Control of the Stefan System and Applications: A Tutorial

Abstract

The so-called Stefan system describes the dynamical model of the liquid–solid phase change in materials ranging from water and ice in the polar caps to metal casting and additive manufacturing (3D printing). The mathematical structure is given by a partial differential equation (PDE) with a moving boundary governed by a scalar ordinary differential equation. Because of the system's moving-boundary nature, control of the Stefan model is unconventional even within the class of otherwise challenging PDE control problems. The second decade of the twenty-first century has witnessed remarkable advances in control design for the Stefan system. Such advances carry significant potential in several areas of technology. In this article, we briefly review the principal literature on control of the Stefan model, along with the associated basics of the PDE analysis of the model and select applications. Principal ideas from our work on control design, stability analysis, and the maintenance of physical phase constraints are given sufficient attention and tutorial treatment so that the article can serve as a self-contained point of entry into the growing subject of boundary control of the Stefan system using the method of PDE backstepping.