Optimal control and homogenization of semi-linear parabolic problem with highly oscillatory coefficients in an oscillating domain

In this article, we explore the homogenization of an optimal control problem driven by a semi-linear parabolic equation within a two-dimensional oscillating domain, denoted as Omega(& varepsilon;). The state equation and cost function in this scenario involve periodic coefficients, A(& varepsilon;) and B-& varepsilon;, which exhibit significant oscillations. The objective of this study is to analyze the limiting behavior of both the optimal control and the corresponding state as the oscillations become increasingly fine. Furthermore, we aim to identify the optimal control problem that encapsulates the effects of these oscillating coefficients and to establish a corrector result for the state variable.