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On the Homogenization of Norm and Time Optimal Controls for Parabolic Equations with Oscillating Coefficients

Year: 2026
Type: preprint
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.
DOI: 10.48550/arxiv.2609.14622
Institutions: University of the Basque Country, Basque Center for Applied Mathematics, Centre National de la Recherche Scientifique, Université de Bordeaux, Institut Universitaire de France, Institut Polytechnique de Bordeaux, Institut de Mathématiques de Bordeaux
Language: English
Open Access status: green
Awards: G1175115578, G1438949037, G2262748287, G2601397997, G300979063, G3428307696, G3480869486, G3681454997, G4033406476, G4042783231, G425668148, G464105488, G5080475149, G5195188938, G594041434, G5943232429, G7084143925
OpenAlex ID: W7213223417
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