Figure 1.
The graph of $g(x)$ with $y_i$, $i=1, 2, 3$, together with the equilibrium $ x^{\ast}$ marked
-
Previous Article
An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization
- DCDS-B Home
- This Issue
- Next Article
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 |
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 $(x^{\ast}-\varepsilon, x^{\ast}+\varepsilon)$.
Keywords:
Stochastic difference equations,
proportional feedback control,
population dynamics models,
Beverton-Holt equation.
Mathematics Subject Classification:
Primary: 39A50, 37H10, 34F05; Secondary: 39A30, 93D15, 93C5.
Citation:
Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete & 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. Google Scholar |
| [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. Google Scholar |
| [3] |
J. A. D. Appleby, X. Mao and A. Rodkina, On stochastic stabilization of difference equations, Discrete Contin. Dyn. Syst., 15 (2006), 843-857. Google Scholar |
| [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. Google Scholar |
| [5] |
E. Braverman, C. Kelly and A. Rodkina, Stabilisation of difference equations with noisy prediction-based control, Physica D, 326 (2016), 21-31. Google Scholar |
| [6] |
E. Braverman and E. Liz, Global stabilization of periodic orbits using a proportional feedback control with pulses, Nonlinear Dynamics, 67 (2012), 2467-2475. Google Scholar |
| [7] |
E. Braverman and E. Liz, On stabilization of equilibria using predictive control with and without pulses, Comput. Math. Appl., 64 (2012), 2192-2201. Google Scholar |
| [8] |
E. Braverman and A. Rodkina, Stabilization of two cycles of difference equations with stochastic perturbations, J. Difference Equ. Appl., 19 (2013), 1192-1212. Google Scholar |
| [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. Google Scholar |
| [10] |
E. Braverman and A. Rodkina, On convergence of solutions to difference equations with additive perturbations, J. Difference Equ. Appl., 22 (2016), 878-903. Google Scholar |
| [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. Google Scholar |
| [12] |
C. W. Clark, Mathematical bioeconomics: The optimal management of renewable resources, 2nd Edition, John Wiley & Sons, Hoboken, New Jersey, 1990.Google Scholar |
| [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. Google Scholar |
| [14] |
E. Liz, How to control chaotic behaviour and population size with proportional feedback, Phys. Lett. A, 374 (2010), 725-728. Google Scholar |
| [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. Google Scholar |
| [16] |
J. G. Milton and J. Bélair, Chaos, noise, and extinction in models of population growth, Theor. Popul. Biol., 37 (1990), 273-290. Google Scholar |
| [17] |
H. Seno, A paradox in discrete single species population dynamics with harvesting/thinning, Math. Biosci., 214 (2008), 63-69. Google Scholar |
| [18] |
L. Shaikhet, Optimal Control of Stochastic Difference Volterra Equations, An Introduction. Studies in Systems, Decision and Control 17, Springer, Cham, 2015.Google Scholar |
| [19] |
A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996.Google Scholar |
| [20] | H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. Google Scholar |
| [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. Google Scholar |
show all references
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. Google Scholar |
| [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. Google Scholar |
| [3] |
J. A. D. Appleby, X. Mao and A. Rodkina, On stochastic stabilization of difference equations, Discrete Contin. Dyn. Syst., 15 (2006), 843-857. Google Scholar |
| [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. Google Scholar |
| [5] |
E. Braverman, C. Kelly and A. Rodkina, Stabilisation of difference equations with noisy prediction-based control, Physica D, 326 (2016), 21-31. Google Scholar |
| [6] |
E. Braverman and E. Liz, Global stabilization of periodic orbits using a proportional feedback control with pulses, Nonlinear Dynamics, 67 (2012), 2467-2475. Google Scholar |
| [7] |
E. Braverman and E. Liz, On stabilization of equilibria using predictive control with and without pulses, Comput. Math. Appl., 64 (2012), 2192-2201. Google Scholar |
| [8] |
E. Braverman and A. Rodkina, Stabilization of two cycles of difference equations with stochastic perturbations, J. Difference Equ. Appl., 19 (2013), 1192-1212. Google Scholar |
| [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. Google Scholar |
