Control and stopping of a diffusion process on an interval

Consider a process X(.)=X(t),0.t<. which takes values in the interval I=(0,1), satisfies a stochastic differential equation dX(t)=.(t)dt+.(t)dW(t),X(0)=x.I and, when it reaches an endpoint of the interval I, it is absorbed there. Suppose that the parameters . and . are selected by a controller at each instant t.[0,.) from a set depending on the current position. Assume also that the controller selects a stopping time . for the process and seeks to maximize Eu(X(.)), where u:[0,1].R is a continuous "reward" function. If .:=infx.I:u(x)=maxu and .:=supx.I:u(x)=maxu, then, to the left of ., it is best to maximize the mean-variance ratio (./.2) or to stop, and to the right of ., it is best to minimize the ratio (./.2) or to stop. Between . and ., it is optimal to follow any policy that will bring the process X(.) to a point of maximum for the function u(.) with probability 1, and then stop.