A characterization of relaxed controls with commensurate delays Original Research Article

The question of how to properly relax optimal control problems involving arbitrary commensurate delays in the controls was studied in a recent paper by characterizing, in terms of projections of a common probability measure, the weak star closure of the space of ordinary controls. In this paper, we provide a characterization where the common probability measure is assumed to be a measurable measure-valued function. Also, we study some implications of this characterization and show how a well-known result, whose traditional role has been to provide a definition of conditional expectations, can be applied to these problems.