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Remarks on local controllability for the Boussinesq system with Navier boundary condition

Cristhian Montoya

Abstract

This note deals with the local exact controllability to a particular class of trajectories for the Boussinesq system with nonlinear Navier-slip boundary conditions and internal controls having vanishing components. Briefly speaking, in two dimensions, the local exact controllability property is obtained using only one control in the heat equation, meanwhile two scalar controls are required in three dimensions.

Remarks on local controllability for the Boussinesq system with Navier boundary condition

Abstract

This note deals with the local exact controllability to a particular class of trajectories for the Boussinesq system with nonlinear Navier-slip boundary conditions and internal controls having vanishing components. Briefly speaking, in two dimensions, the local exact controllability property is obtained using only one control in the heat equation, meanwhile two scalar controls are required in three dimensions.

Paper Structure

This paper contains 3 sections, 5 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. A new Carleman inequality
  3. Local controllability for the Boussinesq system

Key Result

Theorem 1.1

Assume $A\in P^1_{\varepsilon}\cap P^2$ and $(0,\overline{p},\overline\theta)$ satisfying intro_reg_trajectheta--intro_trajectorysys2. There exists a constant $\lambda_0$, such that for any $\lambda\geq\lambda_0$ there exist two constants $C(\lambda)>0$ increasing on $\|A\|_ {P^1_{\varepsilon}\cap P for every $s\geq s_0$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.3