Flow Barriers and Switchability

Abstract

In this chapter, the theory of flow barriers in discontinuous dynamical systems will be systematically presented for the first time, which help one re-think the existing theories of stability and control in dynamical systems. The concept of flow barriers in discontinuous dynamical systems will be introduced, and the passability of a flow to the boundary with flow barriers will be presented. Because the flow barriers exist on the boundary, the switchability of a flow to such a separation boundary will be changed accordingly. The coming and leaving flow barriers in passable flows will be discussed first, and the necessary and sufficient conditions for a flow to pass through the boundary with flow barrier will be developed. Flow barriers for sink and source flows will be also discussed. Once the sink flow is formed, the boundary flow barriers in the sink flow needs to be considered, and such a flow barrier is independent of vector fields in the corresponding domains. Furthermore, when the boundary flow in the sink flow disappears, the vector fields should satisfy the appropriate conditions. Thus, the necessary and sufficient conditions for formations and vanishing of a sink flow will be developed for a discontinuous dynamical system possessing flow barriers on the boundary. A periodically forced friction model will be presented as an example for a better understanding of flow barrier existence in physical problems. The flow barrier theory presented in this chapter will provide a theoretic base for one to further develop control theory and stability.