On the risk-sensitive escape control for diffusion processes pertaining to an expanding construction of distributed control systems

TL;DR

This paper develops a risk-sensitive control framework for a growing distributed diffusion system with degenerate dynamics, focusing on preventing subsystem escape and analyzing control performance under modeling uncertainties.

Contribution

It introduces a novel risk-sensitive escape control approach for expanding degenerate diffusion systems, linking it to stochastic differential games and addressing modeling errors.

Findings

Characterizes risk-sensitive escape control as a stochastic game

Establishes control strategies for degenerate diffusion processes

Analyzes robustness of control under modeling errors

Abstract

In this paper, we consider an expanding construction of a distributed control system, which is obtained by adding a new subsystem one after the other, until all $n$ subsystems, where $n \ge 2$, are included in the distributed control system. It is assumed that a small random perturbation enters only into the first subsystem and is then subsequently transmitted to the other subsystems. Moreover, for any $\ell \in \{2, ..., n\}$, the distributed control system, compatible with the expanding construction, which is obtained from the first $\ell$ subsystems, satisfies an appropriate H\"{o}rmander condition. As a result of this, the diffusion process is degenerate, i.e., the backward operator associated with it is a degenerate parabolic equation. Our main interest here is to prevent the diffusion process (that corresponds to a particular subsystem) from leaving a given bounded open domain. In particular, we consider a risk-sensitive version of the mean escape time criterion with respect to each of the subsystems. Using a variational representation, we characterize the risk-sensitive escape control for the diffusion process as the lower and upper values of an associated stochastic differential game. Finally, we comment on the implication of our results, where one is also interested in evaluating the performance of the risk-sensitive escape control, when there is some modeling error in the distributed control system.