2014/09/09 by Getachew K. Befekadu, Panos J. Antsaklis, Befekadu, Getachew K. +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #34D20 #37H10 #37J25 #49K15 #49L20 #49L25 #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Biology Tumor Growth #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.DS #msc:34D20 #msc:37H10 #msc:37J25 #msc:49K15 #msc:49L20 #msc:49L25
paper · pdf · doi:10.48550/arxiv.1409.2751
12 Pages. (Additional Note: This work is, in some sense, a continuation of our previous paper arXiv:1408.6260.)
openalex publication_date 2014/09/09 · arxiv created 2014/09/10 · arxiv updated 2014/09/12 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28
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 ≥ 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 ℓ ∈ \2, …, n\, the distributed control systems, which is formed by the first ℓ subsystems, satisfies an appropriate Hörmander 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.