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Stabilizing multiple equilibria and cycles with noisy prediction-based control

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

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  • 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.

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


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

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

    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 $

    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

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

    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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