Probabilistic control of chaos: The beta-adic Renyi map under control

We propose a new, probabilistic, approach for the control of chaotic systems. This approach is illustrated by a specific method, for the control of any periodic orbit of the simplest piecewise linear chaotic map, namely the beta-adic Renyi map. As these chaotic maps are structurally stable, they cannot be controlled using conventional control methods without significant change of the controlling system. The chosen periodic orbit of the original system is a global attractor for the probability densities. The generalized spectral decomposition of the associated Frobenius-Perron operator provides a spectral condition of controllability for chaotic dynamical systems.