# Adaptive Finite-Time Synchronization for Complex Dynamical Network with Different Dimensions of Nodes and Time-Varying Outer Coupling Structures

Authors: Lihong Yan; Junmin Li
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, we investigate finite-time synchronization problems of complex dynamical networks with different dimensions of nodes, which contain unknown periodically coupling structures and bounded time-varying delay. Based on finite-time stability theory, the inequality techniques, and the properties of Kronecker production of matrices, some useful finite-time synchronization criteria for complex dynamical network with unknown periodical couplings have been obtained. In addition, with proper adaptive periodical learning law designed, the unknown periodical couplings have been estimated successfully. Finally, some simulation examples are performed to verify the theoretical findings.


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## Body

## 1. Introduction

Complex dynamical networks, which are composed of a large set of nonlinear dynamical systems, wherein the isolated nodes are interdependent and interact with each other, have been applied to scientific research, engineering application and even our daily life [1–3]. The researchers mainly focused on network modeling, collective dynamical behavior analysis and evolution by adopting the knowledge of graph theory, statistical physics, computer simulation methods, and so on. The research of complex network is booming, and many important achievements have been obtained [4–10].Up to now, complex dynamical network is still an active and promising scientific research area, especially synchronization controlling of complex dynamical networks. Many results are mainly focusing on asymptotical or exponential synchronization. However, from the perspective of practical application, finite-time synchronization has greater practical application value; for example, the system convergence time is very important in industrial control. In addition, the existing conclusion shows that the finite-time controller has better robustness and interference suppression performance [9]. Finite-time control methods mainly include sliding mode control [11], optimal control [12], learning control [13, 14], quantized control [15], and pure finite-time control [16, 17]. In recent years, many meaningful advances about finite-time synchronization have been reported in recent years.As one knows, in the process of long-distance information transmission, time delay emerged ubiquitously as for the influence on finite signal transmission and channel bandwidth [10]. In addition, time delays always bring instability or other unpredictable negative effects to the controlled system. Thus, the influence of time delays for synchronization of complex dynamical networks is worthy of consideration. Lu and Chen [8] put forward complete synchronization analyses of networks of the coupled dynamical systems with time-varying couplings based on algebraic graph theory and dynamic system method. By adaptive learning control method, Li et al. [9] presented an intermittent control approach with multiple switched periods for the finite-time synchronization in complex networks with nondelay and time-varying delay couplings. In [12], He et al. presented the guaranteed cost synchronization of complex networks with Markovian jump and mode-dependent mixed time delay. Guo and Li [13] presented a new synchronization algorithm for delayed complex dynamical networks. In 2015, Hao and Li [14] discussed the stochastic synchronization problem of complex dynamical networks with time-varying couplings. In [15], the author considered finite-time synchronization of dynamical networks by designing aperiodically intermittent pinning controllers with logarithmic quantization by using multiple Lyapunov functions and convex combination techniques. Chen et al. [16] presented conclusions about finite-time synchronization of multiweighted complex dynamical networks with and without coupling delay. In [17], the authors investigated adaptive lag synchronization for uncertain complex dynamical network with delay coupling.In practice, the complex dynamical networks such as mobile communication networks and social networks are always time-varying networks. The weights of links are time varying, which results in variations of the network topology and coupling configuration over time [18, 19]. In [7], the author put forward passivity of directed and undirected complex dynamical networks with adaptive coupling weights. Mei et al. [20] investigated finite-time synchronization and parameter identification problem for drive-response dynamical networks with unknown network topological structure and system parameters based on the finite-time stability theory and the adaptive control method. By making use of finite-time stability theory and properties of Wiener process, finite-time topological identification and stochastic synchronization for two complex networks with multiple time delays were obtained in [21]; in addition, Zhao et al. studied finite-time topology identification and stochastic synchronization of complex network with multiple time delays. In 2016, the finite-time synchronization and identification for the uncertain system parameters and topological structure of complex delayed networks with Markovian jumping parameters and stochastic perturbations is studied in [22]. In [23], the author studied finite-time synchronization of multiweighted complex dynamical networks with and without coupling delay on the basis of Dini derivative and some inequality techniques.Stated thus, we can see that the nodes of complex dynamical networks are assumed to have the same state dimension in the above literatures. However, synchronization for the dynamical networks with different dimensions of nodes deserves the same consideration. It is found that the dynamic characteristics of what the complex networks presented are particularly important for the study of networks with different nodes [24, 25], the network may exhibit different dynamical behaviors with nonidentical nodes couplings. In [24], Zhao et al. generalize the master stability function method for local synchronization of networks with identical nodes to the case of nonidentical nodes; a framework for global synchronization of dynamical networks with nonidentical nodes was presented. Wang and Fan [25, 26] discussed stabilization and synchronization of complex dynamical networks with similar nodes. Dai et al. [27] investigated generalized function matrix projective lag synchronization of uncertain complex dynamical networks with different dimension of nodes via adaptive control method combining with time-varying scaling matrix. In 2015, Tan and Tian [28] proposed a class of complex dynamical network modeling in which the nodes have different state dimensions, then with the help of finite-time theory, results of finite-time stabilization and synchronization are derived.Motivated by the above discussion, in this paper, on the basis of adaptive controlling technique, finite-time stability theory, and the inequality techniques, an available control scheme is presented to accomplish the synchronization and estimation of the unknown periodical time-varying outer coupling of complex delayed dynamical networks with different dimensions of nodes and time delay. The rest of this paper is organized as follows: Section2 gives the network model and some relative useful mathematical preliminaries. In Section 3, the finite-time control theory is employed to be addressed the synchronization and estimation of complex networks in detail, and some sufficient conditions are obtained. In Section 4, some representative examples are simulated to demonstrate the effectiveness of the proposed approach. Section 5 gives the conclusions of the paper.


## 2. Problem Formulation and Preliminaries

Consider the following complex dynamical network with different dimensions of nodes with unknown periodical outer coupling matrix and time-varying delay, the differential equation is(1)x˙it=fixit+∑j=1j≠iNcijtΓijxjt-τjt-Γiixit-τit+uit,i=1,2,…,N, where xi(t)=(xi1(t),xi2(t),…,xini(t))T∈Rni is the state variable of node i, fi:Rni→Rni,i=1,2,…,N is a smooth nonlinear function, describes the local dynamics of each node for the network (1), τi(t) is the unknown bounded time-varying delay, Γij∈Rni×nj, i,j=1,2,…,N is the inner coupling matrices, and C(t)=(cij(t))N×N is the unknown periodical outer coupling matrix, which describes the topology structure of the whole network. The matrix C(t) is defined as follows: if there is a connection from node j to node i(i≠j), then cij(t)≠0; otherwise cij(t)=0. The diagonal elements of matrix C(t) satisfy(2)ciit=-∑j=1j≠iNcijt,i=1,2,…,N. Here we need not assume that the coupling matrix C(t) was symmetric.Assume thatC([-τ0,0],Rni) is a Banach space of continuous functions mapping the intervals [τ0,0] into Rni with the norm Φ=sup-t0≤s≤0Φi(s), where 0≤t≤t0. For the network model (1), its initial conditions are given by xi(t)=Φi(t)∈([τ0,0],Rni).The solutionsi(t)∈Rni, i=1,2,…,N, of the following equations is the finite-time synchronization goal orbit:(3)s˙it=fisit,i=1,2,…,N.In the paper, we will design proper controllersui(t),i=1,2,…,N, to realize finite-time synchronization between the complex dynamical networks (1) and (3).For further discussion, the following assumptions and lemmas needed to be introduced firstly.Assumption 1.
There exists a positive definite diagonal matrixL=diag⁡l1,l2,…,lN, such that(4)xit-sitTfixit-fisit≤lixit-sitTxit-sit holds for all xi(t),si(t)∈Rni, where xi(t)≠si(t),i=1,2,…,N.Assumption 2.
In network (1), the unknown time-varying coupling elements cij(t) are periodical parameters, that is, cij(t+T)=cij(t) for t∈[0,+∞), in which T is the known common period of cij(t). Moreover, suppose that there exist a positive constant Bc, such that(5)cijt≤Bc.As one knows that the outer coupling elementscij(t) is periodical, so it must be bounded. Also we suppose the time delay τ(t) and its derivative τ˙(t) are both bounded.Assumption 3.
