Suboptimal boundary controls for elliptic equation in critically perforated domain (Q957580)

The authors consider the elliptic boundary value problem \[ \begin{cases} -\Delta y_\epsilon + y_\epsilon=f_\epsilon &{\text{in}}\,\, \Omega_\epsilon,\\ \partial_\nu y_\epsilon + k_0 y_\epsilon = p_\epsilon &{\text{on}}\,\, \Gamma^N_\epsilon,\\ y_\epsilon = u_\epsilon &{\text{on}}\,\, \Gamma^D_\epsilon,\\ y_\epsilon = 0 &{\text{on }}\,\, \partial\Omega\cap\partial\Omega_\epsilon\end{cases}\tag{*} \] where \(\Omega_\epsilon\) is a finely perforated domain, \(f_\epsilon , p_\epsilon , u_\epsilon\) are given functions, and \(\Gamma^N_\epsilon ,\Gamma^D_\epsilon\) are the Neumann and Dirichlet parts of the holes boundaries. The asymptotic analysis as \(\epsilon\rightarrow 0\) of the problem above has been widely studied in the literature; here the authors consider the optimal control problem \[ \min\left\{\int_{\Omega_\epsilon}|\triangledown y|^2 + |y - z|^2 dx + \int_{\Gamma^N_\epsilon}| p^2 d\sigma + \beta_\epsilon \int_{\Gamma^D_\epsilon} u^2 d\sigma\right\}\leqno(P_\epsilon) \] with \((y, p, u)\) satisfying the state equation (*). The limit optimal control problem as \(\epsilon \rightarrow 0\) is identified and the corresponding optimality conditions are derived.