# Pinning Synchronization of Nonlinearly Coupled Complex Dynamical Networks on Time Scales

Authors: Fang-Di Kong
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we study the synchronization problem for nonlinearly coupled complex dynamical networks on time scales. To achieve synchronization for nonlinearly coupled complex dynamical networks on time scales, a pinning control strategy is designed. Some pinning synchronization criteria are established for nonlinearly coupled complex dynamical networks on time scales, which guarantee the whole network can be pinned to some desired state. The model investigated in this paper generalizes the continuous-time and discrete-time nonlinearly coupled complex dynamical networks to a unique and general framework. Moreover, two numerical examples are given for illustration and verification of the obtained results.


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## Body

## 1. Introduction

Complex networks are an important part of our daily life and in nature, due to many systems in the real world which can be modeled by complex dynamical networks, such as the Internet, World Wide Web, and food webs [1]. There have been a lot of researches on complex networks and neural networks (see, for example, [2–8] and references therein). Synchronization of complex dynamical network has been a hot topic in the past decades [9–16]. When considering synchronization problem of complex dynamical network, the controlled synchronization problem of complex dynamical network is significant. In recent years, the pinning synchronization of complex dynamical networks, which means the network to achieve desired synchronization by applying control to a small fraction of network nodes, has become a topic of great interest; see [17–34]. In particular, Liu and Chen [16] investigated the global synchronization for nonlinearly coupled complex networks. Further, they investigated pinning synchronization for continuous-time nonlinearly coupled networks; see [32]. In [33], a pinning control scheme was developed for continuous-time nonlinearly coupled complex dynamical network, while the results were extended to discrete-time case. In [34], the synchronization of continuous-time dynamical networks with nonlinearly coupling function was considered.In real life, the time domains do not always match the known continuous-time intervals or discrete integer time domains. From practical point of view, it is important to study complex dynamic networks on general time domains. This is the starting point of the present investigation. Recently, synchronization of complex dynamical networks on time scales has attracted considerable attention [35–43], which contains not only synchronization of continuous-time and discrete-time complex dynamical networks but also some continuous-time intervals accompanying some discrete moments.Motivated by the aforementioned discussions, the synchronization of nonlinearly coupled complex dynamical networks on time scales by applying pinning control scheme will be investigated. The objective in this paper is driving the whole network to some desired state by pinning control strategy. By investigating pinning controlled networks on time scales, some sufficient conditions are presented to guarantee the realization of pinning synchronization for nonlinearly coupled complex dynamical networks on time scales. The main contributions of this paper are listed as follows:(i)
The model investigated in this paper generalizes the continuous-time and discrete-time nonlinearly coupled complex dynamical networks to a unique and general framework. Therefore, the obtained results include continuous-time and discrete-time nonlinearly coupled complex dynamical networks as special cases. Moreover, the model investigated in this paper is more general. Our results can be applied to investigate pinning synchronization of nonlinearly coupled complex dynamical networks on a mixed time domain(ii)
Linearly coupled complex dynamical networks on time scale are included as a special case of the present workThe rest of this paper is organized as follows. Some foundational knowledge about time scales and some notations and supporting lemmas are simply outlined in Section2. In Section 3, the pinning synchronization problem of nonlinearly coupled complex dynamical networks on time scales is formulated. In Section 4, the main theorems and some corollaries are established. In Section 5, two numerical examples are given to verify the effectiveness of our results. Finally, conclusions are provided in Section 6.


