Lipschitzian regularity of minimizers for optimal control problems with control-affine dynamics

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 proble m are bounded-a fact which is crucial for the applicability of many necessa ry optimality conditions, like, for example, the Pontryagin Maximum Princip le. As a corollary we obtain conditions for the Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Pr oblem of the Calculus of Variations with higher-order derivatives.