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This paper develops an input-to-state stability (ISS) analysis of the Stefanproblem with respect to an unknown heat loss. The Stefan problem represents aliquid-solid phase change phenomenon which describes the time evolution of amaterial's temperature profile and the liquid-solid interface position. First,we introduce the one-phase Stefan problem with a heat loss at the interface bymodeling the dynamics of the liquid temperature and the interface position. Wefocus on the closed-loop system under the control law proposed in [16] that isdesigned to stabilize the interface position at a desired position for theone-phase Stefan problem without the heat loss. The problem is modeled by a 1-Ddiffusion Partial Differential Equation (PDE) defined on a time-varying spatialdomain described by an ordinary differential equation (ODE) with a time-varyingdisturbance. The well-posedness and some positivity conditions of theclosed-loop system are proved based on an open-loop analysis. The closed-loopsystem with the designed control law satisfies an estimate of {L}_2 norm in asense of ISS with respect to the unknown heat loss. The similar manner isemployed to the two-phase Stefan problem with the heat loss at the boundary ofthe solid phase under the control law proposed in [25], from which we deduce ananalogous result for ISS analysis.

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