Several critical physical properties of a material are controlled by its geometric construction.
Therefore, analyzing the effect of a material’s geometric structure can help to improve some of
its beneficial physical properties and reduce unwanted behavior. This leads to the study of
boundary value problems in complex domains such as perforated domain, thin domain, junctions of
the thin domain of different configuration, domain with rapidly oscillating boundary, networks
This talk will discuss various homogenization problems posed on high oscillating domains. We
discuss in detail one of the articles (see, Journal of Differential Equations 291 (2021): 57-89.),
where the oscillatory part is made of two materials with high contrasting conductivities. Thus the
low contrast material acts as near insulation in-between the conducting materials. Mathematically
this leads to the study of degenerate elliptic PDE at the limiting scale. We also briefly explain
another interesting article (see, ESAIM: Control, Optimisation, and Calculus of Variations 27 (2021): S4.),
where the oscillations are on the curved interface with general cost functional.
In the first part of my talk, I will briefly discuss the periodic unfolding method and its
construction as it is the main tool in our analysis.
The second part of this talk will briefly discuss the boundary optimal control problems subject to
Laplacian and Stokes systems.
In the third part of the talk, we will discuss the homogenization of optimal control problems subject
to a elliptic variational form with high contrast diffusivity coefficients. The interesting result is
the difference in the limit behavior of the optimal control problem, which depends on the control’s
action, whether it is on the conductive part or insulating part. In both cases, we derive the
\two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if
the controls are acting on the conductive region, a complete-scale separation is available, whereas
a complete separation is not visible in the insulating case due to the intrinsic nature of the problem.
In this case, to obtain the limit optimal control problem in the macro scale, two cross-sectional cell
problems are introduced. We obtain the homogenized equation for the state, but the two-scale separation
of the cost functional remains as an open question.