How to minimize the control frequency to sustain transient chaos using partial control

In any control problem it is desirable to apply the control as infrequently as possible. In this paper we address the problem of how to minimize the frequency of control in presence of external perturbations, that we call disturbances, when the goal is to sustain transient chaos. We show here that the partial control method, that allows to find the minimum control required to sustain transient chaos in presence of disturbances, is the key to find such minimum control frequency. We prove first for the paradigmatic tent map of slope greater than 2 that for a constant value of the disturbances, the control required to sustain transient chaos decreases when the control is applied every k iterates of the map. We show that the combination of this property with the fact that the disturbances grow with k implies that there is a minimum control frequency and we provide a procedure to compute it. Finally we give evidence of the generality of this result showing that the same features are reproduced when considering the Hénon map.