Stabilization of Coupled Systems Through Boundary Connection

As a main application of the Riesz basis approach, this chapter presents the Riesz basis property for coupled PDEs, where one subsystem is considered as a controller for other subsystems. The coupling occurs through boundary weak connections in the sense that after differentiating the total energy of the system, only the terms for one subsystem is left on the right side. Section 6.1 presents an analysis for coupled beam and heat equation, where the heat equation is considered as the controller. The Riesz basis property shows that the stability and regularity are transmitted from heat subsystem to less regular beam subsystem through the boundary weak connections. The similar phenomena is observed for coupled heat-Schrodinger system discussed in Sect. 6.2. Section 6.3 discusses the Riesz basis property for interconnected Schrodinger-heat system in a torus region. The Gevrey regularity for the associated \(C_0\)-semigroups has been investigated in this chapter.