Research On Synchronization Control Of Coupled Dynamical Systems

In recent years, collective behaviors of coupled dynamical systems have gained in-creasing research interest in both science and engineering. The study contributes to revealthe nature of some interesting phenomena, such as resonances of cicadas, epidemic spread-ing, etc. The human society can be described by a network, in which each individuality isviewed as a node and there exists an edge between two persons if they know each other. Re-garding the whole system as a network composed of interactively dynamical individualitiesis an effective approach to study system properties, based on which great efforts have beenmade to study communication systems, power systems, transportation systems, metastasissystems, supply chain systems, etc. The complexities of coupled dynamical systems comefrom dynamics of each subsystem as well as their complicate connections.Although processing high complexity, coupled dynamical systems have some collec-tive behaviors under suitable conditions, such as self organization and emergence. Inves-tigating these collective behaviors has both theoretical and practical significance in under-standing and designing complex systems. Synchronization is a typical collective behaviorin nature and has been extensively applied to economy, society, physics and biology, etc.Different from isolated system, the dynamics of coupled systems are also determined bycoupling pattern. However, does dynamic evolution have certain impact on system struc-ture? It is generally known that synchronization can be realized by external control or innercoupling. Is there any relation between them? How do we design effective control methodsfor coupled dynamical systems with different structures? Owing to extremely large size,coupled dynamical systems are often subject to stochastic disturbances, invalid componentsand information transfer delay, etc. How will these noises impact on synchronization perfor-mances? And how do we manage them? To solve above problems, this dissertation mainlyfocuses on synchronization control of coupled dynamical systems, as well as mutual effectsbetween coupling structure and synchronization performance. For some classes of coupleddynamical systems, the authors establish corresponding synchronization criteria by stabilitytheory, stochastic analysis, linear matrix inequality and algebraic graph theory, etc. Further-more, general methods that guarantee synchronization for coupled dynamical systems aresummarized. The main contributions of this dissertation are as follows:A class of dynamical networks with Markovian switching structures and mixed cou-pling delays is considered, whose underlying graph is strongly connected. By algebraicgraph theory and stochastic analysis, the authors study the H∞synchronization perfor- mances of such stochastic system with global controllers and local controllers, respectively.The results include some existing results of H∞synchronization of coupled dynamical sys-tems without stochastic disturbances as special cases.A class of dynamical networks with node delays and coupling delays is studied, whoseunderlying graph contains many communities. Through stability theory and matrix theory,the authors establish cluster synchronization conditions. The controlled objects are detectedby means of spectral analysis, based on which cluster synchronization can be realized inlarge scale network.An array of fuzzy neural networks with Markovian switching parameters and Wienerprocess is analyzed, whose underlying network is weakly connected. Applying stochasticanalysis and algebraic graph theory, the authors develop self synchronization criteria forsuch coupled stochastic systems. Synchronization error is defined by some auxiliary matri-ces and error system is proved to be exponentially stable in the mean square. Furthermore,the relation between pinning synchronization and self synchronization of coupled dynamicalsystems is revealed by these auxiliary matrices.Without control inputs, the authors explore how to synchronize coupled stochastic neu-ral networks by discontinuous coupling, whose underlying network is weakly connected.Two kinds of discontinuous coupling modes are involved, periodically intermittent couplingand impulsive coupling. The results include some existing results of stochastic synchroniza-tion of continuous coupled systems as special cases.According to physical properties of memristors, the authors establish coupled memris-tive neural networks with state-dependent switching. Its exponential synchronization per-formances are analyzed from two aspects, disconnected underlying network and weaklyconnected one. By differential inclusion theory, set valued map theory and generalized sta-bility theory, linear matrix inequality-based synchronization conditions are obtained underthe framework of discontinuous system’s solutions. For the first case, controlled objects aredetected by aggregated approach and pinning synchronization is realized. For the secondcase, weakly connected coupling pattern leads to self synchronization.Finally, conclusion is made and future work is presented. In this dissertation, theresearch on synchronization control of coupled dynamical systems not only enrichessynchronization theories, but also provides reliable bases for practical applications. Nu-merical simulations are given to verify the usefulness and effectiveness of theoretical results.