Numerical Investigation of Soft-to-Hard Convergence: Discretization, Admissibility, and Spectral Regularity
Abstract
This paper presents a numerical investigation of soft-to-hard terminal constraint convergence for optimal control problems governed by a second-order ordinary differential equation and the one-dimensional heat equation. The study complements the analytical convergence theory through comparisons with exact solutions and independently computed numerical approximations. For the heat equation, a fully discrete Crank--Nicolson finite-difference formulation is validated against exact modal solutions, exhibiting approximately second-order convergence in the terminal-state and midpoint-sampled control errors under simultaneous space-time mesh refinement. Numerical experiments demonstrate the predicted first-order penalty convergence for benchmark problems and investigate the admissibility threshold governing finite-energy exact terminal controllability. Additional modal experiments examine the dependence of terminal-error convergence rates on spectral regularity, illustrating intermediate rates consistent with the theoretical estimates and asymptotic predictions. The results provide computational evidence supporting the soft-to-hard convergence framework and clarify the roles of discretization, target admissibility, and spectral regularity.
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BibTeX
@misc{kwon2026numerical,
title = {{Numerical Investigation of Soft-to-Hard Convergence: Discretization, Admissibility, and Spectral Regularity}},
author = {Sung-Sik Kwon and Marcus Nunda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40461},
eprint = {2610.40461},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40461},
}
RIS
TY - UNPB
TI - Numerical Investigation of Soft-to-Hard Convergence: Discretization, Admissibility, and Spectral Regularity
AU - Sung-Sik Kwon
AU - Marcus Nunda
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40461
UR - https://arxiv.org/abs/2610.40461
AB - This paper presents a numerical investigation of soft-to-hard terminal constraint convergence for optimal control problems governed by a second-order ordinary differential equation and the one-dimensional heat equation. The study complements the analytical convergence theory through comparisons with exact solutions and independently computed numerical approximations. For the heat equation, a fully discrete Crank--Nicolson finite-difference formulation is validated against exact modal solutions, exhibiting approximately second-order convergence in the terminal-state and midpoint-sampled control errors under simultaneous space-time mesh refinement. Numerical experiments demonstrate the predicted first-order penalty convergence for benchmark problems and investigate the admissibility threshold governing finite-energy exact terminal controllability. Additional modal experiments examine the dependence of terminal-error convergence rates on spectral regularity, illustrating intermediate rates consistent with the theoretical estimates and asymptotic predictions. The results provide computational evidence supporting the soft-to-hard convergence framework and clarify the roles of discretization, target admissibility, and spectral regularity.
ER -