On the Homogenization of Norm and Time Optimal Controls for Parabolic Equations with Oscillating Coefficients
Abstract
We study homogenization of norm-optimal and time-optimal controls for a one-dimensional heat equation with rapidly oscillating periodic diffusion coefficients. The controls act on a fixed subinterval and are subject to pointwise constraints in space and time.We first derive final-state observability and null-controllability estimates that are uniform with respect to the oscillation scale. Combining these estimates with periodic homogenization and an abstract theory of optimal controls for parabolic systems, we prove convergence of the norm-optimal values and controls to their homogenized counterparts. We then deduce convergence of the optimal times and time-optimal controls. The controls converge weakly-star in the space of essentially bounded functions and strongly in every Lebesgue space with finite exponent, after extension to a common time interval when necessary.
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BibTeX
@misc{barcenapetisco2026homogenization,
title = {{On the Homogenization of Norm and Time Optimal Controls for Parabolic Equations with Oscillating Coefficients}},
author = {Jon Asier Bárcena-Petisco and Marius Tucsnak},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14622},
eprint = {2609.14622},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05750620v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Homogenization of Norm and Time Optimal Controls for Parabolic Equations with Oscillating Coefficients
AU - Jon Asier Bárcena-Petisco
AU - Marius Tucsnak
PY - 2026
DA - 2026/09/13
PB - HAL
DO - 10.48550/arxiv.2609.14622
UR - https://hal.science/hal-05750620v1
AB - We study homogenization of norm-optimal and time-optimal controls for a one-dimensional heat equation with rapidly oscillating periodic diffusion coefficients. The controls act on a fixed subinterval and are subject to pointwise constraints in space and time.We first derive final-state observability and null-controllability estimates that are uniform with respect to the oscillation scale. Combining these estimates with periodic homogenization and an abstract theory of optimal controls for parabolic systems, we prove convergence of the norm-optimal values and controls to their homogenized counterparts. We then deduce convergence of the optimal times and time-optimal controls. The controls converge weakly-star in the space of essentially bounded functions and strongly in every Lebesgue space with finite exponent, after extension to a common time interval when necessary.
ER -