Analysis Of Switching Stability And Synchronization Of Complex Systems

As people know more and more of the real world, people recognize that many phys-ical systems need to be described by switched systems and network-based complex sys-tems. Based on differential and difference equations theory, matrix analysis and graphtheory, this thesis mainly studies the asymptotic stability of a kind of switched positivediscrete-time linear systems and synchronization of two kinds of network-based complexsystems. The main results are as follows.1. Study the asymptotic stability of the associated switched discrete-time system byswitching between the systems produced by (sp) matrices. The concept of an (sp) matrixcharacterizes a kind of asymptotic stable linear time-invariant systems. This thesis firstlydevelops a new definition of the (sp) matrix by means of graph theory. Secondly, based onpartially ordered semigroup and Lie algebraic methods, we establish several new criteriafor absolute asymptotic stability. Generally, a Lie algebraic condition is lack of robustnessproperty, but some of our criteria have this important property to some extent. Thirdly, bycareful analysis for the matrix multiplication, we discover such an arithmetical conditionthat if it is satisfied by the subsystems, then the associate switched system is asymptot-ically stable under arbitrary switching. Furthermore, in case that it is not satisfied, eventhe asymptotic stability under periodic switching can not be assured. This result bringsconvenience to the application of the switched systems of this kind. Taking advantage ofit, we give a new proof, which is simple and direct, for the convergence of the linearizedVicsek's model with leader following. Besides, this result leads to an upper bound esti-mate of the joint spectral radius and the Lyapunov exponent. Lastly, we discuss a kindof higher order difference equations with switching and obtain a sufficient and necessarycondition for its asymptotic stability.2. Study the consensus of the discrete-time Cucker-Smale ?ocking under rootedleadership. The model originates from people's effort to explain the emergent behaviorin ?ocks, such as ?ying birds and fish schools. It also closely relates to many problemsin engineering, such as space ?ight formations and robots systems. The feature of ourproposal, departing from the existing models, is that both the assumption of symmetryand the partial ordering of a hierarchy are dropped. In fact, this is the first time to study the Cucker-Smale model in the absence of both the symmetric and triangle structures.Further, the rooted leadership topology is a necessary condition for the group to convergetowards a single leader's fixed constant velocity. In this thesis the rates of convergenceare established for the ?ocks with fixed and switching topology. It is proved that undercertain conditions on the initial states, the system can achieve velocities consensus, that is,each member follows the leader. Besides, the situation is considered when the membershave free wills, and the convergence of velocities is also established.3. Study the outer synchronization of a kind of coupled networks by Lyapunovmethod. In the case of identical network topologies, we first prove that the outer synchro-nization can be realized using arbitrary coupling strength if the networks are balanced.Then it is shown that the switching can not destroy the outer synchronization for balancednetworks. In the case of nonidentical network topologies, this thesis links the outer syn-chronization and the inner synchronization of drive network. We show that if the drivenetwork can reach inner synchronization and the response network is balanced, then thetwo networks can realize outer synchronization. We also give some numerical examplesto examine the results for the two cases. In case of identical topologies, the examplesindicate that, although the outer synchronization can be obtained by arbitrary couplingstrength, it has real in?uence over the short-time performance of synchronization. How-ever, the transverse Lyapunov exponent is independent of the coupling strength. In otherwords, the strength has nearly no in?uence over the long-time performance of synchro-nization.