[Paper Review] Exit Probabilities for a Chain of Distributed Control Systems with Small Random Perturbations

Getachew K. Befekadu, Panos J. Antsaklis|arXiv (Cornell University)|Aug 26, 2014

Petri Nets in System Modeling13 references3 citations

TL;DR

This paper develops an asymptotic estimate for the exit probability of a diffusion process in a chain of distributed control systems with small random perturbations. By interpreting the exit probability as a value function in stochastic control problems, it derives large deviations-based bounds that characterize how the perturbation propagates through the system, showing exponential decay rates in exit probabilities as the noise intensity ε→0.

ABSTRACT

In this paper, we consider a diffusion process pertaining to a chain of distributed control systems with small random perturbation. The distributed control system is formed by n subsystems that satisfy an appropriate Hormander condition, i.e., the second subsystem assumes the random perturbation entered into the first subsystem, the third subsystem assumes the random perturbation entered into the first subsystem then was transmitted to the second subsystem and so on, such that the random perturbation propagates through the entire distributed control system. Note that the random perturbation enters only in one of the subsystems and, hence, the diffusion process is degenerate, in the sense that the backward operator associated with it is a degenerate parabolic equation. Our interest is to estimate the exit probability with which a diffusion process (corresponding to a particular subsystem) exits from a given bounded open domain during a certain time interval. The method for such an estimate basically relies on the interpretation of the exit probability function as a value function for a family of stochastic control problems that are associated with the underlying chain of distributed control systems.

Motivation & Objective

• To estimate the probability that a diffusion process in a chain of distributed control systems exits a bounded domain under small random perturbations.
• To analyze how random perturbations propagate through a cascade of subsystems, starting from the first and affecting subsequent ones.
• To characterize the exit time behavior of the system using large deviations principles and stochastic control theory.
• To establish uniform asymptotic bounds on exit probabilities as the noise level ε approaches zero.

Proposed method

• Models the system as an n-dimensional diffusion process where only the first subsystem receives direct noise, and subsequent subsystems inherit perturbations through state-dependent dynamics.
• Applies the Hörmander condition to ensure hypoellipticity and well-posedness of the associated degenerate parabolic backward operator.
• Interprets the exit probability function as a value function for a family of stochastic control problems, enabling use of dynamic programming principles.
• Uses the Ventcel-Freidlin asymptotic estimates to derive upper and lower bounds on exit probabilities in the small noise limit.
• Introduces a penalty function Φ_M to construct lower bounds on the exit probability via verification of a dynamic programming inequality.
• Establishes uniform convergence of the exit probability rate function I^ε,ℓ to its zero-noise limit I^0,ℓ over compact subsets of the state space.

Experimental results

Research questions

• RQ1 How does a small random perturbation propagate through a chain of distributed control systems where only the first subsystem is directly affected?
• RQ2 What is the asymptotic behavior of the exit probability for a subsystem in such a system as the noise intensity ε→0?
• RQ3 Can the exit probability of a subsystem be characterized as a value function in a stochastic control framework?
• RQ4 What are the large deviations estimates for the exit time of the system from a bounded domain under small noise?
• RQ5 How does the structure of the chain (e.g., sequential dependency) affect the exit probability rate function?

Key findings

• The exit probability decays exponentially as ε→0, with the rate determined by a large deviations rate function I^ε,ℓ.
• The rate function I^ε,ℓ converges uniformly to its zero-noise limit I^0,ℓ over compact subsets of the state space as ε→0.
• The upper bound on the exit probability is established using Ventcel-Freidlin estimates for degenerate diffusions.
• The lower bound is derived via a penalty function approach and verification of a dynamic programming inequality.
• The exit probability function is shown to be the value function of a stochastic control problem, enabling analytical treatment via dynamic programming.
• The results provide asymptotic information on the time duration the system remains within a prescribed domain before exiting.