This note considers two particular one-dimensional dynamic systems of the form dx(t)=m(x)dt+Bu(t)dt+[v1’/2(x)]dW(t), where W(t) is a Brownian motion process. It first takes m=¸-x(t)/θ and v=N (a positive constant), then chooses m=Nx(t) and v=2Nx2(t). The control that minimizes the expected value of a cost function with quadratic control costs is found on the way and a termination cost which is 0 or •, according as the objective has been attained or not. This objective is either to stay in a given interval for a time τ or to leave the interval before a time τ. Explicit results are presented.