## 2. Preliminaries

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


### 2.1. Foundational Knowledge on Time Scales

Throughout this paper,ℕ, ℤ, and ℝ denote the sets of positive integers, integers, and real numbers, respectively. A time scale is defined as a nonempty closed subset of ℝ and denoted by T.Definition 1 (see [44, 45]).
Lett∈T. Define the forward jump operator σ:T⟶T by σt=infτ∈T:τ>t, while the backward jump operator ρ:T⟶T is defined by ρt=supτ∈T:τ<t. In this definition, we put inf∅=supT and sup∅=infT, where ∅ denotes the empty set. 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 graininess function μ:T⟶0,+∞ is defined by μt≔σt−t. 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 2 (see [44]).
Letf:T⟶ℝ. Define the function fσ:T⟶ℝ by fσt=fσtforallt∈T, i.e., fσ=f∘σ.Definition 3 (see [44, 45]).
Assume thatf:T⟶ℝ and t∈Tκ, then f is called Δ-differentiable at the point t if there exists θ∈ℝ such that for any given ε>0, there is an open neighborhood U of the point t such that
(1)fσt−fs−θσt−s≤εσt−s,s∈U.
In this case,θ is called the Δ-derivative of f at the point t and we denote it by θ=fΔt. Moreover, we say that f is Δ-differentiable (or in short: differentiable) on Tκ provided fΔt exists for all t∈Tκ. The function fΔ:Tκ⟶ℝ is called the Δ-derivative of f on Tκ. If FΔt=ft, t∈Tκ, then for any a,b∈T, the integral is defined as follows:
(2)∫abftΔt=Fb−Fa.Remark 4.
IfT=ℝ, then fσt=ft, fΔt=f′t is the usual derivative, and ∫abftΔt=∫abftdt is the usual integral. If T=ℤ, then fσt=ft+1 is the forward shift, fΔt=Δft=ft+1−ft is the usual forward difference and ∫abftΔt=∑t=ab−1fta<b.Lemma 5 (see [44, 45]).
Iff,g:T⟶ℝ are differentiable at t∈Tκ, then
(3)fgΔt=fΔtgt+fσtgΔt=ftgΔt+fΔtgσt.Lemma 6 (see [44, 45]).
Iff:T⟶ℝ is differentiable at t∈Tκ, then fσt=ft+μtfΔt.Definition 7 (see [44, 45]).
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 these functions is denoted by CrdT;ℝ.Definition 8 (see [44, 45]).
We say that a functionp:T⟶ℝ is regressive (positive regressive) provided
(4)1+μtpt≠01+μtpt>0,forallt∈Tκholds. The set of all regressive (positive regressive) and rd-continuous functions is denoted by RT;ℝR+T;ℝ.Definition 9 (see [44]).
Ifp∈RT;ℝ, then we define the exponential function by
(5)ept,s=exp∫stξμτpτΔτ,foralls,t∈T,where the cylinder transformation ξhz is defined by
(6)ξhz=1hLog1+zh,h>0,z,h=0,where Log is the principal logarithm function.Lemma 10 (see [44]).
Lett0∈T,y,f∈CrdT;ℝ and p∈R+T;ℝ. Then,
(7)yΔt≤ptyt+ft,forallt∈Timplies
(8)yt≤yt0ept,t0+∫t0tept,στfτΔτ,forallt∈T.Lemma 11 (see [43]).
For fixedt0∈T, if p<0 and p∈R+T;ℝ, then ept,t0⟶0 as t⟶+∞,t∈T.
LetA be an m×n-matrix-valued function on T. We say that A is differentiable on T provided each entry of A is differentiable on T. In this case, we put AΔt=aijΔt1≤i≤m,1≤j≤n, where A=aijt1≤i≤m,1≤j≤n.Lemma 12 (see [44]).
If a matrix-valued functionA is differentiable at t∈Tκ, then Aσt=At+μtAΔt.Lemma 13 (see [44]).
Suppose thatA and B are differentiable n×n-matrix-valued functions on T. Then,
(1)
A+BΔ=AΔ+BΔ(2)
αAΔ=αAΔ, where α is a constant(3)
ABΔ=AΔB+AσBΔ=ABΔ+AΔBσ


### 2.2. Notations and Supporting Lemmas

For each intervalI of ℝ, I∩T is denoted by IT. ℝn denotes the n-dimensional Euclidean space with the Euclidean norm ∥·∥. ℝm×n denotes the set of all m×n real matrices. Let IN∈ℝN×N be the N-dimensional identity matrix, diagd1,d2,⋯,dN indicate the diagonal matrix with diagonal entries d1 to dN, AT be the transpose of matrix A, and λmin· and λmax· represent the minimum eigenvalue and the maximum eigenvalue of a real symmetric matrix. For a symmetric matrix P∈ℝn×n, write P>0P<0,P≥0,andP≤0,respectively if P is positive definite (negative definite, positive semidefinite, and negative semidefinite, respectively). For square matrices A and B, the notation A≥BA≤B means that A−B is a positive semidefinite (negative semidefinite) matrix. The symbol ⊗ denotes the Kronecker product.Lemma 14 (see [46]).
For matricesA, B, C, and D with appropriate dimensions, we have the following properties:
(1)
αA⊗B=A⊗αB=αA⊗B, where α is a constant(2)
A+B⊗C=A⊗C+B⊗C(3)
A⊗BT=AT⊗BT(4)
IfA and B are symmetric, then A⊗B is symmetric(5)
For square matricesA and B, every eigenvalue of A⊗B arises as a product of eigenvalues of A and BLemma 15 (see [47]).
LetU=αijN×N, M∈ℝn×n, 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
(9)xTU⊗My=−∑1≤i<j≤Nαijxi−xjTMyi−yj.Lemma 16 (see [26].
SupposeA=aijN×N is a real symmetric and irreducible matrix, in which aij≥0j≠i and aii=−∑j=1,j≠iNaij, 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 17 (see [48]).
IfA,B∈ℝm×m are symmetric, x∈ℝm, x≠0 (m-dimensional zero vector) and 0<a,b∈ℝ, then
(1)
Ak is symmetric for k∈ℕ(2)
A is positive definite ⇔−A is negative definite(3)
A is positive definite ⇔ there exists a positive definite matrix C∈ℝm×m such that A=CC, that is, C is a real square root of A(4)
λminA≤xTAx/xTx≤λmaxA(5)
λmaxaA+bB≤aλmaxA+bλmaxB(6)
ifλ is the eigenvalue of A, then λk is the eigenvalue of Ak, k∈ℕ