Suppose there exist a constantτ,τd, such that(6)0≤τit≤τ0,0≤τi˙t≤τd≤1.Lemma 4 (see [27]).
Assume that there exist a continuous, positive definite functionV(t) which satisfies the following inequality:(7)V˙t≤-αVηt+βVt,∀t≥t0,V1-ηt0≤αβ,where 0<η<1 and α>0, β>0, then the setting time Tf satisfies(8)Tf≤ln⁡1-β/αV1-ηt0βη-1.From the inequality (7), we can easily find that when t0=0, e-β1-ηtV1-η(t)≤V1-η(0)-α/β+(α/β)e-(1-η)βt.Especially, for the time-delay system, we have the following finite-time theorem.Lemma 5 (see [9]).
Suppose that functionV(t) is continuous and nonnegative for t∈[-τ,+∞] and satisfies the following conditions:(9)V˙t≤-αVηt+βVt, where 0<η<1 and α>0,β>0, then the following inequality holds for 0≤t≤Tf,(10)e-β1-ηtV1-ηt≤V¯1-η0-αβ+αβe-1-ηβt for the constant Tf which denotes the settling time, where V¯(0)=sup-τ≤s≤0⁡V(s).Lemma 6 (see [28]).
Letλi,i=1,2,…,m, be eigenvalues of matrix A∈Cm×m and ηi,i=1,2,…,n, be eigenvalues of matrix B∈Cn×n, then eigenvalues of matrix A⊗B are λiηj,i=1,2,…,m;j=1,2,…,n.For convenience, some necessary notations are needed to be introduced.T denotes the transpose of a matrix or a vector. Ini is an identity matrix of size ni. IN are some appropriate identity matrices. The notation A>0(A<0) means that the matrix A is positive (negative) definite. λmax(·) and λmin(·) mean the maximum and minimum eigenvalues of a matrix separately. And the inner coupling matrix Γ is denoted as (11)Γ=Γn1×n1Γn1×n2⋯Γn1×nNΓn2×n1Γn2×n2⋯Γn2×nN⋯⋯⋯⋯ΓnN×n1ΓnN×n2⋯ΓnN×nN.Let ei(t)=xi(t)-si(t), then the dynamical error system is e˙i(t)=x˙i(t)-s˙i(t); substitute (1) and (3) to it, we have(12)e˙it=fixit-fisit+∑j=1j≠iNcijtΓijejt-τjt-Γiieit-τit+ui.Obviously, we can see that finite-time synchronization of complex dynamical network (1) with controller ui equals the finite-time stability of dynamical error system (12). So next we will design proper outer controller and adaptive periodical learning law to promise finite-time stability and unknown coupling structure estimation of complex dynamical network (1).


## 3. Finite-Time Synchronization for Complex Dynamical Networks

In this section, we consider the finite-time synchronization of time-varying delay coupled complex dynamical network (1). In order to achieve the synchronization objective (3), the adaptive controller ui(t) for the ith node is designed as follows:(13)uit=-ε21+σsign⁡eiteitσ+∑i=1N∑j=1N1φij∗∫t-Ttc^ijν+Bc2dν1+σ/2+∑i=1NeiTteit1+σ/2eiteit2-kiteit-∑i=1N∫t-τteiTsQieisds1+σ/2eiteit2,ifeit≠0,uit=0,ifeit=0.ei(t)σ=(ei1(t)σ,ei2(t)σ,…,eini(t)σ)T, sign⁡(ei(t))=(sign⁡ei1t,sign⁡ei2t,…,sign⁡einit)T, 0<σ<1, and ε is an adjusted constant. With the help of the update law design in [20], the adaptive parameter ki(t),i=1,2,…,N, is as follows:(14)k˙it=-ε21+σri1-σ/2kiσsign⁡ki+ri1-ηikieiTteit.The parameters ri,ηi are some positive constants. The time-varying periodical couplings are designed by(15)c^ijt=c^ijt-T+φij∗eiTtΓijejt-τjt,t∈kT,k+1T,k=1,2,…,φijteiTtΓijejt-τjt,t∈0,T,0,t∈-T,0.c^ij(t) is the estimation to the coupling element cij(t) and C^=(c^ij(t)) is the estimated outer coupling matrix. Denote c~ij(t)=cij(t)-c^ij(t) to be the estimation error. φij∗ are some positive constants; φij(t) is a continuous and strictly increasing function for t∈[0,T] and satisfies φij(0)=0,φij(T)=φij∗.With the controller (13) and adaptive update laws (14) and (15), a sufficient condition for the controlled complex network which can realize finite-time synchronization will be presented.Definite 1. Complex dynamical networks (1) and (3) are said to be synchronized in finite time, if for suitable designed feedback controller ui(t), there exist a constant tf>0, any initial time t0, such that each state xi(t) of complex dynamical network (1) will satisfy(16)xit-sit→0,t→tf,xit-sit≡0,t≥tf,i=1,2,…,N.tf depends on the initial state vector value ei(t)=ϕi(t) for t∈[-maxk=0,1,…,N,θ∈R{τk(θ),T},0].Remark 7.
