Finite time pinning function projective synchronization control of complex networks with time-varying delay

Abstract

Finite time control and finite time function projective synchronization for complex networks are studied in this paper. The complex networks with time-varying delay and function projective synchronization are achieved in finite time. The method of pinning control is used to achieve the finite time projective synchronization. A few target nodes are chosen to finish pinning function projective synchronization in a finite time by construct a time-dependent Lyapunov function. An example is provided to illustrate the correctness of the theoretical analysis.

Introduction

With the development of the complex networks theory, scholars use complex networks to describe the diversity and complexity of real society. The complex networks can demonstrate many realistic complex systems such as social networks, biological networks, computer networks, and so on.^1,2 Complex networks can abstractly represent complex system, the complex system consists of many nodes. Different types of nodes represent different networks. When nodes represent computer, the complex network is computer networks. When nodes represent human beings, the complex network is social networks. When nodes represent biology, the complex network is biological networks, etc.^1,2 Complex network is an important tool to study complex system. The study of complex networks can not only reveal the regularity of life and nature, but also solve the problems in many fields of society by using the theory and knowledge of the complex networks.^3,4 Therefore, the complex networks has been widely studied.

Synchronization exists widely in nature, which also in complex network. Synchronization is a kind of clustering behavior, which represents the state of multiple homogeneous systems are consistent or gradually converges to consistent. Synchronization control is one of research content of complex networks. ⁵ The synchronization of the complex network refers to the state of the different dynamic systems gradually achieve complete consistent. Complex network synchronization plays a very important role in the fields of intelligence optimization, nuclear magnetic resonance apparatus, secret communication equipment, information, and signal recognition, etc. The research of complex networks has attracted extensive attention.

There are a large number of nodes in the network, and each node need to controlled to achieve synchronization.^5–7 But pinning control is an effective control method,^8,9 its basic idea is to selectively control several nodes in the network to synchronization, which can greatly reduce the control cost.^10–12 Synchronization control is divided into finite-time control and infinite-time control according to the synchronization time. ¹³ Infinite-time control refers to the state of the system converges to consistent when time tends to infinity, finite-time control refers to the state of the system converges to consistent in fixed time. In recent years, the study of complex network synchronization control has achieved a lot of results. Liu et al. study the finite/fixed-time synchronization problem on complex networks with stochastic disturbances. ¹⁴ Zhang et al. use quantized pinning control to fixed-time synchronization of complex networks. ¹⁵ Wu and Bao study the finite-time synchronization of fractional-order multiplex networks. ¹⁶ Yang et al. propose the finite-time synchronization of delayed complex dynamical. ¹⁷ Zhang et al. study finite-time and fixed-time bipartite synchronization. ¹⁸ Luo et al. study finite-time synchronization control of coupled reaction-diffusion complex network systems. ¹⁹ Xu et al. investigate finite-time synchronization for a class of the complex dynamical network.^20,21 Xu et al. use pinning control method to study finite-time synchronization for networks. ²² Lu et al. investigate finite-time synchronization problem for complex system. ²³ However, the current research on the finite-time pinning synchronization control problem focuses on complete synchronization and rarely research on time-varying delay and function projective synchronization in finite-time by pinning control, which will be the motivation for us to carry out this study.

Inspired by the above discussions, function projective synchronization control is proposed and time-varying delay is discussed by designing Lyapunov function and finite time control theory. The content of this article are summarized as: Firstly, theory of pinning control and finite-time control are presented, by which Lyapunov function is established. Secondly, the finite time pinning function projective synchronization control is designed, sufficient conditions are established to guarantee that the complex networks with time-varying delay achieve finite time pinning function projective synchronization. Theoretical proof and numerical simulation show that the correctness of the theoretical analysis. Thirdly, the time-varying delay exists in complex networks, complex networks with time-varying delay is researched. This paper is organized as follows. In Section “Problem formulation,” we summarize some preliminaries and problem statement. In Section “Main results,” we obtain the main results. The control strategies are given in Section “Finite time pinning function projective synchronization control of complex networks with time-varying delay.” Section “Numerical simulations” is numerical simulations.

Problem formulation

In this article, the pinning control strategies is presented for achieving finite time pinning function projective synchronization control for complex networks, the complex networks containing N nodes can be illustrated as follows,

$$\stackrel{·}{x}(t)=C{x}_i(t)+\sum _{j=1,j\ne i}^{N}Z{\mathrm{\Gamma }}_{\mathrm{ij}}(x_j(t)-x_i(t))+u_i(t)+\mathrm{Df}(x_i(t)),$$

(1)

$x_i(t)=[x_{i1},x_{i2},...,x_{\mathrm{in}}{]}^T\in R^n(i=1,2,...,N)$ represent ith node. Matrix $Z=[z_{\mathrm{ij}}]\in R^{N\times N},z_{\mathrm{ij}}>0$ information interchange between nodes i and j, $z_{\mathrm{ii}}=-\sum _{j=1,j\ne i}^N{z}_{\mathrm{ij}}.$ A diagonal matrix ${\mathrm{\Gamma }}_{\mathrm{ij}}=\mathrm{diag}\{{\mathrm{\Gamma }}_{\mathrm{ij}}^1,{\mathrm{\Gamma }}_{\mathrm{ij}}^2,...,{\mathrm{\Gamma }}_{\mathrm{ij}}^n\}$ describes the all information channels between nodes j and i. ${\mathrm{\Gamma }}_{\mathrm{ij}}^k=1(k=1,2,...,n)$ shows nodes j and i message exchange each other through the kth channel, otherwise ${\mathrm{\Gamma }}_{\mathrm{ij}}^k=0$ . The $u_i(t)$ is pinning controller. Function $f(x_i(t))$ satisfy global Lipschitz condition. The matrix C is defined as $C=\mathrm{diag}\{c_1,c_2,...,c_n\}$ , $D=[d_{\mathrm{ij}}]\in R^{n\times n}$ is real matrix.

The target node $s(t)$ satisfy,

$$\stackrel{·}{s}(t)=\mathrm{Cs}(t)+\mathrm{Df}(s(t)),$$

(2)

the controller $u_i(t)$ is to achieve synchronization control through control target nodes $s(t)$ . The error is defined as $e_i(t)=x_i(t)-q(t)s(t)$ , the $e_i(t)$ represents between the ith node and target node $s(t)$ , $q(t)\mathbf{=}\mathrm{diag}(q_{11}(t),q_{22}(t),\dots ,q_{\mathrm{nn}}(t))$ . When $q_{\mathrm{ii}}(t)(1\le i\le n)$ is a function, synchronization is function projective synchronization. When each element of the matrix q(t) is equal to 1, synchronization is called complete synchronization. The purpose of this article is to design a controller $u_i(t)$ such that $u_i(t)$ can achieve function projective synchronization for complex networks with time-varying delay in finite time by pinning control.

According to equations (1) and (2), the error dynamic equation as follows,

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{e}}_i(t)}& =C{e}_i(t)+\sum _{j=1,j\ne i}^{N}G{\mathrm{\Gamma }}_{\mathrm{ij}}(x_j(t)-x_i(t))+u_i(t)\hfill \\ \multicolumn{1}{c}{}& +\mathrm{DF}(x_i(t);s(t)),\hfill \end{array}$$

(3)

where $e_i(t)=[e_{i1}(t),e_{i2}(t),\dots ,e_{\mathrm{in}}(t){]}^T$ and $F(x_i(t);s(t))=f(x_i(t))-f(s(t))$ . The controller $u_i(t)$ is designed as follows,

$$u_i(t)=-l{d}_i(x_i(t)-s(t)),$$

where $l>0$ is the coupling strength, pinning control gain is $d_i$ . Let U and $U_{\mathrm{pin}}$ be all the collection of all nodes and the collection of the pinning nodes, respectively. When $i\in U_{\mathrm{pin}}$ , $d_i>0$ . Otherwise, $d_i=0$ .

