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Approximate controllability of a bilinear wave equation and minimum time

Karine Beauchard, Thomas Perrin, Eugenio Pozzoli

TL;DR

This work establishes global approximate controllability results for a bilinear Klein-Gordon/wave equation on the torus, with controls supported on only the first (2d+1) Fourier modes. The authors develop a two-pronged strategy: (i) small-time control of the velocity via Lie-bracket approximations that generate new velocity increments while preserving the profile, and (ii) a subsequent transfer of velocity into the profile using dual flows, all within the STAR framework to handle infinite-dimensional dynamics. In low dimensions (d ∈ {1,2}) they prove Tmin = r(W0) for nonzero initial data, while in higher dimensions (d ≥ 3) Tmin can vanish provided the zero-set of the initial state has positive-measure gaps, and GAC is achievable in large time from any W0 ≠ 0. The results synthesize Agrachev–Sarychev Lie-bracket techniques with propagation of well-prepared positive states, yielding both small-time and large-time controllability statements and clarifying the role of the zero-set geometry in controllability times.

Abstract

We study the global approximate controllability (GAC) of a Klein-Gordon wave equation, posed on the torus $\mathbb{T}^d$ of arbitrary dimension $d\in \mathbb{N}^*$, with bilinear control potentials supported on the first $(2d+1)$-Fourier modes. Let $Z(W_0)\subset \mathbb{T}^d$ be the set of essential zeroes of the initial state $W_0\in H^1\times L^2(\mathbb{T}^d)$, and $r(W_0)\geq 0$ be the maximum radius of a ball of $\mathbb{T}^d$ contained in $Z(W_0)$. Due to finite speed of propagation, the minimum control time starting from $W_0$ is necessarily larger than or equal to $r(W_0)$. We prove the following three facts. In low dimensions $d \in \{1,2\}$: the minimum time for GAC from $W_0 \neq 0$ is equal to $r(W_0)$. In any dimensions $d\geq 3$: the minimum time for GAC from $W_0$ is zero if $Z(W_0)$ has zero Lebesgue measure; and the GAC in sufficiently large time from all $W_0\neq 0$. The proof strategy consists in combining Lie bracket techniques \emph{à la Agrachev-Sarychev} with the propagation of well-prepared positive states.

Approximate controllability of a bilinear wave equation and minimum time

TL;DR

This work establishes global approximate controllability results for a bilinear Klein-Gordon/wave equation on the torus, with controls supported on only the first (2d+1) Fourier modes. The authors develop a two-pronged strategy: (i) small-time control of the velocity via Lie-bracket approximations that generate new velocity increments while preserving the profile, and (ii) a subsequent transfer of velocity into the profile using dual flows, all within the STAR framework to handle infinite-dimensional dynamics. In low dimensions (d ∈ {1,2}) they prove Tmin = r(W0) for nonzero initial data, while in higher dimensions (d ≥ 3) Tmin can vanish provided the zero-set of the initial state has positive-measure gaps, and GAC is achievable in large time from any W0 ≠ 0. The results synthesize Agrachev–Sarychev Lie-bracket techniques with propagation of well-prepared positive states, yielding both small-time and large-time controllability statements and clarifying the role of the zero-set geometry in controllability times.

Abstract

We study the global approximate controllability (GAC) of a Klein-Gordon wave equation, posed on the torus of arbitrary dimension , with bilinear control potentials supported on the first -Fourier modes. Let be the set of essential zeroes of the initial state , and be the maximum radius of a ball of contained in . Due to finite speed of propagation, the minimum control time starting from is necessarily larger than or equal to . We prove the following three facts. In low dimensions : the minimum time for GAC from is equal to . In any dimensions : the minimum time for GAC from is zero if has zero Lebesgue measure; and the GAC in sufficiently large time from all . The proof strategy consists in combining Lie bracket techniques \emph{à la Agrachev-Sarychev} with the propagation of well-prepared positive states.
Paper Structure (47 sections, 29 theorems, 86 equations)

This paper contains 47 sections, 29 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. The system
  3. Main results
  4. Detailed results
  5. Proof strategy
  6. Approximating flows along commutators
  7. Propagation from well-prepared states and minimum time
  8. Bibliographical comments
  9. Linear control of wave equations
  10. Exact bilinear control
  11. Approximate bilinear control
  12. Agrachev-Sarychev method
  13. Structure of the article
  14. Well-posedness and preliminaries
  15. Explicit group of isometries
  16. ...and 32 more sections

Key Result

Proposition 1

For every $T>0$, $u:[0,T] \to \mathbb{R}^{2d+1}$ piecewise constant and $W_0 \in H^1 \times L^2(\mathbb{T}^d)$ there exists a unique solution $W \in C^0([0,T],H^1 \times L^2(\mathbb{T}^d)) \cap C^1([0,T],L^2 \times H^{-1}(\mathbb{T}^d))$ of wave_vect associated with the initial condition $W(0)=W_0$.

Theorems & Definitions (56)

  • Proposition 1
  • Definition 2
  • Definition 3: Notations $Z$, $r$
  • Proposition 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Definition 10
  • ...and 46 more