For continence, let us suppose that the coupling node delayτi(t),i=1,2,…,N, is less than the period of time-varying coupling structure T.Theorem 8.
Under Assumptions1–3, if there exist positive constants ri,ηi and positive definite matrices Qi, i=1,2,…,N, such that the following conditions hold:(17)L-H+Qi<0,12δC^TC^⊗ΓTΓ-Q<0,i=1,2,…,N.Q=diag⁡(Q1⊗In1,Q2⊗In2,…,QN⊗InN), H=diag⁡η1,η2,…,ηN, and i,j=n1,n2,…,nN. Then the controlled network (1) with adaptive update laws (12), (13), and (14) warrants finite-time synchronized in a finite time for any given positive initial time t0:(18)tf=4ln⁡1-δ/2εV¯1-σ/20δσ-1, where V¯(0)=1/2ζ2+1/2∑i=1N∑j=1N1/φij∗∫-T0c~ij2(ν)dν+∑i=1N∫-τ0ζiT(s)Qiζi(s)ds+(1/2)∑i=1N(1/ri0)ki(0,ζ)2. ζ=(ζ1,ζ2,…,ζN)T is the initial value of e(t) and satisfies the condition ζ∣V¯1-σ/20<2ε/δ.Proof.
Choose a Lyapunov-Krasovskii like function candidate as follows:(19)Vt=V1t+V2t+V3t+V4t,where(20)V1t=12∑i=1NeiTteit,V2t=12∑i=1N∑j=1N1φij∗∫t-Ttc~ij2τdτ,V3t=∑i=1N∫t-τtteiTsQieisds,V4t=12∑i=1N1riki2.
Obviously, we have(21)V˙t=∑i=14V˙it
Ifei(t)≠0, under the controller (13), the error system of coupling complex dynamical network (1) is reformulated as(22)e˙it=fixit-fisit+∑j=1j≠iNcijtΓijejt-τjt-Γiieit-τit-ε21+σsign⁡eiteitσ-ε21+σ∑i=1N∑j=1N1φij∗∫t-Ttc^ijν+Bc2dν1+σ/2+∑i=1NeiTteit1+σ/2eiteit2-ε∑i=1N∫t-τteiTsQieisds1+σ/2eiteit2-kiteit,
Taking the derivative along the trajectories of the error system respectively, we get(23)V˙1t=∑i=1NeiTte˙it=∑i=1NeiTtfixit-fisit-∑j=1Nc^ijtΓijejt+∑j=1j≠iNcijtΓijejt-τt-Γiieit-τt-∑i=1NeiTtε21+σ∑i=1N∑j=1N1φij∗∫t-Ttc^ijν+Bc2dν1+σ/2+ε∑i=1N∫t-τteiTsQieisds1+σ/2eiteit2-∑i=1NeiTtε21+σsign⁡eiteitσ+∑i=1NeiTteit1+σ/2+kieit≤∑i=1NeiTtli-kieit-∑j=1Nc^ijtΓijejt-τjt+∑j=1NcijtΓijejt-τjt-ε21+σeitσ+1-ε21+σ∑i=1N∑j=1N1φij∗∫t-Ttc^ijν+Bc2dν1+σ/2-ε∑i=1N∫t-τteiTsQieisds1+σ/2
As to the second item, we obtain(24)V˙2t=12∑i=1N∑j=1N1φij∗c~ij2t-c~ij2t-T.
