The problem studied is as follows: when does the full solution of minimizing $x^0 (T)$, given \[\begin{gathered} \dot x(t) = f(x(t),y(t),u(t)),\quad u(t) \in U, \hfill \\ \varepsilon \dot y(t) = g(x(t),y(t),u(t)),\quad 0 \leqq t \leqq T, \hfill \\ \end{gathered} \] with boundary conditions on x and y, converge in some sense to the reduced solution of minimizing $x_0^0 (T_0 )$, given \[\begin{gathered} \dot x_0 (t) = f(x_0 (t),y_0 (t),u_0 (t)),\quad u_0 (t) \in U, \hfill \\ 0 = g(x_0 (t),y_0 (t),u_0 (t)),\quad 0 \leqq t \leqq T_0 , \hfill \\ \end{gathered} \] with boundary conditions on $x_0 $ as $\varepsilon \to 0$? Without the minimization, this is a standard topic in o.d.e. theory which essentially covers the case where $u = u_0 $ is smooth. The corresponding methods need considerable modification for the control problem and, in the end, are closer to those of optimal existence theory. Assuming Lipschitz dependent right sides for the full model, we see that various additional hypotheses give convergence...