arXiv subjects

T. A. Shaposhnikova

Publications and source records attributed to T. A. Shaposhnikova.

3 recordsLinked to original sources

arXiv2026

Homogenization of Dynamic Signorini-Type Problems on Critically Oscillating Boundaries under Time-Periodic Forcing

We study the homogenization, in the critical scaling regime, of a boundary value problem with a nonlinear dynamic Signorini-type condition posed on a rapidly oscillating portion of the boundary. The source term is time-periodic and we look for time-periodic solutions. Using the method of oscillating test functions (Tartar), compactness, and monotonicity arguments, we identify the homogenized problem and the effective nonlinear boundary operator. In contrast with the evolutionary (initial value) setting, the periodic framework eliminates memory effects and yields an instantaneous time-periodic operator defined through a periodic-in-time cell problem.

math.AP↗

arXiv2025

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

Given an optimal control problem on a heterogeneous body with a periodical structure of particles depending on a small parameter e, we study the asymptotic behavior, as e converges to zero, of the optimal control functional and the optimal state when the initial problem is of parabolic type, and when on the particles' boundary, we assume a dynamic condition and the actuation of some controls for some subset of the particles. We show, in the so-called "critical case" (concerning a certain relation between the structure's period, the diameter of the balls, and the growth coefficient of the particles boundary condition), the appearance of some new non-local in time "strange terms", not only in the limit parabolic equation but also in the limit cost functional. Microscopic localized controls generate peculiar terms in both the limit equation and the cost functional that do not appear in the case of controls applied to the entire set of particles or when the boundary condition on the particles is of Robin type.

math.AP↗

arXiv2021

Boundary control and homogenization: optimal climatization through smart double skin boundaries

We consider the homogenization of an optimal control problem in which the control is placed on a part of the boundary and the spatial domain contains a thin layer of "small particles", very close to the controlling boundary, and a Robin boundary condition is assumed on the boundary of those "small particles". This problem can be associated with the climatization modeling of Bioclimatic Double Skin Façades which was developed in modern architecture as a tool for energy optimization. We assume that the size of the particles and the parameters involved in the Robin boundary condition are critical (and so they justify the occurrence of some "strange terms" in the homogenized problem). The cost functional is given by a weighted balance of the distance (in a H^1-type metric) to a prescribed target internal temperature u_{T} and the proper cost of the control (given by its L^2 norm). We prove the (weak) convergence of states u_{ε} and of the controls v_{ε} to some functions which are completely identified: u_0 satisfies an artificial boundary condition on the control boundary and v_0 is the optimal control associated to a limit cost functional J_0 in which the "boundary strange term" on the control boundary arises. This information on the limit problem makes much more manageable the study of the optimal climatization of such double skin structures.

math.AP↗