting topic of study for researchers from diverse backgrounds. Therefore, it is worthwhile to design two types of controllers for study: the linear feedback controller and the nonlinear feedback controller.


### 3.1. The Linear Feedback Controller

Before describing the principal theorem of this section, some important lemmas need to be presented here. The following assumptions are also required:(10)αexp−βT+2h2αexp2h1+6h2T/β<1,Remark 3.
As can be inferred from Theorem2 appearing in [6], the existence and uniqueness of the solution of system (1) is collectively ensured by (D1), (D2), and (D3).Lemma 1.
Letzt stands for the current state of error dynamic (9) and conditions (D1), (D2), (D3), and (D4) are all met. So, the following inequality:(11)zδt≤λzt.
It exists for anyt>0, where(12)λ=1/1−u1,u1=h2ρ+h1ρ1+h2ρexph1ρ,h1=p+cA+W,h2=cB.Proof.
For the deviation termδt=ηq, define a set σ=t/t>0,ρq≤t≤ρq+1, let t∈σ, q∈ℕ, and then one obtains(13)zt=zηq+∫ηqtfzs+cAzs+cBzηq+Wzsds.
Combining (D1) and (13), one has that(14)zt≤zηq+∫ηqtfzs+cAzs+cBzηq+Wzsds=1+h2ρzηq+∫ηqth1zsds,where(15)h1=p+cA+W,h2=cB.
By virtue of the Gronwall–Bellman’s inequality, (14) can evolve into(16)zt≤1+h2ρzηqexph1ρ,
Otherwise, forρq≤t≤ρq+1, similarly, we have(17)zηq≤zt+∫ηqtfzs+cAzs+cBzηq+Wzsds≤zt+h2ρzηq+∫ηqth1zsds≤zt+h2ρzηq+h1ρ1+h2ρzηqexph1ρ≤zt+h2ρ+h1ρ1+h2ρexph1ρzηq≤zt+u1zηq,where u1=h2ρ+h1ρ1+h2ρexph1ρ, and h1,h2 are defined in (15).
Hence, by uniting the aforementioned equation with similar entries, we further get(18)1−u1zηq≤zt.
Accordingly, whenδt=ηq, u1<1 for (D4), it follows that(19)zηq≤1−u101zt≕λzt,where λ=1/1−u1. By this means, (11) is effective for t>0.Remark 4.
By virtue of Lemma1, we establish the link from the deviating argument zδt to the state zt and provide a strong base to prove the subsequent Theorem 1.
Next, we explore the influence of the deviation argument on the robustness of exponentially stable of error system (9).Theorem 1.
If assumption (D1) (D2) (D3) (D4) (D5) hold, and error system (8) is exponential stability, then error system (9) is exponential stability, that is, system (1) is said to be exponential synchronization under the linear-type controller (6), if ρ<ρ¯, where ρ¯ is the only solution of the transcendental equation:(20)k2exp2k1T+αexp−βT=1,where k1=h1+2+λh2, k2=h21+λα/β, λ=1−h2ρ+h1ρ1+h2ρexph1ρ−1 and T>lnα/β. Here, in addition to h1,h2,λ and T, all of them are consistent with those defined in Lemma 1.Proof For convenience, et=et;t0,e0 and zt=zt;t0,z0 are expressed by et and zt, respectively. According to (8) and (9), as well as the initial value e0=z0, one has(21)zt−et=∫t0tfzs−fes+cAzs−es+cBzδt−es+Wzs−esds.
Then,(22)zt−et=∫t0tfzs−fes+cAzs−es+cBzδt−es+Wzs−esds≤∫t0tfzs−fes+cAzs−es+cBzδt−es+Wzs−esds.
In view of (D1) and the norm inequality, for (22), one has(23)zt−et≤∫t0tpzs−es+cAzs−es+cBzδs−es+Wzs−esds≤∫t0tp+cA+Wzs−es+cBzδs−esds≤∫t0tp+cA+Wzs−es+cBzδs−zs+zs−esds≤∫t0tp+cA+Wzs−es+cBzδs−zs+cBzs−esds≤∫t0tp+cA+cB+Wzs−es+cBzδs−zsds≤∫t0th1+h2zs−es+h2zδs−zsds.
By Lemma1, when 0≤t0≤t, then(24)zt−et≤∫t0th1+h2zs−es+h2zδs+h2zsds≤∫t0th1+h2zs−es+h21+λzsds=h1+h2∫t0tzs−esds+h21+λ∫t0tzs−es+esds≤h1+h2∫t0tzs−esds+h21+λ∫t0tzs−es+esds≤h1+2+λh2∫t0tzs−esds+h21+λ∫t0tesds.
Due to the error system (8) is exponential stability, according to Definition 1, on the interval t0−ρ,t0+ρ, it comes to the conclusion that(25)et≤αe0e−βt−t0.
And then,(26)∫t0tetds≤αe0β.
Furthermore,(27)zt−et≤h1+2+λh2∫t0tzs−esds+h21+λe0αβ=k1∫t0tzs−esds+k2e0,where(28)k1=h1+2+λh2,k2=αh21+λβ.
