In this paper, the Rayleigh beam system with two dynamical boundary controls is treated. Theoretically, the well-posedness of the weak solution is obtained. Later, we discretize the system by using the Implicit Euler scheme in time and the P 3 Hermite finite element in space. In addition, we show the decay of the discrete energy and we establish some a priori error estimates. Finally, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]