Identification of a Reflection Boundary Coefficient in an Acoustic Wave Equation by Optimal Control Techniques

Abstract

We apply optimal control techniques to find approximate solutions to an inverse problem for the acoustic wave equation. The inverse problem (assumed here to have a solution) is to determine the boundary refection coefficient from partial measurements of the acoustic signal. The sought reflection coefficient is treated as a control and the goal — quantified by an objective functional — is to drive the model solution close to the experimental data by adjusting this coefficient. The problem is solved by finding the optimal control that minimizes the objective functional. Then by driving the “cost of the control” to zero one proves that the sequence of optimal controls represents a converging sequence of estimates for the solution of the inverse problem. Compared to classical regularization methods (e.g. Tikhonov coupled with optimization schemes), our approach yields: (i) a systematic procedure to solve inverse problems of identification type and (ii) an explicit expression for the approximations of the solution.