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

Abstract

We study the Lagrange Problem of Optimal Control with a functional ∫abL(t, x(t), u(t))dt and control-affine dynamics x = f(t, x)+g(t, x)u and (a priori) unconstrained control u∈Rm. 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.