Set Convergence, Measurability and Existence of Minimizers

This chapter covers basic concepts and definitions concerning set convergence, measurability and multifunctions. Multifunctions are mappings whose values are not single points in the range space but subsets of such points. They are widely encountered in dynamic optimization; for example, it is helpful to interpret a time dependent control constraint set as a multifunction and admissible control functions as selectors of that multifunction. The chapter provides proofs of key properties, with special emphasis on properties relevant to dynamic optimization. A prior study of measurability is an important step in establishing existence of solutions to dynamic optimization problems. The chapter concludes with a theorem concerning existence of minimizers for the generalized problem of Bolza. The underlying optimization problem resembles the problem of Bolza from the classical calculus of variations, involving integral and endpoint costs. But it is in fact a far broader problem formulation, because we depart from the classical framework by allowing the cost integrand and endpoint constraint functions to take values in the extended real line. This formulation subsumes a wide range of problems involving dynamic constraints (in the form of controlled differential equations or differential inclusions), which can be accommodated by the use of extended valued indicator functions added to the cost integrand.