A linear test for global nonlinear controllability

It is known that if a nonlinear control affine system without drift is bracket generating, then its associated sub-Laplacian is invertible under some conditions on the domain. In this note, we investigate the converse. We show how invertibility of the sub-Laplacian operator implies a weaker form of controllability, where the reachable sets of a neighborhood of a point have full measure. From a computational point of view, one can then use the spectral gap of the (infinite-dimensional) self-adjoint operator to define a notion of degree of controllability. An essential tool to establish the converse result is to use the relation between invertibility of the sub-Laplacian to the controllability of the corresponding continuity equation using possibly non-smooth controls. Then using Ambrosio–Gigli–Savaré’s superposition principle from optimal transport theory we relate it to controllability properties of the control system. While the proof can be considered of the Perron–Frobenius type, we also provide a second dual Koopman point of view.