A bang-bang strategy for a finite fuel stochastic control problem

The problem treated is that of controlling a process with values in [0, a]. The non-anticipative controls (µ(t), .(t)) are selected from a set C(x) whenever X(t.) = x and the non-decreasing process A(t) is chosen by the controller subject to the condition where y is a constant representing the initial amount of fuel. The object is to maximize the probability that X(t) reaches a. The optimal process is determined when the function has a unique minimum on [0, a] and satisfies certain regularity conditions. The optimal process is a combination of .timid play' in which fuel is used gradually in the form of local time at 0, and .bold play' in which all the fuel is used at once.