  The periodic unfolding method is one of the latest tool after multi-scale
convergence to study multi-scale problems like homogenization problems. It
provides a good understanding of various micro scales involved in the problem
which can be conveniently and easily applied to get the asymptotic limit. In
this article, we develop {\it the periodic unfolding} for the Heisenberg group
which has a non-commutative group structure. In order to do this, the concept
of the greatest integer part, 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, $L^2$-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 and characterized the interior
periodic optimal control in terms of unfolding operator.

Analysis of PDEs;Optimization and Control