A Linear Test for Global Nonlinear Controllability
Abstract
It is known that if a nonlinear control affine system without drift is bracket generating, then its associated subLaplacian is invertible under some conditions on the domain. In this note, we investigate the converse. We show how invertibility of the subLaplacian 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 (infinitedimensional) selfadjoint 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 subLaplacian to the the controllability of the corresponding continuity equation using possibly nonsmooth controls. Then using AmbrosioGigliSavare'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 PerronFrobenius type, we also provide a second dual Koopman point of view.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.09108
 arXiv:
 arXiv:2405.09108
 Bibcode:
 2024arXiv240509108E
 Keywords:

 Mathematics  Optimization and Control;
 Electrical Engineering and Systems Science  Systems and Control