Stabilization of cycles with stochastic prediction-based and target-oriented control

Abstract

We stabilize a prescribed cycle or an equilibrium of a difference equation using pulsed stochastic control. Our technique, inspired by Kolmogorov's Law of Large Numbers, activates a stabilizing effect of stochastic perturbation and allows for stabilization using a much wider range for the control parameter than would be possible in the absence of noise. Our main general result applies to both Prediction-Based and Target-Oriented Controls. This analysis is the first to make use of the stabilizing effects of noise for Prediction-Based Control; the stochastic version has previously been examined in the literature, but only the destabilizing effect of noise was demonstrated. A stochastic variant of Target-Oriented Control has never been considered, to the best of our knowledge, and we propose a specific form that uses a point equilibrium or one point on a cycle as a target. We illustrate our results numerically on the logistic, Ricker and Maynard Smith models from population biology.

Stabilizing multiple equilibria and cycles with noisy prediction-based control

“…Introduction of noise allowed to lower the level of average control, and this result complements [2]. Compared to [3], the results of the present paper are global, not local, which gives an advantage in practical implementations.…”

“…More sophisticated behaviour of maps with PBC is observed if either g has multiple critical and equilibrium points, or (1.1) is considered with control (1.2) at every k-th step only. This corresponds to pulse control which can be applied in both deterministic and stochastic cases [3,4,9]. Pulse control can be viewed as PBC for the iterated map g k .…”

On target-oriented control of Hénon and Lozi maps

“…Stochastic control is a well-developed area, especially for parametric optimization, optimal stochastic control, dynamic programming etc, see [1] and references therein. Our approach is based on application of the Kolmogorov's Law of Large Numbers, see Lemma 3.2 below, and is closest to the methods developed and applied in [10,11,23] (see also [5,12]). Note that we prove only local stability with any a priori fixed probability from (0, 1).…”

“…Among other works, stabilization of difference equations was studied in [2,5,12], stabilization for certain higher-dimensional models in [23], and finite difference schemes for two-dimensional linear systems in [4,13]. The approach of our paper is closest to [9][10][11], where prediction-based, proportional feedback and target oriented controls with stochastically perturbed parameters were studied for scalar difference equations.…”