Controllability Issues of Linear Ensemble Systems over Multi-dimensional Parameterization Spaces

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 $2\le p\le \mathrm{\infty }$, if its parameterization space contains an open set in ${\mathbb{R}}^d$ for $d\ge 2$.