Stabilisation of difference equations with noisy prediction-based control

TL;DR

This paper investigates how stochastic perturbations, both multiplicative and additive, influence the stability of difference equations under prediction-based control, showing that noise can enhance stability and reduce equilibrium blurring.

Contribution

It provides new sufficient conditions under which stochastic noise improves the stability of difference equations controlled by prediction-based methods.

Findings

Stochastic noise can improve the global asymptotic stability of the equilibrium.

Noise reduces the blurring effect on the equilibrium, which can be minimized by controlling noise intensity.

Derived less restrictive conditions for stability under stochastic perturbations.

Abstract

We consider the influence of stochastic perturbations on stability of a unique positive equilibrium of a difference equation subject to prediction-based control. These perturbations may be multiplicative $$x_{n+1}=f(x_n)-\left( \alpha + l\xi_{n+1} \right) (f(x_n)-x_n), \quad n=0, 1, \dots$$ if they arise from stochastic variation of the control parameter, or additive $$x_{n+1}=f(x_n)-\alpha(f(x_n)-x_n) +l\xi_{n+1}, \quad n=0, 1, \dots $$ if they reflect the presence of systemic noise. We begin by relaxing the control parameter in the deterministic equation, and deriving a range of values for the parameter over which all solutions eventually enter an invariant interval. Then, by allowing the variation to be stochastic, we derive sufficient conditions (less restrictive than known ones for the unperturbed equation) under which the positive equilibrium will be globally a.s. asymptotically stable:i.e. the presence of noise improves the known effectiveness of prediction-based control. Finally, we show that systemic noise has a "blurring" effect on the positive equilibrium, which can be made arbitrarily small by controlling the noise intensity. Numerical examples illustrate our results.