In this study, we investigate the connection between impulse control
and singular control. The dynamic systems are driven by Brownian motion
with drift. For simplicity we consider only one-dimension problems,
where we can perform explicit calculations. We will see that Quasi-Variational
Inequalities (QVI) are the common tool to consider these problems.
The two problems have interesting links. By some aspects, singular
control problems appear as particular cases of impulse control problems;
however an impulse control is a particular case of singular control.
We can, in particular, approximate an optimal singular control by
a minimizing sequence of impulse controls. We show that optimal singular
controls are linked to reflected diffusions. Thanks to the one-dimensionality
we completely solve the QVI by the two band approach.