Quantitative Soft-to-Hard Terminal Constraint Convergence for the Heat Equation
Abstract
We investigate an optimal control problem for the one-dimensional heat equation with a prescribed terminal state. To approximate the hard terminal constraint, we consider quadratic terminal penalization and analyze the convergence of the optimal controls and terminal states as the penalty parameter α→∞. We develop a unified spectral treatment of terminal reachability, minimum-energy control, and quadratic-penalty convergence. Using modal decompositions, we characterize finite-energy admissible terminal targets and construct the corresponding minimum-energy hard-constrained control. For admissible targets, we establish strong convergence of penalized optimal controls and terminal states. Under suitable spectral summability conditions, we obtain regularity-dependent convergence estimates of order O(α−(1/2+θ)) for the terminal-state error, 0≤θ≤1/2, and O(α−θ) for the optimal-control error, 0<θ≤1. The analysis connects these results explicitly with classical Tikhonov regularization in the heat-equation setting. A finite-dimensional prototype illustrates the underlying projection structure, while companion numerical investigations provide supporting evidence for the predicted convergence behavior.
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BibTeX
@misc{kwon2026quantitative,
title = {{Quantitative Soft-to-Hard Terminal Constraint Convergence for the Heat Equation}},
author = {Sung-Sik Kwon},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.14060},
eprint = {2605.14060},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.14060},
}
RIS
TY - UNPB
TI - Quantitative Soft-to-Hard Terminal Constraint Convergence for the Heat Equation
AU - Sung-Sik Kwon
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2605.14060
UR - https://arxiv.org/abs/2605.14060
AB - We investigate an optimal control problem for the one-dimensional heat equation with a prescribed terminal state. To approximate the hard terminal constraint, we consider quadratic terminal penalization and analyze the convergence of the optimal controls and terminal states as the penalty parameter α→∞. We develop a unified spectral treatment of terminal reachability, minimum-energy control, and quadratic-penalty convergence. Using modal decompositions, we characterize finite-energy admissible terminal targets and construct the corresponding minimum-energy hard-constrained control. For admissible targets, we establish strong convergence of penalized optimal controls and terminal states. Under suitable spectral summability conditions, we obtain regularity-dependent convergence estimates of order O(α^−(1/2+θ)) for the terminal-state error, 0≤θ≤1/2, and O(α^−θ) for the optimal-control error, 0<θ≤1. The analysis connects these results explicitly with classical Tikhonov regularization in the heat-equation setting. A finite-dimensional prototype illustrates the underlying projection structure, while companion numerical investigations provide supporting evidence for the predicted convergence behavior.
ER -