Optimal-control methods for two new classes of smart obstacles in time-dependent acoustic scattering

Abstract

Time-dependent acoustic scattering problems involving “smart” obstacles are considered. When hit by an incident acoustic field, smart obstacles react in an attempt to pursue a preassigned goal. Let \(\mathbb{R}^3\) be the three-dimensional real Euclidean space, and let \(\Omega \subset \mathbb{R}^3\) be a bounded simply connected open set with a Lipschitz boundary characterized by a constant acoustic boundary impedance χ, immersed in an isotropic and homogeneous medium that fills \(\mathbb{R}^{3}\backslash\Omega\). The closure of Ω will be denoted as \(\overline{\Omega}\). When hit by an incident field, the obstacle Ω pursues the preassigned goal through the action of a control input acting on its boundary (i.e., a quantity with dimensions of a pressure divided by a time). The obstacles considered in this paper monitor the control input acting on their boundaries in order to achieve one of the following goals: (i) be furtive in a given set of the frequency space, and (ii) appear in a given set of the frequency space and outside a given set of \(\mathbb{R}^3\) containing Ω and Ω G as similar as possible to a “ghost” obstacle Ω G having boundary acoustic impedance χ G . It is assumed that \(\overline{\Omega}\cap\overline{\Omega}_G=\emptyset\) and \(\Omega_G \neq \emptyset\). The problem corresponding to the first goal will be called the definite-band furtivity problem, and the problem corresponding to the second goal will be called the definite-band ghost-obstacle problem. These two goals define two classes of smart obstacles. In this paper, these problems are modeled as optimal-control problems for the wave equation introducing a control input acting on the boundary of Ω for time \(t \in \mathbb{R}\). The cost functionals proposed depend on the value of the control input on the boundary of the obstacle and on the value of the scattered acoustic field generated by the obstacle on the boundary in the “furtivity case”, and on the boundary of a suitable set containing Ω and Ω G in the “ghost-obstacle case”. Under some assumptions, the use of the Pontryagin maximum principle allows us to formulate the first-order optimality conditions for the definite-band furtivity problem and for the definite-band ghost-obstacle problem as exterior problems outside the obstacle for a system of two coupled wave equations. Numerical methods to solve these exterior problems are developed by extending previous work. These methods belong to the class of the operator-expansion methods that are highly parallelizable. Numerical experiments proving the validity of the control problems proposed as mathematical models of the definite-band furtivity problem and definite-band ghost obstacle problem are presented. The numerical results obtained with a parallel implementation of the numerical methods developed are discussed and their properties are established. The speed-up factors obtained using parallel computing are really impressive. The website: http://www.econ.univpm.it/recchioni/w11 contains animations and virtual reality applications relative to the numerical experiments.