By the Gronwall–Bellman’s inequality, whent0+ρ≤t≤t0+2T, we can acquire(29)zt−et≤k2e0exp2k1T.
Sincet0−ρ+T≤t≤t0−ρ+2T, from (27) and (29), one has(30)zt=zt−et+et≤zt−et+et≤k2e0exp2k1T+αe0exp−βT=k2exp2k1T+αexp−βTe0≤h^supt0−ρ≤t≤t0−ρ+Tzt,where h^=k2exp2k1T+αexp−βT.
Denote(31)Hλ=k2exp2k1T+αexp−βT=h21+λα/βexp2h1+2+λh2T+αexp−βT.
By substitutingλ=1 into (31), we can readily get(32)H1=αexp−βT+2h2αexp2h1+6h2T/β<1.
Clearly that,H+∞>1. In addition, Hλ is monotonically increasing for λ. Accordingly, there is only one λ¯∈1,+∞ meeting(33)Hλ¯=1.
Denote(34)Γρ=h2ρ+h1ρ1+h2ρexph1ρ,and identify ρ˜ as the only one positive solution to Γρ=1.
Apparently,(35)λρ=1−Γρ−1∈1,+∞,for ρ∈0,ρ˜. Furthermore, λ is monotonically increasing for ρ. In this sense, there is the only one positive ρ¯∈0,ρ˜ to satisfy(36)λ=λ¯,and ρ¯ is the only one positive solution to (20).
Consequently, forρ<ρ¯,(37)h^=k2exp2k1T+αexp−βT<1.
Picking outξ=−lnh^/T>0, and in consideration of (30), one has(38)supt0−ρ+T≤t≤t0−ρ+2Tzt≤exp−ξTsupt0−ρ≤t≤t0−ρ+Tzt.
Considering the existence and uniqueness of the solutionzt of system (9), when t≥t0−ρ+l−1T, it holds(39)zt,t0,z0=zt,t0−ρ+l−1T,zt0−ρ+l−1T,t0,z0.
From (38) and (39), it follows that(40)supt0−ρ+lT≤t≤t0−ρ+l+1Tzt,t0,z0=supt0−ρ+l−1T+T≤t≤t0−ρ+l−1T+2Tzt;t0−ρ+l−1T,zt0−ρ+l−1T;t0,z0≤exp−ξTsupt0−ρ+l−1T≤t≤t0−ρ+lTzt;t0,z0≤exp−ξlTsupt0−ρ≤t≤t0−ρ+Tzt;t0,z0=Gexp−ξlT,where G=supt0−ρ≤t≤t0−ρ+Tzt;t0,z0.
To go a step further, there is the only scalarl∈ℕ so that t0−ρ+lT≤t≤t0−ρ+l+1T, and one can easily show(41)zt;t0,z0≤Gexp−ξlT≤Gexp−ξt−t0+ξT−ρ≤GexpξTexp−ξt−t0.
Based on Theorem1, one can readily find that an error system (9) is exponentially stable, i.e., the system (1) can achieve exponential synchronization under a designed linear-type controller (6).Remark 5.
Theorem1demonstrates that when an error system (8) is exponentially stable, the perturbed system (9) evoked by the deviation argument can still remain exponentially stable as long as the interval length of the deviating argument δt is less than the estimated upper bound. So, system (1) involving deviating argument is still exponentially synchronous under a designed linear-type controller (6).Remark 6.
As can be seen in Figure1, there is a relationship between the interval length of a deviating argument and time in the proof of Theorem 1.Figure 1
Relationship between interval length of deviating argument and time.


### 3.2. The Nonlinear Feedback Controller

To acquire synchronization, the nonlinear controller is defined as(42)uit=−fxit+fst+R˜eit,i=1,…,N,where R˜∈ℝn×n represents the feedback controller gain matrix.With the nonlinear control law (42), system (5) is reworded to(43)e˙it=c∑j=1Naijejt+c∑j=1Nbijejt+R˜eit,et0=e0∈ℝn.That is,(44)e˙t=cAet+cBet+Ret,et0=e0∈ℝn,where R=In⊗R˜. Here, in addition to R, all of them are consistent with previous notations in (8).In the same way, error system with mixed coupling terms containing deviation arguments is reformulated as(45)z˙t=cAzt+cBzδt+Rzt,zt0=z0∈ℝn,where R=In⊗R˜.To validate Theorem2 more expeditiously, we will introduce Lemma 2. In this subsection, the following assumptions are required:(i)
(D6)ρε1+2ε2expε1ρ<1(ii)
(D7)ε2ρ+ε1ρ1+ε2ρexpε1ρ<1(D8) The following inequality characterize the parameters of the system (44)(46)αexp−βT+2ε2αexp2ε1+6ε2Tβ<1,where T>lnα/β, ε1=cA+R, ε2=cB.Lemma 2.
Letzt stands for the current state of error system (45) and conditions (D1) (D2) (D6) (D7) are all met. So, we have the following inequality:(47)zδt≤κzt,
This exists for anyt>0, where(48)κ=11−υ1,υ1=ε2ρ+ε1ρ1+ε2ρexpε1ρ,ε1=cA+R,ε2=cB.Proof.
