First, an optimal stopping problem of a Markov-Feller process is considered when the controller is allowed to stop the evolution only at the arrival times of a signal. A complete setting and resolution of this problem is discussed, e.g., when the inter-arrival times of the signal are independent identically distributed random variables, and then several extensions to other signals and to other cases of state spaces are also mentioned. These results generalize the works of several authors, where the Markov process was a diffusion process and where the signal arrives at the jump times of a Poisson process.
Next, the impulse control of a Markov-Feller process is considered when the impulses are allowed only when a signal arrives and with a discounted cost. 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.