On Set-Stability,Controllability And Consensus Of Dynamic Networks Over Finite Fields

With the development of network science,more and more researchers have paid their attentions to the dynamic networks over finite fields.Actually,finite fields are usually utilized to model the networks where the agents have limited memory,computation,and communication capabilities,and hence extensively applied in the research of finite automata,sensor networks,quantization problems and so on.For the networks over finite fields(finite-field networks,FFNs),this dissertation focuses on the set-stability,the minimum controllability,stabilization control and the consensus problem of FFNs with time-delays,and the main contents are summarized as follows.(1)With similarity invariants,the set-stability problem is studied for the finite-field networks(FFNs).Basing on the tree-cycle structure,equivalent descriptions of setstability are given in view of the state transition graph and the largest invariant subset,respectively.For a class of special stable sets where the cycle sums consist of self-loops,we derive a necessary and sufficient condition on similarity invariants.Besides,for linear FFN with disturbance,a necessary and sufficient condition for the global stability is obtained.At last,the transient period of the set-stability is estimated by means of a probability-based method and an invariant-subsets-based method,respectively.(2)For the finite-field control networks(FFCNs),the minimum controllability problem(MCP)is studied and the stabilization control is designed.The minimum controllability problem refers to that the minimum set of driving nodes is selected from the network to ensure the system controllability.By applying module theory,a necessary and sufficient condition for controllability is derived.Further,solve the MCPs with single input and multiple inputs respectively by transforming them into the equivalent algebraic problem and the minimum set covering problem,and an upper bound for the minimum of MCP is derived.Fixing the system matrix,a connection between MCPs with single inputs and multiple inputs is revealed.The probability that a feasible solution for MCP occurs is estimated.Then,the stabilization control problem is investigated.By combining module theory and Chinese Remainder Theorem,necessary and sufficient conditions for stabilizability under the direct sum condition and cyclic module condition are obtained,respectively,and the corresponding stabilization controls are designed.(3)For the time-delayed finite-field networks(TDFFNs),the consensus problem and the leader-following consensus problem are investigated.Linear recurring sequence(LRS)theory is applied to explain the dynamics equation of TDFFN and the state transition is described.For the synchronization of TDFFNs,the Smith form criterion and the characteristic polynomial criterion are obtained,respectively.In addition,under the characteristic polynomial condition,a formula on periodic behavior is derived.Also,the leader-following consensus(LFC)problem is studied for the high-ordered TDFFNs.Necessary and/or sufficient conditions are given,respectively,for the LFC under the fixed topology and the switching topologies.