# Synchronization of Complex Dynamical Networks on Time Scales via Pinning Control

Authors: Fang-Di Kong; Jian-Ping Sun
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we are concerned with the synchronization problem of complex dynamical networks on time scales. Some pinning synchronization criteria, which combine main characteristics of time scales with main parameters of the pinning controlled network, are established. A numerical example is also included to verify the effectiveness of the results obtained.


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## Body

## 1. Introduction

Many complex systems in nature and human societies can be modeled by complex dynamical networks with the nodes representing individuals in the system and the edges representing the interactions among them, such as social networks, food webs, the Internet, the World Wide Web, and neural networks (see [1] and the references therein). One of the most ubiquitous and significant phenomena in complex dynamical networks is the synchronization of all dynamical nodes in a network. Over the past decades, the synchronization has attracted considerable attention [2–9]. Control would be a necessary means to guide or force the complex dynamical network to realize synchronization if a given network is not self-synchronized or the synchronized state is not the desired one. At present, the control methods which are often used include adaptive control [10, 11], impulsive control [12], pinning adaptive control [13], pinning impulsive control [14], pinning feedback control [15–25], and so on [26–30]. Since pinning control only needs a small fraction of nodes to be dealt with, the synchronization of complex dynamical networks via pinning control has become a rather significant and interesting topic; see [13–27]. It is necessary to point out that most of the aforementioned discussions were aimed at the synchronization problem of continuous-time and discrete-time complex dynamical networks, respectively.On the one hand, in some real-world systems, the interactions among individuals can take place at any time, maybe some continuous time intervals accompanying some discrete moments. On the other hand, the theory of time scales, a useful tool to deal with continuous and discrete analysis under a unified framework, was introduced by Hilger in his Ph.D. thesis [31]. With the development of the theory of time scales, the synchronization of complex dynamical networks on time scales has received increasing attention [32–40]. For example, in 2016, Liu and Zhang [35] studied the synchronization of linear complex dynamical networks on time scales via pinning impulsive control. In 2018, Lu et al. [39] considered finite-time synchronization of nonlinear complex dynamical networks on time scales via pinning impulsive control. In [40], Xiao, Lewis, and Zeng investigated event-based time-interval pinning control for complex networks on time scales.Since time scale is an arbitrary nonempty closed subset of the real numbers, it has various forms such as the real numbers, the integers, the union of some closed intervals, and the union of some closed intervals and some discrete points. Therefore, complex dynamical networks on time scales have great complexity. Moreover, many existing results for continuous-time or discrete-time complex dynamical networks cannot be simply generalized to complex dynamical networks on time scales [35, 40].Motivated greatly by the abovementioned works, in this paper, we will study a complex dynamical network on time scales by applying pinning feedback control. Some sufficient conditions are derived to guarantee the complex dynamical network to realize synchronization when it is not self-synchronized or the synchronized state is not the desired one. The main contributions of this paper are listed as follows:(i)
In order to overcome the difficulties caused by nonlinear functionf⋅ in the complex dynamical network, we design some appropriate conditions(ii)
The pinning synchronization criteria established in our paper combine main characteristics of time scales with main parameters of the pinning controlled network (such as coupling strengths, coupling configuration matrix, and pinning feedback gain matrix)(iii)
Our results have revealed the discrepancies of the pinning synchronization between continuous-time and discrete-time complex dynamical networksThe rest of this paper is organized as follows. Some notations and supporting lemmas, and some foundational knowledge about time scales are simply enumerated in Section2. In Section 3, the synchronization problem of a complex dynamical network on time scales is formulated. In Section 4, our main results are established. In Section 5, a numerical example is given to verify the effectiveness of the results obtained. Finally, conclusions are provided in Section 6.


## 2. Preliminaries

In this section, we will present some notations and lemmas, and some foundational knowledge on time scales which are needed later.


