Existence, Stability and Controllability of the parabolic-parabolic thermistor model

TL;DR

This paper investigates the mathematical properties of a thermistor system modeled by coupled parabolic equations, establishing well-posedness, stability, and local controllability using advanced Carleman estimates and inverse function techniques.

Contribution

It introduces new Carleman estimates tailored for the coupled parabolic-parabolic thermistor system and proves local null controllability with control acting on a single equation.

Findings

Well-posedness and energy estimates established

Local null controllability proved using Carleman estimates

Control applied to only one equation of the coupled system

Abstract

In this article we establish the well-posedness, energy estimates, stability, and local null controllability for the thermistor system modeled by a parabolic-parabolic system using a control force acting on just one equation of the system. The proof of the controllability is based on appropriate Carleman estimates and Liusternik's inverse function theorem to obtain the local controllability of the nonlinear system. The coupling of the system happens both in the terms of order zero and one, which requires the use of a special Carleman estimate for the system.