The topic of the article is the class of the control systems with a bistable relay element and smooth nonlinear plant running in the forced vibrations mode. Methods were developed for obtaining the sensitivity functions of the periodic motion characteristics (periodic trajectory, algebraic criterion of stability by Lyapunov). These methods can be used for sensitivity investigation of the control systems with the symmetrical pulse wide modulation (PWM), which have widespread technical applications. Sensitivity functions were obtained based on the precision method of investigation of the periodic motion. This ensures validity of the sensitivity functions, which are received due to the developed methods. An example is presented, which illustrates investigation of the sensitivity functions of the periodic trajectory and algebraic criterion of stability. This article is the first work, the aim of which is development of the theory of sensitivity of the relay control systems running in the forced vibrations mode. Besides the solution to the problem, in fact, it contains an algorithm, which allows us to develop methods for obtaining the sensitivity functions of the periodic motion for the systems with non-linear plant of any order. Having obtained the sensitivity functions, we can relatively easily identify the deviations in the output periodic motion characteristics. This allows us to estimate the system's performance. The proposed methods allow us to perform system syntheses and system optimization by setting limits on the periodic motion sensitivity.