In this work, the analytical derivation and the computer implementation of the adjoint method are described. The adjoint method can be effectively used for solving the optimal control problem associated with a large class of nonlinear mechanical systems. As discussed in this investigation, the adjoint method represents a broad computational framework, rather than a single numerical algorithm, in which the control problem for nonlinear dynamical systems can be effectively formulated and implemented employing a set of advanced analytical methods as well as an array of well-established numerical procedures. A detailed theoretical derivation and a comprehensive description of the numerical algorithm suitable for the computer implementation of the methodology used for performing the adjoint analysis are provided in the paper. For this purpose, two important cases are analyzed in this work, namely the design of a feedforward control scheme and the development of a feedback control architecture. In this investigation, the control problem relative to the mechanical vibrations of a nonlinear oscillator characterized by a generalized Van der Pol damping model is considered in order to illustrate the effectiveness of the computational algorithm based on the adjoint method by means of numerical experiments.