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On the Uniform Controllability of the Inviscid and Viscous Burgers-$α$ Systems

Raul K. C. Araújo, Enrique Fernández-Cara, Diego A. Souza

Abstract

This work is devoted to prove the global uniform exact controllability of the inviscid and viscous Burgers-$α$ systems. The state $y$ is the solution to a regularized Burgers equation, where the transport velocity $z$ consists of a filtered version of $y$ - specifically $z=(Id-α^2\partial^2_ {xx})^{-1}y$ with $α>0$ being a small parameter - in place of $y$. First, a global uniform exact controllability result for the nonviscous Burgers-$α$ system with three scalar controls is obtained, using the return method. Then, global exact controllability to constant states of the viscous system is deduced from a local exact controllability result and a global approximate controllability result for smooth initial and target states.

On the Uniform Controllability of the Inviscid and Viscous Burgers-$α$ Systems

Abstract

This work is devoted to prove the global uniform exact controllability of the inviscid and viscous Burgers- systems. The state is the solution to a regularized Burgers equation, where the transport velocity consists of a filtered version of - specifically with being a small parameter - in place of . First, a global uniform exact controllability result for the nonviscous Burgers- system with three scalar controls is obtained, using the return method. Then, global exact controllability to constant states of the viscous system is deduced from a local exact controllability result and a global approximate controllability result for smooth initial and target states.

Paper Structure

This paper contains 17 sections, 15 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Classical Results
  4. Well-Posedness of the Viscous Burgers-$\alpha$ System
  5. Controllability of the inviscid Burgers-$\alpha$ system
  6. Local null controllability
  7. Global exact controllability
  8. Global controllability of the viscous Burgers-$\alpha$ system
  9. Smoothing effect
  10. Uniform approximate controllability
  11. Uniform local exact controllability to the trajectories
  12. Global exact controllability
  13. Additional comments and questions
  14. Passage to the limit when $\alpha \rightarrow 0$
  15. Null controllability with $2$ controls
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\alpha > 0$ and $T>0$ be given. The inviscid Burgers-$\alpha$ system es14 is globally exactly controllable in $C^1$. That is, for any given $y_0, y_T \in C^1([0,L])$, there exist a time-dependent control $p^{\alpha} \in C^0([0,T])$, a couple of boundary controls $(v_l^{\alpha},v_r^{\alpha}) \in Moreover, there exists a positive constant $C>0$ (independent of $\alpha$) such that

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Proposition 1: Theorem $10.19$, ode
  • Proposition 2: Lemma $1$, bardos
  • Theorem 3
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • ...and 12 more