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On the control of some coupled systems of the Boussinesq kind with few controls

Enrique Fernández-Cara, Diego A. Souza

Abstract

This paper is devoted to prove the local exact controllability to the trajectories for a coupled system, of the Boussinesq kind, with a reduced number of controls. In the state system, the unknowns are the velocity field and pressure of the fluid $(y, p)$, the temperature $θ$ and an additional variable $c$ that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by $θ$ and $c$.

On the control of some coupled systems of the Boussinesq kind with few controls

Abstract

This paper is devoted to prove the local exact controllability to the trajectories for a coupled system, of the Boussinesq kind, with a reduced number of controls. In the state system, the unknowns are the velocity field and pressure of the fluid , the temperature and an additional variable that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by and .
Paper Structure (10 sections, 10 theorems, 99 equations)

This paper contains 10 sections, 10 theorems, 99 equations.

Table of Contents

  1. A preliminary result
  2. Proof of Theorem \ref{['teo1']}
  3. Proof of Theorem \ref{['teo2']}
  4. Proof of Theorem \ref{['teo3']}
  5. Final comments and questions
  6. The case $N=2$
  7. Nonlinear $\mathbf{F}$ and geometrical conditions on $\mathcal{O}$
  8. Generalizations to coupled systems with more unknowns
  9. Local null controllability without geometrical hypotheses
  10. Appendix

Key Result

Theorem 1

Assume that the assumptions trajectory--f-case-1 are satisfied. Then there exists $\delta > 0$ such that, whenever $(\mathbf{y}_0,\theta_0,c_0)\in\mathbf{E}\times L^2(\Omega)\times L^2(\Omega)$ and we can find a $L^2$-control $(\mathbf{v}, w_1,w_2)$ with $v_i\equiv v_N\equiv0$ for some $1\leq i<N$ and an associated state $(\mathbf{y},p,\theta,c)$ satisfying

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2
  • proof
  • Theorem 2.3
  • ...and 4 more