Stabilization of difference equations with noisy proportional feedback control

TL;DR

This paper investigates how proportional feedback control can stabilize points in difference equations under noisy conditions, showing that solutions can be confined near the desired point with sufficiently small noise.

Contribution

It provides conditions for stabilization of difference equations with noisy proportional feedback, including both multiplicative and additive noise models.

Findings

Solutions enter a stochastic equilibrium interval.

Small noise levels ensure solutions stay close to the target point.

Stability conditions depend on noise magnitude and control parameters.

Abstract

Given a deterministic difference equation $x_{n+1}= f(x_n)$, we would like to stabilize any point $x^{\ast}\in (0, f(b))$, where $b$ is a unique maximum point of $f$, by introducing proportional feedback (PF) control. We assume that PF control contains either a multiplicative $x_{n+1}= f\left( (\nu + \ell\chi_{n+1})x_n \right)$ or an additive noise $x_{n+1}=f(\lambda x_n) +\ell\chi_{n+1}$. We study conditions under which the solution eventually enters some interval, treated as a stochastic (blurred) equilibrium. In addition, we prove that, for each $\varepsilon>0$, when the noise level $\ell$ is sufficiently small, all solutions eventually belong to the interval $(x^{\ast}-\varepsilon,x^{\ast}+\varepsilon)$.