| [10] |
E. Braverman and A. Rodkina, On convergence of solutions to difference equations with additive perturbations, J. Difference Equ. Appl., 22 (2016), 878-903. Google Scholar |
| [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. Google Scholar |
| [12] |
C. W. Clark, Mathematical bioeconomics: The optimal management of renewable resources, 2nd Edition, John Wiley & Sons, Hoboken, New Jersey, 1990.Google Scholar |
| [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. Google Scholar |
| [14] |
E. Liz, How to control chaotic behaviour and population size with proportional feedback, Phys. Lett. A, 374 (2010), 725-728. Google Scholar |
| [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. Google Scholar |
| [16] |
J. G. Milton and J. Bélair, Chaos, noise, and extinction in models of population growth, Theor. Popul. Biol., 37 (1990), 273-290. Google Scholar |
| [17] |
H. Seno, A paradox in discrete single species population dynamics with harvesting/thinning, Math. Biosci., 214 (2008), 63-69. Google Scholar |
| [18] |
L. Shaikhet, Optimal Control of Stochastic Difference Volterra Equations, An Introduction. Studies in Systems, Decision and Control 17, Springer, Cham, 2015.Google Scholar |
| [19] |
A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996.Google Scholar |
| [20] | H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. Google Scholar |
| [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. Google Scholar |
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 $x_0=0.5$
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 $x_0=0.5$
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, $x_0=1$
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, $x_0=1$
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, $x_0=1$
| [1] |
Ionuţ Munteanu. Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2173-2185. doi: 10.3934/dcds.2019091 |
| [2] |
Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014 |
| [3] |
Hiroaki Morimoto. Optimal harvesting and planting control in stochastic logistic population models. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2545-2559. doi: 10.3934/dcdsb.2012.17.2545 |
| [4] |
Fulvia Confortola, Elisa Mastrogiacomo. Feedback optimal control for stochastic Volterra equations with completely monotone kernels. Mathematical Control & Related Fields, 2015, 5 (2) : 191-235. doi: 10.3934/mcrf.2015.5.191 |
| [5] |
Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052 |
| [6] |
John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843 |
| [7] |
Zhao-Han Sheng, Tingwen Huang, Jian-Guo Du, Qiang Mei, Hui Huang. Study on self-adaptive proportional control method for a class of output models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 459-477. doi: 10.3934/dcdsb.2009.11.459 |
| [8] |
Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060 |
| [9] |
Jacques Henry. For which objective is birth process an optimal feedback in age structured population dynamics?. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 107-114. doi: 10.3934/dcdsb.2007.8.107 |
| [10] |
Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623 |
| [11] |
B. E. Ainseba, W. E. Fitzgibbon, M. Langlais, J. J. Morgan. An application of homogenization techniques to population dynamics models. Communications on Pure & Applied Analysis, 2002, 1 (1) : 19-33. doi: 10.3934/cpaa.2002.1.19 |
| [12] |
Robert Carlson. Myopic models of population dynamics on infinite networks. Networks & Heterogeneous Media, 2014, 9 (3) : 477-499. doi: 10.3934/nhm.2014.9.477 |
| [13] |
MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777 |
| [14] |
G. Buffoni, S. Pasquali, G. Gilioli. A stochastic model for the dynamics of a stage structured population. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 517-525. doi: 10.3934/dcdsb.2004.4.517 |
| [15] |
Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640 |
| [16] |
Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011 |
| [17] |
Eric Ruggieri, Sebastian J. Schreiber. The Dynamics of the Schoener-Polis-Holt model of Intra-Guild Predation. Mathematical Biosciences & Engineering, 2005, 2 (2) : 279-288. doi: 10.3934/mbe.2005.2.279 |
| [18] |
Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019099 |
| [19] |
Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 |
| [20] |
Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations & Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81 |
2018 Impact Factor: 1.008
Tools
Metrics
Other articles
by authors
[Back to Top]