### 2.1. Notations and Supporting Lemmas

First, we define some notations as follows:ℕ0 is the set of all nonnegative integersℤ is the set of all integersℝ is the set of all real numbersℝn is the n-dimensional Euclidean space with the Euclidean norm ⋅ℝm×n is the set of all m×n real matricesIn∈ℝn×n is the n-dimensional identity matrixdiagd1,d2,…,dn is the diagonal matrix with diagonal entries d1 to dnThe superscript “T” stands for the transpose of a matrixFor symmetric matrixP∈ℝn×n, λminP and λmaxP denote the minimum and maximum eigenvalues of P, respectivelyFor symmetric matricesP,Q∈ℝn×n, P≥QP≤Q means that P−Q is positive semidefinite (negative semidefinite)⊗ denotes the Kronecker productLemma 1 (see [23]).
SupposeA=aijn×n is a real symmetric and irreducible matrix, in which aij≥0j≠i and aii=−∑j=1,j≠inaij, and nonzero matrix D=diagd1,d2,…,dn satisfies di≥01≤i≤n. Let B=A−D. Then, all the eigenvalues of B are less than 0.Lemma 2 (see [41]).
IfP,Q∈ℝn×n are symmetric, x∈ℝn is a nonzero vector, and 0<a,b∈ℝ, then(1)
P+Q is symmetric(2)
Pk is symmetric for k=1,2,…,(3)
P is positive definite ⇔−P is negative definite(4)
P is positive definite ⇔ there exists a positive definite matrix S∈ℝn×n such that P=S2(5)
λminP≤xTPx/xTx≤λmaxP(6)
λmaxaP+bQ≤aλmaxP+bλmaxQLemma 3 (see [41, 42]).
For matricesP,Q,R, and S with appropriate dimensions, we have the following properties:(1)
αP⊗Q=P⊗αQ=αP⊗Q,whereαis a constant(2)
P+Q⊗R=P⊗R+Q⊗R(3)
P⊗QR⊗S=PR⊗QS(4)
P⊗QT=PT⊗QT(5)
IfP and Q are symmetric, then P⊗Q is symmetric(6)
For square matricesP and Q, every eigenvalue of P⊗Q arises as a product of eigenvalues of P and QLemma 4 (see [43]).
LetU=αijN×N, M∈ℝn×n, and x=x1T,x2T,…,xNTT, where xi=xi1,xi2,…,xinT∈ℝn and y=y1T,y2T,…,yNTT, where yi=yi1,yi2,…,yinT∈ℝni=1,2,…,N. If  U=UT and each row sum of U is zero, then(1)xTU⊗My=−∑1≤i<j≤Nαijxi−xjTMyi−yj.


### 2.2. Foundational Knowledge on Time Scales

In this section, some foundational definitions and lemmas on time scales are provided. For more details, one can refer to [44, 45].LetT be a time scale; that is, T is a nonempty closed subset of ℝ. For each interval I of ℝ, I∩T is denoted by IT. The properties of T are determined by the following three functions:(1)
The forward jump operatorσt=infs∈T:s>t,t∈T (in this case, we put inf∅=supT, where ∅ denotes the empty set)(2)
The backward jump operatorρt=sups∈T:s<t,t∈T (in this case, we put sup∅=infT, where ∅ denotes the empty set)(3)
The graininess functionμt=σt−t,t∈TFort∈T, if σt>t, we say that t is right-scattered, while if ρt<t, we say that t is left-scattered. Also, if t<supT and σt=t, then t is called right-dense, and if t>infT and ρt=t, then t is called left-dense. The set Tκ is derived from the time scale T as follows: if T has a left-scattered maximum m, then Tκ=T\m. Otherwise, Tκ=T.Definition 1.
Letf:T⟶ℝ. Define the function fσ:T⟶ℝ by fσt=fσtfor allt∈T, i.e., fσ=f°σ.Definition 2.
Assume thatf:T⟶ℝ is a function and let t∈Tκ. Then, f is called differentiable at the point t if there exists a θ∈ℝ such that for any given ε>0, there is an open neighborhood U of t such that(2)fσt−fs−θσt−s≤εσt−s,s∈U.
In this case,θ is called the delta derivative of f at the point t and we denote it by θ=f△t. Moreover, we say that f is delta differentiable (or in short: differentiable) on Tκ provided f△t exists for all t∈Tκ. The function f△:Tκ⟶ℝ is called the (delta) derivative of f on Tκ. If F△t=ft,t∈Tκ, then for any a,b∈T, the integral is defined as follows:(3)∫abft△t=Fb−Fa.Remark 1.
IfT=ℝ, then fσt=ft and f△t=f′t is the usual derivative. If T=ℤ, then fσt=ft+1 is the forward shift and f△t=△ft=ft+1−ft is the usual forward difference.Lemma 5.
Iff,g:T⟶ℝ are differentiable at t∈Tκ, then(4)fg△t=f△tgt+fσtg△t=ftg△t+f△tgσt.Lemma 6.
Iff:T⟶ℝ is differentiable at t∈Tκ, then fσt=ft+μtf△t.Definition 3.
A functionf:T⟶ℝ is called rd-continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T. The set of rd-continuous functions is denoted by CrdT,ℝ.Definition 4.
We say that a functionp:T⟶ℝ is regressive (positively regressive) provided(5)1+μtpt≠0,for allt∈Tκ1+μtpt>0for allt∈T,holds. The set of all regressive (positively regressive) and rd-continuous functions is denoted by ℛT,ℝℛ+T,ℝ.Definition 5.
Ifp∈ℛT,ℝ, then we define the exponential function by(6)ept,s=exp∫stξμτpτΔτ,fors,t∈Twith the cylinder transformation ξhz defined by(7)ξhz=1hLog1+zh,h>0,z,h=0,where Log is the principal logarithm function.Lemma 7.
Lett0∈T,y,f∈CrdT,ℝ, and p∈ℛ+T,ℝ. Then,(8)yΔt≤ptyt+ft,for allt∈T,implies(9)yt≤yt0ept,t0+∫t0tept,στfτ△τ,for allt∈T.Lemma 8 (see [40]).
For fixedt0∈T, if p<0 and p∈ℛ+T,ℝ, then ept,t0⟶0 as t⟶∞,t∈T.Definition 6.
LetA be an m×n-matrix-valued function on T. We say that A is rd-continuous on T if each entry of A is rd-continuous on T. We say that A is differentiable on T provided each entry of A is differentiable on T. In this case, we put A△=aij△1≤i≤m,1≤j≤n, where A=aij1≤i≤m,1≤j≤n.Lemma 9.
IfA is differentiable at t∈Tκ, then Aσt=At+μtA△t.Lemma 10.
SupposeA and B are differentiable n×n-matrix-valued functions on T. Then,(1)
A+B△=A△+B△(2)
αA△=αA△ifαis constant(3)
AB△=A△B+AσB△=AB△+A△BσDefinition 7.
Ann×n-matrix-valued function A on T is called regressive (with respect to T) provided In+μtAt is invertible for all t∈Tκ, and the class of all such regressive and rd-continuous functions is denoted by ℛT,ℝn×n.