For the deviation termδt=ηq, define a set σ=t>0,ρq≤t≤ρq+1, let t∈σ, and q∈ℕ, and then, we have(49)zt=zηq+∫ηqtcAzs+cBzηq+Rzsds.
Combining (D1), we obtain(50)zt≤zηq+∫ηqtcAzs+cBzηq+Rzsds≤zηq+∫ηqtcA+Rzsds+∫ηqtcBzηqds=1+ε2ρzηq+∫ηqtε1zsds,where(51)ε1=cA+R,ε2=cB.
Therefore, for (50), using the Gronwall–Bellman’s inequality, it yields that(52)zt≤1+ε2ρzηqexpε1ρ.
Otherwise, forρq≤t≤ρq+1, similarly, it follows that(53)zηq≤zt+∫ηqtcAzs+cBzηq+Rzsds≤zt+ε2ρzηq+∫ηqtε1zsds≤zt+ε2ρzηq+ε1ρ1+ε2ρzηqexpε1ρ≤zt+ε2ρ+ε1ρ1+ε2ρexpε1ρzηq≤zt+υ1zηq,where υ1=ε2ρ+ε1ρ1+ε2ρexpε1ρ and ε1 and ε2 are defined in (51).
Hence, by uniting the aforementioned equation with similar entries, we can further require(54)1−υ1zηq≤zt.
Accordingly, whenδt=ηq, υ1<1 for (D7), it further derives(55)zηq≤1−υ101zt≕κzt,where κ=1/1−u1−1. By this means, (47) is available for t≥0.Theorem 2.
If assumption (D1) (D2) (D6) (D7) (D8) hold, and error system (44) is exponentially stable, then, error system (45) is exponentially stable, that is, system (1) is said to be exponential synchronization under the nonlinear-type controller (42), if ρ<ρ¯, where ρ¯ is the only solution of the transcendental equation:(56)Λ2exp2Λ1T+αexp−βT=1,where Λ2=αε21+κ/β, κ=1−ε2ρ+ε1ρ1+ε2ρexpε1ρ−1 and T>lnα/β. Here, in addition to ε1,ε2,κ and T, all of them are consistent with those defined in Lemma 2.Proof.
For convenience,et=et;t0,e0 and zt=zt;t0,z0 are expressed by et and zt, respectively. According to (44) and (45), as well as the initial value e0=z0, one has(57)zt−et=∫t0tcAzs−es+cBzδt−es+Rzs−esds.
Then,(58)zt−et=∫t0tcAzs−es+cBzδt−es+Rzs−esds≤∫t0tcAzs−es+cBzδt−es+Rzs−esds.
In view of the norm inequality, for (58), one has(59)zt−et≤∫t0tcAzs−es+cBzδs−es+Rzs−esds=∫t0tcA+Rzs−es+cBzδs−esds=∫t0tcA+Rzs−es+cBzδs−zs+zs−esds≤∫t0tcA+Rzs−es+cBzδs−zs+cBzs−esds≤∫t0tcA+cB+Rzs−es+cBzδs−zsds≤∫t0tε1+ε2zs−es+ε2zδs−zsds.
By Lemma2, when 0≤t0≤t, then(60)zt−et≤∫t0tε1+ε2zs−es+ε2zδs+ε2zsds≤∫t0tε1+ε2zs−es+ε21+κzsds=ε1+ε2∫t0tzs−esds+ε21+κ∫t0tzs−es+esds≤ε1+ε2∫t0tzs−esds+ε21+κ∫t0tzs−es+esds≤ε1+2+κε2∫t0tzs−esds+ε21+κ∫t0tesds.
Due to CDN, (44) is exponential stability, and according to Definition 1, on the interval t0−ρ,t0+ρ, it comes to the conclusion that(61)et≤αe0e−βt−t0.
And then,(62)∫t0tetds≤αe0/β.
Furthermore,(63)zt−et≤ε1+2+κε2∫t0tzs−esds+αε21+κe0β=Λ1∫t0tzs−esds+Λ2e0,where(64)Λ1=ε1+2+κε2,Λ2=αε21+κ/β.
By the Gronwall–Bellman’s inequality, whent0+ρ≤t≤t0+2T, we can expediently acquire(65)zt−et≤Λ2e0exp2Λ1T.
Sincet0−ρ+T≤t≤t0−ρ+2T, from (65) and (66), further we derive(66)zt=zt−et+et≤zt−et+et≤Λ2e0exp2Λ1T+αe0exp−βT=Λ2exp2Λ1T+αexp−βTe0≤ι^supt0−ρ≤t≤t0−ρ+Tzt,where ι^=Λ2exp2Λ1T+αexp−βT.
Denote(67)Ψκ=Λ2exp2Λ1T+αexp−βT=αε21+κexp2ε1+2+κε2Tβ+αexp−βT.
By substitutingκ=1 into (74), we can see that(68)Ψ1=αexp−βT+2ε2αexp2ε1+6ε2Tβ<1.
Clearly that,Ψ+∞>1. In addition, Ψκ is strictly monotonously increasing for κ. Accordingly, there is the only one κ¯∈1,+∞ make(69)Ψκ¯=1.
