    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:
  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. 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. J. A. D. Appleby, X. Mao and A. Rodkina. On stochastic stabilization of difference equations, Discrete Contin. Dyn. Syst., 15 (2006), 843-857. E. Braverman and B. Chan, Stabilization of prescribed values and periodic orbits with regular and pulse target oriented control, Chaos, 24 (2014), 013119, 7pp. E. Braverman, C. Kelly and A. Rodkina, Stabilisation of difference equations with noisy prediction-based control, Physica D, 326 (2016), 21-31. E. Braverman and E. Liz, Global stabilization of periodic orbits using a proportional feedback control with pulses, Nonlinear Dynamics, 67 (2012), 2467-2475. E. Braverman and E. Liz, On stabilization of equilibria using predictive control with and without pulses, Comput. Math. Appl., 64 (2012), 2192-2201. E. Braverman and A. Rodkina, Stabilization of two cycles of difference equations with stochastic perturbations, J. Difference Equ. Appl., 19 (2013), 1192-1212. 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. E. Braverman and A. Rodkina, On convergence of solutions to difference equations with additive perturbations, J. Difference Equ. Appl., 22 (2016), 878-903. 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. C. W. Clark, Mathematical bioeconomics: The optimal management of renewable resources, 2nd Edition, John Wiley & Sons, Hoboken, New Jersey, 1990. 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. E. Liz, How to control chaotic behaviour and population size with proportional feedback, Phys. Lett. A, 374 (2010), 725-728. 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. J. G. Milton and J. Bélair, Chaos, noise, and extinction in models of population growth, Theor. Popul. Biol., 37 (1990), 273-290. H. Seno, A paradox in discrete single species population dynamics with harvesting/thinning, Math. Biosci., 214 (2008), 63-69. L. Shaikhet, Optimal Control of Stochastic Difference Volterra Equations, An Introduction. Studies in Systems, Decision and Control 17, Springer, Cham, 2015. A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. 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. show all references

##### References:
  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. 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. J. A. D. Appleby, X. Mao and A. Rodkina. On stochastic stabilization of difference equations, Discrete Contin. Dyn. Syst., 15 (2006), 843-857. E. Braverman and B. Chan, Stabilization of prescribed values and periodic orbits with regular and pulse target oriented control, Chaos, 24 (2014), 013119, 7pp. E. Braverman, C. Kelly and A. Rodkina, Stabilisation of difference equations with noisy prediction-based control, Physica D, 326 (2016), 21-31. E. Braverman and E. Liz, Global stabilization of periodic orbits using a proportional feedback control with pulses, Nonlinear Dynamics, 67 (2012), 2467-2475. E. Braverman and E. Liz, On stabilization of equilibria using predictive control with and without pulses, Comput. Math. Appl., 64 (2012), 2192-2201. E. Braverman and A. Rodkina, Stabilization of two cycles of difference equations with stochastic perturbations, J. Difference Equ. Appl., 19 (2013), 1192-1212. 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. E. Braverman and A. Rodkina, On convergence of solutions to difference equations with additive perturbations, J. Difference Equ. Appl., 22 (2016), 878-903. 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. C. W. Clark, Mathematical bioeconomics: The optimal management of renewable resources, 2nd Edition, John Wiley & Sons, Hoboken, New Jersey, 1990. 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. E. Liz, How to control chaotic behaviour and population size with proportional feedback, Phys. Lett. A, 374 (2010), 725-728. 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. J. G. Milton and J. Bélair, Chaos, noise, and extinction in models of population growth, Theor. Popul. Biol., 37 (1990), 273-290. H. Seno, A paradox in discrete single species population dynamics with harvesting/thinning, Math. Biosci., 214 (2008), 63-69. L. Shaikhet, Optimal Control of Stochastic Difference Volterra Equations, An Introduction. Studies in Systems, Decision and Control 17, Springer, Cham, 2015. A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. 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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