  Let $A$ and $B$ be invariant linear operators with respect to a decomposition
$\{H_{j}\}_{j\in \mathbb{N}}$ of a Hilbert space $\mathcal{H}$ in subspaces of
finite dimension. We give necessary and sufficient conditions for the
controllability of the Cauchy problem $$ u_t=Au+Bv,\,\,u(0)=u_0,$$ in terms of
the (global) matrix-valued symbols $\sigma_A$ and $\sigma_B$ of $A$ and $B,$
respectively, associated to the decomposition $\{H_{j}\}_{j\in \mathbb{N}}$.
Then, we present some applications including the controllability of the Cauchy
problem on compact manifolds for elliptic operators and the controllability of
fractional diffusion models for H\"ormander sub-Laplacians on compact Lie
groups. We also give conditions for the controllibility of wave and
Schr\"odinger equations in these settings.
