Optimal control of chaotic systems via peak-to-peak maps

In several chaotic systems, the value of a peak (relative maximum) of a variable and its time of occurrence can be predicted, with very good approximation, from the value of the previous peak. This property, which is called peak-to-peak dynamics and which basically relies on the low-dimensionality of the chaotic attractor, allows to approximately model the system in a very compact form by a peak-to-peak map (predicting the next peak value) and a return-time map (predicting the time of occurrence). In this paper, it is shown how these maps can be effectively used to formulate and solve several optimal control problems. The Lorenz system is used as the paradigm to illustrate the methodology.