Stabilization of difference equations with noisy proportional feedback control

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 $\epsilon >0$, when the noise level $\ell$ is sufficiently small, all solutions eventually belong to the interval $(x^{\ast }-\epsilon ,x^{\ast }+\epsilon )$.

20 pages, 19 figures