Synchronization In Finite-Time And Fixed-Time Of Fully Complex-Valued Dynamic Networks

In today's networked era,complex networks are ubiquitous.As a kind of collective behaviors of complex networks,network synchronization has attracted much attention in recent years due to its wide application in power systems,social networks,viewpoint games and other fields.Complex-valued networks,as an important type of network systems,have stronger information storage capability and signal anti-attack ability than realvalued networks.However,the complex-valued characteristics of the network also bring great difficulties to synchronization analysis,especially,how to explore the more practical synchronization problem in finite time and fixed time is full of challenges in the complex field,and few relevant research results have been published so far.In view of this,several theories,including complex function theory,differential inclusion,impulsive differential equation,will be utilized in this thesis to study the finite-time and fixed-time synchronization for three kinds of complex-valued dynamic networks.In the first part,the problem of finite-time and fixed-time synchronization is investigated for a class of fully complex-valued networks.In order to avoid the previous separation methods for complex-valued systems,the definition and related properties of signum function in the complex field are first given.Based on the introduced signum function,complex-valued control strategies are directly designed for the networks,which is completely different from the method of controlling the decomposed real-valued systems in the previous work.Furthermore,by establishing a new inequality for complex variables and using the Lyapunov function method,the synchronization criteria in the form of matrix inequalities are derived and the previous algebraic conditions are simplified.The established theoretical results are demonstrated finally by providing some numerical simulations.The delay phenomenon of signal transmission is considered in the second part,and the finite-time synchronization is considered for a type of fully complex-valued networks with time-varying delays based on intermittent control schemes.First of all,based on the complex-valued signum function introduced above,several intermittent control strategies for network nodes are designed,which are only dependent of the information of the controlled nodes and are different from the centralized control.Secondly,by using measurable selection theory,Lyapunov functional method,and inequality techniques,the finite-time synchronization conditions for complex-valued networks with delays are established.Finally,some numerical examples are provided to illustrate the validity of the obtained results.The finite-time and fixed-time synchronization is explored in the third part for fully complex-valued complex networks with impulsive effects.Firstly,based on the complexvalued signun function defined in the second part,power-law control strategies are designed for the impulsive networks in the complex field.Secondly,by using the principle of comparison,average impulsive theory and mathematical induction,the conditions are obtained based on different norms to respectively realize the finite-time and fixed-time synchronization for the full complex-valued networks with impulsive effects.Moreover,the method of solving the optimal value and the average impulsive theory are applied to optimize the settling time of synchronization,and a more accurate estimate is given.Finally,several numerical simulations are given to verify the correctness of the theoretical results.