In order to realize pinning control for complex network, leaders must directly or indirectly influence all nodes of the network, then network nodes and leaders constitute graph which must contains a directed spanning tree, and the leader is the only root node. For connected undirected networks, there are two ways to choose pinning node: choose randomly and specific selection. Generally speaking, the nodes with large priority pinning can have better control performance for scale-free network. However, select the node with large priority pinning and randomly select the pinning node have the same control performance for stochastic network. When the coupling strength is smaller, the effect of the node with small pinning is better. When the topology of an undirected network is not connected, the network topology is composed of multiple connected subgraphs. The pinning nodes are selected for each subgraph in a random or specific way, then the union of the pining node set of all subgraphs is set of the pinning node.

The error dynamic equation (3) is rewritten as follows,

$${\stackrel{·}{e}}_i(t)=C{e}_i(t)+\sum _{j=1}^{N}Z{\mathrm{\Gamma }}_{\mathrm{ij}}{e}_j(t)-l{d}_i{e}_i(t)+\mathrm{DF}(x_i(t);s(t)),$$

the matrix $Z_d$ is defined as

$$\begin{array}{c}\multicolumn{1}{c}{Z_d=\left[\begin{array}{cccc}{\mathrm{\Gamma }}_{d11}\hfill & z_{12}{\mathrm{\Gamma }}_{d12}\hfill & \cdots \hfill & z_{1N}{\mathrm{\Gamma }}_{d1N}\hfill \\ \multicolumn{1}{c}{z_{21}{\mathrm{\Gamma }}_{d21}}& {\mathrm{\Gamma }}_{d22}\hfill & \cdots \hfill & z_{2N}{\mathrm{\Gamma }}_{d2N}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{z_{N1}{\mathrm{\Gamma }}_{d21}}& z_{N2}{\mathrm{\Gamma }}_{\mathrm{dN}2}\hfill & \cdots \hfill & {\mathrm{\Gamma }}_{\mathrm{dNN}}\hfill \end{array}\right]}\end{array}$$

(4)

the $G_d$ is $N\times N$ dimensions, ${\mathrm{\Gamma }}_{\mathrm{dii}}=\sum _{j=1,j\ne i}^N{z}_{\mathrm{ij}}{\mathrm{\Gamma }}_{\mathrm{dij}},i,j=1,2,\dots ,N$ .

The $e_d(t)=[e_{1d},e_{2d},\dots ,e_{\mathrm{Nd}}{]}^T$ is the synchronization error. If finite time pinning function projective synchronization is achieved, the error $e_d$ of all channels converges to zero in finite time. Therefore, the error dynamic equation of the dth channel as follows,

$$\stackrel{·}{e}(t)=a_k{e}_k(t)+\sum _{\upsilon =1}^n{b}_{k\upsilon }{F}_{\upsilon }+(G_k-\mathrm{cD})e_k(t),$$

(5)

where $k=1,2,\dots ,n,D=\mathrm{diag}\{k_1,k_2,\dots ,k_n\},$

$$\begin{array}{c}\multicolumn{1}{c}{F_{\upsilon }=\left[\begin{array}{c}{f}_{\upsilon }(x_{1\upsilon }(t))-f_{\upsilon }(s_{\upsilon }(t)\hfill \\ \multicolumn{1}{c}{f_{\upsilon }(x_{2\upsilon }(t))-f_{\upsilon }(s_{\upsilon }(t)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{f_{\upsilon }(x_{N\upsilon }(t))-f_{\upsilon }(s_{\upsilon }(t)}\end{array}\right]}\end{array}$$

Some finite-time control theory are introduced before give main results.

• Definition 1: The complex networks (1) is said to be finite time pinning function projective synchronized to the target trajectory $s(t)$ , if there is a moment $T>0$ , thus

$$\underset{t\to T}{\lim }\Vert e_i(t)\Vert =\underset{t\to T}{lim}\Vert x_i(t)-q(t)s(t)\Vert =0.$$

• Lemma 1 ¹⁸ :

$$\stackrel{·}{V}(t)\le -\alpha V^{\beta }(t),\forall t\ge t_0,V(t_0)\ge 0$$

(6)

$\alpha >0,0<\beta <1$ is constant, so $V(t)$ satisfies

$$V^{1-\beta }(t)\le V^{1-\beta }(t_0)-\alpha (1-\beta )(t-t_0),t_0\le t\le T.$$

(7)

And for any $t\ge T$ , $V(t)\equiv 0$ exists, then the stability time T is as follows,

$$T=t_0+\frac{V^{1-b}(t_0)}{\alpha (1-b)}.$$

(8)

• Lemma 2 ¹⁸ :

$$0\le \frac{f_{\beta }(y_1,t)-f_{\beta }(y_2,t)}{y_1-y_2}\le K_{\beta },\beta =1,2,\dots ,n,$$

(9)

for any $y_1,y_2\in R^n,$ β is a positive scalar.

• Lemma 3 ¹⁸ :

$${\eta }^T\lambda +{\lambda }^T\eta \le \epsilon {\eta }^T\eta +\frac{1}{\epsilon }{\lambda }^T\lambda .$$

(10)

• Lemma 4 ¹⁹ :

$${\int }_{{\alpha }_2}^{{\alpha }_1}{x}^T(e)\mathrm{Mx}(e)\mathrm{de}\le \frac{4({\alpha }_2-{\alpha }_1)}{{\pi }^2}{\int }_{{\alpha }_2}^{{\alpha }_1}{\stackrel{·}{x}}^T(e)M\stackrel{·}{x}(e)\mathrm{de}$$

(11)

• Lemma 5 ²⁰ :

$${[{\int }_{h_1}^{h_2}W(v)\mathrm{dv}]}^{T}N[{\int }_{h_1}^{h_2}W(v)\mathrm{dv}]\le (h_2-h_1){\int }_{h_1}^{h_2}{W}^T(v)\mathrm{NW}(v)\mathrm{dv},$$

(12)

where matrix $N>0,h_1$ and $h_2$ are two positive scalars, W is a vector function, $W:[h_1,h_2]\to R^n$ .

Main results

In this section, a finite time control scheme is designed by finite-time control theory and Lyapunov function. The sufficient conditions for synchronization control are derived.

• Theorem 1. Base on Lemma 2, given two positive scalars $e_1,e_2,\hat{e}\in \{e_1,e_2\}$ , if there are scalars $Q_{1d},Q_{2d}$ and matrices $W_d>0,J_d>0,O_d>0,L_d>0,V_{1d},V_{2d},P_{1d},P_{2d},P_{3d},P_{4d},P_{5d},T_d$ , the following linear matrix inequalities hold for $d(d=1,\dots ,n)$ .

$$r_d(\hat{e})=\left[\begin{array}{ccc}{r}_{d11}\hfill & \tilde{e}(-P_{1d}+P_{2d})\hfill & \tilde{e}{P}_{3d}\hfill \\ \multicolumn{1}{c}{*}& r_{d22}\hfill & \tilde{e}{P}_{34}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & \tilde{e}(P_{5d}+P_{5d}^T)\hfill \end{array}\right]>0,$$

(13)

$${\mathrm{\Xi }}_d(\hat{h})=\left[\begin{array}{cccccc}{\mathrm{\Xi }}_{d11}\hfill & {\mathrm{\Xi }}_{d12}\hfill & {\mathrm{\Xi }}_{d13}\hfill & {\mathrm{\Xi }}_{d14}\hfill & \sqrt{n}{V}_{1d}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\mathrm{\Xi }}_{d22}\hfill & {\mathrm{\Xi }}_{d23}\hfill & {\mathrm{\Xi }}_{d24}\hfill & 0\hfill & \sqrt{n}{V}_{2d}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\Pi }_{1d33}\hfill & {\Pi }_{1d34}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & {\Pi }_{1d44}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -Q_{1d}I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & 0\hfill & -Q_{2d}I\hfill \end{array}\right]<0,$$

