  In this paper, we consider the problem of controlling a diffusion process
pertaining to an opioid epidemic dynamical model with random perturbation so as
to prevent it from leaving a given bounded open domain. Here, we assume that
the random perturbation enters only through the dynamics of the susceptible
group in the compartmental model of the opioid epidemic dynamics and, as a
result of this, the corresponding diffusion is degenerate, for which we further
assume that the associated diffusion operator is hypoelliptic. In particular,
we minimize the asymptotic exit rate of such a controlled-diffusion process
from the given bounded open domain and we derive the Hamilton-Jacobi-Bellman
equation for the corresponding optimal control problem, which is closely
related to a nonlinear eigenvalue problem. Finally, we also prove a
verification theorem that provides a sufficient condition for optimal control.