## 2.1. Foundational Knowledge on Time Scales

Throughout this paper,ℕ, ℤ, and ℝ denote the sets of positive integers, integers, and real numbers, respectively. A time scale is defined as a nonempty closed subset of ℝ and denoted by T.Definition 1 (see [44, 45]).
Lett∈T. Define the forward jump operator σ:T⟶T by σt=infτ∈T:τ>t, while the backward jump operator ρ:T⟶T is defined by ρt=supτ∈T:τ<t. In this definition, we put inf∅=supT and sup∅=infT, where ∅ denotes the empty set. 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 graininess function μ:T⟶0,+∞ is defined by μt≔σt−t. 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 2 (see [44]).
Letf:T⟶ℝ. Define the function fσ:T⟶ℝ by fσt=fσtforallt∈T, i.e., fσ=f∘σ.Definition 3 (see [44, 45]).
Assume thatf:T⟶ℝ and t∈Tκ, then f is called Δ-differentiable at the point t if there exists θ∈ℝ such that for any given ε>0, there is an open neighborhood U of the point t such that
(1)fσt−fs−θσt−s≤εσt−s,s∈U.
In this case,θ is called the Δ-derivative of f at the point t and we denote it by θ=fΔt. Moreover, we say that f is Δ-differentiable (or in short: differentiable) on Tκ provided fΔt exists for all t∈Tκ. The function fΔ:Tκ⟶ℝ is called the Δ-derivative of f on Tκ. If FΔt=ft, t∈Tκ, then for any a,b∈T, the integral is defined as follows:
(2)∫abftΔt=Fb−Fa.Remark 4.
IfT=ℝ, then fσt=ft, fΔt=f′t is the usual derivative, and ∫abftΔt=∫abftdt is the usual integral. If T=ℤ, then fσt=ft+1 is the forward shift, fΔt=Δft=ft+1−ft is the usual forward difference and ∫abftΔt=∑t=ab−1fta<b.Lemma 5 (see [44, 45]).
Iff,g:T⟶ℝ are differentiable at t∈Tκ, then
(3)fgΔt=fΔtgt+fσtgΔt=ftgΔt+fΔtgσt.Lemma 6 (see [44, 45]).
Iff:T⟶ℝ is differentiable at t∈Tκ, then fσt=ft+μtfΔt.Definition 7 (see [44, 45]).
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 these functions is denoted by CrdT;ℝ.Definition 8 (see [44, 45]).
We say that a functionp:T⟶ℝ is regressive (positive regressive) provided
(4)1+μtpt≠01+μtpt>0,forallt∈Tκholds. The set of all regressive (positive regressive) and rd-continuous functions is denoted by RT;ℝR+T;ℝ.Definition 9 (see [44]).
Ifp∈RT;ℝ, then we define the exponential function by
(5)ept,s=exp∫stξμτpτΔτ,foralls,t∈T,where the cylinder transformation ξhz is defined by
(6)ξhz=1hLog1+zh,h>0,z,h=0,where Log is the principal logarithm function.Lemma 10 (see [44]).
Lett0∈T,y,f∈CrdT;ℝ and p∈R+T;ℝ. Then,
(7)yΔt≤ptyt+ft,forallt∈Timplies
(8)yt≤yt0ept,t0+∫t0tept,στfτΔτ,forallt∈T.Lemma 11 (see [43]).
For fixedt0∈T, if p<0 and p∈R+T;ℝ, then ept,t0⟶0 as t⟶+∞,t∈T.
LetA be an m×n-matrix-valued function on T. We say that A is differentiable on T provided each entry of A is differentiable on T. In this case, we put AΔt=aijΔt1≤i≤m,1≤j≤n, where A=aijt1≤i≤m,1≤j≤n.Lemma 12 (see [44]).
If a matrix-valued functionA is differentiable at t∈Tκ, then Aσt=At+μtAΔt.Lemma 13 (see [44]).
Suppose thatA and B are differentiable n×n-matrix-valued functions on T. Then,
(1)
A+BΔ=AΔ+BΔ(2)
αAΔ=αAΔ, where α is a constant(3)
ABΔ=AΔB+AσBΔ=ABΔ+AΔBσ


