Convex Duality and Generalized Solutions in Optimal Control Problem for Stopped Processes

The approach consists of imbedding the original control problem tightly in a convex mathematical programming problem on the space of measures and then solving the latter problem by duality in convex analysis. The dual to the control problem is to find the supremum of all smooth subsolutions to the Bellman equation. Because of the effect of stops at the boundary of the domain, a different formulation of strong and weak problems will be adopted to make use of the convex duality method. The results about the decomposition of weak measures provide a clear intepretation for such a effect in the weak formualtion of our control problem