Noisy Prediction-Based Control Leading to Stability Switch

Applying Prediction-Based Control (PBC) $x_{n+1}=(1-\alpha_n)f(x_n)+\alpha_n x_{n}$ with stochastically perturbed control coefficient $\alpha_n=\alpha+\ell \xi_{n+1}$, $n\in \mathbb N$, where $\xi$ are bounded identically distributed independent random variables, we globally stabilize the unique equilibrium $K$ of the equation $x_{n+1}=f(x_n)$ in a certain domain. In our results, the noisy control $\alpha+\ell \xi$ provides both local and global stability, while the mean value $\alpha$ of the control does not guarantee global stability, for example, the deterministic controlled system can have a stable two-cycle, and non-controlled map be chaotic. In the case of unimodal $f$ with a negative Schwarzian derivative, we get sharp stability results generalizing Singer's famous statement `local stability implies global' to the case of the stochastic control. New global stability results are also obtained in the deterministic settings for variable $\alpha_n$ and, generally, continuous but not differentiable at $K$ map $f$.