  High-relative-degree control barrier functions (hi-rel-deg CBFs) play a
prominent role in automotive safety and in robotics. In this paper we launch a
generalization of this concept for PDE control, treating a specific,
physically-relevant model of thermal dynamics where the boundary of the PDE
moves due to a liquid-solid phase change -- the so-called Stefan model. The
familiar QP design is employed to ensure safety but with CBFs that are
infinite-dimensional (including one control barrier "functional") and with safe
sets that are infinite-dimensional as well. Since, in the presence of actuator
dynamics, at the boundary of the Stefan system, this system's main CBF is of
relative degree two, an additional CBF is constructed, by backstepping design,
which ensures the positivity of all the CBFs without any additional
restrictions on the initial conditions. It is shown that the "safety filter"
designed in the paper guarantees safety in the presence of an arbitrary
operator input. This is similar to an automotive system in which a safety
feedback law overrides -- but only when necessary -- the possibly unsafe
steering, acceleration, or braking by a vigorous but inexperienced driver.
Simulations have been performed for a process in metal additive manufacturing,
which show that the operator's heat-and-cool commands to the Stefan model are
being obeyed but without the liquid ever freezing.
