Synchronization Control Of Fractional-order Hybrid Dynamic Network Systems

Fractional-order hybrid dynamic network systems refer to complex networks composed of diverse heterogeneous components that undergo continuous evolution and changes over time,exhibiting hybrid and diverse structural and attribute variations,making them a focal point of research across multiple domains.Representative examples of such hybrid dynamic networks include those with Markov topology switching and parameter jumps,as well as networks with impulse-coupled oscillators or impulsive control mechanisms.Synchronization analysis is a crucial subject in the study of dynamic network systems,and its exploration can effectively enhance the stability and reliability of network systems.The Introduction of fractional calculus into hybrid dynamic network systems can extend the toolkit for network modeling and analysis,enabling a more precise depiction of heterogeneity and long-range memory properties in hybrid networks.Research on the synchronization control of fractional-order hybrid dynamics network systems has already yielded some achievements;however,further investigation is warranted to delve into critical aspects of fractional-order hybrid dynamic network systems.The main research content is as follows:(1)The pinning synchronization problem of fractional-order complex networks under directed fixed topology was investigated.For a class of complex networks without strongly connected structures,a directed acyclic graph condensation theory and a layering algorithm for network partitioning were proposed.The network is divided into several layers,enabling the analysis of synchronization can be carried out layer by layer.Additionally,the Control Rank algorithm which combining degree information and topological properties was introduced to determine the selection of the nodes to be pinned.Furthermore,an energy function was formulated with coupling strength and control gain as variables to minimize energy consumption,while ensuring system synchronization,by considering synchronization criteria for each layer.(2)The quantized synchronization problem of fractional-order dynamic neural networks with reaction-diffusion phenomena under Markov switching topology was investigated.For a class of fractional-order systems with Markov switching characteristics,the continuous frequency distribution model of fractional integrator was introduced,transforming the system into an equivalent integer-order Markov switching model,thus enabling the application of indirect Lyapunov stability theory as the main theoretical tool for synchronization analysis.Secondly,a mode-dependent quantized controller was designed and its effectiveness was verified,demonstrating its advantages in reducing communication rate and channel bandwidth.Moreover,to address spatial reaction-diffusion phenomena in the network and inevitable communication delays while increasing the range of synchronization conditions,the Wirtinger inequality and delay partition technique were utilized,resulting in less conservative sufficient conditions to ensure system synchronization.(3)The asynchronous boundary quantized control problem of fractional-order dynamic neural networks with reaction-diffusion phenomena under Markov parameter jumps was investigated.Considering the spatial and parameter characteristics of the system,an asynchronous boundary quantized control approach was proposed,wherein the controller is applied only at the spatial boundaries,resolving the issue of high cost and operational difficulty when applying controllers at every position in the spatial region in distributed control.Additionally,the adopted control approach is asynchronous,which better matches the mismatch between the system modes and controller modes in practical control scenarios.Furthermore,to mitigate or eliminate chattering phenomena caused by logarithmic quantizer,a hysteresis quantizer was introduced to improve the control performance of the controller.Finally,based on the continuous frequency distribution model and Lyapunov stability theory,sufficient conditions for system synchronization and a mode-dependent control gain matrix were derived.(4)The distributed delayed impulsive control problem of fractional-order reactiondiffusion neural networks with system delays was investigated.For a class of fractionalorder reaction-diffusion neural networks subject to communication delays,based on Laplace and inverse Laplace transforms of fractional calculus,properties of Mittag-Leffler functions,and Lyapunov stability theory,a novel extended Halanay-like inequality was proposed for the first time,resolving the theoretical analysis challenges posed by system delays.Additionally,to achieve the synchronization objective,distributed impulsive control was employed,introducing the concept of control topology to describe the entire controller structure,which considers both local information of nodes and weighted average information from neighboring nodes.Moreover,impulse delays occurring during the control process were also taken into consideration.Finally,under the framework of distributed delayed impulsive control,quasi-synchronization criteria for the system were derived,and numerical simulations were conducted to analyze the influence of system delays and impulse delays on network synchronization.(5)The distributed delayed impulsive control problem of generalized Caputo variableorder fractional reaction-diffusion neural networks with short memory was investigated.Variable-order fractional allows the order of differentiation or integration to vary with different variables and time,and the short memory principle of fractional derivatives effectively addresses the computational cost and efficiency issues caused by the long memory nature of fractional derivatives.Firstly,a fractional-order reaction-diffusion neural network model described by generalized Caputo variable-order partial differential equations was established.Based on the theory of variable-order fractional calculus and the short memory principle of fractional derivatives,a novel Halanay-like inequality was developed,serving as the primary theoretical foundation for synchronization analysis of the system.Moreover,under the framework of distributed delayed impulsive control,by utilizing the direct error method and designing different error vectors,mean synchronization and weighted mean synchronization of the system were achieved under the corresponding derived synchronization conditions.