## 2.2. Notations and Supporting Lemmas

For each intervalI of ℝ, I∩T is denoted by IT. ℝn denotes the n-dimensional Euclidean space with the Euclidean norm ∥·∥. ℝm×n denotes the set of all m×n real matrices. Let IN∈ℝN×N be the N-dimensional identity matrix, diagd1,d2,⋯,dN indicate the diagonal matrix with diagonal entries d1 to dN, AT be the transpose of matrix A, and λmin· and λmax· represent the minimum eigenvalue and the maximum eigenvalue of a real symmetric matrix. For a symmetric matrix P∈ℝn×n, write P>0P<0,P≥0,andP≤0,respectively if P is positive definite (negative definite, positive semidefinite, and negative semidefinite, respectively). For square matrices A and B, the notation A≥BA≤B means that A−B is a positive semidefinite (negative semidefinite) matrix. The symbol ⊗ denotes the Kronecker product.Lemma 14 (see [46]).
For matricesA, B, C, and D with appropriate dimensions, we have the following properties:
(1)
αA⊗B=A⊗αB=αA⊗B, where α is a constant(2)
A+B⊗C=A⊗C+B⊗C(3)
A⊗BT=AT⊗BT(4)
IfA and B are symmetric, then A⊗B is symmetric(5)
For square matricesA and B, every eigenvalue of A⊗B arises as a product of eigenvalues of A and BLemma 15 (see [47]).
LetU=αijN×N, M∈ℝn×n, 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
(9)xTU⊗My=−∑1≤i<j≤Nαijxi−xjTMyi−yj.Lemma 16 (see [26].
SupposeA=aijN×N is a real symmetric and irreducible matrix, in which aij≥0j≠i and aii=−∑j=1,j≠iNaij, 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 17 (see [48]).
IfA,B∈ℝm×m are symmetric, x∈ℝm, x≠0 (m-dimensional zero vector) and 0<a,b∈ℝ, then
(1)
Ak is symmetric for k∈ℕ(2)
A is positive definite ⇔−A is negative definite(3)
A is positive definite ⇔ there exists a positive definite matrix C∈ℝm×m such that A=CC, that is, C is a real square root of A(4)
λminA≤xTAx/xTx≤λmaxA(5)
λmaxaA+bB≤aλmaxA+bλmaxB(6)
ifλ is the eigenvalue of A, then λk is the eigenvalue of Ak, k∈ℕ


## 3. Problem Formulations

Throughout the rest of the paper, letT be a time scale with 0∈T and supT=+∞. In this section, the nonlinearly coupled complex dynamical network on the time scale T will be introduced.In general, the dynamic for each isolated (uncoupled) node of the dynamical network can be described as(10)sΔt=fst,t∈0,+∞T,where st=s1t,s2t,⋯,sntT∈ℝn, sΔ is the Δ-derivative of s on 0,+∞T, f:ℝn⟶ℝn is continuous and of such a nature that existence and uniqueness of solutions to dynamic equation (10) subject to s0=s0s0∈ℝn as well as their dependence on initial values is guaranteed.Suppose that the dynamical network consists ofN identical nodes, with each node being an n-dimensional dynamical system. Then, the nonlinearly coupled dynamical network can be described by
(11)xiΔt=fxit+c∑j=1Ngijhxjt,t∈0,+∞T,i=1,2,⋯,N,where xit=xi1t,xi2t,⋯,xintT∈ℝn is the state vector of the ith node at time t; the constant c>0 represents the coupling strength of network; the nonlinearly coupled function h:ℝn⟶ℝn is continuous, which satisfies standard assumptions on existence and uniqueness of solutions to dynamic equation (11) subject to x0=x0x0∈ℝn as well as their dependence on initial values; the coupling configuration matrix G=gij∈ℝN×N represents the topological structure of the complex network and is defined as follows: if there exists a connection between the node i and the nodej (i≠j), then gij=gji=1; otherwise, gij=gji=0, and gii=−∑j=1,j≠iNgij=−∑j=1,j≠iNgji,i=1,2,⋯,N. Suppose that network (11) is connected in the sense of having no isolated clusters, which means that the coupling configuration matrix G is irreducible.Suppose thatst is a solution of the uncoupled system (10). In order to synchronize the network (11) to the objective state st, we will design a pinning control scheme, if the network (11) cannot synchronize to the objective state st without control. Without loss of generality, we add the controllers to the first l nodes. Hence, we have the pinning-controlled network as follows:
(12)xiΔt=fxit+c∑j=1Ngijhxjt+uit,t∈0,+∞T,i=1,2,⋯,Nwith the feedback controllers given by
(13)uit=−cdihxit−hst,t∈0,+∞T,i=1,2,⋯,N,where di=d>0,i=1,2,⋯,l,,di=0,i=l+1,l+2,⋯,N. Define the matrix: D=diagd1,d2,⋯,dl,⋯,dN.Definition 18 (see [30]).
The network (12) is said to be synchronized by pinning control, if
(14)limt→∞xit−st=0,i=1,2,⋯,N.
We get the following error dynamical network by lettingzit=xit−st∈ℝni=1,2,⋯,N:
(15)ziΔt=fxit−fst+c∑j=1Ngijhxjt−hst−cdihxit−hst,t∈0,+∞T,i=1,2,⋯,N.It follows from Lemma16 that the symmetric matrix G−D is negative definite, and so the maximal eigenvalue λmaxG−D<0. From Lemma 17((1)–(3)), it is easy to prove that the matrix G−D2 is symmetric positive definite.


