We apply optimal control techniques to find approximate solutions to an inverse problem for the acoustic wave equation. The inverse problem (assumed to have a unique solution) is to determine the shape and reflection coefficient of a part of the boundary from partial measurements of the acoustic signal. The sought functions are treated as controls and the goal - quantified by an objective functional - is to drive the model solution close to the experimental data by adjusting these functions. The problem is solved by finding the optimal control pair, which minimizes the objective functional. Then by driving the "cost of the control" to zero one proves that the sequence of optimal controls converges to the solution of the inverse problem.
"Identification of boundary shape and reflectivity in a wave equation by optimal control techniques." Differential Integral Equations 13 (7-9) 941 - 972, 2000. https://doi.org/10.57262/die/1356061205