Self-Triggered Boundary Control of a Class of Reaction-Diffusion PDEs

This paper provides a novel self-triggered boundary control (STBC) strategy for a class of reaction-diffusion PDEs with Robin actuation using infinite-dimensional backstep-ping boundary control. Our goal is to offer a solution for the continuous monitoring of triggering functions in conventional event-triggered control. We propose a method for converting a certain class of continuous-time dynamic event-triggers that require continuous monitoring to self-triggers that proactively compute the time of the next event at the current event time using the knowledge of the available system states and dynamics. We achieve this by designing a positively and uniformly lower-bounded function which, when evaluated at the current event time, outputs the waiting time until the next event. The control input is updated only at events indicated by the self-trigger and is applied in a zero-order hold fashion between two events. We establish the closed-loop system well-posedness under the proposed STBC approach. Furthermore, we prove that the global
$L^{2}$
-exponential convergence to zero under continuous-time event-triggered boundary control (CETBC) is preserved under the proposed STBC approach. We provide a simulation result that validates the theoretical claims.