Strong Legendre regularity and short-time analysis of abnormal curves

Abstract

We study strictly abnormal sub-Riemannian extremals satisfying the Goh condition and a strong (i.e., coercive) generalized Legendre condition, imposed only at the Lebesgue points of the reference control. We prove that such a control is smooth, that the projected extended Hessian of the end-point map is uniformly coercive on the admissible set of variations for short times, and that the abnormal covector gives a quantitative separation of the end-point of the reference arc from the image of the admissible set under the nonlinear end-point map. As a consequence the reference arc is rigid and, for short times, it is a length minimizer: every admissible curve with the same end-points and no greater length is a reparametrization of the reference arc. In particular, among controls of constant speed the reference control is the unique length minimizer. All the estimates are uniform on the admissible set, and no assumption is made on the dimension of the manifold, on the growth vector of the distribution, or on the corank of the differential of the end-point map.

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BibTeX RIS

BibTeX

@misc{boarotto2026strong,
  title = {{Strong Legendre regularity and short-time analysis of abnormal curves}},
  author = {Francesco Boarotto},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10813},
  eprint = {2610.10813},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10813},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Strong Legendre regularity and short-time analysis of abnormal curves
AU  - Francesco Boarotto
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10813
UR  - https://arxiv.org/abs/2610.10813
AB  - We study strictly abnormal sub-Riemannian extremals satisfying the Goh condition and a strong (i.e., coercive) generalized Legendre condition, imposed only at the Lebesgue points of the reference control. We prove that such a control is smooth, that the projected extended Hessian of the end-point map is uniformly coercive on the admissible set of variations for short times, and that the abnormal covector gives a quantitative separation of the end-point of the reference arc from the image of the admissible set under the nonlinear end-point map. As a consequence the reference arc is rigid and, for short times, it is a length minimizer: every admissible curve with the same end-points and no greater length is a reparametrization of the reference arc. In particular, among controls of constant speed the reference control is the unique length minimizer. All the estimates are uniform on the admissible set, and no assumption is made on the dimension of the manifold, on the growth vector of the distribution, or on the corank of the differential of the end-point map.
ER  -

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