Given \((x,u,t,\theta)\to f(x,u,t,\theta)\), f periodic in \(\theta\) with a period \(\omega\) independent of x, u, t, and \((x,u,t)\to L(x,u,t),\) L sufficiently smooth, consider the perturbed problem \(P_{\epsilon}:\) \[ \text{Minimize }\int^{T}_{0}L(x,u,t)dt\text{ subject to } \dot x=f(x,u,t,t/\epsilon),\quad x(0)=x^ 0,\quad u\in L^ 2(0,T). \] The associated averaged problem is \[ \text{Minimize }\int^{T}_{0}dt/\omega \int^{\omega}_{0}L(y(t),\sigma (t,\theta),t)d\theta, \] \[ \text{subject to } \dot y=(1/\omega)\int^{\omega}_{0}f(y(t),\sigma (t,\theta),t,\theta)d\theta,\quad \sigma \in L^ 2[(0,T)\times (0,\omega)]. \] Conditions are given under which, if \(u_ 0\) is the optimal control for the averaged problem and \(u^{\epsilon}(t)=u_ 0(t,t/\epsilon)\), then \(u^{\epsilon}\) is near optimal for the perturbed one with an error on the cost of order \(\epsilon^ 2\). The nonperiodic case is discussed as well.