Inverse scattering and control as constrained optimization problems

A class of inverse scattering and control problems for the Helmholtz equations was reformulated as constrained optimization problems. The cost functional depends on the solution of a boundary value or transmission problem where typically either the boundary or the boundary conditions are not known a priori and are sought to optimize the functional. Through the use of complete families of radiating solutions of the Helmholtz equation the problem is reduced to finite dimensions. Existence (though not uniqueness) of solutions of the finite-dimensional constrained optimization problems and convergence to solutions of the original problem were established in a number of particular cases, including optimizing antenna radiation patterns and reconstructing scattering geometry from measured far-scattered-field data. A general approach suitable for application to a number of similar problems is described and results in particular cases are presented