## 2.1. Notations and Supporting Lemmas

First, we define some notations as follows:ℕ0 is the set of all nonnegative integersℤ is the set of all integersℝ is the set of all real numbersℝn is the n-dimensional Euclidean space with the Euclidean norm ⋅ℝm×n is the set of all m×n real matricesIn∈ℝn×n is the n-dimensional identity matrixdiagd1,d2,…,dn is the diagonal matrix with diagonal entries d1 to dnThe superscript “T” stands for the transpose of a matrixFor symmetric matrixP∈ℝn×n, λminP and λmaxP denote the minimum and maximum eigenvalues of P, respectivelyFor symmetric matricesP,Q∈ℝn×n, P≥QP≤Q means that P−Q is positive semidefinite (negative semidefinite)⊗ denotes the Kronecker productLemma 1 (see [23]).
SupposeA=aijn×n is a real symmetric and irreducible matrix, in which aij≥0j≠i and aii=−∑j=1,j≠inaij, and nonzero matrix D=diagd1,d2,…,dn satisfies di≥01≤i≤n. Let B=A−D. Then, all the eigenvalues of B are less than 0.Lemma 2 (see [41]).
IfP,Q∈ℝn×n are symmetric, x∈ℝn is a nonzero vector, and 0<a,b∈ℝ, then(1)
P+Q is symmetric(2)
Pk is symmetric for k=1,2,…,(3)
P is positive definite ⇔−P is negative definite(4)
P is positive definite ⇔ there exists a positive definite matrix S∈ℝn×n such that P=S2(5)
λminP≤xTPx/xTx≤λmaxP(6)
λmaxaP+bQ≤aλmaxP+bλmaxQLemma 3 (see [41, 42]).
For matricesP,Q,R, and S with appropriate dimensions, we have the following properties:(1)
αP⊗Q=P⊗αQ=αP⊗Q,whereαis a constant(2)
P+Q⊗R=P⊗R+Q⊗R(3)
P⊗QR⊗S=PR⊗QS(4)
P⊗QT=PT⊗QT(5)
IfP and Q are symmetric, then P⊗Q is symmetric(6)
For square matricesP and Q, every eigenvalue of P⊗Q arises as a product of eigenvalues of P and QLemma 4 (see [43]).
LetU=αijN×N, M∈ℝn×n, and x=x1T,x2T,…,xNTT, where xi=xi1,xi2,…,xinT∈ℝn and y=y1T,y2T,…,yNTT, where yi=yi1,yi2,…,yinT∈ℝni=1,2,…,N. If  U=UT and each row sum of U is zero, then(1)xTU⊗My=−∑1≤i<j≤Nαijxi−xjTMyi−yj.


