# American Institute of Mathematical Sciences

August  2017, 22(6): 2067-2088. doi: 10.3934/dcdsb.2017085

## Stabilization of difference equations with noisy proportional feedback control

 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

E. Braverman is a corresponding author. E-mail address: maelena@ucalgary.ca

Received  June 2016 Revised  August 2016 Published  March 2017

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

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

Citation: Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085
##### References:
 [1] J. A. D. Appleby, G. Berkolaiko and A. Rodkina, Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise, Stochastics, 81 (2009), 99-127. [2] J. A. D. Appleby, C. Kelly, X. Mao and A. Rodkina, On the local dynamics of polynomial difference equations with fading stochastic perturbations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 401-430. [3] J. A. D. Appleby, X. Mao and A. Rodkina. On stochastic stabilization of difference equations, Discrete Contin. Dyn. Syst., 15 (2006), 843-857. [4] E. Braverman and B. Chan, Stabilization of prescribed values and periodic orbits with regular and pulse target oriented control, Chaos, 24 (2014), 013119, 7pp. [5] E. Braverman, C. Kelly and A. Rodkina, Stabilisation of difference equations with noisy prediction-based control, Physica D, 326 (2016), 21-31. [6] E. Braverman and E. Liz, Global stabilization of periodic orbits using a proportional feedback control with pulses, Nonlinear Dynamics, 67 (2012), 2467-2475. [7] E. Braverman and E. Liz, On stabilization of equilibria using predictive control with and without pulses, Comput. Math. Appl., 64 (2012), 2192-2201. [8] E. Braverman and A. Rodkina, Stabilization of two cycles of difference equations with stochastic perturbations, J. Difference Equ. Appl., 19 (2013), 1192-1212. [9] E. Braverman and A. Rodkina, Difference equations of Ricker and logistic types under bounded stochastic perturbations with positive mean, Comput. Math. Appl., 66 (2013), 2281-2294. [10] E. Braverman and A. Rodkina, On convergence of solutions to difference equations with additive perturbations, J. Difference Equ. Appl., 22 (2016), 878-903. [11] P. Carmona and D. Franco, Control of chaotic behaviour and prevention of extinction using constant proportional feedback, Nonlinear Anal. Real World Appl., 12 (2011), 3719-3726. [12] C. W. Clark, Mathematical bioeconomics: The optimal management of renewable resources, 2nd Edition, John Wiley & Sons, Hoboken, New Jersey, 1990. [13] C. Kelly and A. Rodkina, Constrained stability and instability of polynomial difference equations with state-dependent noise, Discrete Contin. Dyn. Syst. B, 11 (2009), 913-933. [14] E. Liz, How to control chaotic behaviour and population size with proportional feedback, Phys. Lett. A, 374 (2010), 725-728. [15] E. Liz and A. Ruiz-Herrera, The hydra effect, bubbles, and chaos in a simple discrete population model with constant effort harvesting, J. Math. Biol., 65 (2012), 997-1016. [16] J. G. Milton and J. Bélair, Chaos, noise, and extinction in models of population growth, Theor. Popul. Biol., 37 (1990), 273-290. [17] H. Seno, A paradox in discrete single species population dynamics with harvesting/thinning, Math. Biosci., 214 (2008), 63-69. [18] L. Shaikhet, Optimal Control of Stochastic Difference Volterra Equations, An Introduction. Studies in Systems, Decision and Control 17, Springer, Cham, 2015. [19] A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. [20] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. [21] E. F. Zipkin, C. E. Kraft, E. G. Cooch and P. J. Sullivan, When can efforts to control nuisance and invasive species backfire?, Ecological Applications, 19 (2009), 1585-1595.

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##### References:
 [1] J. A. D. Appleby, G. Berkolaiko and A. Rodkina, Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise, Stochastics, 81 (2009), 99-127. [2] J. A. D. Appleby, C. Kelly, X. Mao and A. Rodkina, On the local dynamics of polynomial difference equations with fading stochastic perturbations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 401-430. [3] J. A. D. Appleby, X. Mao and A. Rodkina. On stochastic stabilization of difference equations, Discrete Contin. Dyn. Syst., 15 (2006), 843-857. [4] E. Braverman and B. Chan, Stabilization of prescribed values and periodic orbits with regular and pulse target oriented control, Chaos, 24 (2014), 013119, 7pp. [5] E. Braverman, C. Kelly and A. Rodkina, Stabilisation of difference equations with noisy prediction-based control, Physica D, 326 (2016), 21-31. [6] E. Braverman and E. Liz, Global stabilization of periodic orbits using a proportional feedback control with pulses, Nonlinear Dynamics, 67 (2012), 2467-2475. [7] E. Braverman and E. Liz, On stabilization of equilibria using predictive control with and without pulses, Comput. Math. Appl., 64 (2012), 2192-2201. [8] E. Braverman and A. Rodkina, Stabilization of two cycles of difference equations with stochastic perturbations, J. Difference Equ. Appl., 19 (2013), 1192-1212. [9] E. Braverman and A. Rodkina, Difference equations of Ricker and logistic types under bounded stochastic perturbations with positive mean, Comput. Math. Appl., 66 (2013), 2281-2294. [10] E. Braverman and A. Rodkina, On convergence of solutions to difference equations with additive perturbations, J. Difference Equ. Appl., 22 (2016), 878-903. [11] P. Carmona and D. Franco, Control of chaotic behaviour and prevention of extinction using constant proportional feedback, Nonlinear Anal. Real World Appl., 12 (2011), 3719-3726. [12] C. W. Clark, Mathematical bioeconomics: The optimal management of renewable resources, 2nd Edition, John Wiley & Sons, Hoboken, New Jersey, 1990. [13] C. Kelly and A. Rodkina, Constrained stability and instability of polynomial difference equations with state-dependent noise, Discrete Contin. Dyn. Syst. B, 11 (2009), 913-933. [14] E. Liz, How to control chaotic behaviour and population size with proportional feedback, Phys. Lett. A, 374 (2010), 725-728. [15] E. Liz and A. Ruiz-Herrera, The hydra effect, bubbles, and chaos in a simple discrete population model with constant effort harvesting, J. Math. Biol., 65 (2012), 997-1016. [16] J. G. Milton and J. Bélair, Chaos, noise, and extinction in models of population growth, Theor. Popul. Biol., 37 (1990), 273-290. [17] H. Seno, A paradox in discrete single species population dynamics with harvesting/thinning, Math. Biosci., 214 (2008), 63-69. [18] L. Shaikhet, Optimal Control of Stochastic Difference Volterra Equations, An Introduction. Studies in Systems, Decision and Control 17, Springer, Cham, 2015. [19] A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. [20] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. [21] E. F. Zipkin, C. E. Kraft, E. G. Cooch and P. J. Sullivan, When can efforts to control nuisance and invasive species backfire?, Ecological Applications, 19 (2009), 1585-1595.
The graph of $g(x)$ with $y_i$, $i=1, 2, 3$, together with the equilibrium $x^{\ast}$ marked
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 $x_0=0.5$
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 $x_0=0.5$
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, $x_0=1$
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, $x_0=1$
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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