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

João Carlos Barreira, Juan Límaco

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.

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

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.

Paper Structure

This paper contains 12 sections, 23 theorems, 199 equations.

Table of Contents

  1. Introduction and main results
  2. Some previous results
  3. Well-posedness results
  4. Carleman estimates
  5. Null controllability of linear system \ref{['lad.Bous.Linear']}
  6. Estimates for the states solutions
  7. Proofs of the main theorems
  8. Development for the proof of the Theorem \ref{['Teo principal control nulo']}
  9. Development for the proof of the Theorem \ref{['Teo extra control nulo']}
  10. Large time null-controllability
  11. Some additional comments and open problems
  12. Existence and uniqueness of solution for (\ref{['lad.boussinesq']})

Key Result

Theorem 1.2

The nonlinear system lad.boussinesq is locally null-controllable at any $T>0$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 30 more