Mathematical Models of Smart Obstacles

We propose mathematical models to describe the behaviour of smart obstacles. In the context of acoustic scattering a smart obstacle in an obstacle that when hit b y an incoming acoustic wave reacts circulating on its boundary a “pressure current” to pursue a given goal. A pressure current is a quantity whose physical dimension is pressure divided by time. The goals considered are: 1) to be undetectable, 2) to appear with a shape and an acoustic boundary impedance different from its actual ones, 3) to appear in a location in space different from its actual one eventually with a shape and an acoustic boundary impedance different from its actual ones. He mathematical models proposed for the smart obstacles are optimal control problem for the wave equation. These optimal control problems are studied analytically and solved quantitatively using ad hoc numerical methods. 1. 0 INTRODUCTION In this paper a mathematical model for an acoustic time dependent scattering problem involving smart obstacles is formulated. Smart obstacles are obstacles that when hit by an incoming acoustic field react in order to pursue an assigned goal. The goal pursued by the smart obstacle considered in this paper is: to appear in a location in space different from its actual location eventually with a shape and boundary impedance different from its actual ones. We call this goal: to appear as a ghost obstacle. The smart obstacle pursues its goal circulating a pressure current (i.e. a quantity whose physical dimension is pressure divided by time) on its boundary. We show that the pressure current necessary to pursue the goal can be determined as the solution of a suitable optimal control problem for the wave equation. The author and its coworkers have studied similar models for several other classes of smart obstacles in acoustic and electromagnetic scattering (see for example (1)-(6) and the website: http://www.econ.univpm.it/recchioni). The obstacles considered pursue one of the following goals: 1. to be undetectable (i.e.: furtivity problem), 2. to appear with a shape and a boundary impedance different from its actual shape and impedance (i.e.: masking problem), 3. to appear in a location in space different from its actual location eventually with a shape and boundary impedance different from its actual ones (i.e.: ghost obstacle problem). Zirilli, F. (2006) Mathematical Models of Smart Obstacles. In Battlefield Acoustic Sensing for ISR Applications (pp. 20-1 – 20-8). Meeting Proceedings RTO-MP-SET-107, Paper 20. Neuilly-sur-Seine, France: RTO. Available from: http://www.rto.nato.int/abstracts.asp. Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 01 OCT 2006 2. REPORT TYPE N/A 3. DATES COVERED 4. TITLE AND SUBTITLE Mathematical Models of Smart Obstacles 5a. CONTRACT NUMBER