Furtivity and masking problems in time dependent acoustic obstacle scattering.

We consider "furtivity" and "masking" problems in time dependent acoustic obstacle scattering. Roughly speaking a "furtivity" ("masking") problem consists in making "undetectable" ("unrecognizable") an object immersed in a medium where an acoustic wave that scatters on the object is propagating. The detection (recognition) of the obstacle must be made through the knowledge of the acoustic field scattered by the object when hit by the propagating wave. These problems are interesting in several application fields. We formulate a mathematical model for the "furtivity" and "masking" problems considered consisting in optimal control problems for the wave equation. Using the Pontryagin maximum principle we show that the solution of these control problems can be characterized as the solution of a suitable exterior problem for a system of two coupled wave equations. The numerical solution of these systems involving partial differential equations in four (space, time) independent variables is a critical issue when reliable and efficient procedures to solve the furtivity or masking problem are required. High performance parallel algorithms are desirable to solve these systems. We suggest a computational method well suited for parallel computing and based on an adapted version of the operator expansion method originally introduced in I and developed by the authors and some co-authors (see 2, 3, 4, 5 and the references therein). Some numerical results in the form of computer animations can be found in the website http://www.econ.unian.it/recchioni/w8. Finally we make some comments on the possible extension of this work to electromagnetic obstacle scattering.