Electron. J. Differential Equations, Vol. 2026 (2026), No. 18, pp. 1-26.

New cost terms through homogenization of an optimal control problem under dynamic boundary conditions on microscopic particles

Jesus Ildefonso Diaz, Tatyana A. Shaposhnikova, Alexander V. Podolskiy

Abstract:
This article concerns optimal control problems in a heterogeneous body with a periodic structure of particles depending on a small parameter \(\varepsilon\). We study the asymptotic behavior, as \(\varepsilon \to 0\), of the optimal control functional and of the optimal state when the initial problem is of parabolic type. We assume a dynamic condition and the effect of some controls for some of the particles on the boundary. In the so-called "critical case", we show the appearance of some new non-local in time "strange terms", in the limit parabolic equation and in the limit cost functional. Microscopic localized controls generate peculiar terms in both the limit equation and the cost functional that do not appear when controls are applied to the entire set of particles, or when the boundary condition on the particles is of Robin type.

Submitted October 17, 2025. Published February 25, 2026.
Math Subject Classifications: 35B27, 35K20, 49K20, 93C20.
Key Words: Homogenization; critical case; optimal control; strange term; dynamic boundary condition; homogenized cost functional.
DOI: 10.58997/ejde.2026.18

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Jesús Ildefonso Díaz
Instituto de Mathematica Interdisciplinar
Universidad Complutense de Madrid, Spain
email: jidiaz@ucm.es
Tatyana A. Shaposhnikova
Lomonosov Moscow State University, Russia
email: shaposh.tan@mail.ru
Alexander V. Podolskiy
Lomonosov Moscow State University, Russia
email: avpodolskiy@yandex.ru

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