Backstepping Control of the One-Phase Stefan Problem

TL;DR

This paper develops a backstepping control method for the one-phase Stefan problem, a moving boundary PDE-ODE system modeling melting, ensuring exponential stability of the interface and temperature distribution.

Contribution

It introduces a novel nonlinear backstepping transformation tailored for moving boundary problems, transforming the coupled PDE-ODE system into a stable target system.

Findings

Proves exponential stability of the controlled system.

Designs a boundary feedback controller satisfying physical constraints.

Ensures stability of the melting interface and temperature distribution.

Abstract

In this paper, a backstepping control of the one-phase Stefan Problem, which is a 1-D diffusion Partial Differential Equation (PDE) defined on a time varying spatial domain described by an ordinary differential equation (ODE), is studied. A new nonlinear backstepping transformation for moving boundary problem is utilized to transform the original coupled PDE-ODE system into a target system whose exponential stability is proved. The full-state boundary feedback controller ensures the exponential stability of the moving interface to a reference setpoint and the ${\cal H}_1$-norm of the distributed temperature by a choice of the setpint satisfying given explicit inequality between initial states that guarantees the physical constraints imposed by the melting process.