Output-feedback event-triggered boundary control of reaction–diffusion PDEs with delayed actuator

We present an output-feedback event-triggered delay-compensated boundary control scheme for a class of reaction–diffusion PDEs under a delayed actuator, where an arbitrarily long time delay exists between the PDE plant and the ODE actuator. After treating the time delay as a transport PDE, the overall plant configuration becomes ODE-PDE-PDE. Combining the PDE and ODE backstepping designs, a three-step backstepping transformation is proposed to build the continuous-in-time control law. A PDE observer is designed to track the PDE states required in the control law using a boundary measurement. Then, a dynamic event-triggering mechanism is designed, based on the evaluation of the overall ODE-PDE-PDE system, to determine the updating times of the control signal. In the resulting output-feedback event-based closed-loop system, a strictly positive lower bound of the minimal dwell time is found, which is independent of initial conditions. As a result, the absence of a Zeno behavior is guaranteed. Besides, exponential convergence to zero of all signals is proved via Lyapunov analysis, including the $H^1$ norm of the transport PDE state, the $L^2$ norm of the reaction–diffusion PDE and observer states, the actuator states, the internal dynamic variable in the event-triggering mechanism, as well as the control input. The effectiveness of the proposed method is illustrated by numerical simulation.