Year: 2026
Type: preprint
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 \(α\to\infty\). 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, \(0\leθ\le1/2\), and \(O(α^{-θ})\) for the optimal-control error, \(0<θ\le1\). 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.
Source: arXiv (Cornell University)
Author: Kwon, Sung-Sik
Institution: North Carolina Central University
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161267274