(14)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathrm{\Omega }}_d(\hat{e})=\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{cccccccc}{\mathrm{\Omega }}_{d11}\hfill & {\Pi }_{1d11}\hfill & {\mathrm{\Omega }}_{d13}\hfill & {\Pi }_{1d14}\hfill & \sqrt{n}{V}_{1d}\hfill & 0\hfill & \hat{e}{T}_{1d}^T\hfill & \hat{e}{T}_{5d}^T\hfill \\ \multicolumn{1}{c}{*}& {\Pi }_{1d22}\hfill & {\mathrm{\Omega }}_{d23}\hfill & {\Pi }_{1d24}\hfill & 0\hfill & \sqrt{n}{V}_{2d}\hfill & \hat{e}{T}_{2d}^T\hfill & \hat{e}{T}_{6d}^T\hfill \\ \multicolumn{1}{c}{*}& {\Pi }_{1d22}\hfill & {\mathrm{\Omega }}_{d23}\hfill & {\Pi }_{1d24}\hfill & 0\hfill & 0\hfill & \hat{e}{T}_{3d}^T\hfill & \hat{e}{T}_{7d}^T\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\Pi }_{1d33}\hfill & {\Pi }_{1d34}\hfill & 0\hfill & 0\hfill & \hat{e}{T}_{4d}^T\hfill & \hat{e}{T}_{8d}^T\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & {\Pi }_{1d44}\hfill & -Q_{1d}I\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -Q_{2d}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & -\hat{e}{X}_{1d}\hfill & -\hat{e}{X}_{2d}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & -\hat{e}{X}_{3d}\hfill \end{array}\right]<0,\hfill \end{array}$$

(15)

where

$$r_{d11}=J_d+\tilde{e}{R}_{1d}+\tilde{e}{P}_{1d}^T$$

$$\begin{array}{c}{r}_{d22}=\tilde{h}(-R_{2d}-R_{2d}^T+R_{1d}+P_{1d}^T)\hfill \\ {\Pi }_{1d11}=J_d+J_d^T-J_d-P_{1d}-P_{1d}^T+a_d(V_{1d}+V_{1d}^T)\hfill \\ +(Q_{1d}+Q_{2d})\sum _{\upsilon =1}^n{b}_{d\upsilon \kappa \upsilon }I+T_{1d}+T_{1d}^T\\ {\Pi }_{1d12}=-V_{1d}^T+a_d{V}_{2d}+T_{2d}\hfill \\ {\Pi }_{1d13}=J_d+P_{1d}-P_{2d}+U_{1d}^T(G_d-\mathrm{cD})+T_{3d}-T_{1d}^T\hfill \\ {\Pi }_{1d14}=-P_{3d}+T_{4d}+T_{5d}^T\hfill \\ {\Pi }_{1d22}=-V_{2d}-V_{2d}^T\hfill \\ \multicolumn{1}{c}{{\Pi }_{1d23}=V_{2d}^T(G_d-\mathrm{cD})-T_{2d}^T}\\ \multicolumn{1}{c}{{\Pi }_{1d24}=Y_{6d}^T}\\ \multicolumn{1}{c}{{\Pi }_{1d33}=-J_d+P_{2d}+P_{2d}^T-P_{1d}-P_{1d}^T-T_{3d}-T_{3d}^T}\\ \multicolumn{1}{c}{{\Pi }_{1d34}=-P_{4d}-T_{4d}+T_{7d}^T}\\ \multicolumn{1}{c}{{\Pi }_{1d44}=-P_5-P_5^T+T_{8d}+T_{8d}^T}\\ \multicolumn{1}{c}{{\Pi }_{2d11}=\hat{e}{P}_{3d}+\hat{e}{L}_{1d}^T+\hat{e}{O}_{3d}}\\ \multicolumn{1}{c}{{\Pi }_{2d12}=\hat{e}{M}_d+\hat{e}(P_{1d}+P_{1d}^T)+\hat{e}{L}_{1d}+\hat{e}{O}_{2d}^T}\\ \multicolumn{1}{c}{{\Pi }_{2d13}=\hat{e}{R}_{4d}^T+\hat{e}{L}_{2d},{\Pi }_{2d14}=\hat{e}(P_{5d}+P_{5d}^T)}\\ \multicolumn{1}{c}{{\Pi }_{2d22}=\hat{e}{O}_{1d}}\\ \multicolumn{1}{c}{{\Pi }_{2d23}=-\hat{e}(J_d^T+P_{1d}-P_{2d}-L_{2d}-L_{3d})}\\ \multicolumn{1}{c}{{\Pi }_{2d24}=\hat{e}{P}_{3d},{\mathrm{\Xi }}_{d11}={\Pi }_{1d11}+{\Pi }_{2d11}}\\ \multicolumn{1}{c}{{\mathrm{\Xi }}_{d12}={\Pi }_{1d12}+{\Pi }_{2d12},{\mathrm{\Xi }}_{d13}={\Pi }_{1d13}+{\Pi }_{2d13}}\\ \multicolumn{1}{c}{{\mathrm{\Xi }}_{d14}={\Pi }_{1d14}+{\Pi }_{2d14},{\mathrm{\Xi }}_{d22}={\Pi }_{1d22}+{\Pi }_{2d22}}\\ \multicolumn{1}{c}{{\mathrm{\Xi }}_{d23}={\Pi }_{1d23}+{\Pi }_{2d23},{\mathrm{\Xi }}_{d24}={\Pi }_{1d24}+{\Pi }_{2d24}}\\ \multicolumn{1}{c}{{\mathrm{\Omega }}_{d11}={\Pi }_{1d11}-\hat{e}{L}_{1d},{\mathrm{\Omega }}_{d13}={\Pi }_{1d13}-\hat{e}{L}_{2d}}\\ \multicolumn{1}{c}{{\mathrm{\Omega }}_{d23}={\Pi }_{1d23}-\hat{e}{L}_{3d}.}\end{array}$$

Then finite time pinning synchronization control is completed.

• Proof. Constructing the following a Lyapunov function,

$$V(t)=\sum _{i=1}^5{V}_i(t)V_6(t),t\in [t_k,t_{k+1}),$$

(16)

