Study of a non-classical perturbed optimal control system (Q1381930)

The paper deals with a perturbed optimal control problem arising in aerodynamics. The state equation is \[ \begin{cases} -\Delta y_\varepsilon(v)=0 & \text{in \(\Omega\);}\\ \partial y_\varepsilon(v)\over\partial\nu+\varepsilon y_\varepsilon(v)=0 & \text{on\;\(\Gamma\)},\qquad\qquad y_\varepsilon(v)\in H^1(\Omega),\end{cases} \] where \(\Gamma=\partial\Omega\) is smooth, and \(\Omega\) locally lies on one side of \(\Gamma\). Denoting by \(T_\varepsilon:L^2(\Gamma)\to L^2(\Gamma)\) the map \(v\mapsto {y_\varepsilon(v)}_\Gamma\), the cost functional is \(J_\varepsilon(v)=\| T_\varepsilon(v)-z\|^2_2\), with \(z\) a given non zero vector in \(L^2(\Gamma)\), and \(v\) varies in some subspace of admissible controls. The paper deals with existence and uniqueness of optimal controls \(v_\varepsilon\), and with their behaviour as \(\varepsilon\to 0\).