Research On The Optimal Control Problem Of Rumor Spreading On Complex Networks

With the popularization of the Internet,social software such as Weibo and WeChat have become the main platforms for people’s information dissemination and communication,and the spread of rumors has also become more rapid.The spread of rumors has a great impact on social order,people’s psychology and life.The research on the dynamics of rumor propagation on complex networks has always been a very hot topic.At present,the research mainly focuses on the propagation behavior of rumors on the Internet and various factors that affect the propagation of rumors.However,it is also necessary to consider the impact of rumor propagation on society.Therefore,the optimal control of the rumor propagation model is very necessary.In order to solve the problem of rumor propagation and control in the network,this paper focuses on the establishment of the rumor propagation model on the complex network and the stability analysis of the rumor propagation model.And the optimal control problem of rumor propagation model is studied.（1）Aiming at the guiding and controlling role of relevant departments on rumors in real life,a rumor propagation model integrating forgetting mechanism,suspicion mechanism,immune mechanism and guidance and control mechanism is established on a uniform network.（2）For the question of whether the system is stable,discuss the existence and uniqueness of the basic reproduction number and the solution of the system,and obtain the stability of the system by using the basic reproduction number,Lyapunov function,Lienard and Chipart theorem,and further study whether the basic reproduction number is the system sufficient and unnecessary conditions for stability,and an example is given to verify the validity of the theoretical results.（3）Aiming at the control problem of the rumor propagation model,a rumor propagation model with control is established.By analyzing the controllability of the model,the dynamic optimal control strategy of the model is studied,and the corresponding optimal control problem is proposed.The total loss caused by the propagation is the smallest.The existence of the solution and the existence of the optimal control are analyzed,the Hamiltonian function is constructed,the necessary conditions for the optimal control are obtained,and an example is given to verify the validity of the theoretical result.