## 2.2. Foundational Knowledge on Time Scales

In this section, some foundational definitions and lemmas on time scales are provided. For more details, one can refer to [44, 45].LetT be a time scale; that is, T is a nonempty closed subset of ℝ. For each interval I of ℝ, I∩T is denoted by IT. The properties of T are determined by the following three functions:(1)
The forward jump operatorσt=infs∈T:s>t,t∈T (in this case, we put inf∅=supT, where ∅ denotes the empty set)(2)
The backward jump operatorρt=sups∈T:s<t,t∈T (in this case, we put sup∅=infT, where ∅ denotes the empty set)(3)
The graininess functionμt=σt−t,t∈TFort∈T, if σt>t, we say that t is right-scattered, while if ρt<t, we say that t is left-scattered. Also, if t<supT and σt=t, then t is called right-dense, and if t>infT and ρt=t, then t is called left-dense. The set Tκ is derived from the time scale T as follows: if T has a left-scattered maximum m, then Tκ=T\m. Otherwise, Tκ=T.Definition 1.
Letf:T⟶ℝ. Define the function fσ:T⟶ℝ by fσt=fσtfor allt∈T, i.e., fσ=f°σ.Definition 2.
Assume thatf:T⟶ℝ is a function and let t∈Tκ. Then, f is called differentiable at the point t if there exists a θ∈ℝ such that for any given ε>0, there is an open neighborhood U of t such that(2)fσt−fs−θσt−s≤εσt−s,s∈U.
In this case,θ is called the delta derivative of f at the point t and we denote it by θ=f△t. Moreover, we say that f is delta differentiable (or in short: differentiable) on Tκ provided f△t exists for all t∈Tκ. The function f△:Tκ⟶ℝ is called the (delta) derivative of f on Tκ. If F△t=ft,t∈Tκ, then for any a,b∈T, the integral is defined as follows:(3)∫abft△t=Fb−Fa.Remark 1.
IfT=ℝ, then fσt=ft and f△t=f′t is the usual derivative. If T=ℤ, then fσt=ft+1 is the forward shift and f△t=△ft=ft+1−ft is the usual forward difference.Lemma 5.
Iff,g:T⟶ℝ are differentiable at t∈Tκ, then(4)fg△t=f△tgt+fσtg△t=ftg△t+f△tgσt.Lemma 6.
Iff:T⟶ℝ is differentiable at t∈Tκ, then fσt=ft+μtf△t.Definition 3.
A functionf:T⟶ℝ is called rd-continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T. The set of rd-continuous functions is denoted by CrdT,ℝ.Definition 4.
We say that a functionp:T⟶ℝ is regressive (positively regressive) provided(5)1+μtpt≠0,for allt∈Tκ1+μtpt>0for allt∈T,holds. The set of all regressive (positively regressive) and rd-continuous functions is denoted by ℛT,ℝℛ+T,ℝ.Definition 5.
Ifp∈ℛT,ℝ, then we define the exponential function by(6)ept,s=exp∫stξμτpτΔτ,fors,t∈Twith the cylinder transformation ξhz defined by(7)ξhz=1hLog1+zh,h>0,z,h=0,where Log is the principal logarithm function.Lemma 7.
Lett0∈T,y,f∈CrdT,ℝ, and p∈ℛ+T,ℝ. Then,(8)yΔt≤ptyt+ft,for allt∈T,implies(9)yt≤yt0ept,t0+∫t0tept,στfτ△τ,for allt∈T.Lemma 8 (see [40]).
For fixedt0∈T, if p<0 and p∈ℛ+T,ℝ, then ept,t0⟶0 as t⟶∞,t∈T.Definition 6.
LetA be an m×n-matrix-valued function on T. We say that A is rd-continuous on T if each entry of A is rd-continuous on T. We say that A is differentiable on T provided each entry of A is differentiable on T. In this case, we put A△=aij△1≤i≤m,1≤j≤n, where A=aij1≤i≤m,1≤j≤n.Lemma 9.
IfA is differentiable at t∈Tκ, then Aσt=At+μtA△t.Lemma 10.
SupposeA and B are differentiable n×n-matrix-valued functions on T. Then,(1)
A+B△=A△+B△(2)
αA△=αA△ifαis constant(3)
AB△=A△B+AσB△=AB△+A△BσDefinition 7.
Ann×n-matrix-valued function A on T is called regressive (with respect to T) provided In+μtAt is invertible for all t∈Tκ, and the class of all such regressive and rd-continuous functions is denoted by ℛT,ℝn×n.


