Optimal Control Problems in a Domain with Oscillating Boundary and Homogenization

Abstract: We consider a two dimensional oscillating domain (comb shape type) $\Omega_{\epsilon}$ consists of a fixed bottom region $\Omega^-$ and an oscillatory (rugose) upper region $\Omega_{\epsilon}^{+}$. We introduce an optimal control problems in $\Omega_{\epsilon}$ for the Laplacian operator. There are mainly two types of optimal control problems; namely distributed control and boundary control. In this talk, first we consider distributed optimal control problem, where the control is supported on the oscillating part $\Omega_{\epsilon}^{+}$ with periodic controls and with Neumann condition on the oscillating boundary $\gamma_{\epsilon}$. Secondly, we introduce boundary optimal control problem, control applied through Neumann boundary condition on the oscillating boundary $\gamma_{\epsilon}$ with suitable scaling parameters. Our main aim is to characterize the controls and study the limiting analysis (as $\epsilon \to 0$) of the optimal solution.