Denote(70)ℑρ=ε2ρ+ε1ρ1+ε2ρexpε1ρ,and identify ρ˜ as the only one positive solution to ℑρ=1.
Apparently,(71)κ=1−ℑρ−1∈1,+∞,for ρ∈0,ρ˜. Furthermore, κ is increase strictly monotonically for ρ. In this sense, there is the only one positive scalar ρ¯∈0,ρ˜ satisfy(72)κ=κ¯,and ρ¯ is the only one positive solution for (56).
Thus,(73)ι^=Λ2exp2Λ1T+αexp−βT<1,for ρ<ρ¯.
Picking outg=−lnι^/T>0, and by (67), one gets(74)supt0−ρ+T≤t≤t0−ρ+2Tzt≤exp−gTsupt0−ρ≤t≤t0−ρ+Tzt.
Considering the existence and uniqueness of solutionzt of the system (45), when t≥t0−ρ+l−1T, it holds(75)zt,t0,z0=zt,t0−ρ+l−1T,zt0−ρ+l−1T,t0,z0.
From (75) and (76), we have(76)supt0−ρ+lT≤t≤t0−ρ+l+1Tzt,t0,z0=supt0−ρ+l−1T+T≤t≤t0−ρ+l−1T+2Tzt;t0−ρ+l−1T,zt0−ρ+l−1T;t0,z0≤exp−gTsupt0−ρ+l−1T≤t≤t0−ρ+lTzt;t0,z0≤exp−glTsupt0−ρ≤t≤t0−ρ+Tzt;t0,z0=Gexp−glT,where G=supt0−ρ≤t≤t0−ρ+Tzt;t0,z0.
Furthermore, there is only scalarl∈ℕ, so that t0−ρ+lT≤t≤t0−ρ+l+1T, and one can easily show that(77)zt;t0,z0≤Gexp−glT≤Gexp−gt−t0+gT−ρ≤GexpgTexp−gt−t0.
By virtue of Theorem2, one can readily deduce that error system (45) is exponentially stable, i.e., system (1) can achieve exponential synchronization under a designed nonlinear-type controller (42).Remark 7.
Theorem2 clearly indicates that when error system (42) is exponentially stable, the corresponding perturbed error system (45) evoked by a deviation argument can still remain exponentially stable as long as the interval length of the deviating argument δt is less than the estimated upper bound. Furthermore, system (1) involving a deviating argument is still exponentially synchronous under a designed nonlinear-type controller (42).


## 3.1. The Linear Feedback Controller

Before describing the principal theorem of this section, some important lemmas need to be presented here. The following assumptions are also required:(10)αexp−βT+2h2αexp2h1+6h2T/β<1,Remark 3.
As can be inferred from Theorem2 appearing in [6], the existence and uniqueness of the solution of system (1) is collectively ensured by (D1), (D2), and (D3).Lemma 1.
Letzt stands for the current state of error dynamic (9) and conditions (D1), (D2), (D3), and (D4) are all met. So, the following inequality:(11)zδt≤λzt.
It exists for anyt>0, where(12)λ=1/1−u1,u1=h2ρ+h1ρ1+h2ρexph1ρ,h1=p+cA+W,h2=cB.Proof.
For the deviation termδt=ηq, define a set σ=t/t>0,ρq≤t≤ρq+1, let t∈σ, q∈ℕ, and then one obtains(13)zt=zηq+∫ηqtfzs+cAzs+cBzηq+Wzsds.
Combining (D1) and (13), one has that(14)zt≤zηq+∫ηqtfzs+cAzs+cBzηq+Wzsds=1+h2ρzηq+∫ηqth1zsds,where(15)h1=p+cA+W,h2=cB.
By virtue of the Gronwall–Bellman’s inequality, (14) can evolve into(16)zt≤1+h2ρzηqexph1ρ,
Otherwise, forρq≤t≤ρq+1, similarly, we have(17)zηq≤zt+∫ηqtfzs+cAzs+cBzηq+Wzsds≤zt+h2ρzηq+∫ηqth1zsds≤zt+h2ρzηq+h1ρ1+h2ρzηqexph1ρ≤zt+h2ρ+h1ρ1+h2ρexph1ρzηq≤zt+u1zηq,where u1=h2ρ+h1ρ1+h2ρexph1ρ, and h1,h2 are defined in (15).
Hence, by uniting the aforementioned equation with similar entries, we further get(18)1−u1zηq≤zt.
Accordingly, whenδt=ηq, u1<1 for (D4), it follows that(19)zηq≤1−u101zt≕λzt,where λ=1/1−u1. By this means, (11) is effective for t>0.Remark 4.
By virtue of Lemma1, we establish the link from the deviating argument zδt to the state zt and provide a strong base to prove the subsequent Theorem 1.
Next, we explore the influence of the deviation argument on the robustness of exponentially stable of error system (9).Theorem 1.
