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

TL;DR

This paper investigates the problem of minimizing the exit rate of a degenerate diffusion process in a chain of distributed control systems with random perturbations, establishing a connection with eigenvalues and optimal control via Hamilton-Jacobi-Bellman equations.

Contribution

It introduces a novel link between the minimum exit rate and principal eigenvalues for degenerate diffusions in distributed control systems, and develops associated Hamilton-Jacobi-Bellman equations.

Findings

Established the connection between exit rate and principal eigenvalue.

Derived Hamilton-Jacobi-Bellman equations for the control problem.

Provided estimates on the exit probability under optimal controls.

Abstract

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.