Stabilization by Unbounded-Variation Noises

TL;DR

This paper demonstrates that deterministic noises with unbounded variation can stabilize dynamical systems, using rough path analysis, and shows cases where they outperform stochastic noises in stabilization.

Contribution

It introduces the concept of asymptotic stability in roughness and shows deterministic unbounded-variation noises can stabilize systems where stochastic noises cannot.

Findings

Deterministic noises with unbounded variation can stabilize system origins.

Asymptotic stability in roughness aligns with uniform almost sure stability.

Deterministic noises can achieve stabilization where stochastic noises fail.

Abstract

In this paper, we claim the availability of deterministic noises for stabilization of the origins of dynamical systems, provided that the noises have unbounded variations. To achieve the result, we first consider the system representations based on rough path analysis; then, we provide the notion of asymptotic stability in roughness to analyze the stability for the systems. In the procedure, we also confirm that the system representations include stochastic differential equations; we also found that asymptotic stability in roughness is the same property as uniform almost sure asymptotic stability provided by Bardi and Cesaroni. After the discussion, we confirm that there is a case that deterministic noises are capable of making the origin become asymptotically stable in roughness while stochastic noises do not achieve the same stabilization results.