Stabilization of difference equations with noisy proportional feedback control

Elena Braverman; Alexandra Rodkina

doi:10.3934/dcdsb.2017085

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2017, Volume 22, Issue 6: 2067-2088. Doi: 10.3934/dcdsb.2017085

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Stabilization of difference equations with noisy proportional feedback control

Elena Braverman^1, , and
Alexandra Rodkina^2,

1.
Dept. of Math. and Stats., University of Calgary, 2500 University Drive N.W. Calgary, AB, T2N 1N4, Canada
2.
Department of Mathematics, the University of the West Indies, Mona Campus, Kingston, Jamaica

Author Bio: E-mail address: alexandra.rodkina@uwimona.edu.jm
E. Braverman is a corresponding author. E-mail address: maelena@ucalgary.ca
E. Braverman is a corresponding author. E-mail address: maelena@ucalgary.ca

Received: May 31, 2016

Revised: July 31, 2016

Published: May 2017

The first author is supported by NSERC grant RGPIN-2015-05976, both authors are supported by AIM SQuaRE program.

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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 $(x^{\ast}-\varepsilon, x^{\ast}+\varepsilon)$.

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Mathematics Subject Classification: Primary:39A50, 37H10, 34F05;Secondary:39A30, 93D15, 93C55.

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Figure 1. The graph of $g(x)$ with $y_i$, $i=1, 2, 3$, together with the equilibrium $ x^{\ast}$ marked

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Figure 2. Solutions of the difference equation with $f$ as in (5.1) and multiplicative stochastic perturbations with $\ell=0.01$ (upper left), $\ell=0.025$ (upper right), where PF control aims at stabilizing $x^*=1.5$, $\nu \approx 0.4685$ and $\ell=0.015$ (two lower rows), with either $x^*=1.125$ (second row, left) or $x^*=1.1$ stabilized (second row, right), and $x^*=0$ is stabilized for $\nu =0.39$ (lower left); for $\nu =0.75$ there is no blurred equilibrium but oscillations (lower right). Everywhere $x_0=0.5$

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Figure 3. Solutions of the difference equation with $f$ as in (5.1) and additive stochastic perturbations with $\ell=0.01$ (upper left), $\ell=0.025$ (upper right), where PF control aims at stabilizing $x^*=1.5$, $\nu \approx 0.4685$ and $\ell=0.015$ (two lower rows), with either $x^*=1.125$ (second row, left) or $x^*=1.1$ stabilized (second row, right), and $x^*=0$ is stabilized for $\nu =0.39$ (lower left); for $\nu =0.75$ there is no blurred equilibrium but oscillations (lower right). Everywhere $x_0=0.5$

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Figure 4. Solutions of the difference equation with $f$ as in (5.2) and multiplicative stochastic perturbations with $\ell=0.01$ (left) and $\ell=0.025$ (right), where PF control aims at stabilizing $x^*=2.5$, $\nu \approx 0.253555$. In both figures, five runs are illustrated, $x_0=1$

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Figure 5. Solutions of the difference equation with $f$ as in (5.2) and multiplicative stochastic perturbations with $\ell=0.01$, where we stabilize the maximum $\approx 2.877$ (left), the zero equilibrium with $\nu=0.23$ (middle) and obtain a blurred cycle for $\nu=0.35$ (right). Everywhere we present five runs, $x_0=1$

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Figure 6. Solutions of the difference equation with $f$ as in (5.2) and additive stochastic perturbations with $\ell=0.01$, where $x^*=2.5$ is stabilized (left), or there are sustainable blurred oscillations for $\nu=0.35$ (right). In each figure, we present five runs, $x_0=1$

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