A windowed averaged scheme is defined for general control systems. The same method is used to average costs in optimal control problems (OCPs). A numerical parameter α can be computed, which expresses the distance between the original system and the averaged system in a weak sense. The value of α is well defined for any system. Then, if we use the optimal control of the averaged OCP in the original OCP, the suboptimality of the control is bounded by an expression of the form C α 2 . This bound holds for any optimal control problem which satisfies the Legendre-Clebsch condition and such that the convexity parameter satifies a general inequation with respect to the numerical parameter α.