Neural-Spectral and Newton-HUM Methods for Reconstructing Controls of Wave Equations
Abstract
We develop a framework for constructing and rigorously validating controls for semilinear wave equations with localized interior control. We analyze the continuous energy-space problem and certify exact control for finite-dimensional Galerkin systems. Under explicit smallness and neighborhood conditions, our a posteriori theorem proves the existence of an exact Galerkin control near a computed candidate. It combines a validated terminal-state error, a bounded right inverse of the derivative of the control-to-terminal-state map, and a Lipschitz bound for that derivative. Under additional uniform right-inverse and neighborhood conditions, we also show that Newton correction converges with quadratic decay of terminal-state errors. We use residual estimates and interval arithmetic to enclose the nonlinear trajectory and its linearized controllability Gramian. For the specified eight-mode tanh problem, we rigorously bound a source-response operator to prove contraction of the exact source-frozen iteration. We compare deterministic controls based on the Hilbert Uniqueness Method with neural approximations of temporal modal coefficients. Two Newton corrections reduce the computed terminal-state errors of all six tested neural candidates below 10−8. For the three certified candidates, we obtain smaller bounds on the distance to an exact Galerkin steering control by refining the validation mesh while keeping the computed controls fixed.
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BibTeX
@misc{nguyen2026neuralspectral,
title = {{Neural-Spectral and Newton-HUM Methods for Reconstructing Controls of Wave Equations}},
author = {Tan-Phuc Nguyen and Minh-Binh Tran and Son Tu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.28029},
eprint = {2606.28029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.28029},
}
RIS
TY - UNPB
TI - Neural-Spectral and Newton-HUM Methods for Reconstructing Controls of Wave Equations
AU - Tan-Phuc Nguyen
AU - Minh-Binh Tran
AU - Son Tu
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2606.28029
UR - https://arxiv.org/abs/2606.28029
AB - We develop a framework for constructing and rigorously validating controls for semilinear wave equations with localized interior control. We analyze the continuous energy-space problem and certify exact control for finite-dimensional Galerkin systems. Under explicit smallness and neighborhood conditions, our a posteriori theorem proves the existence of an exact Galerkin control near a computed candidate. It combines a validated terminal-state error, a bounded right inverse of the derivative of the control-to-terminal-state map, and a Lipschitz bound for that derivative. Under additional uniform right-inverse and neighborhood conditions, we also show that Newton correction converges with quadratic decay of terminal-state errors. We use residual estimates and interval arithmetic to enclose the nonlinear trajectory and its linearized controllability Gramian. For the specified eight-mode tanh problem, we rigorously bound a source-response operator to prove contraction of the exact source-frozen iteration. We compare deterministic controls based on the Hilbert Uniqueness Method with neural approximations of temporal modal coefficients. Two Newton corrections reduce the computed terminal-state errors of all six tested neural candidates below 10^−8. For the three certified candidates, we obtain smaller bounds on the distance to an exact Galerkin steering control by refining the validation mesh while keeping the computed controls fixed.
ER -