The authors study the existence and uniqueness of -periodic solutions for the equation
where is a -periodic function given by
with These problems arise in different physical situations such as in the stabilization of matter-wave breathers in Bose-Einstein condensates, in the propagation of guided waves in optical fibers and in the electromagnetic trapping of a neutral atom near a charged wire. If the parameters are fixed, and an interesting question is how to control the switching times in order to get periodic states with a particular amplitude. This question is studied in the paper as well as the stability properties (in the linear sense) of the -periodic solutions.