Synchronization Control Of Complex Dynamical Network Systems

Representing a complex system as a network consisting of numerous coupled individuals is a key approach to describe and understand different behaviors of a complex system. Complex network has become highly active, with many new systems being proposed in application areas ranging from physics to biology to social science and computer science. Synchronization problem, as a common phenomenon of a population of dynamically interacting units, is one of the most demonstrating topics in complex networks. Ubiquitous in various kinds of complex networks, synchronization is closely related to multiple external flock behaviors. Therefore it is potentially of great significance to investigate synchronization control problem of dynamical systems on complex networks. The main objective of this dissertation is to investigate the synchronization control of a class of continuous-time coupled dynamical networks. We introduce several effective controllers for synchronization of distinct networks, which primarily encompass static coupled networks, time-varying networks and general uncertain delayed coupled networks. The main content of this dissertation are summarized as follows:1. A class of random pseudofractal network models (RPNs) is proposed for modeling the network topology in the sequel. RPNs are generated and evolved according to the rules of fractal growth and competition. Aspects of degree distribution, clustering coefficient and average path length of the RPNs can be obtained by taking continuous approximation. In addition, analytical and numerical results indicate that the PRN model possess a power-law distribution with the exponents tuned by model parameters between 2 and 3, and small-world effect, consistent with most real networks.2. An adaptive pinning control scheme is proposed for synchronizing coupled weighted dynamical networks. Number estimation and selection strategy of pinned nodes, as two fundamental pinning design problems, are solved. To avoid the prediction of linear feedback gains in pinning process, an adaptive pinning scheme are proposed, with the criteria of local and global synchronization being given by construction of Lyapunov functions. In particular, adaptive pinning and linear pinning, in the aspect of pinning strategy, are proven to be equivalent. As a result, number estimation and selection strategy of pinned nodes can be solved by using matrix theory and linear matrix inequality theory, which also set the foundation for subsequent discussion. Firstly, necessary and sufficient condition of the number of pinned nodes is given in form of eigenvalues of coupling matrix. Secondly, LMI-based criteria are given to determine the pinned nodes. As an extension to networks with general topology, several pinning and searching strategies are introduced respectively according to degrees of nodes.3. Synchronization control of fast switching coupled network is studied, with the specific task of dictating multiple mobile agents in a plane towards a desired orbit. The problem in this dissertation is investigated by means of pinning, where each agent is associated with a chaotic oscillator coupled with those of neighboring ones. In addition, the pinning strategy is achieved by exerting common linear feedback on a small fraction of agents that are chosen randomly. We explore aspects of pinning probability, feedback gains and agent density under the assumption of fast switching constraint. Particularly, we show that in cases where network synchronization region is unbounded, synchronization of the dynamical network is exclusively determined by pinning density. However, such control strategy works little for or even weakens the synchronizability for networks with single bounded synchronization region.4. Synchronization controllers for uncertain dynamical networks with delays are designed. We consider linearly-coupled networks with uncertain delay, typified by bounded entries of coupling matrix. Synchronization schemes based on linear feedback and adaptive control are demonstrated respectively. For a nonlinearly-coupled network with completely unknown or partially known nodes, two adaptive synchronization controllers are proposed respectively. Several criteria guaranteeing synchronization of these systems are established by employing the Lyapunov-Krasovskii functionals as well as numerical results for validation. In particular, we show that such controllers are highly robust against uncertain parameters, such as node functions, coupling matrix and time delays.