Approximation of boundary control problems on curved domains

Abstract

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 Ωh (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 Ωh and we study the influence of the replacement of Ω by Ωh on the solutions of the control problems. Our goal is to compare the optimal controls defined on Γ = δΩ with those defined on Γh = δΩh 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 [1], the results for nonconvex domains are included in a work in progress.