Optimal control problems in radiation and scattering

The problem of determining the surface current on a closed piecewise smooth curve which optimizes some functional of the far field is considered. In particular, the existence of square integrable Dirichlet data on the boundary which optimizes power radiated in angular sectors is proven. Moreover a Galerkin method for approximating the optimal solution is developed and estimates of the rates of convergence are presented. The results are applied in the specific example of a circular cylinder and numerical evaluation of both the optimal surface current and far field are given.