Approximate controllability of a bilinear wave equation and minimum time

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.