Stabilization of difference equations with noisy proportional feedback control

Abstract

Given a deterministic difference equation $x_{n+1}= f(x_n)$ with a continuous $f$ increasing on $[0, b]$, $f(0) \geq 0$, we would like to stabilize any point $x^{\ast}\in (f(0), f(b))$, by introducing the 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 $(x^{\ast}-\varepsilon, x^{\ast}+\varepsilon)$.