Pinning Control In Dynamical System: Time-varying Topologies And Pinned Vertex Set

In this paper, we study pinning control problem of coupled dynamical systems. We start from the network topological properties, and establish tractable sufficient conditions for stability of the coupled system with the conclusions in Gragh Theory, Algebra and Stochastic Process. Firstly, we consider deterministic directed time varying topologies. According to the detailed complex network topology, selective pinning based on the connectivity degrees has better performance than totally randomly pinning. By adding an extra virtual static agent to the original network, the stability of system is conducted to consensus problem of the extended network, then we give the sufficient conditions for stability of networks with deterministic time-varying topologies and pinned vertex set. Secondly, with the same method, we conclude the consensus conditions for independent stochastically switching and Markovian switching topologies. Thirdly, we study pinning control problem of nonlinearly coupled dynamical systems with stochastically switching couplings and stochastically selected controller-node set. By introducing the function class QUAD, we consider two scenarios:1.lf each subsystem in the switching system with fixed couplings can be stabilized by fixed pinning nodes and the Markovian switching is sufficiently slow, the system can be stabilized; 2.If the system with average coupling matrix and pinning gains can be stabilized and the Markovian switching is sufficiently fast, the system can be stabilized. In the end, we not only give some numerical examples to illustrate the theoretical results, but also study the correlation between the convergent speed and parameters or variables. In addition, we have deeper understanding on RWP.