  We present an adaptive event-triggered boundary control scheme for a
parabolic PDE-ODE system, where the reaction coefficient of the parabolic PDE,
and the system parameter of a scalar ODE, are unknown. In the proposed
controller, the parameter estimates, which are built by batch least-square
identification, are recomputed and the plant states are resampled
simultaneously. As a result, both the parameter estimates and the control input
employ piecewise-constant values. In the closed-loop system, the following
results are proved: 1) the absence of a Zeno phenomenon; 2) finite-time exact
identification of the unknown parameters under most initial conditions of the
plant (all initial conditions except a set of measure zero); 3) exponential
regulation of the plant states to zero. A simulation example is presented to
validate the theoretical result.
