Input-to-State Stability for the Control of Stefan Problem with Respect to Heat Loss

This paper develops an input-to-state stability (ISS) analysis of the Stefan problem with respect to an unknown heat loss. The Stefan problem represents a liquid-solid phase change phenomenon which describes the time evolution of a material's temperature profile and the liquid-solid interface position. First, we introduce the one-phase Stefan problem with a heat loss at the interface by modeling the dynamics of the liquid temperature and the interface position. We focus on the closed-loop system under the control law proposed in [16] that is designed to stabilize the interface position at a desired position for the one-phase Stefan problem without the heat loss. The problem is modeled by a 1-D diffusion Partial Differential Equation (PDE) defined on a time-varying spatial domain described by an ordinary differential equation (ODE) with a time-varying disturbance. The well-posedness and some positivity conditions of the closed-loop system are proved based on an open-loop analysis. The closed-loop system with the designed control law satisfies an estimate of {L}_2 norm in a sense of ISS with respect to the unknown heat loss. The similar manner is employed to the two-phase Stefan problem with the heat loss at the boundary of the solid phase under the control law proposed in [25], from which we deduce an analogous result for ISS analysis.