Stabilizing multiple equilibria and cycles with noisy prediction-based control

Elena Braverman; Alexandra Rodkina

doi:10.3934/dcdsb.2021281

Article Contents

2022, Volume 27, Issue 10: 5419-5446. Doi: 10.3934/dcdsb.2021281

This issue Previous Article Exponential decay for quasilinear parabolic equations in any dimension Next Article On weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effects

Stabilizing multiple equilibria and cycles with noisy prediction-based 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

Received: April 30, 2020

Revised: June 30, 2021

Early access: November 2021

Published: October 2022

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 39A50, 39A30, 93D15; Secondary: 37H10, 37H30, 93C55, 39A33.

Citation:

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Figure 1. The second iteration of the Ricker map for $ r = 2.7 $

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Figure 2. The fourth iteration of the Ricker map for $ r = 2.6 $

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Figure 3. 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 $

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Figure 4. 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

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Figure 5. The graph of the map defined in (4.1), together with $ y = x $

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Figure 6. 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

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