where

$$V\prime (t)=\sum _{i=1}^5{V}_i(t),$$

$$\begin{array}{cc}\multicolumn{1}{c}{V_1(t)}& =\sum _{d=1}^n{\theta }_d^T(t)P_d{\theta }_d(t),\hfill \\ \multicolumn{1}{c}{V_2(t)}& =\sum _{d=1}^n(t_{k+1}-t)\underset{t_k}{\overset{t}{\int }}{\left[\begin{array}{c}{\stackrel{·}{\theta }}_d(s)\hfill \\ \multicolumn{1}{c}{{\theta }_d(s)}\end{array}\right]}^T{X}_d\left[\begin{array}{c}{\stackrel{·}{\theta }}_d(s)\hfill \\ \multicolumn{1}{c}{{\theta }_d(s)}\end{array}\right]\mathrm{ds},\hfill \\ \multicolumn{1}{c}{V_3(t)}& =\sum _{d=1}^n(t_{k+1}-t)({\theta }_d(t)-{\theta }_d(t_k){)}^T{M}_d({\theta }_d(t)-{\theta }_d(t_k)),\hfill \\ \multicolumn{1}{c}{V_4(t)}& =\sum _{d=1}^n(t_{k+1}-t){\varphi }^T(t)R_d\varphi (t),\hfill \\ \multicolumn{1}{c}{V_5(t)}& =\sum _{d=1}^n(t_{k+1}-t)(t-t_k){\left[\begin{array}{c}{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{{\theta }_d(t_k)}\end{array}\right]}^T{H}_d\left[\begin{array}{c}{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{{\theta }_d(t_k)}\end{array}\right],\hfill \\ \multicolumn{1}{c}{V_6(t)}& =[V_1(t)+V_2(t)+V_3(t)+V_4(t)+V_5(t){]}^b,\hfill \\ \multicolumn{1}{c}{\varphi (t)}& ={[{\theta }_d^T(t),{\theta }_d^T(t_k),\underset{t_k}{\overset{t}{\int }}{\theta }_d^T(s)\mathrm{ds}]}^T,\hfill \\ \multicolumn{1}{c}{X_d}& =\left[\begin{array}{cc}{X}_{1d}\hfill & X_{2d}\hfill \\ \multicolumn{1}{c}{*}& X_{3d}\hfill \end{array}\right],H_d=\left[\begin{array}{cc}{H}_{1d}\hfill & H_{2d}\hfill \\ \multicolumn{1}{c}{*}& H_{3d}\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{R_d}& =\left[\begin{array}{ccc}{R}_{1d}+R_{1d}^T\hfill & -R_{1d}+R_{2d}\hfill & R_{3d}\hfill \\ \multicolumn{1}{c}{*}& -R_{2d}-R_{2d}^T+R_{1d}+R_{1d}^T\hfill & R_{4d}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & R_{5d}+R_{5d}^T\hfill \end{array}\right].\hfill \\ \multicolumn{1}{c}{V(t)}& \ge V_1(t)+V_5(t)\ge {\left[\begin{array}{c}{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{{\theta }_d(t_k)}\\ \multicolumn{1}{c}{{\int }_{t_k}^t{\theta }_d(s)\mathrm{ds}}\end{array}\right]}^T\hfill \\ \multicolumn{1}{c}{}& (\left[\begin{array}{ccc}{P}_d\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& 0\hfill & 0\hfill \end{array}\right]+(t_{k+1}-t)R_d)\times \left[\begin{array}{c}{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{{\theta }_d(t_k)}\\ \multicolumn{1}{c}{{\int }_{t_k}^t{\theta }_d(s)\mathrm{ds}}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& \ge {\left[\begin{array}{c}{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{{\theta }_d(t_k)}\\ \multicolumn{1}{c}{{\int }_{t_k}^t{\theta }_d(s)\mathrm{ds}}\end{array}\right]}^T\left(\frac{h_2-(t_{k+1}-t)}{h_2-h_1}\right)\times r_d(h_1)\hfill \\ \multicolumn{1}{c}{}& +\frac{(t_{k+1}-t)-h_1}{h_2-h_1}{r}_d(h_2)\left[\begin{array}{c}{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{{\theta }_d(t_k)}\\ \multicolumn{1}{c}{{\int }_{t_k}^t{\theta }_d(s)\mathrm{ds}}\end{array}\right].\hfill \end{array}$$

Then,

The $r_d(h_1)>0$ and $r_d(h_2)>0$ base on equation (11). So Lyapunov function $V(t)$ is reasonable. Taking the derivative of equation (12),

$${\stackrel{·}{V}}_{1}t=\sum _{d=1}^{n}2e_d^T(t)P_d{\stackrel{·}{e}}_d(t),$$

(17)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\stackrel{·}{V}}_{2}t=\sum _{d=1}^n\hfill \\ \multicolumn{1}{c}{}& \{-\underset{t_k}{\overset{t}{\int }}{\left[\begin{array}{c}{\stackrel{·}{e}}_d(s)\hfill \\ \multicolumn{1}{c}{e_d(s)}\end{array}\right]}^T{X}_d\left[\begin{array}{c}{\stackrel{·}{e}}_d(s)\hfill \\ \multicolumn{1}{c}{e_d(s)}\end{array}\right]\mathrm{ds}+(t_{k+1}-t){\left[\begin{array}{c}{\stackrel{·}{e}}_d(s)\hfill \\ \multicolumn{1}{c}{e_d(s)}\end{array}\right]}^T{X}_d\left[\begin{array}{c}{\stackrel{·}{e}}_d(s)\hfill \\ \multicolumn{1}{c}{e_d(s)}\end{array}\right]\},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{V}}_3(t)}& =\sum _{d=1}^n\{-{(e_d(t)-e_d(t_k))}^T{M}_d(e_d(t)-e_d(t_k))\hfill \\ \multicolumn{1}{c}{}& +2(t_{k+1}-t)(e_d(t)-e_d(t_k{))}^T{M}_d{\stackrel{·}{e}}_d(t)\},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{V}}_4(t)}& =\sum _{d=1}^n\{-{\left[\begin{array}{c}{e}_d(t)\hfill \\ \multicolumn{1}{c}{e_d(t_k)}\\ \multicolumn{1}{c}{{\int }_{t_k}^t{e}_d(s)\mathrm{ds}}\end{array}\right]}^T{R}_d\left[\begin{array}{c}{e}_d(t)\hfill \\ \multicolumn{1}{c}{e_d(t_k)}\\ \multicolumn{1}{c}{{\int }_{t_k}^t{e}_d(s)\mathrm{ds}}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& +2(t_{k+1}-t){\left[\begin{array}{c}{e}_d(t)\hfill \\ \multicolumn{1}{c}{e_d(t_k)}\\ \multicolumn{1}{c}{{\int }_{t_k}^t{e}_d(s)\mathrm{ds}}\end{array}\right]}^T{R}_d\left[\begin{array}{c}{\stackrel{·}{e}}_d(t)\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{e_d(t)}\end{array}\right]\},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{V}}_5(t)}& =\sum _{d=1}^n\{2(t_{k+1}-t){\left[\begin{array}{c}{e}_d(t)\hfill \\ \multicolumn{1}{c}{e_d(t_k)}\end{array}\right]}^T{H}_d\left[\begin{array}{c}{\stackrel{·}{e}}_d(t)\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& +(t_{k+1}+t_k-2t){\left[\begin{array}{c}{e}_d(t)\hfill \\ \multicolumn{1}{c}{e_d(t_k)}\end{array}\right]}^T{H}_d\left[\begin{array}{c}{\stackrel{·}{e}}_d(t)\hfill \\ \multicolumn{1}{c}{e_d(t_k)}\end{array}\right]\}.\hfill \end{array}$$

(21)

The vector is defined as follows,

$$\zeta t={[e_d^T(t){\stackrel{·}{e}}_d^T(t){\theta }_d^T(t_k){(\underset{t_k}{\overset{t}{\int }}{e}_d(s)\mathrm{ds})}^T]}^T.$$

According to Lemma 1 and $V_2(t)$ ,

$$\begin{array}{cc}\multicolumn{1}{c}{}& -\underset{t_k}{\overset{t}{\int }}{\left[\begin{array}{c}{\stackrel{·}{e}}_d(s)\hfill \\ \multicolumn{1}{c}{e_d(s)}\end{array}\right]}^T{X}_d\left[\begin{array}{c}{\stackrel{·}{e}}_d(s)\hfill \\ \multicolumn{1}{c}{e_d(s)}\end{array}\right]\mathrm{ds}\le 2\zeta (t{)}^T{Y}_d^T\left[\begin{array}{cccc}I\hfill & 0\hfill & -I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & I\hfill \end{array}\right]\zeta (t)\hfill \\ \multicolumn{1}{c}{}& +(t-t_k)\zeta (t{)}^T{Y}_d^T{X}_d^{-1}{Y}_d\zeta (t),\hfill \end{array}$$

where

$$Y_d=\left[\begin{array}{cccc}{Y}_{11}\hfill & Y_{2d}\hfill & Y_{3d}\hfill & Y_{4d}\hfill \\ \multicolumn{1}{c}{Y_{5d}}& Y_{6d}\hfill & Y_{7d}\hfill & Y_{8d}\hfill \end{array}\right].$$

According to equation (18),

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{V}}_{2}t}& \le (t_{k+1}-t){\left[\begin{array}{c}{\stackrel{·}{e}}_d(t)\hfill \\ \multicolumn{1}{c}{e_d(t)}\end{array}\right]}^T{X}_d\left[\begin{array}{c}{\stackrel{·}{e}}_d(t)\hfill \\ \multicolumn{1}{c}{e_d(t)}\end{array}\right]+2\zeta (t{)}^T{Y}_d^T\hfill \\ \multicolumn{1}{c}{}& \left[\begin{array}{cccc}I\hfill & 0\hfill & -I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & I\hfill \end{array}\right]\zeta (t)+(t-t_k)\zeta (t{)}^T{Y}_d^T{X}_d^{-1}{Y}_d\zeta (t).\hfill \end{array}$$

(22)

Furthermore,

$$2{\zeta }^T(t)U_d^T(a_d{e}_d(t)+\sum _{\upsilon =1}^n{b}_{d\upsilon }{F}_{\upsilon }+G_d-\mathrm{cD})e_d(t_k)-{\stackrel{·}{e}}_d^T(t))=0,$$

(23)

where $U_d={\left[\begin{array}{ccccc}{U}_{1d}\hfill & U_{2d}\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]}^T$ is a real matrix.

