In this paper, we address nonradiating and cloaking problems exploiting the surface equivalence principle, by imposing at any arbitrary boundary the control of the admittance discontinuity between the overall object (with or without cloak) and the background. After a rigorous demonstration, we apply this model to a nonradiating problem, appealing for anapole modes and metamolecules modeling, and to a cloaking problem, appealing for non-Foster metasurface design. A straightforward analytical condition is obtained for controlling the scattering of a dielectric object over a surface boundary of interest. Previous quasistatic results are confirmed and a general closed-form solution beyond the subwavelength regime is provided. In addition, this formulation can be extended to other wave phenomena once the proper admittance function is defined (thermal, acoustics, elastomechanics, etc.).