Sliding mode control for a nonlinear phase-field system

TL;DR

This paper develops sliding mode control strategies for a nonlinear phase-field system, proving well-posedness and demonstrating finite-time convergence to desired states using different feedback laws.

Contribution

It introduces novel SMC laws for phase-field models, including both non-local and local control approaches, with rigorous theoretical analysis.

Findings

Proved well-posedness and regularity of controlled phase-field systems.

Demonstrated finite-time convergence to the sliding manifold.

Compared effectiveness of different feedback control laws.

Abstract

In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state-feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.