Finite dimensional optimal control problem in inverse acoustics applications

A non linear, finite dimensional optimal control problem is considered, which consists of determining the shape of an acoustic scatterer from information about the scattering amplitude. The shape parameters, which describe the unknown obstacle surface, are the control variables. Three aspects of the proposed method of solution are relevant as applications of optimal control theory: i) the finite dimensional formulation, obtained from the properties of complete families in state space; ii) the minimization of a one term cost function, the `boundary defect', where some constraints, treated as a penalty term by previous methods, are now translated into the action of an approximate back propagation (ABP) operator acting on the far field coefficients, the latter being estimated from the scattering amplitude; iii) the computationally efficient structure of the minimization algorithm, where data back propagation and shape parameter update occur in two separate stages. Some details of the physical problem are provided. The corresponding model and the solution algorithm are described. Existence and uniqueness of the optimal control are considered. Several numerical results are presented, which comply with an error estimate based on the approximation scheme.