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

In order to derive the synchronization criteria for the pinning-controlled network (12), we make the following assumptions:(A1) (see [18]). The function f:ℝn⟶ℝn is assumed to satisfy Lipschitz condition, that is, there exists a constant l1>0 such that
(16)fx−fy≤l1x−yholdsforanyx,y∈ℝn.(A2) (see [9, 21, 49]). There exists a constant l2>0, such that
(17)x−yThx−hy≥l2x−yTx−y,foranyx,y∈ℝn.(A3). There exists a constant matrixK∈ℝn×n, such that
(18)fx−fyThx−hy≥x−yTKx−y,foranyx,y∈ℝn.(A4) (see [18]). The function h:ℝn⟶ℝn is assumed to satisfy Lipschitz condition, that is, there exists a constant l3>0 such that
(19)hx−hy≤l3x−yholdsforanyx,y∈ℝn.We have the following theorems and corollaries, which give some sufficient conditions to guarantee synchronization of the network (12) by pinning control.Theorem 19.
Suppose that assumptions (A1), (A2), (A3), and (A4) hold. If there existsθ:T⟶ℝ with θ∈R+T;ℝ and limt→∞eθt,0=0 such that the following condition is satisfied:
(20)2l1InN+2cl2G−D⊗In+μtl12InN+2cG−D⊗K+c2l32λmaxG−D2InN≤θtInN,t∈0,+∞T,then, the network (12) is synchronized by pinning control.Proof. Consider the Lyapunov functionVt=∑i=1NziTtzit,t∈0,+∞T. By Lemmas 5, 6, 12, and 13, calculating the Δ-derivative of Vt along the trajectories of the error dynamical network (15), one has
(21)VΔt=∑i=1NziTΔtzit+ziTσtziΔt=∑i=1NziTΔtzit+ziTt+μtziTΔtziΔt=∑i=1N2ziTtziΔt+μtziΔtTziΔt=∑i=1N2ziTtfxit−fst+c∑j=1Ngijhxjt−hst−cdihxit−hst+μtfxit−fstT+c∑j=1Ngijhxjt−hstT−cdihxit−hstTfxit−fst+c∑j=1Ngijhxjt−hst−cdihxit−hst=∑i=1N2ziTtfxit−fst+2cziTt∑j=1Ngijhxjt−hst−2cdiziTthxit−hst+μtfxit−fstTfxit−fst+cfxit−fstT∑j=1Ngijhxjt−hst−cdifxit−fstThxit−hst+c∑j=1Ngijhxjt−hstTfxit−fst+c2∑j=1Ngijhxjt−hstT∑j=1Ngijhxjt−hst−c2di∑j=1Ngijhxjt−hstThxit−hst−cdihxit−hstTfxit−fst−c2dihxit−hstT∑j=1Ngijhxjt−hst+c2di2hxit−hstThxit−hst≤∑i=1N2zitfxit−fst+2c∑j=1NgijziTthxjt−hst−2cdiziTthxit−hst+μtfxit−fst2+c∑j=1Ngijfxit−fstThxjt−hst−cdifxit−fstThxit−hst+c∑j=1Ngijhxjt−hstTfxit−fst+c2∑j=1Ngjihxjt−hstT∑j=1Ngijhxjt−hst−c2di∑j=1Ngijhxjt−hstThxit−hst−cdihxit−hstTfxit−fst−c2dihxit−hstT∑j=1Ngijhxjt−hst+c2di2hxit−hstThxit−hst.By assumption (A1), the following inequalities can be obtained:(22)VΔt≤∑i=1N2l1∥zit∥2+V1t+V2t+μt∑i=1Nl12∥zit∥2+V3t+V4t+V5t+V6t+V7t+V8t,where V1t=2c∑i=1N∑j=1NgijziTthxjt−hst, V2t=−2c∑i=1NdiziTthxit−hst, V3t=2c∑i=1N∑j=1Ngijfxit−fstThxjt−hst, V4t=−2c∑i=1Ndifxit−fstThxit−hst, V5t=c2∑i=1N∑j=1Ngjihxjt−hstT∑j=1Ngijhxjt−hst, V6t=−c2∑i=1Ndi∑j=1Ngijhxjt−hstThxit−hst, V7t=−c2∑i=1Ndihxit−hstT∑j=1Ngijhxjt−hst, and V8t=c2∑i=1Ndi2hxit−hstThxit−hst.We introduce the notations(23)zt=z1Tt,z2Tt,⋯,zNTtT,Hzt=hx1t−hstT,hx2t−hstT,⋯,hxNt−hstTT,Fzt=fx1t−fstT,fx2t−fstT,⋯,fxNt−fstTT.By assumption (A2) and Lemma15, we have
