Local null controllability of the complete N-dimensional Ladyzhenskaya-Boussinesq model

TL;DR

This paper studies the controllability of the N-dimensional Ladyzhenskaya-Boussinesq system, demonstrating local and large-time null controllability using Carleman estimates and inverse mapping techniques.

Contribution

It establishes controllability results for a complex coupled nonlinear PDE system with controls supported on small subsets.

Findings

Proves local null controllability of the Ladyzhenskaya-Boussinesq system.

Establishes large time null controllability under certain conditions.

Uses classical Carleman estimates and inverse mapping theorem in the proof.

Abstract

This work investigates both local null controllability and large time null controllability for a class of complete Ladyzhenskaya Boussinesq systems, where the controls are distributed and supported on small subsets of the domain. The proof of local null controllability relies on classical techniques, including Carleman estimates and Liusternik Inverse Mapping Theorem. Nevertheless, the presence of nonlinearities in both the velocity and temperature equations necessitates careful treatment.