Backstepping control of the Stefan problem with flowing liquid

This paper presents a backstepping control design for the one-phase Stefan problem with flowing liquid. The problem is modeled by a 1-D diffusion-advection Partial Differential Equation (PDE) defined on a time-varying spatial domain described by an ordinary differential equation (ODE). Based on our recent contribution, a nonlinear backstepping transformation for moving boundary problems is utilized, and the associated boundary feedback controller is designed. We show that for the closed-loop system a set of physical inequalities is guaranteed, provided that the setpoint and initial values verify some (reasonable) constraints. These constraints also ensure the exponential convergence of the moving interface to a setpoint and exponential stability (in the ℋ
₁
norm) of the temperature equilibrium profile. A numerical study shows the performance of the proposed method.