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

This paper addresses the optimal control problem for a stochastic evolution system perturbed by a Markov-modulated Poisson process within a diffusion approximation framework. The considered system captures complex dynamics involving continuous evolution and discrete, state-dependent jumps, enabling the modeling of systems with regime-switching behavior or infrequent but significant events. The control function is constructed by minimizing a quality criterion through a stochastic optimization procedure. To analyze the asymptotic behavior of the system as the perturbation parameter vanishes, we derive the generator of the process and solve a corresponding singular perturbation problem. This allows us to prove the weak convergence of the stochastic system to a diffusion process. Furthermore, we establish sufficient conditions under which the control strategy converges almost surely to an optimal control. The obtained result makes it possible to study the rate of convergence of evolution under the optimal control for problems with a Markov-modulated Poisson perturbation.