Infinite-Dimensional Linear Control Problems
In this chapter we show nonoccurrence of gap for two large classes of infinite-dimensional linear control systems in a Hilbert space with nonconvex integrands. These classes are identified with the corresponding complete metric spaces of integrands which satisfy a growth condition common in the literature. For most elements of the first space of integrands (in the sense of Baire category) we establish the existence of a minimizing sequence of trajectory-control pairs with bounded controls. We also establish that for most elements of the second space (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.