  This paper introduces the first observer-based periodic event-triggered
control (PETC) and self-triggered control (STC) for boundary control of a class
of parabolic PDEs using PDE backstepping control. We introduce techniques to
convert a certain class of continuous-time event-triggered control into PETC
and STC, eliminating the need for continuous monitoring of the event-triggering
function. For the PETC, the event-triggering function requires only periodic
evaluations to detect events, while the STC proactively computes the time of
the next event right at the current event time using the system model and the
continuously available measurements. For both strategies, the control input is
updated exclusively at events and is maintained using a zero-order hold between
events. We demonstrate that the closed-loop system is Zeno-free. We offer
criteria for selecting an appropriate sampling period for the PETC and for
determining the time until the next event under the STC. We prove the system's
global exponential convergence to zero in the spatial $L^2$ norm for both
anti-collocated and collocated sensing and actuation under the PETC. For the
STC, local exponential convergence to zero in the spatial $L^2$ norm for
collocated sensing and actuation is proven. Simulations are provided to
illustrate the theoretical claims.
