Characteristic properties of electromagnetic radiation are investigated for radiating systems in which variation of parameters is described by model smooth functions permitting derivation of exact solutions. Picked as examples of radiating systems are a moving charge with continuously variable velocity (with respect to magnitude and direction), a dipole with continuously variable dipole moment, and a charge uniformly moving in a medium with permittivity continuously varying in space or time. Exact expressions are derived for the electromagnetic field and spectral energy density of radiation in all problems under consideration. It is shown that the common property of electromagnetic radiation in such systems is exponential decline of the spectrum in the high frequency region.