If assumption (D1) (D2) (D3) (D4) (D5) hold, and error system (8) is exponential stability, then error system (9) is exponential stability, that is, system (1) is said to be exponential synchronization under the linear-type controller (6), if ρ<ρ¯, where ρ¯ is the only solution of the transcendental equation:(20)k2exp2k1T+αexp−βT=1,where k1=h1+2+λh2, k2=h21+λα/β, λ=1−h2ρ+h1ρ1+h2ρexph1ρ−1 and T>lnα/β. Here, in addition to h1,h2,λ and T, all of them are consistent with those defined in Lemma 1.Proof For convenience, et=et;t0,e0 and zt=zt;t0,z0 are expressed by et and zt, respectively. According to (8) and (9), as well as the initial value e0=z0, one has(21)zt−et=∫t0tfzs−fes+cAzs−es+cBzδt−es+Wzs−esds.
Then,(22)zt−et=∫t0tfzs−fes+cAzs−es+cBzδt−es+Wzs−esds≤∫t0tfzs−fes+cAzs−es+cBzδt−es+Wzs−esds.
In view of (D1) and the norm inequality, for (22), one has(23)zt−et≤∫t0tpzs−es+cAzs−es+cBzδs−es+Wzs−esds≤∫t0tp+cA+Wzs−es+cBzδs−esds≤∫t0tp+cA+Wzs−es+cBzδs−zs+zs−esds≤∫t0tp+cA+Wzs−es+cBzδs−zs+cBzs−esds≤∫t0tp+cA+cB+Wzs−es+cBzδs−zsds≤∫t0th1+h2zs−es+h2zδs−zsds.
By Lemma1, when 0≤t0≤t, then(24)zt−et≤∫t0th1+h2zs−es+h2zδs+h2zsds≤∫t0th1+h2zs−es+h21+λzsds=h1+h2∫t0tzs−esds+h21+λ∫t0tzs−es+esds≤h1+h2∫t0tzs−esds+h21+λ∫t0tzs−es+esds≤h1+2+λh2∫t0tzs−esds+h21+λ∫t0tesds.
Due to the error system (8) is exponential stability, according to Definition 1, on the interval t0−ρ,t0+ρ, it comes to the conclusion that(25)et≤αe0e−βt−t0.
And then,(26)∫t0tetds≤αe0β.
Furthermore,(27)zt−et≤h1+2+λh2∫t0tzs−esds+h21+λe0αβ=k1∫t0tzs−esds+k2e0,where(28)k1=h1+2+λh2,k2=αh21+λβ.
By the Gronwall–Bellman’s inequality, whent0+ρ≤t≤t0+2T, we can acquire(29)zt−et≤k2e0exp2k1T.
Sincet0−ρ+T≤t≤t0−ρ+2T, from (27) and (29), one has(30)zt=zt−et+et≤zt−et+et≤k2e0exp2k1T+αe0exp−βT=k2exp2k1T+αexp−βTe0≤h^supt0−ρ≤t≤t0−ρ+Tzt,where h^=k2exp2k1T+αexp−βT.
Denote(31)Hλ=k2exp2k1T+αexp−βT=h21+λα/βexp2h1+2+λh2T+αexp−βT.
By substitutingλ=1 into (31), we can readily get(32)H1=αexp−βT+2h2αexp2h1+6h2T/β<1.
Clearly that,H+∞>1. In addition, Hλ is monotonically increasing for λ. Accordingly, there is only one λ¯∈1,+∞ meeting(33)Hλ¯=1.
Denote(34)Γρ=h2ρ+h1ρ1+h2ρexph1ρ,and identify ρ˜ as the only one positive solution to Γρ=1.
Apparently,(35)λρ=1−Γρ−1∈1,+∞,for ρ∈0,ρ˜. Furthermore, λ is monotonically increasing for ρ. In this sense, there is the only one positive ρ¯∈0,ρ˜ to satisfy(36)λ=λ¯,and ρ¯ is the only one positive solution to (20).
Consequently, forρ<ρ¯,(37)h^=k2exp2k1T+αexp−βT<1.
Picking outξ=−lnh^/T>0, and in consideration of (30), one has(38)supt0−ρ+T≤t≤t0−ρ+2Tzt≤exp−ξTsupt0−ρ≤t≤t0−ρ+Tzt.
Considering the existence and uniqueness of the solutionzt of system (9), when t≥t0−ρ+l−1T, it holds(39)zt,t0,z0=zt,t0−ρ+l−1T,zt0−ρ+l−1T,t0,z0.
From (38) and (39), it follows that(40)supt0−ρ+lT≤t≤t0−ρ+l+1Tzt,t0,z0=supt0−ρ+l−1T+T≤t≤t0−ρ+l−1T+2Tzt;t0−ρ+l−1T,zt0−ρ+l−1T;t0,z0≤exp−ξTsupt0−ρ+l−1T≤t≤t0−ρ+lTzt;t0,z0≤exp−ξlTsupt0−ρ≤t≤t0−ρ+Tzt;t0,z0=Gexp−ξlT,where G=supt0−ρ≤t≤t0−ρ+Tzt;t0,z0.
To go a step further, there is the only scalarl∈ℕ so that t0−ρ+lT≤t≤t0−ρ+l+1T, and one can easily show(41)zt;t0,z0≤Gexp−ξlT≤Gexp−ξt−t0+ξT−ρ≤GexpξTexp−ξt−t0.