From the estimation of unknown coupling structure items (15), we have(25)∑i=1N∑j=1N1φij∗c~ij2t-c~ij2t-T=-∑i=1N∑j=1N1φij∗c^ijt-c^ijt-T2-2∑i=1N∑j=1N1φij∗cijt-c^ijt-T·c^ijt-c^ijt-T=-2∑i=1N∑j=1N1φij∗eiTtΓijejt-∑i=1N∑j=1N1φij∗c^ijt-c^ijt-T2≤-2∑i=1N∑j=1Nc~ijteiTtΓijejt-τjt.
So(26)V˙2t≤-∑i=1N∑j=1Nc~ijteiTtΓijejt-τjt.
For the third item, according to the boundness of derivation of time-varying delayτ(t), we can yield(27)V˙3t=∑i=1NeiTtQieit-1-τ˙teiTt-τtQieit-τt≤∑i=1NeiTtQieit-1-τdeiTt-τtQieit-τt.
Combining with the adaptive update laws of the feedback gain (14), we have(28)V˙4t=-ε∑i=1Nkiri121+σri1-σ/2kiσsign⁡ki+∑i=1Nkiriri1-ηikieiTteit=-ε∑i=1N121+σkiσ+1riσ+1+∑i=1Nki-ηieiTteit.
Based on Assumption1, with the help of properties of Kronecker production, substitute (23), (26), (27), and (28) into the derivative term (21); we can further get(29)V˙t≤∑i=1NeiTtli-ηiIni+Qieit+∑j=1Nc^ijtΓijejt-τjt-ε21+σeitσ+1-ε21+σ∑i=1N∑j=1N1φij∗∫t-Ttc^ijν+Bc2dν1+σ/2-ε∑i=1N∫t-τteiTsQieisds1+σ/2-∑i=1N1-τdeiTt-τtQieit-τt-ε∑i=1N121+σkiσ+1riσ+1=eTtli-ηiIni+Qi⊗INet+eTtC^⊗Γet-τt-ε21+σeitσ+1-ε21+σ∑i=1N∑j=1N1φij∗∫t-Ttc^ijν+Bc2dν1+σ/2-ε∑i=1N∫t-τteiTsQieisds1+σ/2-1-τdeTt-τtQet-τt-ε∑i=1N121+σkiσ+1riσ+1.
the vectorse(t)=(e1(t),e2(t),…,eN(t))T, e(t-τ(t))=(e1(t-τ1(t)),e2(t-τ2(t)),…,eN(t-τN(t)))T. With Young’s inequality, the item can be rewritten as(30)eTtC^⊗Γet-τt≤δ2eTtet+12δeTt-τtC^⊗ΓT·C^⊗Γet-τt.
With Assumption3, we can obtain(31)V˙t≤eTtli-ηi+δ2Ini+Qi⊗INet-121+σeitσ+1-ε∑i=1N121+σkiσ+1riσ+1-ε21+σ∑i=1N∑j=1N1φij∗∫t-Ttc^ijν+Bc2dν1+σ/2-ε∑i=1N∫t-τteiTsQieisds1+σ/2+eTt-τt12δC^⊗ΓT·C^⊗Γ-Qet-τt
Base on the properties of Kronecker product of Lemma6, one gets(32)C^⊗ΓT·C^⊗Γ=C^TC^⊗ΓTΓ.
According to Assumption3, we can see that(33)-∑i=1N∫t-τteiTsQieisds1+σ/2≤-∑i=1N∫t-τtteiTsQieisds1+σ/2. As one can see, the estimation error of unknown coupling structure elements c~ij(t)=cij(t)-c^ij(t), so c~ij2(t)≤(c^ijt+Bc)2. Based on Assumption 1, with the help of the inequality x1q+x2q+⋯+xnq≤x12+x22+⋯+xn2q/2, we can further get(34)V˙t≤eTtli-ηiIni+Qi⊗INet-εV1+σ/2t+eTt-τt12δC^TC^⊗ΓTΓ-Qet-τt+δ2eTtet.Q=diag⁡(Q1⊗In1,Q2⊗In2,…,Qi⊗InN). Combine the conditions(35)li-ηiIni+Qi<0,i=1,2,…,N,12δC^TC^⊗ΓTΓ-Q<0. So when condition (17) of Theorem 8 is satisfied, can we have(36)V˙t≤-εV1+σ/2t+δ2eTtet≤-εV1+σ/2t+δ2Vt.
