Optimal control on perforated domains (Q1279951)

The paper considers a family \(\{P_\varepsilon\}\) of optimal control problems in perforated domains \[ P_\varepsilon= \begin{cases} I_\varepsilon= \int_{\Omega_\varepsilon} \langle B_\varepsilon(x)\nabla u_\varepsilon, u_\varepsilon\rangle dx+ N\int_{\Omega_\varepsilon} \theta^2_\varepsilon dx\to \min,\\ \text{div }A_\varepsilon(x)\nabla u_\varepsilon= f+ \theta_\varepsilon\quad\text{in }\Omega_\varepsilon,\\ u_\varepsilon= 0\quad\text{on }\partial\Omega,\;\langle A_\varepsilon(x)\nabla u_\varepsilon, n_\varepsilon\rangle= 0\quad\text{on }\partial S_\varepsilon,\\ \theta_\varepsilon\in U_{\text{ad}},\end{cases} \] where \(\Omega\subset \mathbb{R}^n\) is a bounded Lipschitz domain, \(S_\varepsilon\) are families of closed subsets of \(\Omega\), \(\Omega_\varepsilon= \Omega\setminus S_\varepsilon\), \(\partial\Omega\) and \(\partial S_\varepsilon\) are boundaries of \(\Omega\) and \(S_\varepsilon\), respectively, \(n_\varepsilon\) is the normal to \(\partial S_\varepsilon\), \(U_{\text{ad}}\subset L_2(\Omega)\) is the set of admissible controls. Provided that the characteristic functions \(\chi_\varepsilon\) of \(\Omega_\varepsilon\) converges weakly as \(\varepsilon\to 0\) to a function \(\chi_0\), \(\chi_0(x)\geq\delta>0\) for a.e. \(x\in\Omega\), it is shown that there exists a limit problem \(P_0\) of the same type as \(P_\varepsilon\). A formal description of the limit matrices \(A_0\) and \(B_0\), and exact limit relationships between optimal solutions \(\theta^*_\varepsilon\) (of the problem \(P_\varepsilon\)) and \(\theta^*_0\) (of the problem \(P_0\)) are given.