Stability Analysis And Cooperativity Control Of Fractional-order Complex Networks

With the rapid development of information technology,the world has entered a new era,the network era.A variety of complex networks can be seen in every corner of human life,such as the Internet,the road transportation network,the power network,the mobile communication network,the interpersonal network,and so on.Naturally,the research on complex networks has become a hot research topic in the current world and has attracted researchers from various fields such as the mathematical science,life science,and engineering science.It should be noted that compared to the integer-order calculus,the fractional-order calculus shows greater advantages in the accurate modeling of systems and processes in the real world,therefore it has become a research hotspot for scholars at home and abroad to study the dynamical characteristics and control problems of fractional-order complex networks by introducing the fractional-order calculus into the modeling of complex networks.Considering the great research value and the potential of the fractional-order complex networks,this article studies the topology identification and the cooperativity control problems of the fractional-order complex networks based on the fractional-order calculus theory and the graph theory.Meanwhile,to improve the qualitative theory for dynamic analysis of the fractional-order complex networks,the article also focuses on the study about the stability of fractional-order differential equations.Specifically,the paper has done the following work:1.The stability of fractional-order differential equations is studied.The fractionalorder Lyapunov method is improved so that it can be used to analyze the dynamical behaviors of more general fractional-order complex networks.Firstly,the article takes the Caputo fractional-order differential equations as the research objects,analyzes the stability of the equilibrium points of them,and constructs more generalized stability conditions.Then,the article expands the definition of the Caputo fractional-order derivative,proposes a class of fractional-order differential operators with the short-term memory property,studies the stability of the equilibrium points of the short memory fractional-order differential equations based on the proposed operators,and presents the related stability criteria.Finally,the article conducts some numerical simulations to verify the correctness of the proposed stability results.2.Via using the adaptive-control-based method,the problem about topology identification of fractional-order complex networks is researched.Firstly,to ensure the efficient estimation of unknown topology in complex networks,an auxiliary network composed of isolated nodes and a regulation protocol are constructed.Next,through strict mathematical derivation,the article proves the realizability of the topology identification for complex networks under the designed regulation protocol.That is,the validity of the regulation protocol is verified theoretically.Then,based on the results of realizability analysis,two algorithms are proposed to identify the unknown topology of complex networks.Finally,the effectiveness of the two algorithms is verified by numerical simulations.3.The cooperativity control of fractional-order complex networks with switching communication topologies is studied.Firstly,to reduce the the negative effects of switching phenomena on the cooperative behaviors of networks,a control strategy with information storage mechanisms is designed.Under the control strategy,the state information of network nodes and their neighbors is stored when they can communicate with each other.When the network nodes are disconnected from some of their neighbors due to the communication topologies switch,the stored historical state information is used as control inputs to compensate the dynamical behaviors of the network nodes.Secondly,with the Lyapunov stability theory and the linear matrix inequality(LMI)techniques,some sufficient conditions are constructed to ensure that the synchronization errors of fractional-order switching complex networks under the designed control strategy can be ultimately bounded.Finally,the effectiveness of the control strategy is experimentally verified with numerical simulations.4.The synchronization control of fractional-order complex networks affected by impulsive disturbances and time-varying communication delays is studied.Firstly,a pinning control strategy with the event triggering feature is designed to effectively control the complex networks,reduce the control cost,and to better utilize resources.Then some sufficient conditions are established by using the linear matrix inequality(LMI)techniques to ensure the synchronization of the network under the proposed control strategy.Meanwhile,Zeno behaviors are excluded via a rigorous proof.Moreover,a pinning scheme is proposed that some delay-coupled nodes should be pinned first.Finally,a numerical example is provided to verify the effectiveness of the control strategy and the correctness of the presented synchronization conditions.