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

TL;DR

This paper introduces a stochastic control method to stabilize cycles and equilibria in difference equations, leveraging noise to expand control parameter ranges, with novel applications to Prediction-Based and Target-Oriented Controls.

Contribution

It is the first to utilize noise for stabilization in Prediction-Based Control and proposes a new stochastic variant of Target-Oriented Control.

Findings

Stochastic control broadens the range of stabilizable parameters.

Numerical tests confirm stabilization in biological models.

First application of noise effects in Prediction-Based Control.

Abstract

We stabilize a prescribed cycle or an equilibrium of the difference equation using pulsed stochastic control. Our technique, inspired by the 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 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 demonstrate our results numerically on the logistic, Ricker and Maynard Smith models from population biology.