Generalized Hamiltonian gradient flow of contact type II: singularities, cut locus, and controllability
Abstract
We study singularities, cut loci and controllability for time-dependent contact Hamilton--Jacobi equations. A contact version of Arnaud's theorem identifies the superdifferential graph of an evolved profile with the contact Hamiltonian image of the differential graph of its positive Lax--Oleinik regularization. This representation defines intrinsic singular characteristics and determines their initial velocity through a minimum-energy selection on the space-time superdifferential. The curves are continuous and locally 1/2-Hölder, but they need not be locally Lipschitz. The exponent 1/2 is optimal even for a quadratic Hamiltonian. The intrinsic characteristic issued from a cut point is singular at every later time in the short-time interval. We also establish the contact Lax--Oleinik commutator identities, which identify the superlevel and sublevel sets of the cut time with the corresponding terminal momentum graphs through bi-Lipschitz maps. For reachable terminal data we characterize the associated calibrated sets, and we describe all continuous initial data that reach a prescribed semiconcave profile. The proofs use Herglotz' variational principle and short-time estimates for contact fundamental solutions.
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BibTeX
@misc{cai2026generalized,
title = {{Generalized Hamiltonian gradient flow of contact type II: singularities, cut locus, and controllability}},
author = {Qingyang Cai and Wei Cheng and Tianqi Shi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01911},
eprint = {2611.01911},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01911},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalized Hamiltonian gradient flow of contact type II: singularities, cut locus, and controllability
AU - Qingyang Cai
AU - Wei Cheng
AU - Tianqi Shi
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01911
UR - https://arxiv.org/abs/2611.01911
AB - We study singularities, cut loci and controllability for time-dependent contact Hamilton--Jacobi equations. A contact version of Arnaud's theorem identifies the superdifferential graph of an evolved profile with the contact Hamiltonian image of the differential graph of its positive Lax--Oleinik regularization. This representation defines intrinsic singular characteristics and determines their initial velocity through a minimum-energy selection on the space-time superdifferential. The curves are continuous and locally 1/2-Hölder, but they need not be locally Lipschitz. The exponent 1/2 is optimal even for a quadratic Hamiltonian. The intrinsic characteristic issued from a cut point is singular at every later time in the short-time interval. We also establish the contact Lax--Oleinik commutator identities, which identify the superlevel and sublevel sets of the cut time with the corresponding terminal momentum graphs through bi-Lipschitz maps. For reachable terminal data we characterize the associated calibrated sets, and we describe all continuous initial data that reach a prescribed semiconcave profile. The proofs use Herglotz' variational principle and short-time estimates for contact fundamental solutions.
ER -