According to Lemma 2,

$$F_{\upsilon }^T{F}_{\upsilon }\le {\kappa }_{\upsilon }{e}_{\upsilon }^T(t)e_{\upsilon }(t).$$

Basis on above inequality and Lemma 1, then,

$$\begin{array}{cc}\multicolumn{1}{c}{}& 2(e_d^T(t)U_1^T+{\stackrel{·}{e}}_d^T(t)U_2^T)\underset{v=1}{\sum ^n}B{F}_{\upsilon }\le {\mathrm{nQ}}_{1d}^{-1}{e}_d^T(t)U_{1d}{U}_{1d}^T{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{}& +{\mathrm{nQ}}_{2d}^{-1}{e}_d^T(t)U_{2d}{U}_{2d}^T{\stackrel{·}{e}}_d^T(t)+(Q_{1d}+Q_{2d})\underset{v=1}{\sum ^n}{b}_{d\upsilon }{\kappa }_{\upsilon }{e}_{\upsilon }^T(t)e_{\upsilon }(t)\hfill \end{array}$$

(24)

Base on equations (17)–(24) and Lemmas 1 and 2,

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}\prime (t)}& \le \sum _{d=1}^n\{2e_d^T(t)P_d{e}_d(t)-(t_{k+1}-t)[{\stackrel{·}{e}}_d^T(t)X_{1d}{\stackrel{·}{e}}_d^T(t)\hfill \\ \multicolumn{1}{c}{}& +2{\stackrel{·}{e}}_d^T(t)X_{2d}^T{\stackrel{·}{e}}_d^T(t)+e_d^T{X}_{3d}{e}_d(t)]+2e_d^T(t)Y_{1d}^T{e}_d(t)\hfill \\ \multicolumn{1}{c}{}& +2{\stackrel{·}{e}}_d^T(t)Y_{2d}^T{\theta }_d(t)+e_d^T(t_k)Y_3^T{e}_d(t)+\underset{t_k}{\overset{t}{\int }}{e}_d(s){\mathrm{dsY}}_{4d}^T{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{}& -2(e_d^T(t)Y_{1d}^T+{\stackrel{·}{e}}_d^T(t)Y_{2d}^T+e_d^T(t_k)Y_3^T+\underset{t_k}{\overset{t}{\int }}{e}_d^T(s){\mathrm{dsY}}_{4d}^T)e_d(t_k)\hfill \\ \multicolumn{1}{c}{}& +2(e_d^T(t)Y_5^T+{\stackrel{·}{e}}_d^T(t)Y_{6d}^T+e_d^T(t_k)Y_7^T+\underset{t_k}{\overset{t}{\int }}{e}_d(s){\mathrm{dsY}}_{8d}^T)\underset{t_k}{\overset{t}{\int }}{\theta }_d(s)\mathrm{ds}\hfill \\ \multicolumn{1}{c}{}& -e_d^T(t)M_d{\theta }_d(t)+2e_d^T(t)M_d{e}_d(t_k)-e_d^T(t_k)M_d{e}_d(t_k)\hfill \\ \multicolumn{1}{c}{}& -e_d^T(t)(R_{1d}+R_{1d}^T)e_d(t)-2e_d^T(t)(R_{2d}-R_{1d})e_d(t_k)\hfill \\ \multicolumn{1}{c}{}& -2e_d^T(t)R_{3d}e\underset{t_k}{\overset{t}{\int }}{\theta }_s(s)\mathrm{ds}-2e_d^T(t_k)R_{4d}\underset{t_k}{\overset{t}{\int }}{e}_s(s)\mathrm{ds}\hfill \\ \multicolumn{1}{c}{}& -{(\underset{t_k}{\overset{t}{\int }}{e}_s(s)\mathrm{ds})}^T(R_5+R_5^T)\underset{t_k}{\overset{t}{\int }}{e}_s(s)\mathrm{ds}+2a_d{\theta }_d^T(t)U_1^T{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{}& +2a_d\stackrel{.}{{\theta }_d^T}(t)U_2^T{\theta }_d(t)+2{\theta }_d^T(t)U_1^T(G_d-\mathrm{cD}){\theta }_d(t_k)\hfill \\ \multicolumn{1}{c}{}& +2\stackrel{.}{{\theta }_d^T}(t)U_2^T(G_d-\mathrm{cD}){\theta }_d(t_k)-2{\theta }_d^T(t)U_1^T{\stackrel{.}{\theta }}_d(t)\hfill \\ \multicolumn{1}{c}{}& -2\stackrel{.}{{\theta }_d^T}(t)U_1^T{\stackrel{.}{\theta }}_d(t)+{\mathrm{nQ}}_{1d}^{-1}{\theta }_d^T(t)U_{1d}{U}_{1d}^T{\theta }_d(t)\hfill \\ \multicolumn{1}{c}{}& +{\mathrm{nQ}}_{2d}^{-1}{\theta }_d^T(t)U_{2d}{U}_{2d}^T{\theta }_d(t)+(Q_{1d}+Q_{2d})\sum _{\upsilon =1}^n{b}_{d\upsilon }{\kappa }_{\upsilon }{\theta }_{\upsilon }^T(t){\theta }_{\upsilon }(t)\hfill \\ \multicolumn{1}{c}{}& +(t-t_k)s(t{)}^T{Y}_d^T{X}_d^{-1}{Y}_{d}s(t).\hfill \end{array}$$

So,

$$\begin{array}{cc}\multicolumn{1}{c}{}& \stackrel{·}{V}\prime (t)\le \sum _{d=1}^n\hfill \\ \multicolumn{1}{c}{}& \{\frac{t_{k+1}-t}{h_k}({\Pi }_{1d}+{\Pi }_{2d}(h_k))+\frac{t-t_k}{h_k}({\Pi }_{1d}+{\Pi }_{3d}(h_k))\},\hfill \end{array}$$

(25)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\Pi }_{1d}=\left[\begin{array}{cccc}{\Pi }_{d11}\hfill & {\Pi }_{d12}\hfill & {\Pi }_{d13}\hfill & {\Pi }_{d14}\hfill \\ \multicolumn{1}{c}{*}& {\Pi }_{d22}\hfill & {\Pi }_{d23}\hfill & {\Pi }_{d24}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\Pi }_{d33}\hfill & {\Pi }_{d34}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & {\Pi }_{d44}\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{}& {\Pi }_{2d}(h_k)=h_k\left[\begin{array}{cccc}{\Theta }_{d11}\hfill & {\Theta }_{d12}\hfill & R_{4d}^T+H_2\hfill & R_{5d}+R_{5d}^T\hfill \\ \multicolumn{1}{c}{*}& X_{1d}\hfill & {\Theta }_{d23}\hfill & R_{3d}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & 0\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{}& {\Pi }_{3d}(h_k)=h_k\left[\begin{array}{cccc}-H_{1d}\hfill & 0\hfill & -H_{2d}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& 0\hfill & -H_{3d}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & 0\hfill \end{array}\right]+(t-t_k)h_k{Y}_d^T{X}_d^{-1}{Y}_d,\hfill \\ \multicolumn{1}{c}{}& {\Theta }_{d11}=R_{3d}+H_1+X_{3d},\hfill \\ \multicolumn{1}{c}{}& {\Theta }_{d12}=M_d+R_{1d}+R_{1d}^T+H_{1d}+X_{2d}^T,\hfill \\ \multicolumn{1}{c}{}& {\Theta }_{d13}=M_d^T-R_{2d}+R_{1d}+H_{3d}+H_{2d}.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\multicolumn{1}{c}{{\Pi }_{1d}+{\Pi }_{2d}(h_k)}& =\frac{h_2-h_k}{h_2-h_1}({\Pi }_{1d}+{\Pi }_{2d}(h_2))\hfill \\ \multicolumn{1}{c}{}& +\frac{h_k-h_1}{h_2-h_1}({\Pi }_{1d}+{\Pi }_{2d}(h_1)),\hfill \end{array}$$

(26)

It can be concluded that $\underset{1d}{\Pi }+\underset{2d}{\Pi }(h_2)>0$ and $\underset{1d}{\Pi }+\underset{2d}{\Pi }(h_1)<0$ , which base on equation (13). Thus, $\underset{1d}{\Pi }+\underset{2d}{\Pi }(h_k)<0$ .

