Sampled-Data Boundary Feedback Control for Distributed PDE-ODE Cascade Systems

This paper investigates the problem of boundary control for parabolic PDE-ODE cascade systems with distributed connections. A novel sampled-data boundary controller is proposed, and a new analytical framework is developed to establish the stability of the resulting closed-loop system. The framework includes the following three key features: 1) A state transformation is introduced for the concerned PDE-ODE system with non-strict feedback structure using a newly devised backstepping-forwarding technique. 2) A small gain theorem based method and a piecewise continuous Lyapunov approach are utilized to overcome the limitations of existing approaches which are applicable only to pure PDEs. 3) The input-to-state stability (ISS) approach is employed to derive an exponentially stable estimate of the closed-loop system’s state in L² norm. Simulations validate the effectiveness of our approach.