Multiplicative control in the scattering problem for the Helmholtz equation

Under consideration is the problem of optimal multiplicative control for the Helmholtz equation. Scattering of the harmonic waves by a certain obstacle with impedance boundary is studied. We look for the value of impedance such that the distribution of the acoustic pressure on a certain sphere is maximally close to a given one. The main result consists in the description of the structure of the solution set, substantiation of the system of optimality, and derivation of the bang-bang principle for the optimal control.