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Cost of controllability of the Burgers' equation linearized at a steady shock in the vanishing viscosity limit

Vincent Laheurte

Abstract

We consider the one-dimensional Burgers' equation linearized at a stationary shock, and investigate its null-controllability cost with a control at the left endpoint. We give an upper and a lower bound on the control time required for this cost to remain bounded in the vanishing viscosity limit, and construct an admissible control with an explicit limit behavior. We also provide an extension of the analysis to the case where the control acts on both endpoints. The proof relies on complex analysis and adapts methods previously used to tackle the same issue with a constant transport term.

Cost of controllability of the Burgers' equation linearized at a steady shock in the vanishing viscosity limit

Abstract

We consider the one-dimensional Burgers' equation linearized at a stationary shock, and investigate its null-controllability cost with a control at the left endpoint. We give an upper and a lower bound on the control time required for this cost to remain bounded in the vanishing viscosity limit, and construct an admissible control with an explicit limit behavior. We also provide an extension of the analysis to the case where the control acts on both endpoints. The proof relies on complex analysis and adapts methods previously used to tackle the same issue with a constant transport term.

Paper Structure

This paper contains 21 sections, 17 theorems, 216 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Shock profiles for Burgers' equation
  3. Vanishing viscosity controllability problem
  4. Control of the limit system
  5. Related problems
  6. Main results
  7. Spectral analysis of the operator
  8. Proof of the upper bound on $T_{\rm{unif}}$
  9. Scheme of the proof
  10. Controllability of the first mode. Proof of Lemma \ref{['lem1']}
  11. Controllability of the other modes. Proof of Lemma \ref{['lem2']}
  12. Proof of the lower bound on $T_{\rm{unif}}$
  13. The two-boundary control case
  14. Setting and expected results
  15. Statement of the result
  16. ...and 6 more sections

Key Result

Theorem 1.1

For every $\sigma\in(-L,L),$, there exists a minimal time $T_{\rm{unif}} < +\infty$ such that the system cont asym is uniformly null-controllable for any time $T>T_{\rm{unif}}$. If $\sigma\ge 0$, this minimal time verifies Otherwise, if $\sigma<0$, it instead verifies

Figures (1)

  • Figure 1: Evolution of the solution with this choice of control, for $\sigma=0$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 21 more