Solvability of a class of phase field systems related to a sliding mode control problem

TL;DR

This paper investigates a phase-field system related to sliding mode control, proving existence, regularity, and uniqueness of solutions, and analyzing their dependence on initial data.

Contribution

It introduces a novel phase-field model with a maximal monotone nonlinearity for sliding mode control and establishes key mathematical properties of the solutions.

Findings

Existence and regularity of solutions are proven.

Solutions depend continuously on initial data.

Uniqueness of solutions is established under certain conditions.

Abstract

We consider a phase-field system of Caginalp type perturbed by the presence of an additional maximal monotone nonlinearity. Such a system arises from a recent study of a sliding mode control problem. We prove existence and regularity of the solutions. Moreover, under further assumptions, we show the continuous dependence on the initial data and the uniqueness of the solution.