A Linear Test for Global Nonlinear Controllability

TL;DR

This paper explores how the invertibility of the sub-Laplacian operator relates to a weaker form of controllability in nonlinear control systems, linking spectral properties to controllability measures.

Contribution

It establishes a novel connection between sub-Laplacian invertibility and controllability, using optimal transport theory and spectral analysis to define controllability degrees.

Findings

Invertibility of the sub-Laplacian implies full measure of reachable sets.

Spectral gap can be used to quantify controllability.

The approach links PDE spectral properties to control system behavior.

Abstract

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 the controllability of the corresponding continuity equation using possibly non-smooth controls. Then using Ambrosio-Gigli-Savare'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.