Based on finite-time stabilization theorems Lemmas4 and 5 of the error system, can we obtain that the controlled complex dynamical network (1) can realize finite-time synchronization less than tf≤t0+4ln⁡(1-(δ/2ε)V¯(1-σ)/2(t0))/δ(σ-1) for any given initial time t0. So based on the theory of finite-time synchronization, can we know that the synchronization of complex dynamical network (1) is realized.Remark 9.
Theorem8 gives a matrix dependent condition, according to the second item of condition (17), if we choose proper parameter δ, by calculation of (1/2δ)C^TC^⊗ΓTΓ, the matrix Q can be decided, that is, the blocking matrix Qi is known, then substitute Qi to the first item of condition (17), and the diagonal matrix H will be given. So we can see matrix calculation plays an important role in this process; in fact, the number of nodes for the dynamical network is usually large in real network, the calculation will be a little bit complex, and next we will present a simplified conclusion with the properties of eigenvalue of matrices showed in Corollary 11.Remark 10.
The adjustable parametersε and σ can not only adjust the time of the synchronization time tf itself, but also ensure that the periodic coupling structure matrix of the system C^(t) be estimated in the limited time T<tf by selecting the appropriate parameters. It means that at least one periodical coupling structure matrix has been estimated successfully when the finite-time synchronization of complex dynamical network (1) with controller (17) is realized. If the outer coupling structure of one period is known, the coupling structure of all periods is naturally determined.Condition (17) of Theorem 8 is time-delay dependent, which has lower conservatism than the delay independent condition. Combining with Lemma 6, rewriting the theorem with the eigenvalue form of matrix, then Corollary 11 will be obtained.Corollary 11.
Under Assumptions1–3, if there exist positive constants ri,ηi and positive definite matrices Qi, i=1,2,…,N, such that the following conditions hold:(37)ηmin>lmax+λmaxQi,λminQi>12δλmaxC^TC^λmaxΓTΓ,i=1,2,…,N.ηmin=mini=1,…,N{ηi} and lmax=maxi=1,…,N{li}. Then the controlled network (1) with the controller (13) and adaptive update laws (14) and (15) warrants finite-time synchronized in a finite time for any given positive initial time t0:(38)tf=4ln⁡1-δ/2εV¯1-σ/20δσ-1,where V¯(0)=1/2ζ2 + 1/2∑i=1N∑j=1N1/φij∗∫-T0c~ij2(ν)dν + ∑i=1N∫-τ0ζiT(s)Qiζi(s)ds + 1/2∑i=1N1/ri0ki(0,ζ)2. ζ=(ζ1,ζ2,…,ζN)T is the initial value of e(t) and satisfies the condition ζ∣V¯1-σ/20<2ε/δ.In addition, if the feedback gain is constant, the adaptive finite-time controller becomes linear finite-time controller, so the following results will be presented.Corollary 12.
Under Assumptions1–3, if there exist large enough positive constant ki and positive definite matrices Qi,i=1,2,…,N, the following conditions hold:(39)kmin>lmax+λmaxQi,λminQi>12δλmaxC^TC^λmaxΓTΓ,i=1,2,…,N.Then the controlled network (1) with the periodical adaptive update laws (15) and the controller(40)uit=-ε21+σsign⁡eiteitσ+∑i=1N∑j=1N1φij∗∫t-Ttc^ijν+Bc2dν1+σ/2+∑i=1NeiTteit1+σ/2eiteit2-kieit-∑i=1N∫t-τteiTsQieisds1+σ/2eiteit2,ifeit≠0,uit=0,ifeit=0 warrants finite-time synchronized in a finite time tf:(41)tf=4ln⁡1-δ/2εV¯1-σ/20δσ-1,where V¯(0)=1/2ζ2 + 1/2∑i=1N∑j=1N1/φij∗∫-T0c~ij2(ν)dν + ∑i=1N∫-τ0ζiT(s)Qiζi(s)ds. ζ is the initial value of e(t) and satisfies the condition ζ∣V¯1-σ/20<2ε/δ.Proof.
