Mathematical models of 'active' obstacles in acoustic scattering.

We consider the following problem: given an acoustic incident field that hits a bounded obstacle Omega characterized by a constant acoustic boundary impedance X and immersed in an isotropic and homogeneous medium that fills R-3\Omega choose a pressure current, acting on the boundary of Omega, in order to pursue a given goal that involves the field scattered by Omega when hit by the incident field. Obstacles of this kind are called active obstacles. Let us give two examples of goals that can be pursued. Let D and D-G be bounded sets contained in R-3 and characterized by a non-negative constant boundary acoustic impedance X' and XG respectively, such that D subset of Omega and D-G boolean AND Omega = phi, D-G not equal phi. The problems considered consist in choosing a pressure current defined on the boundary of Omega in order to make the wave scattered by Omega when hit by the incident acoustic field to appear like the wave scattered by D or by D-G in the same circumstances outside a bounded region that contains Omega and D or Omega and DG respectively. The problem involving D is called "masking problem", the problem involving D-G is called "ghost obstacle problem." In this paper the two problems proposed are modeled as optimal control problems for the wave equation. Using the Pontryagin maximum principle the first order optimality conditions associated to these two problems are formulated as exterior problems defined outside Q for a system of two coupled wave equations. These exterior problems are well suited to be solved with an adapted version of the operator expansion method proposed in [5]. Finally the numerical results obtained on some test cases are discussed.