Control problem for the Markov-modulated Poisson process in the averaging schema

This paper discusses the control problem for the stochastic evolution system under Markov-modulated Poisson perturbations in the averaging schema. Such a perturbation process combines a Poisson process with a Markov process that modulates the intensity of jumps. Formulating the problem allows us to model systems with transitions between regimes or significant rare jumps. In this problem, the control function is determined by the condition of the extremum of the quality criterion function. The asymptotic properties of the stochastic evolution and control coupled process are being investigated initially. For this purpose, we build the generator for the limit process, solving the singular perturbation problem of the original process. The obtained result allows us to formulate and prove the theorem about weak convergence of the evolution process to the averaged one. The solution of a deterministic differential equation determines the limit process for a stochastic evolution system at increasing time intervals. Then, the obtained result allows us to formulate the conditions that ensure the weak convergence of the control function to optimal control.