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A new look at the controllability cost of linear evolution systems with a long gaze at localized data

Roberta Bianchini, Vincent Laheurte, Franck Sueur

Abstract

We revisit the classical issue of the controllability/observability cost of linear first order evolution systems, starting with ODEs, before turning to some linear first order evolution PDEs in several space dimensions, including hyperbolic systems and pseudo-differential systems obtained by linearization in fluid mechanics. In particular we investigate the cost of localized initial data, and in the dispersive case, of initial data which are semi-classically microlocalized.

A new look at the controllability cost of linear evolution systems with a long gaze at localized data

Abstract

We revisit the classical issue of the controllability/observability cost of linear first order evolution systems, starting with ODEs, before turning to some linear first order evolution PDEs in several space dimensions, including hyperbolic systems and pseudo-differential systems obtained by linearization in fluid mechanics. In particular we investigate the cost of localized initial data, and in the dispersive case, of initial data which are semi-classically microlocalized.
Paper Structure (16 sections, 17 theorems, 178 equations)

This paper contains 16 sections, 17 theorems, 178 equations.

Table of Contents

  1. Reminder on the cost of controllability of linear ODEs
  2. Case of a transport PDE
  3. A localization in space and amplitude of the cost
  4. Microlocalized cost
  5. Semiclassical Wigner matrix transform
  6. Pseudodifferential operators.
  7. Statement of the result
  8. Proof of Theorem \ref{['PDO transport']}
  9. Application to the incompressible Euler equations linearized at a Couette flow
  10. Case of systems without crossing modes
  11. Statement of the result
  12. Proof of Theorem \ref{['thm1 micro']}
  13. Diagonalisation of the propagation term. Proof of Lemma \ref{['lem:diag']}
  14. Block-diagonalisation of the amplification term. Proof of Lemma \ref{['lem:block diag']}
  15. Estimate of the off-diagonal energy terms. Proof of Lemma \ref{['lem:off-diag']}
  16. ...and 1 more sections

Key Result

Lemma 1.3

Let $A$ and $B$ some continuous time-dependent fields of matrices. Then there is a unique global solution $G$ of the differential Lyapunov equation: Moreover at any time $t$, the matrix $G(t)$ is symmetric and nonnegative. Finally, for any time $T$, where, for any time $T$, the matrix-valued function $R(\cdot,T)$ is the solution to the linear system

Theorems & Definitions (42)

  • Lemma 1.3
  • proof
  • Theorem 1.4
  • proof : Proof of Theorem \ref{['thm0']}
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 32 more