The impulse control of a Markov–Feller process is considered when the impulses are allowed only when a signal arrives. This is referred to as an impulse control problem with constraint. A detailed setting is described, a characterization of the optimal cost is obtained using previous results of the authors on optimal stopping problems with constraint, and an optimal impulse control is identified.
Menaldi, Jose L. and Robin, Maurice, "On Some Impulse Control Problems with Constraint" (2017). Mathematics Faculty Research Publications. 62.