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On some impulse control problems with constraint. (English) Zbl 1381.49037

In this paper, the authors give a comprehensive analysis about an impulse control problem of a discounted-cost type associated to a considerably general Markov-Feller process \(\{x_t\}\) with the key property that impulses are allowed to take place only when a given signal “arrives”. The signal is modeled as a stochastic process \(\{y_t\}\) that does not necessarily satisfy the Markov property and that could be dependent to \(\{x_t\}\) in some sense. Once the process \(\{y_t\}\) is provided, the “arrival” of the signal means either scenarios: (a) \(y_t=0\), or a more restricted case \(y_t=0\) and \(t>0\). Both cases are analyzed in detail.
Optimality results are described through the use of the well-known dynamic programming method; in particular, an associated HJB equation is described whose solution is shown to exist in a suitable sense and it coincides with the optimal value of the control problem. Finally, a specific optimal control of feedback-type is supplied based on the use of the so-named continuation region.
The paper is well-written and easy to follow.

MSC:

49N25 Impulsive optimal control problems
93E20 Optimal stochastic control
49J40 Variational inequalities
60J60 Diffusion processes
60J75 Jump processes (MSC2010)
49L20 Dynamic programming in optimal control and differential games
49N35 Optimal feedback synthesis
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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