Approximation of boundary control problems on curved domains

Abstract

In this paper we consider boundary control problems associated to a semilinear elliptic equation defined in a curved domain [omega]. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of [omega] by an appropriate domain [omega]h (typically polygonal) is required. Here we do not consider the numerical approximation of the control problems. Instead, we formulate the corresponding infinite dimensional control problems in [omega]h, and we study the influence of the replacement of [omega] by [omega]h on the solutions of the control problems. Our goal is to compare the optimal controls defined on T=e[omega] with those defined on Th=e[omega]h and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates.