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On the control of the Burgers-alpha model

Fágner D. Araruna, Enrique Fernández-Cara, Diego A. Souza

Abstract

This work is devoted to prove the local null controllability of the Burgers-$α$ model. The state is the solution to a regularized Burgers equation, where the transport term is of the form $zy_x$, $z=(Id-α^2\frac{\partial^2}{\partial x^2})^{-1}y$ and $α>0$ is a small parameter. We also prove some results concerning the behavior of the null controls and associated states as $α\to 0^+$.

On the control of the Burgers-alpha model

Abstract

This work is devoted to prove the local null controllability of the Burgers- model. The state is the solution to a regularized Burgers equation, where the transport term is of the form , and is a small parameter. We also prove some results concerning the behavior of the null controls and associated states as .
Paper Structure (9 sections, 9 theorems, 98 equations)

This paper contains 9 sections, 9 theorems, 98 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Local null controllability of the Burgers-$\alpha$ mo-del
  4. Large time null controllability of the Burgers-$\alpha$ system
  5. Controllability in the limit
  6. Additional comments and questions
  7. A boundary controllability result
  8. No global null controllability?
  9. The situation in higher spatial dimensions. The Leray-$\alpha$ system

Key Result

Theorem 1

For each $T>0$, the system CPalpha is locally null-controllable at time $T$. More precisely, there exists $\delta>0$$($independent of $\alpha$$)$ such that, for any $y_0\in H^1_0(0,L)$ with $\|y_0\|_{\infty} \leq \delta$, there exist controls $v_\alpha\in L^\infty((a,b)\times(0,T))$ and associated s

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof
  • Lemma 1
  • proof : Proof of Lemma \ref{['est-L-infty']}
  • Proposition 2
  • proof
  • Remark 1
  • ...and 5 more