A singularly perturbed control system involving two ordinary differential equations is studied. These equations are used to model a system with a slow variable and a fast variable. The main goal is to examine the convergence of an optimal cost associated with the equations under study. A perturbed and reduced control system is also considered. The existence of optimal solutions for this system is proved. The convergence of the optimal cost of the full control system and the reduced one are determined. The rate of convergence can also be studied if the limit of the reduced control system satisfies an extra regularity condition.