Approximation of boundary control problems on curved domains

The boundary control problems associated to a semilinear elliptic equation defined in a curved domain Ω are considered. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of Ω by an appropriate domain Ω_h (typically polygonal) is required. Here, we do not consider the numerical approximation of the control problems. Instead of it, we formulate the corresponding infinite dimensional control problems in Ω_h and we study the influence of the replacement of Ω by Ω_h on the solutions of the control problems. Our goal is to compare the optimal controls defined on Γ = δΩ with those defined on Γ_h = δΩ_h and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates. The results for convex domains are given in, the results for nonconvex domains are included in a work in progress.