Riccati-feedback Control of a Two-dimensional Two-phase Stefan Problem

TL;DR

This paper develops a feedback control method for a complex two-dimensional Stefan problem using Riccati equations, mesh-movement, and adaptive schemes, demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a novel Riccati-feedback control approach for the two-dimensional Stefan problem with detailed discretization and numerical solution techniques.

Findings

The feedback control approach is applicable and makes the large-scale Stefan problem computationally feasible.

Numerical experiments confirm the effectiveness of the control method.

The influence of controller parameters on performance is analyzed.

Abstract

We discuss the feedback control problem for a two-dimensional two-phase Stefan problem. In our approach, we use a sharp interface representation in combination with mesh-movement to track the interface position. To attain a feedback control, we apply the linear-quadratic regulator approach to a suitable linearization of the problem. We address details regarding the discretization and the interface representation therein. Further, we document the matrix assembly to generate a non-autonomous generalized differential Riccati equation. To numerically solve the Riccati equation, we use low-rank factored and matrix-valued versions of the non-autonomous backward differentiation formulas, which incorporate implicit index reduction techniques. For the numerical simulation of the feedback controlled Stefan problem, we use a time-adaptive fractional-step-theta scheme. We provide the implementations for the developed methods and test these in several numerical experiments. With these experiments we show that our feedback control approach is applicable to the Stefan control problem and makes this large-scale problem computable. Also, we discuss the influence of several controller design parameters, such as the choice of inputs and outputs.