Investigation Of Stabilization And Synchronization Control For Complex Dynamical Networks With Similar Nodes

From the engineering view of point, the stabilization property of complex dynamical networks is prerequisite towards effective applications. But the realist networks do not always realize stabilization by themselves. Thus, it is very necessary to investigate the stabilization problem for complex dynamical networks via control strategies. It should also be noticed that synchronization phenomena are ubiquitous in natural and artificial networks, and some synchronous phenomena are very useful for our life. In the past ten or more years, as an important collective behaviors of complex dynamical networks, synchronization has attracted extensive attention from various fields of science and engineering throughout the world.Like to like. In real world, there exists a class of complex dynamical networks whose nodes possess some special similarities in their inner dynamics. For example, the large-scale electric power networks with identical generator units, the wireless sensor networks composed of the same structure sensors and so on are the networks whose nodes possess some inner similarities. These special similarities have direct effect on the stability and synchronization of the networks. From the view of theoretical analysis, these similarities will help to simplify the stabilization and synchronization controllers. It is worth pointing out that similar nodes may be with different state dimensions and may be different from each other.In fact, the complex dynamical network with different-dimensional similar nodes can be more coincident with the real-world networks. It should also be noted that time-varying coupling coefficients, coupling delays or nonlinear coupling functions exist in most realist networks, especially in biological networks and engineering networks.Inspired by the above discussions, this dissertation investigates the stabilization and synchronization problems for the complex dynamical networks with similar nodes and different kinds of outer and inner couplings. Lyapunov stability theory of ordinary differential equations, Lyapunov-Krasovskii stability theorem of delayed systems and Barbalat’s lemma of non-autonomous systems are the main mathematical tools used in the theoretical analysis of this dissertation.The main research work and results are listed in the followings.1. Complex networks and their characters, research meaning and development overview are briefly introduced. Besides, the research contents and categories of complex dynamical networks are also briefly introduced. Furthermore, the definitions, research meaning and research contents of stabilization and synchronization for the complex dynamical networks are introduced, and some necessary topics to be studied in this dissertation are presented.2. The definitions of similar nodes are proposed for the affine linear and nonlinear isolated nodes, respectively.3. Stabilization control problems for two classes of dissipatively coupled complex dynamical networks with similar nodes are investigated, respectively. Firstly, the asymptotic stabilization controllers are synthesized for the complex dynamical networks with nonlinear similar nodes and nonlinear delayed coupling functions. Then, for the time-varying and dissipatively coupled complex dynamical network model with nonlinear similar nodes and nonlinear coupling functions, two kinds of asymptotic stabilization controllers are designed according to its certain or uncertain common bound of the outer coupling coefficients, respectively. Especially, when the common bound is unknown, adaptive control mechanism is introduced and there is only one adaptive law needed. The stabilization strategies proposed here do not depend on the outer coupling information of the network, so they are general. Numerical simulations have verified the effectiveness of the theoretical results mentioned above.4. Synchronization controllers are proposed for three classes of non-delayed coupling complex dynamical networks with different-dimensional similar nodes, which are listed as follows.(1) Decentralized dynamical compensation asymptotic synchronization controllers are proposed for the complex dynamical networks with linear similar nodes and nonlinearly coupled functions.(2) Two kinds of decentralized dynamical compensation asymptotic synchronization controllers are synthesized for the complex dynamical networks with nonlinear similar nodes and nonlinearly coupling functions.(3) To well describe the realist networks, a non-dissipatively coupled time-varying complex dynamical network model with nonlinear similar nodes and nonlinearly coupled functions is introduced. For this kind of networks, the definitions of exponential synchronization and asymptotic synchronization are proposed based on the nodes’ trajectories in the network. Furthermore, synchronization control strategies are proposed for this kind of networks according to the known or unknown common bound of the outer coupling coefficients, respectively. When the common bound is known, the decentralized state-feedback controllers synthesized here can guarantee the network realizing exponential synchronization. And when the common bound is unknown, the adaptive decentralized state-feedback controllers designed here can make the network achieve asymptotic synchronization. Corresponding simulation examples have verified the effectiveness of the theoretical results mentioned above.5. Synchronization controllers are designed for the delayed coupling complex dynamical networks with different-dimensional nonlinear similar nodes, which are listed as follows.(1) Decentralized dynamical compensation and delay-independent synchronization controllers are synthesized for the time-invariant complex dynamical networks with nonlinear similar nodes and constant coupling delays, which can guarantee the networks achieving asymptotic synchronization.(2) To well describe the real-world networks, a general non-dissipatively coupled time-varying complex dynamical network model is introduced here, in which, different time-varying coupling delays for different nodes are considered. Furthermore, decentralized state-feedback delay-independent exponential synchronization controllers or adaptive asymptotic synchronization controllers are proposed for the networks with the known or unknown common bound of their coupling coefficients, respectively.(3) Considering the limitation of the delay-independent synchronization controllers, delay-dependent exponential or adaptive asymptotic synchronization controllers are also designed for the non-dissipatively coupled time-varying complex dynamical networks with nonlinear different-dimensional similar nodes and different time-varying coupling delays according to their certain or uncertain common bound, respectively. Corresponding numerical simulations have verified the effectiveness of the theoretical results mentioned above.This dissertation was supported by the National Science Foundation of China (61273219), the National Science Foundation of Guangdong Province of China (S2013010015768) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20134420110003).