Similar to equation (25),

$$\begin{array}{cc}\multicolumn{1}{c}{{\Pi }_{1d}+{\Pi }_{3d}(h_k)}& =\frac{h_2-h_k}{h_2-h_1}({\Pi }_{1d}+{\Pi }_{3d}(h_2))\hfill \\ \multicolumn{1}{c}{}& +\frac{h_k-h_1}{h_2-h_1}({\Pi }_{1d}+{\Pi }_{3d}(h_1)),\hfill \end{array}$$

(27)

Base on schur complement to equation (14), $\underset{1d}{\Pi }+\underset{3d}{\Pi }(h_2)<0$ and $\underset{1d}{\Pi }+\underset{3d}{\Pi }(h_1)>0$ , and $\underset{1d}{\Pi }+\underset{3d}{\Pi }(h_k)<0$ .

Based on equations (24)–(26), $\stackrel{·}{V}\prime (t)<0.$ So, there is a scalar $\tilde{\omega }$ , then

$$\stackrel{·}{V}\prime (t)+\varpi V\prime (t)<0,t\in [t_k,t_{k+1}),$$

(28)

so

$$\begin{array}{c}V\prime (t)<e^{-{\varpi }_k(t-t_k)}V\prime (t_k).\\ \begin{array}{cc}\multicolumn{1}{c}{V_6(t)}& ={[V_1(t)+V_2(t)+V_3(t)+V_4(t)+V_5(t)]}^b,\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{V}}_6(t)}& =b{[V_1(t)+V_2(t)+V_3(t)+V_4(t)+V_5(t)]}^{b-1}\hfill \\ \multicolumn{1}{c}{}& [{\stackrel{·}{V}}_1(t)+{\stackrel{·}{V}}_2(t)+{\stackrel{·}{V}}_3(t)+{\stackrel{·}{V}}_4(t)+{\stackrel{·}{V}}_5(t)],\hfill \\ \multicolumn{1}{c}{\stackrel{·}{V}(t)}& =\stackrel{·}{V}\prime (t)+{\stackrel{·}{V}}_6(t)=\stackrel{·}{V}\prime (t)+b{V}^{b-1}(t)\stackrel{·}{V}\prime (t)\hfill \\ \multicolumn{1}{c}{}& =\stackrel{·}{V}\prime (t)[1+b{V}^{b-1}(t)]\le -\beta V^{b-1}(t),\hfill \end{array}\end{array}$$

(29)

So $\stackrel{·}{V}(t)<-\beta V^b(t)$ .

According to Lemma 1, the stability of the kth channel is achieved, thus finite time pinning function projective synchronization control is achieved.

• Remark 1. The novel controller is designed to assure the finite time pinning synchronization. The upper bound of maximum sampling and the coupling strength can be calculated by using the LMI box. Base on Jensen’s inequality, Schur theorem and Wirtinger’s inequality, the finite time controller is designed to achieve function projective synchronization control by pinning control.

Finite time pinning function projective synchronization control of complex networks with time-varying delay

In this section, finite time pinning function projective synchronization of complex networks with time-varying delay is investigated. Time-varying delay exists widely in information transmission, the time-varying delays are more widespread than constant delay. The time-varying delay to be considered in complex networks, the time-varying delay is $\delta (t)$ . $x_i(t-\delta (t))$ and $z(t-\delta (t))$ are the state of the ith node and the target node, respectively. The upper bound and lower bound of the sampling intervals are $h{\prime }_2$ and $h{\prime }_1$ .

$$h{\prime }_1=h_1+\delta (t)\le t_{k+1}-t_k+\delta (t)\le h_2+\delta (t)=h{\prime }_2,$$

where $h{\prime }_2$ is the maximum interval between $t_{k+1}$ and $t_k-\delta (t)$ . Designing the following the controller of the finite time synchronization,

$$U_i(t)=-c{d}_i[x_i(t_k-\delta (t))-H(t)s(t_k-\delta (t))].$$

(30)

The equation of state for the ith node,

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{x}}_i(t)}& =A{x}_i(t)+\sum _{j=1}^{N}G{T}_{\mathrm{ij}}[x_j(t_k-\delta (t))-x_i(t_k-\delta (t)]\hfill \\ \multicolumn{1}{c}{}& -c{d}_i[x_i(t_k-\delta (t))-H(t)z(t_k-\delta (t))]+\mathrm{Bf}(u_i(t)),\hfill \end{array}$$

(31)

which

$$\stackrel{·}{e}(t)<a_d{e}_d(t)+\sum _{Q=1}^n{b}_{d\theta }{F}_Q+(G_d-\mathrm{CD})e_d(t_k-\delta (t)).$$

Let $\omega (t)=t-t_k+\delta (t)$ and $\stackrel{·}{\omega }(t)=1+\stackrel{·}{\delta }(t),t\ne t_k$ . Then

$$\stackrel{·}{e}(t)=a_d{e}_d(t)+\sum _{d=1}^n{b}_{d\theta }{F}_{\theta }+(G_d-\mathrm{CD})e_d(t-\omega (t)),$$

(32)

In the following context, a novel condition is obtained to guarantee finite time pinning function projective synchronization.

• Theorem 2. According to Lemma 2, delay $\delta (t)>0$ and $h{\prime }_2$ is the maximum time interval, which $h{\prime }_2$ between $t_k-\delta (t)$ and $t_{k+1}$ , if there is the following matrices,

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\Omega }}_d=\left[\begin{array}{ccccccc}{\mathrm{\Omega }}_{d11}\hfill & {\mathrm{\Omega }}_{d12}\hfill & {\mathrm{\Omega }}_{d13}\hfill & W_{3d}\hfill & 0\hfill & U_{1d}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\mathrm{\Omega }}_{d22}\hfill & {\mathrm{\Omega }}_{d23}\hfill & 0\hfill & 0\hfill & 0\hfill & U_{2d}^T\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -\frac{{\pi }^2}{4}{s}_d\hfill & \frac{{\pi }^2}{4}{s}_d\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & {\mathrm{\Omega }}_{d44}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & {\mathrm{\Omega }}_{d55}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -{\rho }_{1d}I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & 0\hfill & -{\rho }_{1d}I\hfill \end{array}\right]}\end{array}$$

(33)

where

$$\begin{array}{c}{\mathrm{\Omega }}_{d11}=W_{3d}+a_d(U_{1d}+U_{1d}^T),\hfill \\ {\mathrm{\Omega }}_{d12}=W_{1d}-U_{1d}^T+a_d(U_{2d}+U_{2d}^T),\hfill \\ {\mathrm{\Omega }}_{d13}=U_{1d}^T(G_d-\mathrm{cD}),\hfill \\ {\mathrm{\Omega }}_{d22}=W_{3d}+h{\prime }_2{S}_d+(h\prime -\delta (t))Z_{2d}-U_{2d},\hfill \\ {\mathrm{\Omega }}_{d23}=U_{2d}^T(G_d-\mathrm{cD}),\hfill \\ {\mathrm{\Omega }}_{d44}=-W_{2d}-W_{3d}+Z_{1d}-Z_{2d}-\frac{{\pi }^2}{4}{S}_d,\hfill \\ {\mathrm{\Omega }}_{d55}=-Z_{1d}-Z_{2d}.\end{array}$$

Therefore, for any initial condition, finite time pinning control can be completed.

