Including stochastics in Prediction-Based Control of difference systems: Stabilizing and destabilizing by noise

A chaotic system $X_{n+1}=F\left(X_n\right)$ in certain cases can be stabilized with taking a weighted average of the state variable and the next-stage position $X_{n+1}=U{X}_n+(I-U)F\left(X_n\right)$ corresponding to a modified Prediction-Based Control with a diagonal non-scalar matrix $U$ ensuring stability of the controlled system. Sharp constants are determined for the values on the diagonal providing local stabilization. Introducing small additive noise in the control parameters keeps convergence of all solutions to the equilibrium. As noise increases, we distinguish between the cases when noise can stabilize the system while control with mean parameters does not, and those when noise destabilizes. Noise bounds allowing to lower average control intensity are determined. In the case of real eigenvalues of the Jacobian matrix being less than −1, these conditions are universal, while for complex eigenvalues, they are sharp.