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

TL;DR

The paper analyzes a parabolic-parabolic thermistor PDE system coupling temperature and electric potential, proving well-posedness, energy estimates, and exponential stability for small data. It develops Carleman estimates for the linearized adjoint, establishing null controllability, and then extends to the nonlinear system via Liusternik's inverse mapping theorem to obtain local null controllability with a single interior control. Additionally, it discusses large-time controllability and outlines open problems, including higher-dimensional extensions and boundary-control variants. The methods combine Faedo-Galerkin existence, Carleman-based observability, and a fixed-point framework to address nonlinear coupling with order-zero and order-one terms.

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.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Definition 1.1
• Theorem 1.3
• proof
• Theorem 4.1: Liusternik
• proof
• Lemma 4.1: Fursikov's function
• proof
• Proposition 4.1