Approximate controllability of a bilinear wave equation and minimum time

TL;DR

This paper investigates the approximate controllability and minimum control time for a bilinear wave equation on a torus, revealing dimension-dependent results and employing Lie bracket and propagation techniques.

Contribution

It establishes the exact minimum control time in low dimensions and conditions for zero minimum time in higher dimensions, advancing control theory for wave equations.

Findings

Minimum control time equals the zero set radius in 1D and 2D.

Zero minimum time for GAC in dimensions ≥3 with zero Lebesgue measure zero set.

GAC achieved in large time from all initial states.

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{\`a la Agrachev-Sarychev} with the propagation of well-prepared positive states.