## 3. Problem Formulations

In the remainder of this paper, we always assume thatT is a time scale with 0∈T and supT=∞.Suppose that a complex dynamical network onT consists of N identical nodes, with each node being an n-dimensional dynamical system. This complex dynamical network can be described as(10)xi△t=fxit+∑j=1,j≠iNcijgijΓxjt−xit,t∈0,∞T,i=1,2,…,N,where xit=xi1t,xi2t,…,xintT∈ℝn is the state vector of the ith node at time t, xi△ is the delta derivative of xi on 0,∞T, f:ℝn⟶ℝn is a vector function, the constant cij>01≤i,j≤N,i≠j represents the coupling strength between node i and node j, Γ∈ℝn×n is the inner coupling matrix, and the coupling configuration matrix G=gijN×N represents the topological structure of the complex dynamical network and is defined as follows: if there exists a connection between node i and node ji≠j, then gij=gji=1; otherwise, gij=gji=0; the diagonal elements of G are defined as(11)gii=−∑j=1,j≠iNgij=−∑j=1,j≠iNgji,i=1,2,…,N,and cii1≤i≤N satisfies(12)ciigii+∑j=1,j≠iNcijgij=ciigii+∑j=1,j≠iNcjigji=0,i=1,2,…,N.Then, network (10) can be equivalently written in the following form:(13)xi△t=fxit+∑j=1NcijgijΓxjt,t∈0,∞T,i=1,2,…,N.In what follows, we always assume that network (13) is connected in the sense of having no isolated clusters, which means that the symmetric matrix G is irreducible, and the following condition is satisfied:(14)ciigiiΓ∈ℛT,ℝn×n,i=1,2,…,N.Now, our goal is to control network (13) onto a solution of the uncoupled system(15)s△t=fst,t∈0,∞T,that is,(16)limt⟶∞xit−st=0,i=1,2,…,N.Here, we assume thatf is continuous and of such a nature that existence and uniqueness of solutions to dynamic equation (15) subject to s0=s0 as well as their dependence on initial values is guaranteed.To achieve the goal, we apply the pinning control strategy on a fraction of the nodes in network (13). Without loss of generality, let the first l nodes be selected to be pinned. The pinning controlled network can be described as follows:(17)xi△t=fxit+∑j=1NcijgijΓxjt+uit,t∈0,∞T,i=1,2,…,l,xi△t=fxit+∑j=1NcijgijΓxjt,t∈0,∞T,i=l+1,l+2,…,N,where uit=−ciidiΓxit−st,t∈0,∞T, di>0, and ciigii−diΓ∈ℛT;ℝn×n,i=1,2,…,l.Remark 2.
IfT=ℝ, then network (13) is reduced to the continuous-time network:(18)x˙it=fxit+∑j=1NcijgijΓxjt,t∈0,∞,i=1,2,…,N,and pinning controlled network (17) can be described by(19)x˙it=fxit+∑j=1NcijgijΓxjt+uit,t∈0,∞,i=1,2,…,l,x˙it=fxit+∑j=1NcijgijΓxjt,t∈0,∞,i=l+1,l+2,…,N,where uit=−ciidiΓxit−st,t∈0,∞, di>0,i=1,2,…,l, and st is a solution of the system(20)s˙t=fst,t∈0,∞.Remark 3.
IfT=ℤ, then network (13) is reduced to the discrete-time network:(21)Δxit=fxit+∑j=1NcijgijΓxjt,t∈ℕ0,i=1,2,…,N,and pinning controlled network (17) can be described by(22)Δxit=fxit+∑j=1NcijgijΓxjt+uit,t∈ℕ0,i=1,2,…,l,Δxit=fxit+∑j=1NcijgijΓxjt,t∈ℕ0,i=l+1,l+2,…,N,where uit=−ciidiΓxit−st,t∈ℕ0, di>0,i=1,2,…,l, and st is a solution of the system(23)Δst=fst,t∈ℕ0.
Letxt=x1Tt,x2Tt,…,xNTtT∈ℝnN, Fxt=fTx1t,fTx2t,…,fTxNtT∈ℝnN, St=sTt,sTt,…,sTt︸NT∈ℝnN, A=cijgij∈ℝN×N, and D=diagc11d1,c22d2,…,clldl,0,…,0︸N−l. Then, we can write pinning controlled network (17) as(24)x△t=Fxt+A⊗Γxt−D⊗Γxt+D⊗ΓSt=Fxt+A−D⊗Γxt+D⊗ΓSt,t∈0,∞T,and obtain the following error dynamical network:(25)z△t=Fxt−FSt+A−D⊗Γzt,t∈0,∞T,where zt=z1Tt,z2Tt,…,zNTtT∈ℝnN, zit=xit−st∈ℝni=1,2,…,N, and FSt=fTst,fTst,…,fTst︸NT∈ℝnN.
By [17], we know that A is a symmetric and irreducible matrix. So, it follows from Lemmas 1 and 2 that λmaxA−D<0 and A−D2 is symmetric positive definite.


