Safe Stabilization of the Stefan Problem with a High-Order Moving Boundary Dynamics by PDE Backstepping
Shumon Koga, Miroslav Krstic
TL;DR
This work tackles the safe stabilization of a Stefan-type PDE with a moving boundary evolving under a second-order dynamics, formulated as $\varepsilon \ddot s(t) = -\dot s(t) - \beta T_x(s(t),t)$, and aims to regulate the interface to a chosen setpoint while preserving the physical validity of the liquid region through $T(x,t)\ge T_m$. It employs PDE backstepping to design a boundary heat-flux control $q_c(t)$ that stabilizes the moving boundary and enforces the positivity safety constraint, with a Lyapunov-based analysis proving global exponential stability in the $H_1$-norm; the approach also provides the explicit control law $q_c(t)= - \frac{kc_2}{\alpha} \int_0^{s(t)}(T-T_m)dx - \frac{k}{\beta}[ c_1 (s-s_r) + c_2 \varepsilon \dot s]$ under a positivity condition. A higher-order extension to third-order moving boundary dynamics is outlined, including a corresponding extended control law with an extra term in $\ddot s$ and updated gain relations, maintaining safety and stability guarantees under analogous assumptions. Numerical simulations using Zinc parameters validate monotone convergence of the interface to the setpoint, a transient peak in boundary temperature above $T_m$, and strictly positive boundary heat flux, demonstrating practical feasibility. The work advances safe boundary control for phase-change models and lays the groundwork for future enhancements such as safety-filtering via QP, delays, and observer-based output feedback.
Abstract
This paper presents a safe stabilization of the Stefan PDE model with a moving boundary governed by a high-order dynamics. We consider a parabolic PDE with a time-varying domain governed by a second-order response with respect to the Neumann boundary value of the PDE state at the moving boundary. The objective is to design a boundary heat flux control to stabilize the moving boundary at a desired setpoint, with satisfying the required conditions of the model on PDE state and the moving boundary. We apply a PDE backstepping method for the control design with considering a constraint on the control law. The PDE and moving boundary constraints are shown to be satisfied by applying the maximum principle for parabolic PDEs. Then the closed-loop system is shown to be globally exponentially stable by performing Lyapunov analysis. The proposed control is implemented in numerical simulation, which illustrates the desired performance in safety and stability. An outline of the extension to third-order moving boundary dynamics is also presented. Code is released at https://github.com/shumon0423/HighOrderStefan_CDC2025.git.