Null controllability for a parabolic-elliptic coupled system

TL;DR

This paper establishes the null controllability of certain parabolic-elliptic systems using Carleman estimates and fixed-point methods, demonstrating controllability through localized distributed controls and asymptotic analysis.

Contribution

It introduces a novel approach combining Carleman estimates and fixed-point reformulation to prove null controllability for coupled parabolic-elliptic systems.

Findings

Null controllability is achieved for specific parabolic-elliptic systems.

Control is distributed, localized, and affects only one PDE.

Solutions can be approximated as limits of similar parabolic systems.

Abstract

In this paper, we prove the null controllability of some parabolic-elliptic systems. The control is distributed, locally supported in space and appears only in one PDE. The arguments rely on fixed-point reformulation and suitable Carleman estimates for the solutions to the adjoint system. Under appropriate assumptions, we also prove that the solution can be obtained as the asymptotic limit of some similar parabolic systems.