HIGH CONTRASTING DIFFUSION IN HEISENBERG GROUP: HOMOGENIZATION OF OPTIMAL CONTROL VIA UNFOLDING

Abstract

The periodic unfolding method is one of the latest tools for studying multiscale problems like homogenization after the development of multiscale convergence in the 1990s. It provides a good understanding of various microscales involved in the problem, which can be conveniently and easily applied to get the asymptotic limit. In this article, we develop the periodic unfolding for the Heisenberg group, which has a noncommutative group structure. The concept of greatest integer part and fractional part for the Heisenberg group has been introduced corresponding to the periodic cell. Analogous to the Euclidean unfolding operator, we prove the integral equality, L2-weak compactness, unfolding gradient convergence, and other related properties. Moreover, we have the adjoint operator for the unfolding operator, which can be recognized as an average operator. As an application of the unfolding operator, we have homogenized the standard elliptic PDE with oscillating coefficients. We have also considered an optimal control problem with the state equation having high contrasting diffusivity coefficients. The high contrasting coefficients are an added difficulty in the analysis. Moreover, we have characterized the interior periodic optimal control in terms of the unfolding operator, which helps us to analyze the asymptotic behavior.