  We address an open problem in ensemble control: Whether there exist
controllable linear ensemble systems over multi-dimensional parameterization
spaces? We provide a negative result: Any real-analytic linear ensemble system
is not $\mathrm{L}^p$-controllable, for $2\leq p \leq \infty$, if its
parameterization space contains an open set in $\mathbb{R}^d$ for $d \geq 2$.
