Synchronization Of Networks With Switching Topology And Time-delays Over Finite Fields

Finite field networks are a kind of networked control systems consisting of multi-agents in a finite field,which are considered due to the fact that networks often have limited memory,communication and computation capabilities.The concept of synchronization is a generalization of network consensus,and the study of synchronization provides new theoretical insights into networks with limited memory,communication and computation capabilities.This paper focuses on the synchronization of networks with switching topology and time-delays over finite fields,and proposes necessary and sufficient conditions for such networks to achieve synchronization.Firstly,according to finite-field networks without switching topology and time-delays,this paper defines the synchronization of networks over finite fields,and derives a corollary of network synchronization by using the adjacency matrix of networks,which makes it easier to determine the synchronization of networks.Secondly,we study the switching networks over finite fields and propose several necessary and sufficient conditions for the synchronization of such networks by using the definition of synchronization,adjacency matrix and transition graphs of networks respectively.Based on the transition graphs of switching networks,Fermat’s little theorem and number theory,we give the correspondence between the number of circles in the transition graphs,their length and the prime numbers of finite fields.Finally,networks with time-delays over finite fields are studied,and some necessary and sufficient conditions for the synchronization of such networks are derived by using the equivalent algebraic form of finitefield networks.Subsequently,based on the transition graphs and adjacency matrix associated with switching networks with time-delays over finite-fields,we present two necessary and sufficient conditions to determine the synchronization of such networks.Some numerical examples are given to verify the validity of the obtained results.