• Proof. The following Lyapunov function is contracted,

$$V(t)=V_1(t)+V_2(t)+V_3(t),$$

where

$$\begin{array}{l}{\tilde{V}}_1(t)=\sum _{d=1}^n\left\{{\tilde{e}}_d^T(t){\tilde{W}}_{1d}{\tilde{e}}_d(t)+{\int }_{t-\delta (t)}^t{\tilde{e}}_d^T(s){\tilde{W}}_{2d}{\tilde{e}}_d(s)ds+\tau (t){\int }_{\delta (t)}^0{\int }_{t+\phi }^t\widetilde{\stackrel{·}{e}}(s){\tilde{W}}_{3d}\widetilde{\stackrel{·}{e}}(s)dsd\phi \right\},\\ {\tilde{V}}_2(t)=\sum _{d=1}^n{\int }_{t-h{\prime }_2}^{t-\delta (t)}{\tilde{e}}_d^T(s){\tilde{Z}}_{1d}{\tilde{e}}_d(s)ds+{\tilde{V}}_{Z_{2d}}(t),\\ {\tilde{V}}_3(t)=\sum _{d=1}^n\left\{{h^{\prime }}_2{\int }_{t-\delta (t)}^t\widetilde{\stackrel{·}{e}}(s){\tilde{S}}_d\widetilde{\stackrel{·}{e}}(s)ds+\frac{{\pi }^2}{4}{\int }_{t_k-\delta (t)}^{t-\delta (t)}{[{\tilde{e}}_d^T(s)-{\tilde{e}}_d^T(t_k-\delta (t))]}^T\times {\tilde{S}}_d[{\tilde{e}}_d^T(s)-{\tilde{e}}_d^T(t_k-\delta (t))]ds\right\},\end{array}$$

with

$${\tilde{V}}_{z2d}(t)=\sum _{d1}^n[h{\prime }_2-\delta (t)]{\int }_{t_k-\delta (t)}^{-\delta (t)}{\int }_{t+\varphi }^t\widetilde{\stackrel{·}{e_d^T}}(s)Z_{2d}\widetilde{\stackrel{·}{e}}(s)\mathrm{dsd}\varphi ,$$

Taking the time derivative of $\tilde{V}(t)$ ,

$$\begin{array}{l}{\stackrel{·}{\stackrel{~}{V}}}_1(t)=\sum _{d=1}^n\{2{\tilde{e}}_d^T(t){\tilde{W}}_{1d}{\stackrel{·}{\stackrel{~}{e}}}_d(t)+{\tilde{e}}_d^T(t){\tilde{W}}_{2d}{\stackrel{·}{\stackrel{~}{e}}}_d(t)-{\tilde{e}}_d^T(t-\delta (t)){\tilde{W}}_{2d}{\tilde{e}}_d(t-\delta (t))\delta (t)\\ +\stackrel{·}{\delta }(t){\int }_{\delta (t)}^0{\int }_{t+\varphi }^t{\widetilde{\stackrel{·}{e}}}_d^T(t)W_{3d}{\widetilde{\stackrel{·}{e}}}_d(t)dsd\varphi +\delta (t){\widetilde{\stackrel{·}{e}}}_d^T(t){\tilde{W}}_{3d}{\widetilde{\stackrel{·}{e}}}_d(t)-\delta (t){\int }_{t-\delta (t)}^t{\widetilde{\stackrel{·}{e}}}_d^T(t){\tilde{W}}_{3d}{\widetilde{\stackrel{·}{e}}}_d(t)ds,\\ {\stackrel{·}{\stackrel{~}{V}}}_2(t)=\sum _{d=1}^n\{{\tilde{e}}^T[t-\delta (t)]{\tilde{Z}}_{1d}\tilde{e}[t-\delta (t)]-{\tilde{e}}^T[t-{h^{\prime }}_2(t)]{\tilde{Z}}_{1d}\tilde{e}[t-{h^{\prime }}_2(t)]\\ +[{h^{\prime }}_2(t)-\delta (t)]{{\widetilde{\stackrel{·}{e}}}_d}^T(t){\tilde{Z}}_{2d}{\widetilde{\stackrel{·}{e}}}_d(t)\},\\ {\stackrel{·}{\stackrel{~}{V}}}_3(t)=\sum _{d=1}^n\{h{\prime }_2{\widetilde{\stackrel{·}{e}}}_d^T(t){\tilde{S}}_d\widetilde{\stackrel{·}{e}}(t)-h\prime {\widetilde{\stackrel{·}{e}}}_d^T(t)(t_k-\delta (t)){\tilde{S}}_d\widetilde{\stackrel{·}{e}}(t_k-\delta (t))\stackrel{·}{\delta }(t)\\ +\frac{{\pi }^2}{4}{[{\tilde{e}}_d(t-\delta (t))-{\tilde{e}}_d(t_k-\delta (t))]}^T\times {\tilde{S}}_d[{\tilde{e}}_d(t-\delta (t))-{\tilde{e}}_d(t_k-\delta (t))](1-\stackrel{·}{\delta }(t))\}.\end{array}$$

Base on Jensen’s inequality

$$\begin{array}{cc}\multicolumn{1}{c}{}& -\delta (t){\int }_{t-\delta (t)}^t\widetilde{\stackrel{·}{e_d^T}}(s){\tilde{W}}_{3d}{\widetilde{\stackrel{·}{e}}}_d(s)\mathrm{ds}\hfill \\ \multicolumn{1}{c}{}& \le -{\int }_{t-\delta (t)}^t\widetilde{\stackrel{·}{e_d^T}}(s){\tilde{W}}_{3d}{\widetilde{\stackrel{·}{e}}}_d(s)\mathrm{ds}{\int }_{t-f(t)}^t{\widetilde{\stackrel{·}{e}}}_d(s)\mathrm{ds}\hfill \\ \multicolumn{1}{c}{}& \le -[{\tilde{e}}_d^T(t)-{\tilde{e}}_d^T(t-\delta (t)){]}^T{\tilde{W}}_{3d}[{\tilde{e}}_d^T(t)-{\tilde{e}}_d^T(t-\delta (t))]\hfill \\ \multicolumn{1}{c}{}& -(h{\prime }_2-\delta (t)){\int }_{t-h{\prime }_2}^{t-\delta (t)}\widetilde{\stackrel{·}{e_d^T}}(s){\tilde{Z}}_{2d}{\widetilde{\stackrel{·}{e}}}_d(s)\mathrm{ds}\hfill \\ \multicolumn{1}{c}{}& \le -{\int }_{t-h{\prime }_2}^{t-\delta (t)}\widetilde{\stackrel{·}{Q_d^T}}(s)\mathrm{ds}{\tilde{Z}}_{2d}{\int }_{t-h{\prime }_2}^{t-\delta (t)}\widetilde{\stackrel{·}{e_d^T}}(s){\tilde{Z}}_{2d}{\widetilde{\stackrel{·}{e}}}_d(s)\mathrm{ds}\hfill \\ \multicolumn{1}{c}{}& \le -{\int }_{t-h{\prime }_2}^{t-\delta (t)}\widetilde{\stackrel{·}{e_d^T}}(s)\mathrm{ds}{\tilde{Z}}_{2d}{\int }_{t-h{\prime }_2}^{t-\delta (t)}{\widetilde{\stackrel{·}{e}}}_d(s)\mathrm{ds}\hfill \\ \multicolumn{1}{c}{}& \le -[{\tilde{e}}_d^T(t-\delta (t))-{\tilde{e}}_d^T(t-h{\prime }_2){]}^T{\tilde{Z}}_{2d}[{\tilde{e}}_d^T(t-\delta (t))-{\tilde{e}}_d^T(t-{h\prime }_2)].\hfill \end{array}$$

The vector is defined as follows

$$\tilde{S}t=[{\tilde{e}}_d^T(t)\widetilde{\stackrel{·}{e_d^T}}(t){\tilde{e}}_d^T(t-\delta (t)){\tilde{e}}_d^T(t-\delta (t)){\tilde{e}}_d^T(t-h{\prime }_2){]}^T.$$

Moreover

$$2q^T{\tilde{U}}_d^T[a_d{\tilde{e}}_d(t)+\sum _{\theta =1}^n{b}_{d\theta }{F}_{\theta }+({\tilde{G}}_d-\tilde{D}){\tilde{e}}_d(t_k-\delta (t))-\widetilde{\stackrel{·}{e_d^T}}(t)]=0,$$

where ${\tilde{U}}_d=[{\tilde{U}}_{1d}{\tilde{U}}_{2d}000{]}^T$ is real matrix.

