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

Abstract

This paper develops an input-to-state stability (ISS) analysis of the one-phase Stefan problem with a prescribed heat loss at the liquid-solid interface. We focus on the closed-loop system under the control law proposed in [7] which 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.