Choose the Lyapunov function candidate as follows:(42)Vt=12∑i=1NeiTteit+12∑i=1N∑j=1N1φij∗∫t-Ttc~ij2τdτ+12∑i=1N∫t-τtteiTtsQieisds.Qi, i=1,2,…,N, are any positive definite symmetric matrices.
With similar methods, the derivative ofV(t) will be less than(43)V˙t≤eTtli-kiIni+Qiet-εV1+σ/2t+eTt-τt12δC^TC^⊗ΓTΓ-Qet-τt+δ2Vt.So when the following condition is satisfied,(44)li-kiIni+Qi<0,i=1,2,…,N,12δC^TC^⊗ΓTΓ-Q<0.
That is, when condition (39) is satisfied, one can obtain(45)V˙t≤-εV1+σ/2t+δ2Vt.
That is, when the condition (39) is satisfied, the controlled complex dynamical network (1) will be finite-time synchronized in a finite-time tf:(46)tf=4ln⁡1-δ/2εV¯1-σ/20δσ-1.Remark 13.
Choosing appropriate parameters can help us to realize finite-time synchronization and unknown outer coupling estimation, so we put forward a calculation scheme to prepare the conditions of Corollary11 as follows:
( 1 ) Choose a proper parameter δ, for the given dynamical network model, calculate the eigenvalues of matrices ΓTΓ and C^TC^, and then decide the values of matrices Qi such that λmin(Qi)>1/2δλmax(C^TC^)λmax(ΓTΓ).
( 2 ) Calculate the Lipschitz constant, decide the parameters li, and then combine with the values of Qi, to decide the parameters ηi, such that ηmin>lmax+λmax(Qi).
( 3 ) With the initial matrix Φ∗, determine the exponential parameter σ of the controller and the feedback parameters k and to calculate periodical coupling matrices Cij(t).


## 4. Numerical Simulations

In this section, we will illustrate the effectiveness of the proposed approaches to achieve finite-time synchronization of complex networks (1) with different chaotic models as the node dynamics with unknown time-varying coupling and time delay.Without loss of generality, we choose five different chaotic dynamical as the nodes of complex dynamical networks (1). The system function is as follows:(47)x˙it=fixit+∑j=1j≠iNcijtΓijxjt-τjt-Γiixit-τit+uit,i=1,2,…,5.The five nodes of the controlled network (1) separately are chaotic systems with different dimensions, including Sprott-O, hyperchaotic lu¨, Lorenz, duffing, and Chen systems. Also the synchronization orbits are the following five isolate dynamical systems.The single Sprott-O system can be described as(48)x˙11x˙12x˙13=x12x11-x13x11+x11x13+2.7x12.The single hyperchaoticlu¨ system function is as follows:(49)x˙21x˙22x˙23x˙24=36x21-x22+x24-x21x23+20x22x21x22-3x23x21x23+x24.The single Lorenz system can be described as follows:(50)x˙31x˙32x˙33=10x32-x3128x31-x31x33-x32x31x32-83x33.The single Duffing system is(51)x˙41x˙42=x42-0.25x41-x413+11cos⁡t.The Chen system of drive network can be described as follows:(52)x˙51x˙52x˙53=35x52-x51-7x51-x51x53+28x52x51x52-3x53.The outer coupling matrix which depicts the structure of network (1) is a 5×5 matrix, where we choose c11(t)=-2.5-1.4sin⁡π/3t-1.3cos⁡(2π/3)t, c12(t)=1+0.4sin⁡π/3t, c13(t)=0, c14(t)=0.5+sinπ/3t, c15(t)=1-1.3cos⁡2π/3t; c21(t)=1, c22(t)=-5.3-cos⁡πt, c23(t)=1.3+cos⁡πt, c24(t)=1, c25(t)=2; c31(t)=1-sin⁡π/4t, c32(t)=1+0.5cos⁡π/4t, c33(t)=-2.7-0.5cos⁡π/4t+sin⁡π/4t, c34(t)=0, c35(t)=0.7; c41(t)=1+0.5cos⁡π/4t, c42(t)=sin⁡πt, c43(t)=cos⁡πt, c44(t)=-1-sin⁡πt-cos⁡πt-0.5cos⁡π/4t, c45(t)=0; c51(t)=cos⁡2π/3t, c52(t)=cos⁡π/4t,c53t=1, c54(t)=1, c55(t)=-2-cos⁡2π/3t-cos⁡π/4t. we can calculate that the bound of Cij(t)≤Bc=6.3.By calculation, we get the common periodT=12 of cij(t); the parameters Φ∗=φij∗ are designed