We study the periodic oscillations of a 1st order delayed linear system with relay output and proportional$+$integral feedback and describe the behavior of the general solutions of the closed loop. We present results on control of the oscillations and on rejection of the perturbations. For the system under study, we first exhibit a countable set of periodic limit cycles. We show that in the particular case where only proportional control is used, any solution tends in finite time towards one of the limit cycles (whose determination depends on the initial conditions). All the cycles are orbitally unstable except one of them, the only slowly oscillating one. Finally, we provide some well-posedness and ultimate boundedness results for a time-varying perturbed version of the system under study. The given estimates show that the proportional$+$integral feedback law permits to reject various parametric perturbations.