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 L-2 norm. Numerical simulation is provided to illustrate the good performance of the proposed control law.