Sampled-Data Boundary Stabilization of PDE-ODE Cascade Systems with Long Delays

This paper investigates the sampled-data feedback stabilization of a PDE-ODE cascade system in which both the actuator and the input experience arbitrarily long delays. The backstepping-forwarding technique is employed to effectively transform the original system with delays into a target system, governed by a linear differential equation with sampled and delayed inputs. A new sampled-data boundary controller is then proposed, simultaneously compensating for delays and sampling effects. The stability of the target system is established via a small-gain-based analysis, which ensures exponential stability of the original closed-loop system through the input-to-state stability (ISS) framework. Theoretical findings are supported by simulations.