A Linear Test for Global Nonlinear Controllability
<|reference_start|>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 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.<|reference_end|>
@article{elamvazhuthi2024a,
    title={A Linear Test for Global Nonlinear Controllability},
    author={Karthik Elamvazhuthi},
    journal={arXiv preprint arXiv:2405.09108},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.09108},
    primaryClass={math.OC cs.SY eess.SY}
}