(24)V1t=2c∑i=1N∑j=1NziTtgijhxjt−hst=2czTtG⊗InHzt=−2c∑1≤i<j≤Ngijzit−zjtThxit−hst−hxjt−hst=−2c∑1≤i<j≤Ngijxit−xjtThxit−hxjt≤−2cl2∑1≤i<j≤Ngijxit−xjtTxit−xjt=−2cl2∑1≤i<j≤Ngijzit−zjtTzit−zjt=2cl2zTtG⊗Inzt.By assumption (A2), we get(25)V2t=−2c∑i=1NdiziTthxit−hst≤−2cl2∑i=1NdiziTtzit=−2cl2zTtD⊗Inzt.By assumption (A3) and Lemma15, we have
(26)V3t=2c∑i=1N∑j=1Ngijfxit−fstThxjt−hst=2cFTztG⊗InHzt=−2c∑1≤i<j≤Ngijfxit−fst−fxjt−fstThxit−hst−hxjt−hst=−2c∑1≤i<j≤Ngijfxit−fxjtThxit−hxjt≤−2c∑1≤i<j≤Ngijxit−xjtTKxit−xjt=−2c∑1≤i<j≤Ngijzit−zjtTKzit−zjt=2czTtG⊗Kzt.By assumption (A3), we get(27)V4t=−2c∑i=1Ndifxit−fstThxit−hst≤−2c∑i=1NdiziTtKzit=−2czTtD⊗Kzt.By calculating, one has(28)V5t=c2∑i=1N∑j=1Ngjihxjt−hstT∑j=1Ngijhxjt−hst=c2HTztG2⊗InHzt,V6t=−c2∑i=1Ndi∑j=1Ngjihxjt−hstThxit−hst=−c2HTztGD⊗InHzt,V7t=−c2∑i=1Ndihxit−hstT∑j=1Ngijhxjt−hst=−c2HTztDG⊗InHzt,V8t=c2∑i=1Ndi2hxit−hstThxit−hst=c2HTztD2⊗InHzt.Hence, by Lemmas14((1), (2), (4), (5)) and 17 (4) and assumption (A4), we have
(29)V5t+V6t+V7t+V8t=c2HTztG2−GD−DG+D2⊗InHzt=c2HTztG−D2⊗InHzt≤c2λmaxG−D2HTztHzt=c2λmaxG−D2∑i=1Nhxit−hstThxit−hst=c2λmaxG−D2∑i=1Nhxit−hst2≤c2l32λmaxG−D2∑i=1Nzit2.In view of (22), (24), (25), (26), (27), (29), and Lemma 14((1), (2)), one can get
(30)VΔt≤2l1zTtzt+2cl2zTtG−D⊗Inzt+μtl12zTtzt+2czTtG−D⊗Kzt+c2l32λmaxG−D2zTtzt=zTt2l1InN+2cl2G−D⊗In+μtl12InN+2cG−D⊗K+c2l32λmaxG−D2InNzt.From (20), we have
(31)VΔt≤θtVt,t∈0,+∞T.By Lemma10, it implies that
(32)Vt≤V0eθt,0,t∈0,+∞T.Sinceθ∈R+T;ℝ and limt→∞eθt,0=0, Vt⟶0 as t⟶∞. Thus, the network (12) is synchronized by pinning control.Theorem 20.
Letμt≤μ∗ for all t∈T. Suppose that assumptions (A1), (A2), (A3), and (A4) hold. Then, the network (12) is synchronized by pinning control, if there exists a constant α<0 with α∈R+T;ℝ such that
(33)2l1InN+2cl2G−D⊗In+μ∗l12InN+2cG−D⊗K+c2l32λmaxG−D2InN≤αInNholds.Proof.
Lemma11 yields limt→∞eαt,0=0. Choose the Lyapunov function Vt=∑i=1NziTtzit,t∈0,+∞T. Note that
(34)VΔt=∑i=1N2ziTtziΔt+μtziΔtTziΔt≤∑i=1N2ziTtziΔt+μ∗ziΔtTziΔt.
The rest proof of the theorem is analogy with the proof of Theorem19.Theorem 21.
Letμt≤μ∗ for all t∈T. Suppose that assumptions (A1), (A2), (A3), and (A4) hold, and K is symmetric in assumption (A3). Then, the network (12) is synchronized by pinning control, if λ=λmax2l1InN+2cl2G−D⊗In+μ∗l12InN+2cG−D⊗K+c2l32λmaxG−D2InN<0 and λ∈R+T;ℝ.Proof.
Choose the Lyapunov functionVt=∑i=1NziTtzit,t∈0,+∞T. Taking into account the inequality (34), by analogy with the proof of Theorem 19, one has
(35)VΔt≤zTt2l1InN+2cl2G−D⊗In+μ∗l12InN+2cG−D⊗K+c2l32λmaxG−D2InNzt.
Hence, by Lemma17(4), the following estimate holds:
(36)VΔt≤λmax2l1InN+2cl2G−D⊗In+μ∗l12InN+2cG−D⊗K+c2l32λmaxG−D2InNzTtzt=λzTtzt=λVt,t∈0,+∞T.
