  We consider stochastic control with discretionary stopping for the drift of a
diffusion process over an infinite time horizon. The objective is to choose a
control process and a stopping time to minimize the expectation of a convex
terminal cost in the presence of a fixed operating cost and a control-dependent
running cost per unit of elapsed time. Under appropriate conditions on the
coefficients of the controlled diffusion, an optimal pair of control and
stopping rules is shown to exist. Moreover, under the same assumptions, it is
shown that the optimal control is a constant which can be computed fairly
explicitly; and that it is optimal to stop the first time an appropriate
interval is visited. We consider also a constrained version of the above
problem, in which an upper bound on the expectation of available stopping times
is imposed; we show that this constrained problem can be reduced to an
unconstrained problem with some appropriate change of parameters and, as a
result, solved by similar arguments.
