  Quasi-variational inequalities (QVIs) of obstacle type in many cases have
multiple solutions that can be ordered. We study a multitude of properties of
the operator mapping the source term to the minimal or maximal solution of such
QVIs. We prove that the solution maps are locally Lipschitz continuous and
directionally differentiable and show existence of optimal controls for
problems that incorporate these maps as the control-to-state operator. We also
consider a Moreau--Yosida-type penalisation for the QVI wherein we show that it
is possible to approximate the minimal and maximal solutions by sequences of
minimal and maximal solutions (respectively) of certain PDEs, which have a
simpler structure and offer a convenient characterisation in particular for
computation. For solution mappings of these penalised problems, we prove a
number of properties including Lipschitz and differential stability. Making use
of the penalised equations, we derive (in the limit) C-stationarity conditions
for the control problem, in addition to the Bouligand stationarity we get from
the differentiability result.
