We report on recent works carried out in collaboration with Lorenzo Ferreri and Raphaël Prunier regarding some inverse obstacle problems arising in optimal control theory. More specifically, we show that for a wide class of bilinear optimal control problems, the optimal controls are characteristic functions of subsets, and we investigate the regularity of these subsets by relying on the theory of free-boundary problems. A typical result we obtain is the $C^2$ regularity of the boundary of these sets. These information have interesting consequences regarding the convergence of numerical methods.