# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021281
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Stabilizing multiple equilibria and cycles with noisy prediction-based 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

Received  May 2020 Revised  July 2021 Early access November 2021

Fund Project: E. Braverman is a corresponding author. The first author is supported by NSERC grant RGPIN-2020-03934

Pulse stabilization of cycles with Prediction-Based Control including noise and stochastic stabilization of maps with multiple equilibrium points is analyzed for continuous but, generally, non-smooth maps. Sufficient conditions of global stabilization are obtained. Introduction of noise can relax restrictions on the control intensity. We estimate how the control can be decreased with noise and verify it numerically.

Citation: Elena Braverman, Alexandra Rodkina. Stabilizing multiple equilibria and cycles with noisy prediction-based control. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021281
##### References:

show all references

##### References:
The second iteration of the Ricker map for $r = 2.7$
The fourth iteration of the Ricker map for $r = 2.6$
A bifurcation diagram for the second iterate of the Ricker map with $r = 2.7$, $\alpha \in (0.1, 0.3)$ and (left) no noise, (right) $\ell = 0.15$
A bifurcation diagram for the third iterate of the Ricker map with $r = 3.5$ for $\alpha\in (0.75, 0.9)$ and (left) without noise, (right) $\ell = 0.06$. The last bifurcation leading to two stable equilibrium points occurs for $\alpha \approx 0.88$ in the deterministic case and $\alpha<0.86$ in the stochastic case
The graph of the map defined in (4.1), together with $y = x$
A bifurcation diagram for the map defined in \protect{(4.1)} with $\alpha \in (0.45, 0.65)$ and (left) no noise, we have two stable equilibrium points starting from $\alpha\approx 0.605$, (right) for Bernoulli noise with $\ell = 0.04$, the last bifurcation happens for smaller $\alpha\approx 0.535$. Here the two attractors correspond to two stable equilibrium points with separate basins of attraction, not to a cycle
 [1] 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 [2] Ionuţ Munteanu. Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2173-2185. doi: 10.3934/dcds.2019091 [3] 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 [4] 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 [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] 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 [7] John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843 [8] 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 [9] Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060 [10] Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2037-2053. doi: 10.3934/dcdsb.2020365 [19] Xiujuan Wang, Mingshu Peng. Rich dynamics in some generalized difference equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3205-3212. doi: 10.3934/dcdss.2020191 [20] 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

2020 Impact Factor: 1.327

## Tools

Article outline

Figures and Tables