By Lemma10, we have Vt≤V0eλt,0,t∈0,+∞T. Since λ<0 and λ∈R+T;ℝ, it follows from Lemma 11 that limt→∞eλt,0=0. Therefore, Vt⟶0 as t⟶∞. This completes the proof.Corollary 22.
Letμt≤μ∗ for all t∈T. Suppose that assumptions (A1), (A2), (A3), and (A4) hold, and K is symmetric in assumption (A3). If inequality 2l1+2cl2λmaxG−D+μ∗l12+2cλmaxG−D⊗K+c2l32λmaxG−D2<0 holds, and 2l1+2cl2λmaxG−D+μ∗l12+2cλmaxG−D⊗K+c2l32λmaxG−D2∈R+T;ℝ, then the network (12) is synchronized by pinning control.Proof.
By Lemmas14((4), (5)) and 17((5)) and Theorem 21, it is not hard to see the result holds when μ∗>0 and μ∗=0.Next, let us consider the particular case when the functionhx is linear: hx=Γx, where Γ∈ℝn×n. The network (11) can be written as
(37)xiΔt=fxit+c∑j=1NgijΓxjt,t∈0,+∞T,i=1,2,⋯,N.We write the network (12) as the following
(38)xiΔt=fxit+c∑j=1NgijΓxjt+uit,t∈0,+∞T,i=1,2,⋯,N,where uit=−cdiΓxit−st,t∈0,+∞T,di=d>0,i=1,2,⋯,l,di=0,i=l+1,l+2,⋯,N, and st is a solution of system: sΔt=fst,t∈0,+∞T. Meanwhile, the assumption (A3) can be described as(A3′). There exists a constant matrix K¯∈ℝn×n, such that
(39)x−yTΓTfx−fy≥x−yTK¯x−y,foranyx,y∈ℝn.Corollary 23.
Letμt≤μ∗ for all t∈T and Γ be symmetric positive definite. Suppose that assumptions (A1) and (A3′) hold. The network (38) is synchronized by pinning control if there exists a constant β<0 with β∈R+T;ℝ such that the following inequality is satisfied:
(40)2l1InN+2cλminΓG−D⊗In+μ∗l12InN+2cG−D⊗K¯+c2λmaxΓ2λmaxG−D2InN≤βInN.Proof.
SinceΓ is symmetric positive definite, one has 0<λminΓ. By Lemma 17(3), Γ2 is also symmetric positive definite. From Lemma 17((4), (6)), it is easy to see that
(41)x−yTΓx−y≥λminΓx−yTx−y,foranyx,y∈ℝn,Γx−y≤λmaxΓx−y,foranyx,y∈ℝn.
That is, assumptions (A2) and (A4) hold, wherel2=λminΓ and l3=λmaxΓ. Now, the result follows from Theorem 20, immediately.By Theorem21 and Corollary 23, we have the following corollary.Corollary 24.
Letμt≤μ∗ for all t∈T and Γ be symmetric positive definite. Suppose that assumptions (A1) and (A3′) hold, and K¯ is symmetric in assumption (A3′). The network (38) is synchronized by pinning control if γ=λmax2l1InN+2cλminΓG−D⊗In+μ∗l12InN+2cG−D⊗K¯+c2λmaxΓ2λmaxG−D2InN<0 and γ∈R+T;ℝ.Obviously, the research of synchronization problem for nonlinearly coupled complex dynamical networks on time scales is more general. It contains continuous-time and discrete-time nonlinearly coupled complex dynamical networks. In addition, it contains linearly coupled complex dynamical networks on time scales.Remark 25.
According to the previous works [12, 26, 31, 35, 39, 40], the synchronization problem of complex dynamical networks with delay on time scales by pinning control strategy can be investigated, but there are some challenges for the effects of time delays and the various forms of time scales.


