  We consider a stochastic impulse control problem that is motivated by
applications such as the optimal exploitation of a natural resource. In
particular, we consider a stochastic system whose uncontrolled state dynamics
are modelled by a non-explosive positive linear diffusion. The control that can
be applied to this system takes the form of one-sided impulsive action. The
objective of the control problem is to maximise a discounted performance
criterion that rewards the effect of control action but involves a fixed cost
at each time of a control intervention. We derive the complete solution to this
problem under general assumptions. It turns out that the solution can take four
qualitatively different forms, several of which have not been observed in the
literature. In two of the four cases, there exist only $\varepsilon$-optimal
control strategies. We also show that the boundary classification of 0 may play
a critical role in the solution of the problem. Furthermore, we develop a way
for establishing the strong solution to a stochastic impulse control problem's
optimally controlled SDE.
