  We consider a stochastic system whose uncontrolled state dynamics are
modelled by a general one-dimensional It\^{o} diffusion. The control effort
that can be applied to this system takes the form that is associated with the
so-called monotone follower problem of singular stochastic control. The control
problem that we address aims at maximising a performance criterion that rewards
high values of the utility derived from the system's controlled state but
penalises any expenditure of control effort. This problem has been motivated by
applications such as the so-called goodwill problem in which the system's state
is used to represent the image that a product has in a market, while control
expenditure is associated with raising the product's image, e.g., through
advertising. We obtain the solution to the optimisation problem that we
consider in a closed analytic form under rather general assumptions. Also, our
analysis establishes a number of results that are concerned with analytic as
well as probabilistic expressions for the first derivative of the solution to a
second order linear non-homogeneous ordinary differential equation. These
results have independent interest and can potentially be of use to the solution
of other one-dimensional stochastic control problems.
