The behavior of smart obstacles in electromagnetic scattering: mathematical models as optimal control problems

Abstract

We consider a bounded obstacle characterized by a boundary electromagnetic impedance contained in the three dimensional real Euclidean space filled with a homogeneous isotropic medium. When an incoming electromagnetic field illuminates the obstacle a scattered field is generated. A smart obstacle is an obstacle that in the scattering process, circulating a surface electric current density on its boundary, tries to achieve a given goal. We consider four possible goals: making the obstacle undetectable (i.e.: furtivity problem), making the obstacle to appear with a shape and impedance different from its actual ones (i.e.: masking problem), making the obstacle to appear in a location different from its actual one eventually with a shape and impedance different from its actual ones (i.e.: ghost obstacle problem) and finally one of the previous goals limited to a given subset of the frequency space (i.e.: definite band problems). We consider the problem of determining the optimal electric current density to achieve the given goal. The relevance in many application fields (i.e. stealth technology, electromagnetic noise control, etc.) of these problems is well known. The previous problems are modelled as optimal control problems for the Maxwell equations. Some numerical results on test problems obtained solving the optimal control problems proposed are shown.