Based on Theorem1, one can readily find that an error system (9) is exponentially stable, i.e., the system (1) can achieve exponential synchronization under a designed linear-type controller (6).Remark 5.
Theorem1demonstrates that when an error system (8) is exponentially stable, the perturbed system (9) evoked by the deviation argument can still remain exponentially stable as long as the interval length of the deviating argument δt is less than the estimated upper bound. So, system (1) involving deviating argument is still exponentially synchronous under a designed linear-type controller (6).Remark 6.
As can be seen in Figure1, there is a relationship between the interval length of a deviating argument and time in the proof of Theorem 1.Figure 1
Relationship between interval length of deviating argument and time.


## 3.2. The Nonlinear Feedback Controller

To acquire synchronization, the nonlinear controller is defined as(42)uit=−fxit+fst+R˜eit,i=1,…,N,where R˜∈ℝn×n represents the feedback controller gain matrix.With the nonlinear control law (42), system (5) is reworded to(43)e˙it=c∑j=1Naijejt+c∑j=1Nbijejt+R˜eit,et0=e0∈ℝn.That is,(44)e˙t=cAet+cBet+Ret,et0=e0∈ℝn,where R=In⊗R˜. Here, in addition to R, all of them are consistent with previous notations in (8).In the same way, error system with mixed coupling terms containing deviation arguments is reformulated as(45)z˙t=cAzt+cBzδt+Rzt,zt0=z0∈ℝn,where R=In⊗R˜.To validate Theorem2 more expeditiously, we will introduce Lemma 2. In this subsection, the following assumptions are required:(i)
(D6)ρε1+2ε2expε1ρ<1(ii)
(D7)ε2ρ+ε1ρ1+ε2ρexpε1ρ<1(D8) The following inequality characterize the parameters of the system (44)(46)αexp−βT+2ε2αexp2ε1+6ε2Tβ<1,where T>lnα/β, ε1=cA+R, ε2=cB.Lemma 2.
Letzt stands for the current state of error system (45) and conditions (D1) (D2) (D6) (D7) are all met. So, we have the following inequality:(47)zδt≤κzt,
This exists for anyt>0, where(48)κ=11−υ1,υ1=ε2ρ+ε1ρ1+ε2ρexpε1ρ,ε1=cA+R,ε2=cB.Proof.
For the deviation termδt=ηq, define a set σ=t>0,ρq≤t≤ρq+1, let t∈σ, and q∈ℕ, and then, we have(49)zt=zηq+∫ηqtcAzs+cBzηq+Rzsds.
Combining (D1), we obtain(50)zt≤zηq+∫ηqtcAzs+cBzηq+Rzsds≤zηq+∫ηqtcA+Rzsds+∫ηqtcBzηqds=1+ε2ρzηq+∫ηqtε1zsds,where(51)ε1=cA+R,ε2=cB.
Therefore, for (50), using the Gronwall–Bellman’s inequality, it yields that(52)zt≤1+ε2ρzηqexpε1ρ.
Otherwise, forρq≤t≤ρq+1, similarly, it follows that(53)zηq≤zt+∫ηqtcAzs+cBzηq+Rzsds≤zt+ε2ρzηq+∫ηqtε1zsds≤zt+ε2ρzηq+ε1ρ1+ε2ρzηqexpε1ρ≤zt+ε2ρ+ε1ρ1+ε2ρexpε1ρzηq≤zt+υ1zηq,where υ1=ε2ρ+ε1ρ1+ε2ρexpε1ρ and ε1 and ε2 are defined in (51).
Hence, by uniting the aforementioned equation with similar entries, we can further require(54)1−υ1zηq≤zt.
Accordingly, whenδt=ηq, υ1<1 for (D7), it further derives(55)zηq≤1−υ101zt≕κzt,where κ=1/1−u1−1. By this means, (47) is available for t≥0.Theorem 2.
If assumption (D1) (D2) (D6) (D7) (D8) hold, and error system (44) is exponentially stable, then, error system (45) is exponentially stable, that is, system (1) is said to be exponential synchronization under the nonlinear-type controller (42), if ρ<ρ¯, where ρ¯ is the only solution of the transcendental equation:(56)Λ2exp2Λ1T+αexp−βT=1,where Λ2=αε21+κ/β, κ=1−ε2ρ+ε1ρ1+ε2ρexpε1ρ−1 and T>lnα/β. Here, in addition to ε1,ε2,κ and T, all of them are consistent with those defined in Lemma 2.Proof.
For convenience,et=et;t0,e0 and zt=zt;t0,z0 are expressed by et and zt, respectively. According to (44) and (45), as well as the initial value e0=z0, one has(57)zt−et=∫t0tcAzs−es+cBzδt−es+Rzs−esds.
Then,(58)zt−et=∫t0tcAzs−es+cBzδt−es+Rzs−esds≤∫t0tcAzs−es+cBzδt−es+Rzs−esds.