## 4. Pinning Synchronization Criteria for Complex Dynamical Networks on Time Scales

To derive the main results, first, we introduce a definition.Definition 8.
(see [2, 17, 46]). A function ϕ:ℝn⟶ℝn is said to be increasing if(26)x−yTϕx−ϕy≥0,for allx,y∈ℝn.
Throughout this section, we always assume that the functionf:ℝn⟶ℝn satisfies Lipschitz condition; that is, there exists a constant L>0 such that fx−fy≤Lx−y holds for any x,y∈ℝn.Theorem 1.
Suppose that there exists a constantμ∗≥0 such that μt≤μ∗ for all t∈T, Γ is symmetric and Γf⋅ is increasing. Then, the pinning controlled network (17) is synchronized, if there exists a constant function α∈ℛ+T,−∞,0 such that(27)2LInN+2A−D⊗Γ+μ∗L2InN+A−D2⊗Γ2≤αInNholds.Proof.
Construct the Lyapunov functionVt=zTtzt,t∈0,∞T. In view of Lemmas 3, 5, 6, 9, and 10, we can obtain the Δ-derivative of Vt along the trajectory (25):(28)V△t=zTt△zt+zTσtz△t=zTt△zt+zTt+μtzTt△z△t=2zTtz△t+μtz△tTz△t≤2zTtz△t+μ∗z△tTz△t=2zTtFxt−FSt+A−D⊗Γzt+μ∗Fxt−FStT+A−D⊗ΓztTFxt−FSt+A−D⊗Γzt=V1t+V2t+μ∗V3t+V4t+V5t,t∈0,∞T,where(29)V1t=2zTtFxt−FSt,V2t=2zTtA−D⊗Γzt,V3t=Fxt−FStTFxt−FSt,V4t=2zTtA−D⊗ΓFxt−FSt,V5t=zTtA−D2⊗Γ2zt.
On the one hand, sincef satisfies Lipschitz condition, we get(30)V1t=2zTtFxt−FSt=2∑i=1NziTtfxit−fst≤2∑i=1NziTtfxit−fst=2∑i=1Nxit−stTfxit−fst≤2L∑i=1Nxit−st2=2L∑i=1Nxit−stTxit−st=2LzTtzt,t∈0,∞T,(31)V3t=Fxt−FStTFxt−Fst=∑i=1Nfxit−fstTfxit−fst=∑i=1Nfxit−fst2≤L2∑i=1Nxit−st2=L2∑i=1Nxit−stTxit−st=L2zTtzt,t∈0,∞T.
On the other hand, sinceΓf⋅ is increasing, by Lemma 4, we have(32)V4t=2zTtA−D⊗ΓFxt−FSt=2zTtA⊗ΓFxt−FSt−zTtD⊗ΓFxt−FSt=−2∑1≤i<j≤Ncijgijzit−zjtTΓfxit−fst−fxjt−fst+∑i=1lziTtciidiΓfxit−fst=−2∑1≤i<j≤Ncijgijxit−xjtTΓfxit−fxjt+∑i=1lciidixit−stΓfxit−fst=−2∑1≤i<j≤Ncijgijxit−xjtTΓfxit−Γfxjt+∑i=1lciidixit−stTΓfxit−Γfst,≤0,t∈0,∞T.
So, it follows from (28), (30)–(32), and condition (27) that(33)V△t≤2LzTtzt+2zTtA−D⊗Γzt+μ∗L2zTtzt+zTtA−D2⊗Γ2zt=zTt2LInN+2A−D⊗Γ+μ∗L2InN+A−D2⊗Γ2zt≤αVt,t∈0,∞T,which together with Lemma 7 implies that(34)Vt≤V0eαt,0,t∈0,∞T.
At the same time, it follows fromα∈ℛ+T,−∞,0 and Lemma 8 that(35)limt⟶∞eαt,0=0.
In view of (34) and (35), we know that Vt⟶0 as t⟶∞. This indicates that pinning controlled network (17) is synchronized.Theorem 2.
Suppose that there exists a constantμ∗≥0 such that μt≤μ∗ for all t∈T, Γ is symmetric, and Γf⋅ is increasing. Then, pinning controlled network (17) is synchronized, if(36)β≔λmax2LInN+2A−D⊗Γ+μ∗L2InN+A−D2⊗Γ2<0,andβ∈ℛ+T,ℝ.Proof.
Construct the Lyapunov functionVt=zTtzt,t∈0,∞T. Similar to the proof of Theorem 1, we have(37)V△t≤zTt2LInN+2A−D⊗Γ+μ∗L2InN+A−D2⊗Γ2zt,t∈0,∞T.
So, by (37) and Lemma 2, we get(38)V△t≤λmax2LInN+2A−D⊗Γ+μ∗L2InN+A−D2⊗Γ2zTtzt=βzTtzt=βVt,t∈0,∞T,which together with Lemma 7 implies that(39)Vt≤V0eβt,0,t∈0,∞T.
