  This paper presents a control design for the one-phase Stefan problem under
actuator delay via a backstepping method. The Stefan problem represents a
liquid-solid phase change phenomenon which describes the time evolution of a
material's temperature profile and the interface position. The actuator delay
is modeled by a first-order hyperbolic partial differential equation (PDE),
resulting in a cascaded transport-diffusion PDE system defined on a
time-varying spatial domain described by an ordinary differential equation
(ODE). Two nonlinear backstepping transformations are utilized for the control
design. The setpoint restriction is given to guarantee a physical constraint on
the proposed controller for the melting process. This constraint ensures the
exponential convergence of the moving interface to a setpoint and the
exponential stability of the temperature equilibrium profile and the delayed
controller in the ${\cal H}_1$ norm. Furthermore, robustness analysis with
respect to the delay mismatch between the plant and the controller is studied,
which provides analogous results to the exact compensation by restricting the
control gain.