as(53)Φ∗=φij∗=0.50.30.721.10.11.20.40.5-1.90.90.80.5-0.71.10.51.5-0.52.11.81.21.41.82.0-2.2, and φij(t)=(t/12)φij∗ for i,j=1,2,3,4,5. The time delays are chosen as τj(t)=0.1+0.2sin(jt),j=1,2,…,5. And the bound of time delay is τ=0.3.The inner coupling matrices are designed as(54)Γ12=10000100001-1,Γ14=10010-1,Γ24=10010-110,Γ32=Γ52=Γ12, Γ21=Γ23=Γ25=Γ12T, Γ34=Γ54=Γ14, Γ41=Γ43=Γ45=Γ14T, Γ42=Γ24T, Γ11=Γ33=Γ55=I3, Γ22=I4, Γ44=I2, Γij=Γji=I3, i,j=1,3,5.Choosing the initial values of complex dynamical networks (49) randomly in [-5,5], since t0=0, the finite-time synchronization of the complex dynamical network (1) with controller (12) is achieved in a finite time tf=4ln⁡1-δ/2εV¯(1-σ)/2(0)/δ(σ-1), where V¯(0)=1/2ζ2 + 1/2∑i=1N∑j=1N1/φij∗∫-T0c~ij2(ν)dν + ∑i=1N∫-τ0ζiT(s)Qiζi(s)ds + 1/2∑i=1N1/ri(0)ki(0,ζ)2. The error curves of states of the complex dynamical network (1) are showed in Figures 1–5, which indicates that the finite-time synchronization of the complex dynamical network (49) is achieved with the controllers (13), (14) and the adaptive periodical undate law (15).Figure 1
The error evolution of the first node.Figure 2
The error evolution of the second node.Figure 3
The error evolution of the third node.Figure 4
The error evolution of the fourth node.Figure 5
The error evolution of the fifth node.Simultaneously, Figure6 displays the time evolution curves of the feedback gains ki(t), i=1,2,…,5, with the initial values ki(0)=0.2; as times goes on, the five adaptive feedback gains converge to the determined value k1=8.17, k2=4.77, k3=3.57; k4=2.23, k5=0.48 gradually. And the unknown adaptive outer coupling structure c^ij are presented in Figure 7, from which we can see that the outer coupling connection matrix is periodical absolutely.Figure 6
The adaptive update of feedback gainski, i=1,2,3,4,5.Figure 7
The adaptive undate of feedback gainsc^ij(t), i,j=1,2,3,4,5.


## 5. Conclusions

In past twenty years, thanks to the concerted efforts of physicists, mathematicians, and control scientists, complex networks have enjoyed unprecedented prosperity. As one can see, the synchronization results of complex dynamical networks have been applied to automatic control, secure communication, and other fields. For better understanding the principles and strategies of synchronization phenomena, in this paper, we investigate finite-time synchronization problems of complex dynamical networks with different number of nodes, which contains unknown periodically coupling structures and bounded time-varying delay. Based on finite-time stability theory and Lyapunov-Krasovskii method, some sufficient finite-time synchronization criteria for complex dynamical network with unknown periodical couplings have been presented.Also some illustrative examples with their numerical simulations based on the different numbers of nodes are provided to demonstrate the effectiveness and feasibility of the proposed synchronization methods. In the simulations, the authors take heterogeneity and periodical connections of the dynamical nodes into account, which is much more correspondence with the actual complex network system greatly, also which shows that finite-time control techniques have better robustness and disturbance rejection properties. In the future, the author will take factors such as switching, uncertainties, and noise disturbance into account, to study the systems’ finite-time synchronization.



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