Sliding mode control for a generalization of the Caginalp phase-field system

TL;DR

This paper develops and analyzes sliding mode control strategies for a generalized Caginalp phase-field system, focusing on controlling thermal displacement and phase distribution with proven existence, uniqueness, and finite-time convergence.

Contribution

It introduces two novel control laws for the generalized system and provides rigorous mathematical proofs of their well-posedness and finite-time reaching properties.

Findings

Solutions reach the sliding manifold in finite time under certain conditions.

Existence and uniqueness of solutions are established for both control problems.

Regularity results for the controlled system are provided.

Abstract

In the present paper, we present and solve the sliding mode control (SMC) problem for a second-order generalization of the Caginalp phase-field system. This generalization, inspired by the theories developed by Green and Naghdi on one side, and Podio-Guidugli on the other, deals with the concept of thermal displacement, i.e., a primitive with respect to the time of the temperature. Two control laws are considered: the former forces the solution to reach a sliding manifold described by a linear constraint between the temperature and the phase variable; the latter forces the phase variable to reach a prescribed distribution $\varphi^*$. We prove existence, uniqueness as well as continuous dependence of the solutions for both problems; two regularity results are also given. We also prove that, under suitable conditions, the solutions reach the sliding manifold within finite time.