Partial control of chaotic transients using escape times

The partial control technique allows one to keep the trajectories of a dynamical system inside a region where there is a chaotic saddle and from which nearly all the trajectories diverge. Its main advantage is that this goal is achieved even if the corrections applied to the trajectories are smaller than the action of environmental noise on the dynamics, a counterintuitive result that is obtained by using certain safe sets. Using the Henon map as a paradigm, we show here the deep relationship between the safe sets and the sets of points with different escape times, the escape time sets. Furthermore, we show that it is possible to find certain extended safe sets that can be used instead of the safe sets in the partial control technique. Numerical simulations confirm our findings and show that in some situations, the use of extended safe sets can be more advantageous.