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

On the minimum exit rate for a diffusion process pertaining to a chain of distributed control systems with random perturbations