  We consider a quasi-variational inequality governed by a moving set. We
employ the assumption that the movement of the set has a small Lipschitz
constant. Under this requirement, we show that the quasi-variational inequality
has a unique solution which depends Lipschitz-continuously on the source term.
If the data of the problem is (directionally) differentiable, the solution map
is directionally differentiable as well. We also study the optimal control of
the quasi-variational inequality and provide necessary optimality conditions of
strongly stationary type.
