Carleman Estimates for Wave Equations on the Half-Line: AI-Assisted Weights and Applications
Abstract
We develop an AI-assisted search-and-certification workflow for constructing Carleman weights for wave equations on semi-infinite domains. Starting from a weighted cross-term identity used to derive the Carleman estimate, we derive explicit analytical screening conditions and use an AI system to propose symbolic candidate weights subject to those conditions. The search identifies a logarithmic weight structure with concrete parameter values; the subsequent human verification by the authors then retains this structure, derives the admissible parameter range, and rigorously certifies the selected weight by verifying the required quantitative conditions. Based on the identified and certified Carleman weight, the subsequent proofs, analysis, and applications are carried out entirely by the authors: we establish a global Carleman estimate for the wave operator on the half-line and derive a weighted conditional lateral Cauchy stability result. The proposed weight is further used in a finite-depth one-sided reconstruction problem for semilinear wave equations on the half-line, where the initial-time flux induces an endpoint graph stabilizer and a contractive frozen nonlinear reconstruction map. Numerical experiments support the predicted contraction behavior and show improved robustness to noisy data from the Carleman weighting, with the induced endpoint graph term providing further stabilization at higher noise levels.
Related
- Carleman estimates for a nonconservative Timoshenko beam involving Dirichlet-Neumann boundary conditions and application to exact controllability
- Global Hölder estimates for linearized Monge-Ampère equations in divergence form with applications to dual semigeostrophic equations and periodic homogenization
- Strichartz estimates for Klein-Gordon equations on asymptotically hyperbolic manifolds with applications
- On ABP Estimates for a Class of Fully Nonlinear Singular-Degenerate Elliptic Equations and Applications
- Carleman Estimates for Critical Electromagnetic Schrödinger Operators on Conic Manifolds
BibTeX
@misc{an2026carleman,
title = {{Carleman Estimates for Wave Equations on the Half-Line: AI-Assisted Weights and Applications}},
author = {Chenyang An and Yu Wang and Qihao Ye and Zhongqiang Zhang and Qiao Zhuang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06283},
eprint = {2609.06283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06283},
}
RIS
TY - UNPB
TI - Carleman Estimates for Wave Equations on the Half-Line: AI-Assisted Weights and Applications
AU - Chenyang An
AU - Yu Wang
AU - Qihao Ye
AU - Zhongqiang Zhang
AU - Qiao Zhuang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06283
UR - https://arxiv.org/abs/2609.06283
AB - We develop an AI-assisted search-and-certification workflow for constructing Carleman weights for wave equations on semi-infinite domains. Starting from a weighted cross-term identity used to derive the Carleman estimate, we derive explicit analytical screening conditions and use an AI system to propose symbolic candidate weights subject to those conditions. The search identifies a logarithmic weight structure with concrete parameter values; the subsequent human verification by the authors then retains this structure, derives the admissible parameter range, and rigorously certifies the selected weight by verifying the required quantitative conditions. Based on the identified and certified Carleman weight, the subsequent proofs, analysis, and applications are carried out entirely by the authors: we establish a global Carleman estimate for the wave operator on the half-line and derive a weighted conditional lateral Cauchy stability result. The proposed weight is further used in a finite-depth one-sided reconstruction problem for semilinear wave equations on the half-line, where the initial-time flux induces an endpoint graph stabilizer and a contractive frozen nonlinear reconstruction map. Numerical experiments support the predicted contraction behavior and show improved robustness to noisy data from the Carleman weighting, with the induced endpoint graph term providing further stabilization at higher noise levels.
ER -