At the same time, it follows fromβ<0, β∈ℛ+T,ℝ, and Lemma 8 that(40)limt⟶∞eβt,0=0.
In view of (39) and (40), we know that Vt⟶0 as t⟶∞. This completes the proof.Corollary 1.
Suppose that there exists a constantμ∗≥0 such that μt≤μ∗ for all t∈T, Γ is symmetric positive definite, and Γf⋅ is increasing. Then, pinning controlled network (17) is synchronized, if(41)γ≔2L+2λmaxA−DλminΓ+μ∗L2+λmaxA−D2λmaxΓ2<0,andγ∈ℛ+T,ℝ.Proof.
SinceA−D is symmetric negative definite, Γ, Γ2, and A−D2 are symmetric positive definite, and by Lemmas 2 and 3, we get(42)λmax2LInN+2A−D⊗Γ+μ∗L2InN+A−D2⊗Γ2≤2L+2λmaxA−DλminΓ+μ∗L2+λmaxA−D2λmaxΓ2=γ.
Similar to the proof of Theorem2, we can prove that pinning controlled network (17) is synchronized.Corollary 2.
Suppose that there exists a constantμ∗≥0 such that μt≤μ∗ for all t∈T, cij=c, Γ=In, and f⋅ is increasing. Then, pinning controlled network (17) is synchronized, if(43)ξ≔2L+2cλmaxG−D1+μ∗L2+c2λmaxG−D12<0,andξ∈ℛ+T,ℝ,where D1=diagd1,d2,…,dl,0,…,0︸N−l.WhenT=ℝ, Theorem 2 yields the following result immediately.Corollary 3.
LetΓ be symmetric. Then, pinning controlled network (19) is synchronized if(44)λmaxLInN+A−D⊗Γ<0.Corollary 4.
Letcij=c and Γ be symmetric positive definite. Then, pinning controlled network (19) is synchronized if(45)c>−LλmaxG−D1λminΓ,where D1=diagd1,d2,…,dl,0,…,0︸N−l.Proof.
In view of Lemma1, it is easy to know that λmaxG−D1<0. Now, the result follows from Lemmas 2, 3, and Corollary 3.
WhenT=ℤ, Theorem 2 yields the following result immediately.Corollary 5.
Suppose thatΓ is symmetric and Γf⋅ is increasing. Then, pinning controlled network (22) is synchronized if(46)−1<λmax2L+L2InN+2A−D⊗Γ+A−D2⊗Γ2<0.Corollary 6.
Suppose thatcij=c, Γ is symmetric positive definite, and Γf⋅ is increasing. Then, pinning controlled network (22) is synchronized if(47)−1<2L+L2+2cλmaxG−D1λminΓ+c2λmaxG−D12λmaxΓ2<0,where D1=diagd1,d2,…,dl,0,…,0︸N−l.Remark 4.
By comparingCorollaries 4 and 6, we find that the pinning synchronization criteria for discrete-time complex dynamical networks are different from those for continuous-time complex dynamical networks. For example, to achieve pinning synchronization, in the discrete-time case, the coupling strength needs to have an upper bound, while there is no requirement of upper bound for the coupling strength in the continuous-time case.
Obviously, the research of the synchronization problem for complex dynamical networks on time scales is more general. On the one hand, it provides a unified framework for continuous-time and discrete-time complex dynamical networks. On the other hand, it can give us a better insight into the differences of the pinning synchronization between continuous-time and discrete-time complex dynamical networks.


