  For parabolic PDEs, we present a new certainty equivalence-based adaptive
boundary control scheme with a least-squares identifier of an event-triggering
type, where the triggering is based on the size of the regulation error (as
opposed to the identifier updates being triggered by the estimation error, or
the control changes being triggered by the regulation error). The scheme
guarantees exponential convergence of the state to zero in the L2 norm and a
finite-time convergence of the parameter estimates to the true values of the
unknown parameters. The scheme is developed for a specific benchmark problem
with Dirichlet actuation, where the only unknown parameters are the reaction
coefficient and the high-frequency gain. For this specific problem, no existing
adaptive scheme can handle the unknown high-frequency gain. An illustrative
example allows the comparison with other adaptive control design methodologies.
