Noisy prediction-based control leading to stability switch

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Applying Prediction-Based Control (PBC) xn+1=(1−αn)f(xn)+αnxn with stochastically perturbed control coefficient αn=α+ℓξn+1, n∈N, where ξ are bounded identically distributed independent random variables, we globally stabilize the unique equilibrium K of the equation xn+1=f(xn) in a certain domain. In our results, the noisy control α+ℓξ provides both local and global stability, while the mean value α 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 αn and, generally, continuous but not differentiable at K map f.