  In this paper, we consider the problem of minimizing the exit rate with which
a diffusion process pertaining to a chain of distributed control systems, with
random perturbations, exits from a given bounded open domain. In particular, we
consider a chain of distributed control systems that are formed by $n$
subsystems (with $n \ge 2$), where the random perturbation enters only in the
first subsystem and is then subsequently transmitted to the other subsystems.
Furthermore, we assume that, for any $\ell \in \{2, \ldots, n\}$, the
distributed control systems, which is formed by the first $\ell$ subsystems,
satisfies an appropriate H\"ormander condition. As a result of this, the
diffusion process is degenerate, in the sense that the infinitesimal generator
associated with it is a degenerate parabolic equation. Our interest is to
establish a connection between the minimum exit rate with which the diffusion
process exits from the given domain and the principal eigenvalue for the
infinitesimal generator with zero boundary conditions. Such a connection allows
us to derive a family of Hamilton-Jacobi-Bellman equations for which we provide
a verification theorem that shows the validity of the corresponding optimal
control problems. Finally, we provide an estimate on the attainable exit
probability of the diffusion process with respect to a set of admissible
(optimal) Markov controls for the optimal control problems.
