Stability estimates for solutions to inverse extremal problems for the Helmholtz equation

Inverse problems for the Helmholtz equation of the acoustic scattering on a three-dimensional inclusion are considered. Using an optimization method, we reduce these problems to inverse extremal problems in which the role of controls is played by a variable refraction index and boundary source density. Solvability of these problems is proved and some optimality systems are obtained that describe necessary optimality conditions. Basing on the analysis of the optimality systems, sufficient conditions on the input data are deduced that guarantee the uniqueness and stability of optimal solutions.