Finite-Time Bounded Synchronization of the Growing Complex Network with Nondelayed and Delayed Coupling

1School of Finance, Nanjing Audit University, Jiangsu 211815, China 2Jiangsu Key Laboratory of Financial Engineering, Nanjing Audit University, Jiangsu 211815, China 3Department of Mathematics and Finance, Hanjiang Normal University, Hubei 442000, China 4School of Engineering Management, Nanjing Audit University, Jiangsu 211815, China 5College of Information Science and Technology, Donghua University, Shanghai 201620, China 6College of Electronic and Electrical Engineering, Shanghai University of Engineering Science, Shanghai 201621, China


Introduction
In recent years, as finite-time control is an important indicator of test control of complex networks, which not only has important theoretical significance, but also has important practical value in practical engineering, finite-time control of complex networks has become one of the most interesting subjects in control theory of complex networks.To get fast convergence speed, much finite-time stabilization theory is proposed to investigate finite-time control of complex networks [1][2][3][4][5][6][7][8][9][10].For example, in [11], Shen et al. investigated finite-time synchronization control of uncertain Markov jump neural networks by input constraints.In [12], Zhang et al. discussed finite-time stabilization of time-varying nonlinear systems by state feedback control.In [13], Wang et al. discussed finite-time global synchronization of Markovian jump complex networks by partially unknown transition rates with feedback control.In [14], Liu et al. considered finite-time synchronization of Markovian switching neutral complex networks based on pinning controller.In [15], Shi studied finite-time control of linear systems with timevarying sampling, and so on.It is noticed that most of the studies on finite-time synchronization of dynamical network have been mainly focused on static networks.
In addition, as bounded control of complex system is relevant in practical applications, for example, power networks cannot achieve complete synchronization, so it is desirable to obtain conditions within given bound such that the rotor phase differences between the generators remain [16].In recent years, bounded control of complex dynamical network concerns the existence of bounded synchronization region globally stabilizing the complex system, which has grown very quickly, and it is one of the important issues in control theory and applications [17][18][19][20][21].
In [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], the authors studied the numbers of the nodes in the networks invariant about synchronization of complex dynamical networks.As the structures of many complex systems are typically dynamic, some new nodes can enter the network as time goes on [22].Therefore, how to achieve synchronization of a growing dynamical network with bounded error is a very interesting and indeed important subject for research.Motivated by the existing works, the aim of this paper is to discuss finite-time synchronization of the growing complex dynamical network with nondelayed and delayed coupling.A new finite-time stabilization theory is proposed to investigate finite-time bounded synchronization of complex dynamical networks.Finite-time stabilization theory presented in this paper extends the conclusions of literatures [11][12][13][14][15] using finite-time stabilization theory.
The rest of this paper is organized as follows: Section 2 presents model and preliminaries.Section 3 gives the sufficient conditions of finite-time bounded synchronization.Section 4 presents an example and relates simulation results.Section 5 gives the conclusions of this paper.

Model and Preliminaries
Firstly, we introduce the network model and give some useful mathematical preliminaries.

Finite-Time Bounded Synchronization of Complex Networks
In this section, we study finite-time bounded synchronization between two networks.We can give the following main result.
Remark 7. In the network, when the number of nodes   increases, the value of  will also increase, and the suitable  value can always be found.

Conclusion
In this paper, finite-time synchronization for a class of the growing complex dynamical network with nondelayed and delayed coupling was investigated.A new finite-time bounded stable theory is proposed; synchronization criterion was derived to ensure the realization of finite-time bounded synchronization.Finally, numerical simulation was given to verify the effectiveness of the proposed schemes.