We study the Lagrange Problem of Optimal Control with a functional integral(a)(b) L (t, x (t), u (t)) dt and control-affine dynamics (x) over dot = f (t, x) + g (t, x)u and (a priori) unconstrained control u is an element of R-m. We obtain conditions under which the minimizing controls of the problem are bounded-a fact which is crucial for the applicability of many necessary optimality conditions, like, for example, the Pontryagin Maximum Principle. As a corollary we obtain conditions for the Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Problem of the Calculus of Variations with higher-order derivatives.