In view of the norm inequality, for (58), one has(59)zt−et≤∫t0tcAzs−es+cBzδs−es+Rzs−esds=∫t0tcA+Rzs−es+cBzδs−esds=∫t0tcA+Rzs−es+cBzδs−zs+zs−esds≤∫t0tcA+Rzs−es+cBzδs−zs+cBzs−esds≤∫t0tcA+cB+Rzs−es+cBzδs−zsds≤∫t0tε1+ε2zs−es+ε2zδs−zsds.
By Lemma2, when 0≤t0≤t, then(60)zt−et≤∫t0tε1+ε2zs−es+ε2zδs+ε2zsds≤∫t0tε1+ε2zs−es+ε21+κzsds=ε1+ε2∫t0tzs−esds+ε21+κ∫t0tzs−es+esds≤ε1+ε2∫t0tzs−esds+ε21+κ∫t0tzs−es+esds≤ε1+2+κε2∫t0tzs−esds+ε21+κ∫t0tesds.
Due to CDN, (44) is exponential stability, and according to Definition 1, on the interval t0−ρ,t0+ρ, it comes to the conclusion that(61)et≤αe0e−βt−t0.
And then,(62)∫t0tetds≤αe0/β.
Furthermore,(63)zt−et≤ε1+2+κε2∫t0tzs−esds+αε21+κe0β=Λ1∫t0tzs−esds+Λ2e0,where(64)Λ1=ε1+2+κε2,Λ2=αε21+κ/β.
By the Gronwall–Bellman’s inequality, whent0+ρ≤t≤t0+2T, we can expediently acquire(65)zt−et≤Λ2e0exp2Λ1T.
Sincet0−ρ+T≤t≤t0−ρ+2T, from (65) and (66), further we derive(66)zt=zt−et+et≤zt−et+et≤Λ2e0exp2Λ1T+αe0exp−βT=Λ2exp2Λ1T+αexp−βTe0≤ι^supt0−ρ≤t≤t0−ρ+Tzt,where ι^=Λ2exp2Λ1T+αexp−βT.
Denote(67)Ψκ=Λ2exp2Λ1T+αexp−βT=αε21+κexp2ε1+2+κε2Tβ+αexp−βT.
By substitutingκ=1 into (74), we can see that(68)Ψ1=αexp−βT+2ε2αexp2ε1+6ε2Tβ<1.
Clearly that,Ψ+∞>1. In addition, Ψκ is strictly monotonously increasing for κ. Accordingly, there is the only one κ¯∈1,+∞ make(69)Ψκ¯=1.
Denote(70)ℑρ=ε2ρ+ε1ρ1+ε2ρexpε1ρ,and identify ρ˜ as the only one positive solution to ℑρ=1.
Apparently,(71)κ=1−ℑρ−1∈1,+∞,for ρ∈0,ρ˜. Furthermore, κ is increase strictly monotonically for ρ. In this sense, there is the only one positive scalar ρ¯∈0,ρ˜ satisfy(72)κ=κ¯,and ρ¯ is the only one positive solution for (56).
Thus,(73)ι^=Λ2exp2Λ1T+αexp−βT<1,for ρ<ρ¯.
Picking outg=−lnι^/T>0, and by (67), one gets(74)supt0−ρ+T≤t≤t0−ρ+2Tzt≤exp−gTsupt0−ρ≤t≤t0−ρ+Tzt.
Considering the existence and uniqueness of solutionzt of the system (45), when t≥t0−ρ+l−1T, it holds(75)zt,t0,z0=zt,t0−ρ+l−1T,zt0−ρ+l−1T,t0,z0.
From (75) and (76), we have(76)supt0−ρ+lT≤t≤t0−ρ+l+1Tzt,t0,z0=supt0−ρ+l−1T+T≤t≤t0−ρ+l−1T+2Tzt;t0−ρ+l−1T,zt0−ρ+l−1T;t0,z0≤exp−gTsupt0−ρ+l−1T≤t≤t0−ρ+lTzt;t0,z0≤exp−glTsupt0−ρ≤t≤t0−ρ+Tzt;t0,z0=Gexp−glT,where G=supt0−ρ≤t≤t0−ρ+Tzt;t0,z0.
Furthermore, there is only scalarl∈ℕ, so that t0−ρ+lT≤t≤t0−ρ+l+1T, and one can easily show that(77)zt;t0,z0≤Gexp−glT≤Gexp−gt−t0+gT−ρ≤GexpgTexp−gt−t0.
By virtue of Theorem2, one can readily deduce that error system (45) is exponentially stable, i.e., system (1) can achieve exponential synchronization under a designed nonlinear-type controller (42).Remark 7.
Theorem2 clearly indicates that when error system (42) is exponentially stable, the corresponding perturbed error system (45) evoked by a deviation argument can still remain exponentially stable as long as the interval length of the deviating argument δt is less than the estimated upper bound. Furthermore, system (1) involving a deviating argument is still exponentially synchronous under a designed nonlinear-type controller (42).


## 4. Simulations

Two illustrative examples will be enumerated to show the validity of conclusions obtained above in this section.Example 1.
Here considering a CDN with linear control law, which consists of two nodes:(78)x˙it=fxit+c∑j=12aijxjt+uit,xt0=x0∈ℝn,s˙t=fst,uit=W˜eit.