## 5. Numerical Examples

In this section, two numerical examples are given to verify the theoretical results established above.Example 1.
Consider 2-dimensional nonlinearly coupled complete network with ten nodes on time scaleT, which is described by
(42)xiΔt=fxit+c∑j=110gijhxjt,t∈0,+∞T,i=1,2,⋯,10,where T=⋃k=0∞k,k+0.7,k∈ℕ∪0, in this case the graininess function of T is given by
(43)μt=0,t∈⋃k=0∞k,k+0.7,0.3,t=k+0.7,
xit=xi1t,xi2tT,i=1,2,⋯,10,fxit=f1xi1t,f2xi2tT with fjxijt=0.01tanhxijt+xijt,j=1,2,hxit=h1xi1t,h2xi2tT with hjxijt=0.01tanhxijt+0.1xijt,j=1,2,c=0.2, and the corresponding matrix G is
(44)G=−91111111111−91111111111−91111111111−91111111111−91111111111−91111111111−91111111111−91111111111−91111111111−9.
Note that each isolated node of network (42) is a 2-dimensional nonlinear system described by
(45)sΔt=s1Δts2Δt=0.01tanhs1t+s1t0.01tanhs2t+s2t,t∈0,+∞T.
It is not difficult to obtain thatf· and h· satisfy assumptions (A1), (A2), (A3), and (A4) with l1=0.02, l2=0.1, l3=0.11, and constant matrix
(46)K=0.001000.001.
The objective here is to synchronize the network (42) to the solution s=0 (2-dimensional zero vector) of (45) by pinning control. From Figure 1, we see the complex dynamical network (42) without control cannot reach synchronization with s=0. Now, we apply pinning control to the network (42) with matrix D=diag3,3,3,3,3,3,3,0,0,0. By calculations, all conditions of Theorem 21 are fulfilled. Then by Theorem 21, the network (42) can achieve synchronization under the above pinning control strategy. Figure 2 also shows that the synchronization is realized.Example 2.
Consider 2-dimensional nonlinearly star-shaped network with five nodes on time scaleT, which is described by
(47)xiΔt=fxit+c∑j=15gijhxjt,t∈0,+∞T,i=1,2,⋯,5,where T=0.1ℤ; in this case, the graininess function is μt=0.1, xit=xi1t,xi2tT,i=1,2,⋯,5,fxit=f1xi1t,f2xi2tT with fjxijt=0.01tanhxijt+xijt,j=1,2,hxit=Γxit with Γ=diag0.05,0.05,c=0.6, and the corresponding matrix G is
(48)G=−411111−100010−100100−101000−1.
Note that each isolated node of network (47) is a 2-dimensional nonlinear system described by
(49)sΔt=s1Δts2Δt=0.01tanhs1t+s1t0.01tanhs2t+s2t,t∈0,+∞T.
Γ be symmetric positive definite. It is not difficult to obtain that assumptions (A1) and (A3′) are satisfied with l1=0.02 and constant matrix
(50)K¯=0.0005000.0005.
The objective here is to synchronize the network (47) to the solution s=0 (2-dimensional zero vector) of (49) by pinning control. From Figure 3, we see the complex dynamical network (47) without control cannot reach synchronization with s=0. Now, we apply pinning control to the network (47) with matrix D=diag2.5,2.5,2.5,2.5,0. By calculations, all conditions of Corollary 24 are fulfilled. By Corollary 24, the network (47) can achieve synchronization under the above pinning control strategy. Figure 4 also shows that the synchronization is realized.Figure 1
Errorzi1 and zi2 of the system (42) without control.Figure 2
Synchronization errorzi1 and zi2 of the system (42).Figure 3
Errorzi1 and zi2 of the system (47) without control.Figure 4
Synchronization errorzi1 and zi2 of the system (47).


## 6. Conclusions

In this paper, we have investigated the synchronization problem of nonlinearly coupled complex dynamical networks on time scales by pinning control strategy. Some pinning synchronization criteria have been established which guarantee that the nonlinearly coupled complex dynamical networks on time scales can be pinned to some desired state. Two numerical examples have been given to verify the effectiveness of the obtained results.



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