What is Impulse control, quasi-variational inequality?

Impulse control acts on a state by discrete interventions: at stopping times $\tau_1 < \tau_2 < \dots$ it moves $X$ by jumps $\xi_i$, each paying a cost with a fixed part $K > 0$. Its value function satisfies a quasi-variational inequality: $\max(\mathcal LV - rV - f,\ \mathcal MV - V) = 0$ (for a cost-minimisation, with signs reversed), where $\mathcal MV(x) = \sup_\xi\{V(x + \xi) - K - \text{cost}(\xi)\}$ is the value of acting now: the obstacle depends on the unknown $V$ itself.