In the case of deviation argument, system (78) turns into(79)x˙it=fxit+c∑j=12aijxjt+c∑j=12bijxjδt+uit,xt0=x0∈ℝn,s˙t=fst,uit=W˜eit,where i∈1,2, and xit=xi1t,xi2tT∈ℝ2 is the state vectors of i−th nodes for the CDN. let eit=xit−st, one can see that error system (78) and error system (79) turns into, respectively,(80)e˙t=Fet+cAet+Wet,et0=e0∈ℝn,(81)e˙t=Fet+cAet+cBeδt+Wet,et0=e0∈ℝn.
Let coupling matricesA=−552−2, B=−0.0010.0010.002−0.002, and K=−5.500−5.3. The coupling strength is designed as c=0.1. The activation function is f⋅=tanh⋅. Two nodes and isolated nodes of the initial value can be designed as x1=−1.1,1.2T, x2=1.7,−1.4T, and st=0.2,0.1T.
As shown in Figure2, error system (80) is exponentially stable when α=1.1 and β=0.8.
Fix two consequences:ρq=q/20, ηq=2q+1/40, and q∈ℕ. Let T=0.2≥ln1.1/0.8=0.1191, T=0.2≥ln1.1/0.8=0.1191; then by calculation, we can obtain h1=7.2616 and h2=3.1623×10−4.
Based on inequality (D4), it is expedient to calculateρ=0.0781. According to (30), we can obtain λ=6.8815 and ρ¯=0.0705.
Selectingδt=ρ=1/20<ρ¯=0.0705, by conducting simple calculations, we can find(82)120×7.2616+2×3.1623×10−4×exp7.2616×120=0.5221<1,and (D3) is satisfied.
It is obvious that all the requirements appearing are each fulfilled in Theorem1. In view of Theorem 1 and Definition 1, we are able to deduce that error system (81) is exponential stability, that is, system (79) is exponential synchronization. As shown in Figure 3, the simulated findings closely match the theory.Figure 2
The convergent behavior of the error system under a nonlinear control law.Figure 3
The convergent behavior of error system (80) under a linear control law.Example 2.
We discuss a CDN, consisting of two nodes, with nonlinear control law whose dynamics are described as(83)x˙it=fxit+c∑j=12aijxjt+wit,xt0=x0∈ℝn,s˙t=fst,uit=−fxit+fst+eR˜it.
Subsequently, in the presence of deviating argument, system (83) becomes(84)x˙it=fxit+c∑j=12aijxjt+c∑j=12bijxjδt+wit,xt0=x0∈ℝn,s˙t=fst,wit=−fxit+fst+R˜eit,,where i∈1,2, and xit=xi1t,xi2tT∈ℝ2, is the state vectors.
Leteit=xit−st, one can see that error system (83) and error system (84) are simplified as, respectively,(85)e˙t=cAet+Ret,et0=e0∈ℝn,(86)e˙t=cAet+cBeδt+Ret,et0=e0∈ℝn.
Let coupling matricesA=−224−4, B=−0.0030.0030.002−0.002, and K=−4.100−4.7. The coupling strength is designated as c=0.05 for i=1,2. The activation function is f⋅=tanh⋅. Two nodes and isolated nodes of the initial value can be designed as x1=−2,2T, x2=3,−1T, and st=1,0T.
As shown in Figure4, the error system (85) is exponential stability when α=1.1 and β=0.5.
Fix two consequences:ρq=q/20, ηq=2q+1/40, q∈ℕ. Let T=0.4≥ln1.1/0.5=0.1906, then by calculation, we can obtain h1=5.0162, h2=2.5495×10−4.
Based on inequality (D7), it is expedient to calculateρ=0.1131. According to (56), we can obtain λ=2.2011, ρ¯=0.0748.
Selectingδt=ρ=1/20<ρ¯=0.0748, by conducting simple calculations, we can find(87)120×5.0162+2×2.5495×10−4×exp5.0162×120=0.3223<1,and (D6) is satisfied.
Clearly, all the requirements outlined in Theorem2 are each fulfilled. In the light of Theorem 2 and Definition 1, we are able to deduce that error system (86) is exponentially stable, that is to say, system (84) can achieve exponential synchronization when ρ=1/20. As indicated in Figure 5, the simulated findings closely correlate with the theory.
Figure6 depicts that error system (86) is instable when ρ=1/2. Moreover, in this situation, the parameters are not appropriate for Theorem 2. Therefore, system (84) is not exponential synchronization.Figure 4
The evolvement behavior of error system (81) under a linear control law.Figure 5
The evolvement behavior of error system (86) under a nonlinear control law.Figure 6
The instable behavior of error system (86) with ρ=1/2 under a nonlinear control law.


## 5. Concluding Remarks

The robustness of the complex system to control laws evoked by deviating arguments is investigated. In this paper, two categories of control laws are provided and some corresponding sufficient criteria are put forward to prove the synchronization of CDNs with deviation argument. The findings show that a complex system containing a deviation function will keep exponential synchronization continuously as long as the interval length of the deviation function is lower than the derived upper limit. In view of the analysis and methodology discussed in this paper, more complex models will be considered for further topics.



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