Control of Two-Phase Stefan Problem via Single Boundary Heat Input

This paper presents the control design of the two-phase Stefan problem via a single boundary heat input. The two-phase Stefan problem is a representative model of liquid-solid phase transition by describing the time evolutions of the temperature profile which is divided by subdomains of liquid and solid phases as the liquid-solid moving interface position. The mathematical formulation is given by two diffusion partial differential equations (PDEs) defined on a time-varying spatial domain described by an ordinary differential equation (ODE) driven by the Neumann boundary values of both PDE states, resulting in a nonlinear coupled PDE-ODE-PDE system. As an extension from our previous study on the one-phase Stefan problem, we design a state feedback control law to stabilize the interface position to a desired setpoint by employing the backstepping method. We prove that the closed-loop system under the control law ensures some conditions for model validity and the global exponential stability estimate is shown in L2 norm. Numerical simulation is provided to illustrate the good performance of the proposed control law.