Feedback control of the two-phase Stefan problem, with an application to the continuous casting of steel

A full-state feedback control law is derived that stabilizes the two-phase Stefan problem with respect to a reference solution using control of the Neumann boundary condition. Stability and convergence are shown via a Lyapunov functional on the error system with moving boundaries. A second control law is also derived, for which stability is proved and convergence is conjectured due to the clearly convergent simulation results. A simple Dirichlet controller is also considered, and is used to design a boundary-output-based estimator that, in combination with full-state feedback controllers, yields a plausible output feedback control law with boundary sensing and actuation. The performance of the control laws is demonstrated using numerical simulation.