  In this paper, we study the asymptotic of exit problem for controlled Markov
diffusion processes with random jumps and vanishing diffusion terms, where the
random jumps are introduced in order to modify the evolution of the controlled
diffusions by switching from one mode of dynamics to another. That is,
depending on the state-position and state-transition information, the dynamics
of the controlled diffusions randomly switches between the different drift and
diffusion terms. Here, we specifically investigate the asymptotic exit problem
concerning such controlled Markov diffusion processes in two steps: (i) First,
for each controlled diffusion model, we look for an admissible Markov control
process that minimizes the principal eigenvalue for the corresponding
infinitesimal generator with zero Dirichlet boundary conditions -- where such
an admissible control process also forces the controlled diffusion process to
remain in a given bounded open domain for a longer duration. (ii) Then, using
large deviations theory, we determine the exit place and the type of
distribution at the exit time for the controlled Markov diffusion processes
coupled with random jumps and vanishing diffusion terms. Moreover, the
asymptotic results at the exit time also allow us to determine the limiting
behavior of the Dirichlet problem for the corresponding system of elliptic
partial differential equations containing a small vanishing parameter.
