The modern electrodynamic theory of gratings represents a unique collection of analytical and experimental results providing a rather full physical picture of the processes in resonant wave scattering under conditions of complex dispersion laws (including nonclassical ones). By now the reliable mathematical methods oriented to both qualitative and numerical analysis are developed and evaluated [1–4]. A large body of anomalous and resonant phenomena are revealed and investigated [1, 5, 6]. Justified recommendations have been elaborated for their use in spectroscopy, antenna engineering, optics, electronics, acoustics, solid state physics, millimeter and submillimeter radio engineering, etc. In total, all these results constitutes the necessary basis for posing and solving practically important inverse problems. However, a number of challenging mathematical problems arises in their solution [7]. Sometimes they are significantly different from that in direct problems. Certain of them are considered in the present work devoted to the synthesis of one-dimensional periodic perfectly conducting gratings.