Safe Trajectory Tracking of the Stefan Problem with Second-Order Moving Boundary Dynamics

Abstract

This paper considers a safe trajectory tracking of the Stefan problem with a second-order moving boundary dynamics. The model is given by a parabolic Partial Differential Equation (PDE) defined on a time-varying domain of moving boundary governed by a second-order Ordinary Differential Equation (ODE) associated with the Neumann boundary condition. A feedforward control is designed by a series expansion approach to solve the inverse Stefan problem under given reference trajectory of the moving boundary, and the convergence of infinite series is proven. A trajectory tracking controller is derived based on an energy-shaping, which ensures the safety of the model constraint in the closed-loop system. The closed-loop system is also shown to be globally exponentially stable with respect to the tracking error by performing PDE backstepping transformation and Lyapunov analysis. Numerical simulation illustrates an effective tracking performance of the proposed method under a sinusoidal reference trajectory. Code is released at https://github.com/shumon0423/StefanTracking_ACC2026.git.