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 Hörmander sub-Laplacians on compact Lie groups. We also give conditions for the controllability of wave and Schrödinger equations in these settings.