Some Problems Of Study On Adaptive Control Of Synchronization In Fractional-order Complex Dynamical Networks

In recent years, research on complex dynamical networks has attracted increasing attention from various fields, such as mathematics, physics, system sciences, control sciences, and nonlinear dynamics, due to its broad application prospect. For one thing, the couplings between neighboring nodes increase the complexity of collective behaviors, which stimulates much interest of modeling, analyzing, and predicting dynamical processes on complex networks. From a control perspective, one may utilize the couplings to control the network dynamics, i.e., targeting a network's dynamics towards a desired final state, by driving only a few suitable nodes with external signals. Among them, synchronization in complex networks, as an typical and significant collective behavior, has received increasing attention and become a hot topic due to its potential applications in many fields, including secure communization, smart grid, neural networks, robot rendezvous problem,etc.So far, the great majority of research activities on synchronization have focused on the integer-order real-variable dynamical networks, in which node dynamics are described by the classical ordinary differential equations with real-variable. However, due to the limited theories for fractional-order dynamical systems,the research on fractional-order real-variable or complex-variable is still a challenging topic and the existing works is still few. In particular, the adaptive control has made great progress on synchronization of integer-order complex dynamical networks. It should be noted that the incorporation of weakly singular kernels by fractional-order derivative makes the Lyapunov-based approach for adaptive synchronization of integer-order complex dynamical networks no longer applicable, which is also the "bottleneck" preventing synchronization of fractional-order networks being solved by adaptive control. Therefore, by using the fractional Lyapunov functional method combined with the fractional-order techniques, the fractional-order adaptive control is further studied, and an effective solution for synchronization of some kinds of fractional-order dynamical networks. The main work of this dissertation is summarized as follows:(1) The stability and synchronization control problems of a class of fractional-order time-varying systems, where the linear and nonlinear parts can be separated. By constructing quadratic Lyapunov functions and utilizing a new property for Caputo fractional derivative,some sufficient conditions are derived for the global asymptotical stability and stabilization of the system by using linear state feedback control and adaptive control. Furthermore, by establishing bidirectional link between the drive-response fractional-order systems and designing adaptive law tune its gain, some sufficient criteria are given for reaching synchronization. Finally, some illustrative examples and numerical simulations are given to validate the effectiveness of the proposed stabilization and synchronization controllers, and correctness of theoretical results.(2) The mutual synchronization between two controlled interdependent networks is studied. First, we propose the general model of controlled interdependent networks with time-varying internetwork delays coupling. Second, by constructing suitable Lyapunov functions and utilizing inequality techniques, some sufficient conditions are established for reaching the mutual synchronization by using the designed nonlinear controllers and adaptive laws. Then, the model of interdependent networks and the corresponding theological analysis are extended to the case, where node dynamics and the adaptive laws are described by fractional-order differential equations. At the same time, by using fractional Lyapunov functional method combined with fractional inequality techniques, Mittag-Leffler function, and Laplacetrans form, the rigorous theological proofs are given for projective synchronization by the designed adaptive controllers. Finally, some illustrative examples and numerical simulations verify the effectiveness and correctness of the corresponding synchronization criteria.(3) We study fractional-order decentralized adaptive control of synchronization in fractional-order complex dynamical networks with linear and diffusive coupling. Based on local information among neighboring nodes, several novel fractional-order decentralized adaptive strategies are designed to tune all or only a small fraction of the coupling gains respectively. By constructing quadratic Lyapunov functions and utilizing fractional inequality techniques, Mittag-Leffler function, and Laplace transform,some sufficient conditions are derived for reaching network synchronization by using the proposed adaptive laws. Finally, numerical simulation results agree well with the theoretical analysis.(4) We combine decentralized adaptive control and fractional-order techniques to investigate the synchronization of fractional-order complex-variable dynamical networks.First, the model of fractional-order dynamical networks is extended to the case where the state of each node is a complex variable. A new lemma is proposed for estimating the Caputo fractional derivatives of Hermite quadrtic Lyapunov functions. Based on local information among neighboring nodes, some effective fractional-order decentralized adaptive laws are designed to tune the coupling gains of network nodes. This method is further extended to the case where only a small fraction of coupling gains are choosen to be adapted. By constructing suitable Lyapunov functions and utilizing the proposed lemma, several sufficient criteria are obtained to realize network synchronization by using the proposed adaptive laws. Finally, theoretical analysis and numerical simulations show that synchronization is asymptotically achieved and the adaptive gains asymptotically converge to constant values.(5) The complex projective synchronization in drive-response networks coupled with 1 + N fractional-order complex-variable dynamical systems is investigated. Based on the local mismatch with the desired state and between neighboring nodes, an effective fractional-order fully decentralized adaptive scheme is proposed to tune both coupling weights and feedback gains. By introducing pinning control to the proposed adaptive scheme,several fractional-order fully decentralized adaptive pinning strategies are designed to tune only a small fraction of coupling weights and feedback gains. By constructing suitable Lyapunov functions and utilizing the new lemma, some sufficient criteria are obtained to realize the complex projective synchronization by using the proposed adaptive laws. Finally,numerical simulation results show that synchronization is asymptotically achieved by the designed adaptive strategies.