Weak compactness of conditional probability measures and its application to stochastic linear control problems

Abstract

Abstract In this paper we consider a class of stochastic linear Ito differential equations with control parameters. It is shown (theorem 2.1) that the set of conditional probability measures, generated by the differential system in response to a class of bounded measurable controls with values in a compact convex subset U of a finite dimensional Euclidean space Rv , is equivalent to those corresponding to controls taking values only on the boundary of the set U. Further this set of conditional probability measures is shown (theorem 2.2) to be weakly compact. These results are then used to prove the existence (theorem 2.4) of an optimal control that satisfies the so called ’ bang-bang ’ principle (Hermes and LaSalle, p. 46).