Base on equations (23), (27) and Lemma 2,

$$\widetilde{\stackrel{·}{V}}(t)=\sum _{d=1}^n{\tilde{S}}^T(t){\mathrm{\Omega }}_d\tilde{S}(t),$$

where

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\Omega }}_d=\left[\begin{array}{ccccc}{\mathrm{\Omega }}_{d11}\hfill & {\tilde{\mathrm{\Omega }}}_{d12}\hfill & {\tilde{\mathrm{\Omega }}}_{d13}\hfill & {\tilde{W}}_{3d}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\mathrm{\Omega }}_{d22}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -\frac{{\pi }^2}{4}{S}_d\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & {\tilde{\mathrm{\Omega }}}_{d44}\hfill & Z_{2d}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & {\tilde{\mathrm{\Omega }}}_{d55}\hfill \end{array}\right],}\end{array}$$

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathrm{\Omega }}_{d11}={\tilde{\mathrm{\Omega }}}_{d11}+{\mathrm{nP}}_{1d}^{-1}{U}_{1d}^T{\tilde{U}}_{1d},\hfill \\ \multicolumn{1}{c}{}& {\mathrm{\Omega }}_{d22}={\tilde{\mathrm{\Omega }}}_{d22}+{\mathrm{nP}}_{2d}^{-1}{{\tilde{U}}^T}_{2d}{\tilde{U}}_{2d}.\hfill \end{array}$$

The $\stackrel{·}{V}(t)<0.$ Then finite time pinning function projective synchronization control of complex networks is achieved.

Numerical simulations

In this section, an example is given to prove the validity of a theoretical test. Lorenz equation (31) is chosen as the node of complex networks,

$$\{\begin{array}{c}\stackrel{·}{x}=-\sigma (x-y)\hfill \\ \multicolumn{1}{c}{\stackrel{·}{y}=\mathrm{rx}-y-\mathrm{xz}}\\ \multicolumn{1}{c}{\stackrel{·}{z}=-\mathrm{bz}+\mathrm{xy}}\end{array}$$

(34)

where x, y, z are state variable, $x,y,z\in R^{\eta }$ respectively. The parameters are chosen as $r=28,\sigma =10,b=\frac{13}{5},\alpha =0.5,\beta =15$ , the initial condition is chosen as $x=0.5,y=0.5,z=0.5$ . The trajectory of Lorenz system is shown in Figure 1.

Figure 1.

The trajectory of Lorenz system.

The matrices for equation (1) are chosen as follows,

$$C=\left[\begin{array}{ccc}-1.256\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -6.35\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -3.56\hfill \end{array}\right],$$

$$D=\left[\begin{array}{ccc}3.126\hfill & 2.365\hfill & -2.356\hfill \\ \multicolumn{1}{c}{-2.365}& 1.35\hfill & -1.286\hfill \\ \multicolumn{1}{c}{1.365}& 3.365\hfill & -0.56\hfill \end{array}\right],$$

$$Z=\left[\begin{array}{cccc}-2.3654\hfill & 2.3654\hfill & 0.2341\hfill & 2.3654\hfill \\ \multicolumn{1}{c}{1.897}& -2.3985\hfill & 0.2358\hfill & 0.8529\hfill \\ \multicolumn{1}{c}{5.3691}& 3.6871\hfill & -4.3691\hfill & 2.5934\hfill \\ \multicolumn{1}{c}{4.3652}& 0.3253\hfill & 0.3694\hfill & 2.3674\hfill \end{array}\right],$$

The node 1and node 3 are pinned and pinning matrix is chosen as

$$K=\left[\begin{array}{cccc}5.36\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 6.35\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \end{array}\right].$$

Due to Lorenz system is a three-dimensional system, there is three information interchange channels between node i and j,

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathrm{\Gamma }}_{12}=\mathrm{diag}\left\{010\right\},{\mathrm{\Gamma }}_{13}=\mathrm{diag}\left\{100\right\},{\mathrm{\Gamma }}_{14}=\mathrm{diag}\left\{001\right\},\hfill \\ \multicolumn{1}{c}{}& {\mathrm{\Gamma }}_{21}=\mathrm{diag}\left\{000\right\},{\mathrm{\Gamma }}_{23}=\mathrm{diag}\left\{000\right\},{\mathrm{\Gamma }}_{24}=\mathrm{diag}\left\{001\right\},\hfill \\ \multicolumn{1}{c}{}& {\mathrm{\Gamma }}_{31}=\mathrm{diag}\left\{010\right\},{\mathrm{\Gamma }}_{32}=\mathrm{diag}\left\{001\right\},{\mathrm{\Gamma }}_{34}=\mathrm{diag}\left\{101\right\},\hfill \\ \multicolumn{1}{c}{}& {\mathrm{\Gamma }}_{41}=\mathrm{diag}\left\{100\right\},{\mathrm{\Gamma }}_{42}=\mathrm{diag}\left\{001\right\},{\mathrm{\Gamma }}_{43}=\mathrm{diag}\left\{010\right\}.\hfill \end{array}$$

The matrices $Z_d(d=1,2,3)$ as follows,

$$\begin{array}{cc}\multicolumn{1}{c}{}& Z_1=\left[\begin{array}{cccc}-0.356\hfill & 0\hfill & 0\hfill & 1.5751\hfill \\ \multicolumn{1}{c}{2.3652}& -2.6583\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{5.3694}& 0\hfill & 3.6541\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 3.2561\hfill & 5.3698\hfill & -0.6589\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{}& Z_2=\left[\begin{array}{cccc}-5.3698\hfill & 2.3654\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{2.3654}& -1.3654\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1.8963\hfill & 5.8521\hfill & 2.3659\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0.3698\hfill & -2.3681\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{}& Z_3=\left[\begin{array}{cccc}-1.3654\hfill & 0\hfill & 0\hfill & 5.3698\hfill \\ \multicolumn{1}{c}{1.3256}& -2.3658\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -2.3651\hfill & -3.3658\hfill \\ \multicolumn{1}{c}{4.2356}& 0\hfill & 0\hfill & -5.3629\hfill \end{array}\right].\hfill \end{array}$$

The initial value of the target node is $s_0=[0.50.50.5{]}^T$ , the error $e_{\mathrm{id}}(t)=x_{\mathrm{id}}(t)-h_{\mathrm{id}}(t)s_{0d}$ , $h_{\mathrm{id}}=\sin (\mathrm{it}),$ where $d=1,2,3,i=1,2,3,4.$ Figures 2–4 show that the synchronization error converge to zero in finite time.

Figure 2.

Synchronization error of the first channel and target node.

Figure 3.

Synchronization error of the second channel and target node.

Figure 4.

Synchronization error of the third channel and target node.

Conclusions

Finite time pinning function projective synchronization control for complex networks is investigated, the time-varying delay is considered in network synchronization. The complex networks can be achieved function projective synchronization in finite time by control target nodes. A finite time Lyapunov function has been established to accomplish pinning function projective synchronization. Sufficient conditions have been derived to ensure the finite time pinning synchronization for complex networks with a target node. A discrete controller have been constructed by the method of finite time and the pinning control.