Nonoccurrence of the Lavrentiev Phenomenon in Optimal Control

Abstract

In this chapter we study nonoccurrence of the Lavrentiev phenomenon for a large class of nonconvex optimal control problems which is identified with the corresponding complete metric space of integrands $\mathcal{M}$ which satisfy a growth condition common in the literature and are Lipschitzian on bounded sets. We establish that for most elements of $\mathcal{M}$ (in the sense of Baire category) the infimum on the full admissible class of trajectory-control pairs is equal to the infimum on a subclass of trajectory-control pairs whose controls are bounded by a certain constant.