## 5. A Numerical Example

To verify the effectiveness of the results established in Section4, we give a numerical example in this section.Example 1.
Consider the following complex dynamical network with ten nodes onT:(48)xi△t=xi1Δtxi2Δt=0.1tanhxi1t0.1tanhxi2t+∑j=110cijgijxjt,t∈0,∞T,i=1,2,…,10,where(49)G=−41100101001−40000111010−40011100000−31010010001−30010110100−30001011100−51101110101−61001000011−30000111000−3.
Note that each isolated node of network (48) is a system described by(50)s△t=s1Δts2Δt=0.1tanhs1t0.1tanhs2t,t∈0,∞T.
Obviously,G is a symmetric and irreducible matrix. Since fuv=0.1tanhu0.1tanhv and Γ=I2, it is easy to know that f⋅ satisfies Lipschitz condition with L=0.1 and Γf⋅=f⋅ is increasing. Our objective is to synchronize network (48) onto the solution s=0,0T of system (50) by applying pinning control strategy. For convenience, let cij=c=0.08 in this example. Now, we consider the following three cases.Case 1.
LetT=∪k=0∞k,k+0.6.
From Figure1, we find that network (48) cannot synchronize onto s=0,0T without control.
In this case, since(51)μt=0,t∈∪k=0∞k,k+0.6,0.4,t=k+0.6,k∈ℕ0,it is easy to verify that ciigiiΓ=0.08giiI2∈ℛT,ℝ2×2. Now, we apply pinning control to network (48) with l=5 and feedback gain matrix D=0.08diag5,7,5,2,8,0,0,0,0,0. By direct calculations, we know that ciigii−diΓ=0.08gii−diI2∈ℛT,ℝ2×2,i=1,2,…,5,(52)β≔λmax2LInN+2A−D⊗Γ+μ∗L2InN+A−D2⊗Γ2=λmax2×0.1I20+20.08G−D⊗I2+0.40.12I20+0.08G−D2⊗I2=−0.0023<0and β∈ℛ+T,ℝ. So, all the conditions of Theorem 2 are fulfilled. Hence, it follows from Theorem 2 that network (48) can realize pinning synchronization. In fact, Figure 2 also shows that the pinning synchronization is achieved.Figure 1
Errorszi1 and zi2 of network (48) without control, cij=c=0.08, T=∪k=0∞k,k+0.6.Figure 2
Synchronization errorszi1 and zi2 of network (48), cij=c=0.08, D=0.08diag5,7,5,2,8,0,0,0,0,0, T=∪k=0∞k,k+0.6.Case 2.
LetT=ℝ.
From Figure3, we find that network (48) cannot synchronize onto s=0,0T without control.
In this case, sinceμt≡0, it is easy to verify that ciigiiΓ=0.08giiI2∈ℛT,ℝ2×2,i=1,2,…,10. Now, we apply pinning control to network (48) with l=6 and feedback gain matrix D=0.08diag6,8,5,3,8,9,0,0,0,0. By direct calculations, we know that ciigii−diΓ=0.08gii−diI2∈ℛT,ℝ2×2,i=1,2,…,6, and λmaxLInN+A−D⊗Γ=λmax0.1I20+0.08G−D⊗I2=−0.0299<0. So, all the conditions of Corollary 3 are fulfilled. Hence, it follows from Corollary 3 that network (48) can realize pinning synchronization. In fact, Figure 4 also shows that the pinning synchronization is achieved.Figure 3
Errorszi1 and zi2 of network (48) without control, cij=c=0.08, T=ℝ.Figure 4
Synchronization errorszi1 and zi2 of network (48), cij=c=0.08, D=0.08diag6,8,5,3,8,9,0,0,0,0, T=ℝ.Case 3.
LetT=ℤ.
From Figure5, we find that network (48) cannot synchronize onto s=0,0T without control.
In this case, sinceμt≡0, it is easy to verify that ciigiiΓ=0.08giiI2∈ℛT,ℝ2×2,i=1,2,…,10. Now, we apply pinning control to network (48) with l=6 and feedback gain matrix D=0.08diag6,8,5,3,8,9,0,0,0,0. By direct calculations, we know that ciigii−diΓ=0.08gii−diI2∈ℛT,ℝ2×2,i=1,2,…,6, and(53)−1<λmax2L+L2InN+2A−D⊗Γ+A−D2⊗Γ2=λmax2×0.1+0.12I20+20.08G−D⊗I2+0.08G−D2⊗I2=−0.0329<0.
So, all the conditions of Corollary5 are fulfilled. Hence, it follows from Corollary 5 that network (48) can realize pinning synchronization. In fact, Figure 6 also shows that the pinning synchronization is achieved.Figure 5
Errorszi1 and zi2 of network (48) without control, cij=c=0.08, T=ℤ.Figure 6
Synchronization errorszi1 and zi2 of network (48), cij=c=0.08, D=0.08diag6,8,5,3,8,9,0,0,0,0, T=ℤ.Remark 5.
To illustrate Remark4, we choose cij=c=0.165 in network (48). By numerical simulations, we find that network (48) can still realize pinning synchronization when T=ℝ, see Figure 7. However, Figure 8 shows that network (48) cannot realize pinning synchronization when T=ℤ.Figure 7
Synchronization errorszi1 and zi2 of network (48), cij=c=0.165, D=0.165diag6,8,5,3,8,9,0,0,0,0, T=ℝ.Figure 8
Errorszi1 and zi2 of network (48), cij=c=0.165, D=0.165diag6,8,5,3,8,9,0,0,0,0, T=ℤ.


## 6. Conclusions

In this paper, we have investigated the synchronization problem of a complex dynamical network on time scales by pinning control strategy. The pinning synchronization criteria established combine main characteristics of time scales with main parameters of the pinning controlled network (such as the coupling strengths, the coupling configuration matrix, and the pinning feedback gain matrix). Our results have revealed the discrepancies of the pinning synchronization between continuous-time and discrete-time complex dynamical networks. A numerical example has also been given to verify the effectiveness of the theoretical results.



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*Source: 5544063-2021-03-20.xml*