State and output feedback boundary control for a coupled PDE–ODE system

This note is devoted to stabilizing a coupled PDE–ODE system with interaction at the interface. First, a state feedback boundary controller is designed, and the system is transformed into an exponentially stable PDE–ODE cascade with an invertible integral transformation, where PDE backstepping is employed. Moreover, the solution to the resulting closed-loop system is derived explicitly. Second, an observer is proposed, which is proved to exhibit good performance in estimating the original coupled system, and then an output feedback boundary controller is obtained. For both the state and output feedback boundary controllers, exponential stability analyses in the sense of the corresponding norms for the resulting closed-loop systems are provided. The boundary controller and observer for a scalar coupled PDE–ODE system as well as the solutions to the closed-loop systems are given explicitly.